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Stability conditions of an electrified miscible viscous fluid sheet
Journal of Colloid and Interface Science 259 (2003) 186–199 www.elsevier.com/locate/jcis
Stability conditions of an electrified miscible viscous fluid sheet Galal M. Moatimid Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt Received 6 February 2002; accepted 20 November 2002
Abstract The Kelvin–Helmholtz problem of viscous fluids under the influence of a normal periodic electric field in the absence of surface charges is studied. The system is composed of a streaming dielectric fluid sheet of finite thickness embedded between two different streaming finite dielectric fluids. The interfaces permit mass and heat transfer. Because of the complexity of the considered system, a mathematical simplification is adopted. The weak viscous effects are taken into account so that their contributions are incorporated into the boundary conditions. Therefore, the equations of motion are solved in the absence of viscous effects. The boundary value problem leads to two simultaneous Mathieu equations of damped terms having complex coefficients. The symmetric and antisymmetric deformations reduced the coupled Mathieu equations to a single Mathieu equation. The classical stability criterion is found to be substantially modified due to the effect of mass and heat transfer. The analytical results are numerically confirmed. It is found that the sheet thickness and mass and heat transfer parameters have a dual influence on the stability criteria. It is also found that the field frequency has a stabilizing influence especially at small values of the wave number. In contrast to the case of a pure inviscid fluid, it is found that the uniform normal electric field plays a dual role in the stability criteria. This role depends on the choice of the numerical values of the physical parameters of the system under consideration. 2003 Elsevier Science (USA). All rights reserved. Keywords: Electrohydrodynamic; Viscous fluids; Kelvin–Helmholtz; Periodic forces; Mass and heat transfer; Interfacial; Stability
1. Introduction The stability of thin films has received increasing interest due to their technological applications in many industries such as the petroleum industry. The fluid layer may be considered as a thin fluid sheet such as a bilayer lipid membrane. Studies on the stability of a fluid layer can contribute to a better understanding of colloidal and biological systems. Stability is used to rupture biological and artificial membranes and thin films which are involved in the mechanisms of a wide variety of biological and engineering processes and is important for practical applications. Typical examples are membrane fusion, membrane rupture and mechanical destruction, cell division and fusion, and colloid stability [1]. If the layer is stressed by an electric field, then the classical stability conditions will be considerably modified. El-Shehawey et al. [2] studied the linear electrohydrodynamic stability of inviscid fluids. The dielectric fluids are stressed by gravitational forces and a tangential periodic electric field. They found that this field has a stabilizing effect except at resonance points. Furthermore, the tangenE-mail address: [email protected].
tial periodic field cannot stabilize a system which is unstable under the influence of a uniform electric field. Mohamed et al. [3] investigated the electrohydrodynamic stability of a fluid sheet at finite thickness. They found that when there are no surface charges present at the interfaces, the normal electric field showed a destabilizing influence. Also, in the presence of surfaces charges, the field is still destabilizing but its effect is partially shielded in some situations. Parhi and Nath [4] gave a new analytical criterion for the Rayleigh– Taylor instability of a three-layer viscous, stratified, incompressible steady flow, when the top and the bottom layers extend to infinity while the middle layer has a small thickness. They recovered the results of the corresponding problem for two fluid layers. The problem of parametric resonance arises in many branches of physics and engineering. One of the important problems is that of dynamic instability, which is the response of mechanical and elastic systems to time-varying loads, especially periodic loads. There are cases in which the introduction of a small vibration loading can stabilize a system which is statically unstable or destabilize a system which is statically stable. There are many books devoted to the analysis and applications of parametric excitation. McLachlan [5]
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00164-9
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
discussed the theory and applications of Mathieu functions, while Magnus and Winkler [6] discussed Hill’s equation and its engineering vibration problem. The treatment of the parametric excitation system having many degrees of freedom and distinct natural frequencies are usually operated on using the multiple time scales method as given by Nayfeh and Mook [7]. The influence of periodic forces on the stability of flow is a relatively new topic in the theory of hydrodynamic stability. The mathematical analysis is rather difficult, because the method of normal modes is not applicable. The linearized partial differential equations have coefficients which vary with time so that the exponential time dependence of the perturbation is not separable. The stability of a viscous Rayleigh–Taylor fluid results in a transcendental dispersion relation (see Chandrasekhar [8]). When different layers of a stratified fluid are in horizontal motion, another type of instability is obtained. The instability of the plane interface between two superposed fluids with a relative horizontal velocity is called the Kelvin–Helmholtz instability. In the Kelvin–Helmholtz model, the effect of streaming is destabilizing in the linear sense [8]. Mohamed et al. [9] studied the electroviscoelastic stability of a Kelvin fluid layer in the presence of field periodicity. They found that the surface elevations are governed by two transcendental coupled equations of Mathieu type. El-Dib and Matoog [10] investigated the stability of streaming in an electrified Maxwell fluid sheet influenced by a vertical periodic field in the absence of surface charges. Their calculations show that the increase in the sheet thickness plays a destabilizing effect in the presence or absence of the field frequency as well as being a function of the field intensity. A dual role of the stratified viscosities, in the presence or the absence of the field frequency, is observed in their analysis. These studies in interfacial instabilities are based on the assumption that the fluids are immiscible. Thus, there is no mass transfer across the interface. The immiscibility condition concerns the limited case of infinite latent heat. Ordinarily, since the latent heat is very large, it is a very good approximation to treat the fluids as immiscible when thermal effects are very small. However, when there is a strong temperature gradient in the fluid, thermal effects on the interfacial waves can be appreciable. Therefore, there is significant mass transfer across the interface, and in turn transfer of heat in the fluid has to be taken into consideration. The mechanism of heat and mass transfer across an interface (gas–liquid interface, say) is of great importance in numerous industrial and environmental processes. These include the design of many types of contacting equipment, e.g., boilers, condensers, evaporators, gas absorbers, pipelines, chemical reactors, nuclear reactors, and are relevant to other problems such as aeration of rivers and the sea. Hsieh [11] formulated the general problem of interfacial fluid flow with mass and heat transfer, which he applied to discuss the Kelvin– Helmholtz instability in the linear theory. He showed that the classical stability criterion is substantially modified due to the effect of mass and heat transfer. He [12] employed
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the multiple time scales analysis to investigate the nonlinear Rayleigh–Taylor stability with mass and heat transfer. He found that when the heat transfer rate is strong enough, the classically unstable system is stabilized by the nonlinear effect and the effect of mass and heat transfer across the interface. Moatimid [13] studied the problem of electrohydrodynamic instability of two superposed viscous, miscible, streaming fluids. His investigation includes the stability analysis of the presence of the periodic electric field as well the constant one. He found that the presence of surface charges plays a dual role in the stability criteria. He also found that the mass and heat transfer has a destabilizing effect whether the electric field is static or periodic. Recently, the stability of a basic flow of streaming magnetic fluids in the presence of an oblique periodic magnetic field has been investigated by Moatimid [14]. He found that the mass and heat transfer parameter has a destabilizing influence regardless of the field mechanism. He also found that the field frequency plays a dual role in the stability picture. The problem of hydrodynamic stability of two-phase fluids, acted upon by an external electric field in different situations in the presence of mass and heat transfer, separated by a single interface, has been extensively studied [15–17]. Moatimid [15,16] showed that, in contrast to the case of the uniform electric fields, the periodic one—frequently— shows the effect of mass and heat transfer parameter on the stability conditions. The analysis of Elhefnawy and Moatimid [17] indicates that the instability criterion is independent of mass and heat transfer parameters, but it is different from that in the same problem without mass and heat transfer. Their analysis showed the destabilizing effect of mass and heat transfer. To the best of my knowledge, no attempt has been made to examine—analytically—the hydrodynamic stability of three-phase fluids separated by double interfaces. Therefore, we have considered the Kelvin– Helmholtz instability problem with mass and heat transfer in a plane geometry. The model consists of three-phase fluids which are divided by double interfaces. The system is excited by an external periodic normal electric field distribution. Hsieh’s [11] simplified formulation of a single interface is extended to adopt double interfaces. The analysis is based on the linear theory of stability. Because of the complexity in solving the present boundary-value problem, only the symmetric and antisymmetric deformations of the surface deflections are considered. Therefore, we have divided the work into various sections for the clarity of presentation as follows: The formulation of the problem is presented in Section 2. The basic equations are introduced in Section 3. The perturbation equations are formulated in Section 4. Section 5 is devoted to presenting the boundary conditions, which include Maxwellian, kinematical, and stress tensor conditions. The derivation of the characteristic equations are presented in Section 6. The treatment of the symmetric and antisymmetric deformations of the surface deflections are introduced in Section 7, the analysis includes the case of a uniform normal electric field as well as a periodic one. Section 8 is de-
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voted to a discussion of the marginal state analysis. Finally, the outlines of the problem are introduced as a conclusion in Section 9.
2. Formulation of the problem The derivation of the dynamical system for a viscous fluid is presented in this section. We consider three fluid layers confined between two infinitely long horizontal parallel planes of thickness 2b. It has been shown by Yih [18] for stratified fluids that stability or instability of threedimensional disturbances can be determined from that of the corresponding two-dimensional disturbances. Hence, it is sufficient to consider only two-dimensional disturbances. Therefore, without any loss of generality, only Cartesian coordinates (x, y) are used with the y-axis being taken vertically. The z-axis is ignored. A horizontal fluid sheet of finite thickness 2a and density ρ (2) occupies the region −a < y < a. This layer is assumed to be an incompressible, viscous, and dielectric fluid. Its viscosity coefficient and dielectric constant are µ(2) and (2) , respectively. The middle plane of the fluid sheet is taken to be y = 0. This layer is embedded in two fluids, the upper fluid having density ρ (1) and bounded by the conducting plane y = b (b > a) which is raised to the periodic potential (φ0 cos ωt). The lower fluid with density ρ (3) is bounded from below by an earthed conducting plane y = −b. Both fluids are viscous and incompressible. The superscripts (1), (2), and (3) refer to quantities in the upper, middle, and lower fluid layers, respectively. Also, we consider that the primary flow state is given (j ) by three uniform velocities v0 (j = 1, 2, 3) along the positive x-direction. The temperatures at y = b, a, −a, and −b are taken as τ1 , τ2 , τ3 , and τ4 , in order. Gravity is taken to be in the negative y-direction. A sketch of the system under consideration is given in Fig. 1. As a result of the potential difference between the planes y = ±b, a normal electric field is produced in the three
regions. This field may be represented by (j )
E(j ) = −E0 cos ωtey
(j = 1, 2, 3),
(2.1)
where ω is the frequency of the applied normal electric field and ey is the unit vector along the y-direction. Consequently, the potential φ0 may be written as φ0 = a E0(1) − 2E0(2) + E0(3) − b E0(1) + E0(3) cos ωt. (2.2) The problem considered here is the stability of a dielectric miscible viscous fluid layer sandwiched between two dielectric miscible viscous fluids and stressed by a normal periodic electric field. The analysis of the present study focuses on the case where the interfaces are ideally conducting and have no surface charges. Two forces must be accounted for by the stress tensor σij : First, the surface force, which results from the Newtonian effect, is given by [8] ∂vj ∂vi (vis) + , σij = −P δij + µ (2.3) ∂xj ∂xi where P is the hydrostatic pressure and δij is the Kronecker delta. Second, the body force, which is caused by the electrical forces, is given by [19] 1 1 ∂ F = ρf E − E 2 ∇ + ∇ E 2 ρ (2.4) , 2 2 ∂ρ where ρf is the free charge density, is the dielectric constant, ρ is the mass density of the fluid, and ∇ is the gradient operator. The first term is the free charge force density, the second term is due to inhomogeneities in the dielectric, and the last one results from changes in the material density. This last term is called the electrostriction force density. Manipulation of Eq. (2.4), which incorporates the irrotational nature of the electric field intensity, shows that the stress tensor representation of combined free charge and polarization force densities is 1 ∂ (ele) , σij = Ei Ej − δij Ek Ek − ρ (2.5) 2 ∂ρ
Fig. 1. Schematic diagram of the system under consideration.
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where ∂σij . Fj = ∂xi
(2.6)
The resultant stress tensor acting on the system is then given by σij = σij(vis) + σij(ele) .
(2.7)
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during perturbation [19]. A number of simplifications of Maxwell’s equations are appropriate to the description of electrodynamic phenomena. The fluids under consideration are poor conductors compared with fluids such as mercury and other liquid metals and so induced magnetic effects are very small. In electrodynamics, it is generally assumed that the quasi-static approximation is valid [21]. Thus, the electrical equations are
3. Basic equations
∇ ∧ E = 0,
(3.6)
The equation that governs the behavior of the viscous fluid is [8] ∂v + (v · ∇)v = −∇P + µ∇ 2 v − ρgey , ρ (3.1) ∂t
∇ · E = 0.
(3.7)
associated with incompressibility condition ∇ · v = 0,
(3.2)
where v = (u, v, 0) is the fluid velocity. Because of the intricacy of the problem for the generalization to the Kelvin–Helmholtz flow of a viscous dielectric fluid, we confine the analysis to weak viscous effects [20]. If the viscous forces are very small in comparison with nonviscous ones, then viscosity produces only a thin weak vertical layer at the surfaces of separation, while the motion remains irrotational throughout the bulk of the fluids. Thus, the derivation in this problem deals completely with potential flow so that the complicated manipulation of the boundarylayer equations for the weak vortical flow can be avoided. In the present analysis, viscous effects are extremely weak and they can be incorporated into the boundary conditions. Therefore, and in view of the weak viscous effects considered here, the governing equation for the bulk of a fluid phases becomes ∂v + (v · ∇v) = −∇P − ρgey . ρ (3.3) ∂t Since the configuration must satisfy this equation of motion in the equilibrium state, the equilibrium solution then becomes (j )
P0
= ρ (j ) gy + C (j ) (t)
(j = 1, 2, 3),
(3.4)
where C (j ) (t) is an arbitrary time-dependent function of time. From the continuity of the normal stresses at the interfaces, one gets C
(j +1)
(j +1)
(j +1)
(t) − C (t) = (−) (ρ − ρ 2 1 (j ) (j )2 (j +1) + E0 − (j +1) E0 cos2 ωt 2 (j = 1, 2).
(j )
(j )
)ga
(3.5)
No volume charges are assumed to be present in the bulk of the fluids. Also, because of the continuity of the electric field, no surface charges are present at the interface in the equilibrium state and the electric field will, therefore, vanish
Therefore, one can construct an electrostatic potential φ (j ) (x, y, t) such that (j )
E(j ) = E0 cos ωt ey − ∇φ (j )
(j = 1, 2, 3).
(3.8)
Therefore, the electrostatic potential satisfies the Laplace equation ∇ 2 φ (j ) = 0 (j = 1, 2, 3).
(3.9)
To facilitate the stability analysis, a simplified formulation of the interfacial flow problems with mass and heat transfer has been employed by Hsieh [11,22]. In his formulation, the effect of mass and heat transfer on the dynamics of the system is expressed through an interfacial condition which makes use of the equilibrium temperature distribution of the system. In this work, we extend Hsieh’s [11] simplified formulation to include the double-interface problem. Therefore, the influence of mass and heat transfer is considered only through the interfacial boundary conditions.
4. Perturbation equations In what follows, the equations used to assess the linear stability of the uniform flow equations are presented; to this end, the same coordinate system as illustrated in Fig. 1 is used. All disturbances are assumed to be two dimensional. In addition, it is assumed that the fluid interfaces have surface tension coefficient Tj (j +1) , where the label j refers to the upper fluid, and that there are no surfactants present there. The amplitude of waves formed on the fluid sheet is assumed to be small. For a small departure from the equilibrium state, every physical perturbed quantity may be expressed in the form F1 (x, y; t) = f1 (y, t)eikx ,
(4.1)
where k is the wave number and is assumed to be real and positive, and F1 stands for each of φ1 , P1 , and v1 . These, expansions are introduced into the governing equations and the relevant boundary conditions. The linearized terms in these perturbed quantities are only maintained in view of the linear stability theory [8]. To perform a linear stability analysis of the present problem, the interfaces between the three fluids will be
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assumed to be perturbed about their equilibrium locations to cause displacements of the material particles of the fluid system. Consider the effect of small wave disturbances to the interfaces y = ±a, propagating in the positive x-direction. Assuming that the surface deflections are given by y = (−)j +1 a + ξj
(j = 1, 2),
(4.2)
5.1. Maxwell’s electric conditions Because of wave propagation, attention is restricted to the normal electric field where no surface charges are present on the interfaces. It follows that the equilibrium electric field intensity satisfies the relation [21] (j +1)
(j ) E0 − (j +1) E0 (j )
where ξj = γj (t)eikx ,
(4.3)
γ1 (t) and γ2 (t) are arbitrary time-dependent functions which determine the behavior of the amplitude of the disturbances on the interfaces. These functions are determined after applying the appropriate boundary conditions. According to linear perturbation theory [8], the unit vectors normal to the interfaces are given by ∂ξj ex + ey (j = 1, 2), nj = − (4.4) ∂x where ex is the unit vector along the x-direction. The equations of motion and the relevant boundary conditions will be solved for these perturbations under the assumption that the perturbations are small; that is, all equations will be linearized in the perturbation quantities. Actually, the equations governing the perturbation quantities are readily found to satisfy Laplace’s equations (j )
(4.5)
(j )
(4.6)
∇ 2 P1 (x, y; t) = 0, ∇ 2 φ1 (x, y; t) = 0 (j = 1, 2, 3).
As a result of the perturbations P1 , v1 , and φ1 and in view of the dependence given by Eq. (4.1), we have the solutions in the three fluid layers as ρ (j ) d (j ) (j ) + ikv0 P1 = − k dt (j ) × C (t) cosh ky + D (j ) (t) sinh ky eikx , (4.7) (j ) v1 = C (j ) (t) sinh ky + D (j ) (t) cosh ky eikx ,
(4.8)
(j ) φ1 = A(j ) (t) cosh ky + B (j ) (t) sinh ky eikx (j = 1, 2, 3), A(j ) (t),
C (j ) (t),
(j )
(j )
y = (−)j +1 a
= 0,
(j = 1, 2).
(5.2)
In addition, the jump in the tangential component of the electric field is zero across the interfaces. This requires (j ) (j +1) ∂φ ∂φ ∂ξi (j ) (j +1) E0 − E0 = 0, cos ωt + 1 − 1 ∂x ∂x ∂x y = (−)j +1 a (j = 1, 2).
(5.3)
The electric part is not completely specified until the electrical properties of the external electrodes are given. In an experimental apparatus, these electrodes provide the electric potential difference that induces surface charges on the interfaces. It is, therefore, reasonable to assume that they are perfectly conducting so as to form an equipotential. Then the boundary conditions on the electric field at y = ±b, as a consequence of the fact that the electric field is zero inside the conductor, are the same as condition (5.3). Therefore, at the rigid boundaries, one gets (j )
∂φ1 = 0, ∂x
y = (−)j +1 b
(j = 1, 2).
(5.4)
Applying the boundary conditions (5.2), (5.3), and (5.4) on the general solution (4.9), one gets (1) E0 (2) ( − (1) ) ∗ × (2) tanh k(b − a) sinh 2ka + (3) cosh 2ka γ1 sinh k(b − y) cos ωt eikx , + ( (3) − (2) ) (1)γ2 cosh k(b − a) (5.5)
φ1 = −
where and are arbitrary functions of time which are to be determined by making use of the appropriate boundary conditions.
The solutions given in the previous section must satisfy certain boundary conditions through which the arbitrary functions A(j ) (t), B (j ) (t), C (j ) (t), and D (j ) (t) may be identical. For the problem under consideration, the appropriate boundary conditions may be listed as follows.
(5.1)
(j +1)
∂φ1 ∂φ − (j +1) 1 ∂y ∂y
D (j ) (t)
5. Boundary conditions and solutions of the perturbed functions
(j = 1, 2).
To verify this fact, the continuity of the normal component of electric displacement at the surfaces of separation is applicable, and this requires that
(1)
(4.9) B (j ) (t),
=0
(2)
φ1 =
(2) E0 (2) ( − (1)) (2) tanh k(b − a) cosh k(y + a) ∗ + (3) sinh k(y + a) γ1 − ( (3) − (2) ) (2) tanh k(b − a) cosh k(y − a) − (1) sinh k(y − a) γ2 cos ωt eikx , (5.6)
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
(3) φ1
E0(3) (2) = − ∗ ( − (3) ) (2) × tanh k(b − a) sinh 2ka + (1) cosh 2ka γ2 sinh k(b + y) + ( (1) − (2)) (3) γ1 cos ωt eikx , cosh k(b − a) (5.7)
As indicated by Hsieh [11], we may expand his formula as χ1 (y) =
K (1) (τ2 − τ1 ) K (2) (τ3 − τ2 ) − , b−y a+y
(5.10)
χ2 (y) =
K (2) (τ3 − τ2 ) K (3) (τ4 − τ3 ) − . a−y b+y
(5.11)
where ∗ = (2) ( (1) + (3) ) tanh k(b − a) cosh 2ka + (2)2 tanh2 k(b − a) + (1) (3) sinh 2ka. 5.2. Kinematical conditions Hsieh [11] formulated the interfacial boundary conditions for a single interface. These conditions are necessary to obtain the unknowns coming from solutions of the hydrodynamic equations of motion. They include the conservation of mass and energy across the interface, accounting for interfacial transfer of mass due to heat transfer and latent heat effects. Hsieh [11] assumed that the amount of the released latent heat depends mainly on the instantaneous position of the interface. Hsieh’s [11] interfacial conditions, for a single interface, must be expanded to adopt the double interfaces in the problem under consideration. Therefore, let us define the function Sj (x, y; t) = y + (−)j a − ξj
(j = 1, 2),
(5.8)
where Sj (x, y; t) = 0 describes the wave like profile of the disturbed interfaces. The kinematical boundary conditions at the interfaces are as follows: (i) The interfacial conditions of energy transfer may be expressed as ∂Sj (j ) + v1 · ∇Sj = χj (ξj ) (j = 1, 2), Lj ρ (j ) (5.9) ∂t where Lj is the latent heat of transformation from the fluid of density ρ (j ) to the fluid of density ρ (j +1) , and χj (ξj ) is the perturbed heat flux at the interface used for evaporation or condensation. χj is a function of the instantaneous profile of the interface and is determined from the heat transfer relation at equilibrium. As shown by Hsieh [22], the heat fluxes have to be determined from equations governing the heat transfer in the fluids, thus completely coupling the dynamics and the thermal exchanges in the entire flow region. In this simplified version, the expression is that χj simply a function of ξj , and moreover, χj is to be determined from the heat exchange relations in the equilibrium state. Let us consider the heat fluxes in the positive y-direction in regions (3), (2), and (1) as given by K (3)(τ4 − τ3 )/(b − a), K (2) (τ3 − τ2 )/(2a), and K (1) (τ2 − τ1 )/(b − a), where K (1) , K (2) , and K (3) are the heat conductivities of the three-phase fluids, in order.
191
The perturbed heat flux at the interface results in evaporation or condensation at the troughs or crests of the interfacial wave, where the interface is located deep in the hotter vapor or cooler liquid, respectively. Hsieh [11] used a convenient way of representing the perturbed heat flux by a Taylor series approximation about the point of equilibrium with respect to interfacial displacements. Hsieh [11] argued that only the linear terms of the series should enter the linear stability analysis and, therefore, truncated the series beyond the second terms. Therefore, the perturbed heat flux χj (ξj ) may be expanded in the form χj (ξj ) = χj (−1)j +1 a + αj (j +1 )Lj ξj , (5.12) where αj (j +1) is the measure of the rate of interfacial heat and mass transfer across the interfaces. The basic state of heat flux to the interface that is conducted through the vapor is assumed to be equal to the heat conducted from the interface into the liquid. Therefore, at the basic state of the unperturbed interface, no heat is used for evaporation or condensation. It follows that χj ((−)j +1 a) represents the net heat flux from the interface into the fluid regions (j ) and (j + 1). Since it is an equilibrium state, we have χj (−)j +1 a = 0. (5.13) It follows that K (1) (τ2 − τ1 ) K (2) (τ3 − τ2 ) K (3)(τ4 − τ3 ) = = . b−a 2a b−a (5.14) Hence we set 1 G 1 + αj (j +1) = (5.15) (j = 1, 2). Lj b − a 2a G=
If liquid (3) is hotter than liquid (2) at the same time that liquid (2) is hotter than liquid (1) (i.e., τ4 > τ3 > τ2 > τ1 ), then G and Lj are simultaneously positive. On the other hand, if (τ4 < τ3 < τ2 < τ1 ), then G and Lj are both negative. Therefore, in both cases αj (j +1) is always positive. These parameters represent the contribution of heat and mass transfer to the problem. Thus, correlation of experimental data would be greatly facilitated by this simplification. (ii) The conservation of mass across the interfaces [22] leads to (j ) (j +1) (j ) ∂Sj (j +1) ∂Sj + v1 · ∇Sj = ρ + v1 · ∇Sj , ρ ∂t ∂t y = (−)j +1 a
(j = 1, 2).
(5.16)
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(iii) At the rigid boundaries, the normal velocities must vanish and so one gets (1)
y = b,
(5.17a)
(3)
y = −b.
(5.17b)
v1 = 0, v1 = 0,
Applying the foregoing boundary conditions (5.9), (5.16), and (5.17a), (5.17b) to the general solutions given by Eqs. (4.7) and (4.8), one gets the special solutions of the perturbed (j ) (j ) quantities v1 and P1 as follows: sinh k(b − y) d α12 + ikv0(1) + (1) γ1 eikx , (5.18) v1(1) = sinh k(b − a) dt ρ
d 1 α12 (2) (2) sinh k(y + a) + ikv0 + (2) γ1 v1 = sinh 2ka dt ρ
α23 d (2) + ikv0 + (2) γ2 eikx , − sinh k(y − a) dt ρ (5.19) (3) v1
(1) P1
P1(2)
sinh k(b + y) d α23 (3) + ikv0 + (3) γ2 eikx , = sinh k(b − a) dt ρ
(5.20)
ρ (1) d 2 α12 d (1) = + 2ikv0 + (1) k dt 2 dt ρ
cosh k(b − y) ikx α12 e , + ikv0(1) ikv0(1) + (1) γ1 sinh k(b − a) ρ (5.21) d2 ρ (2) α12 d (2) =− + 2ikv + 0 k sinh 2ka dt 2 ρ (2) dt
α12 (2) (2) + ikv0 ikv0 + (2) γ1 cosh k(y + a) ρ α23 d d2 (2) + 2ikv0 + (2) − dt dt 2 ρ
α23 (2) (2) + ikv0 ikv0 + (2) γ2 cosh k(y − a) eikx , ρ (5.22)
and (3) P1
ρ (3) d 2 α23 d (3) =− + 2ikv + 0 k dt 2 ρ (3) dt
α23 cosh k(b + y) ikx (3) (3) e . + ikv0 ikv0 + (3) γ2 sinh k(b − a) ρ (5.23)
5.3. Stress tensor conditions At the fluid interfaces, two sets of stress tensor conditions determine the shape of the surfaces as follows: (i) The normal component of the stress tensor is discontinuous by the amount of the surface tension Tj (j +1) . Therefore, the balance at the dividing surfaces gives (j +1) (j ) P1 − P1 + (ρ (j ) − ρ (j +1) )g − k 2 Tj (j +1) ξj (j ) (j +1) (j ) (j ) ∂φ1 (j +1) (j +1) ∂φ1 − cos ωt + E0 E0 ∂y ∂y (j ) (j +1) (j ) ∂v1 (j +1) ∂v1 +2 µ −µ = 0, ∂y ∂y y = (−)j +1 a
(j = 1, 2),
(5.24)
where Tj (j +1) is the surface tension coefficient of the surfaces separating fluid j from fluid j + 1. (ii) The tangential component of the stress tensor is continuous across the interfaces. The balance of the stresses due to perturbations along the interfaces requires that (j ) (j +1) (j ) (j ) ∂φ1 (j +1) (j +1) ∂φ1 E0 − E0 cos ωt ∂x ∂x ∂ξj (j ) (j )2 (j +1)2 + cos2 ωt E0 − (j +1) E0 ∂x 2 2 (j ) (j +1) (j ) ∂ 2 (j +1) ∂ 2 + k v1 − µ + k v1 = 0, +µ ∂y 2 ∂y 2 y = (−)j +1 a
(j = 1, 2).
(5.25)
6. The characteristic equations The aim in this section is to complete the boundary-value problem. It consists of (1) a homogenous set of equations of motion, (2) a collection of appropriate boundary conditions, and (3) corresponding consistent solutions. All these components explain the time-dependent functions governing surface wave propagation. Equations that determine the surface deflections are called the characteristic equations. Because there are no surface charges on the unperturbed interfaces y = ±a, condition (5.1) will be taken into account in driving the characteristic equations. For inviscid fluids it is known that, in the absence of surface charges, conditions (5.2) and (5.25) are equivalent, but this is not so in the presence of viscosity. Therefore, there exists a contribution for condition (5.25), given by dγj + αj (j +1) (ν (j ) − ν (j +1) ) dt (j ) (j +1) γj = 0 (j = 1, 2), + ik µ(j ) v0 − µ(j +1) v0
(µ(j ) − µ(j +1) )
(6.1) where ν (j ) = µ(j ) /ρ (j ) (the kinematic viscosity) is used.
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199 (j )
(j )
(j )
Substituting the solutions φ1 , v1 , and P1 (j = 1, 2, 3) into the normal stress condition (5.24), after straightforward calculations, one obtains the coupled equations d 2 γ1 dγ1 + (f11 + il11 ) 2 dt dt
(2)2 + s11 + k11E0 cos2 ωt + ih11 γ1 dγ2 + (f22 + il22 ) dt
2 + s22 + k22E0(2) cos2 ωt + ih22 γ2 = 0,
(6.2)
dγ2 d 2 γ2 + (f12 + il12 ) 2 dt dt
(2)2 + s12 + k12E0 cos2 ωt + ih12 γ2 dγ1 + (f21 + il21 ) dt
(2)2 + s21 + k21E0 cos2 ωt + ih21 γ1 = 0,
which are two coupled Mathieu equations having damping terms and complex coefficients. These coefficients are defined in Appendix A. By making use of these equations, the stability behavior of the fluid sheet is controlled. The aim of this linear eigenvalue problem is to determine the surface elevation amplitudes γ1 and γ2 as time-dependent functions for the given physical geometrical and flow properties. The combination of the system of equations (6.1), (6.2), and (6.3) will lead to a simplification of the characteristic equations.
7. The symmetric and antisymmetric analysis The solution of the coupled Mathieu equations (6.2) and (6.3) can be simplified by considering the symmetric and antisymmetric deformations of the surface deflections ξ1 and ξ2 . Therefore, the variables ξ1 and ξ2 may be connected by ξ2 = J ξ1 = ξ,
where the constant coefficients a’s, b’s, c’s, and d’s are given by 1 mf1j + (µ(2j −1) − µ(2) ) aj = m (5−2j ) × µ(5−2j )v0 − µ(2)v0(2) h1j + J mf2(3−j ) + (µ(2) − µ(5−2j ) ) (2j −1) × µ(2)v0(2) − µ(2j −1)v0 h2(3−j ) , bj = l1j + J l2(3−j ), cj =
(6.3)
(7.1)
where J = 1 refers to the antisymmetric deformation, while J = −1 defines the symmetric one. In solving the characteristic equations (6.2) and (6.3), in view of symmetric and antisymmetric deformations, we use definition (7.1), keeping in mind the condition (6.1), which allows us to eliminate the imaginary part of γj . This procedure leads to
193
1 ms1j + αj (j +1) (ν (2j −1) − ν (2) ) m (5−2j ) (2) × µ(5−2j ) v0 − µ(2) v0 h1j + J ms2(3−j ) + α(3−j )(4−j )(ν (2) − ν (5−2j ) ) (2j −1) (2) h2(3−j ) , × µ(2) v0 − µ(2j −1) v0
and dj = k1j + J k2(3−j ), with (2) (1) (3) (2) m = k µ(2)v0 − µ(1) v0 µ(3) v0 − µ(2) v0 . Therefore, Eqs. (7.2) and (7.3) allow the boundary-value problem to be obtained entirely in terms of the unknown time-dependent function γ . It is easy to show that this system of equations, (7.2) and (7.3), may be reduced to a single Mathieu equation with a real damped term. The elimination of the constants b1 and b2 from these equations yields the damped Mathieu equation dγ ∗ d 2γ (2)2 + b0 + c0∗ E0 cos2 ωt γ = 0, + a0∗ 2 dt dt where a0∗ , b0∗ , and c0∗ are given by a1 b2 − a2 b1 , b2 − b1 d1 b2 − d2 b1 . c0∗ = b2 − b1
a0∗ =
b0∗ =
c1 b 2 − c2 b 1 , b2 − b1
(7.4)
and
The following analysis may be classified to include the case of a uniform normal electric field as well as the periodic one. 7.1. Stability analysis in the presence of a uniform electric field
d 2γ dγ (2)2 + c1 + d1 E0 cos2 ωt γ = 0, (7.2) + (a1 + ib1 ) 2 dt dt
For a uniform normal electric field, the periodicity of the field will be absent. Therefore, wave propagation is excited by using the electro-capillarity technique. The damped Mathieu equation (7.4) then becomes
2 d 2γ dγ + c2 + d2 E0(2) cos2 ωt γ = 0, (7.3) + (a2 + ib2 ) 2 dt dt
2 dγ ∗ d 2γ + b0 + c0∗ E0(2) γ = 0, + a0∗ 2 dt dt
(7.5)
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which is a linear differential equation with constant coefficients. It can be satisfied by a growth rate solution having the form γ (t) = exp(σ0 + iω0 )t,
(7.6)
where σ0 and ω0 are assumed to be real constants. This assumption imposes the following stability conditions: (2)2
E0
>
1 ∗2 a0 − 4b0∗ , ∗ 4c0
c0∗ > 0,
(7.7)
<
1 ∗2 a0 − 4b0∗ , ∗ 4c0
c0∗ < 0.
(7.8)
or (2)2
E0
In contrast to the linear stability theory of a pure inviscid fluid, it is found that the normal electric field intensity plays a dual role on the stability criteria. This role depends on the sign of the constant c0∗ . The goal, in what follows, is to determine a numerical calculation for the stability picture in the case of a uniform normal electric field. The theoretical analysis includes the two modes of surface deformation: the symmetric deformation (J = −1) and the antisymmetric one (J = 1). Since the calculations are too extensive, the following numerical calculations considered only the antisymmetric deformation (J = 1) as an example. In spite of this, Fig. 2b includes the case of a symmetric mode (J = −1). To screen the examination of the uniform normal electric field on the stability criteria, numerical calculations for the stability conditions (7.7) and (7.8) are made. The results are 2 displayed in Figs. 2a, 2b, and 3 to indicate the (E0(2) − k) plane. The sample chosen in these figures makes the system statically stable in the absence of the electric field. The plane 2 (E0(2) − k) is partitioned by the transition curve (7.8) into two regions. The calculations show that the constant c0∗ is negative. Therefore, the upper region of the transition curve (7.8) is unstable while the lower one is stable. The upper unstable region is labeled by the letter U , and the lower stable one is referred to by the letter S. The presence of the uniform electric field plays a destabilizing role, which is an early phenomena through the linear stability theory found by Melcher [21] and by several other researchers for inviscid flow. The same sense is obtained by El-Dib and Matoog [10]. Melcher [21] demonstrated that in the linear stability theory, the tangential field has a stabilizing effect (it increases the surface tension effect), while the normal field has a destabilizing influence (it decreases the surface tension effect). The examination of the influence of the mass and heat transfer parameters and the sheet thickness on the stability criteria for surface wave propagation is the aim of the present calculations. These calculations are depicted in Figs. 2 and 3, (2)2
respectively. In Fig. 2a, the plane (E0 − k) displays the natural stability curve (7.8) at the fixed value α12 = 1 g/cm3 s for various values of the parameter α23 as shown
(a)
(b) (2)2
− k) for a system having Fig. 2. (a) The stability diagram (E0 ρ (1) = 0.24 g/cm3 , ρ (2) = 0.65 g/cm3 , ρ (3) = 1.3 g/cm3 , a = 0.1 cm, (1) (2) (3) b = 1 cm, v0 = 5 cm/s, v0 = 10 cm/s, v0 = 15 cm/s, (1) = 3.3, (2) (3) = 30, = 18, T12 = 29 dynes/cm, T23 = 35 dynes/cm, µ(1) = 2.5 g/cm s, µ(2) = 4 g/cm s, µ(3) = 6 g/cm s, g = 980.665 cm/s2 , J = 1, α12 = 1 g/cm3 s, and different values of α23 . The curves are drawn according to relation (7.8). (b) Stability diagram for the same system as considered in Fig. 2a, but in the case of a symmetric deformation (J = −1).
in the figure. It is apparent from inspection of this graph that increasing the values of the parameter α23 increases the stabilizing influence of the normal electric field. Figure 2b represents the case of a symmetric deformation (J = −1). The numerical calculations show that the constant c0∗ is also negative. The destabilizing influence of the mass and heat transfer parameter α23 is also seen in Fig. 2a. A comparison between Fig. 2a and Fig. 2b shows that the instability of the system is enhanced under the influence of the symmetric deformations. The influence of the sheet thickness on the stability picture is displayed in Fig. 3. The calculations on this figure are made for the same values of the parameters α12 and α23 (= 1g/cm3 s). It is apparent from inspection of this graph that increasing the sheet thickness increases the destabilizing effect of the normal field. Thus the increase of sheet thickness has a destabilizing influence, which has not been observed before for Newtonian fluids. In pure inviscid fluids, El-Shehawey et al. [2] demonstrated that the increase of the inviscid fluid layer has a stabilizing influence. Therefore, one can see that the flow has undergone a dramatic change in its behavior due to the presence of the
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
Fig. 3. Stability diagram for the same system as considered in Fig. 2a, except α12 = α23 = 1 g/cm3 s and for different values of a. The curves are drawn according to relation (7.8).
195
Fig. 4. Stability diagram for the same system as considered in Fig. 1, but when ω = 5 Hz. The figure indicates the transition curves (7.14).
viscous factor. The same result is also found in the presence of the viscoelastic factor [10]. 7.2. Stability behavior due to the periodicity of the field In the presence of the periodicity of the normal electric field, the stability picture has changed dramatically. In this case, Eq. (7.4) may be written as dγ d 2γ + (b0 + c0 cos τ )γ = 0, + a0 dτ 2 dτ where τ = 2ωt is used, and a0 = a0∗ /2ω, (2)2
c0 = c0∗ E0
b0 =
b0∗
2 + c0∗ E0(2) /2 4ω2 ,
(7.9) Fig. 5. Stability diagram for the same system as considered in Fig. 4, except α12 = 1 g/cm3 s, α23 = 2 g/cm3 s and for different values of ω. The figure indicates the transition curves (7.14).
and
/8ω2 .
Consider a0 > 0 and b0 > 0. If we write γ (τ ) = e−a0 τ/2 η(τ ),
(7.10)
Eq. (7.9) is transformed to d 2η + (d0 + c0 cos τ )η = 0, dτ 2
(2)
In terms of the electric field E0 , conditions (7.13) may be rewritten in the form (7.11)
which is the standard form the Mathieu’s equation, with d0 = b0 − a02 /4.
periodic 4π . Jordan and Smith [23] deduced the stability criterion (b0 − 1/4)2 − c02 − a02 4 > 0. (7.13)
(7.12)
Jordan and Smith [23] try to construct part of a diagram of the stable regions for Eq. (7.9). Certainly, whenever d0 and c0 have values such that only bounded solutions of Eq. (7.11) exist, then only bounded solutions of Eq. (7.9) exist, by Eq. (7.11). Also, by Eq. (7.10), even some exponential growth in η is allowed, with γ still remaining bounded. Therefore, the regions of Eq. (7.9) will be the extensions of those for Mathieu’s equation. Jordan and Smith [23] have examined particularly the first unstable region for Eq. (7.11), which occurs near d0 = 1/4 or near b0 = (1/4)(1 + a02 ). On the boundaries of this unstable region, there are periodic solutions of Eq. (7.11) of
(2)2
E0
> E1∗
(2)2
or E0
< E2∗ ,
(7.14)
provided that E1∗ > E2∗ , where 2 4 ∗ E1,2 = ∗ −2 b0∗ − ω2 ± b0∗ − ω2 − 3a0∗2 ω2 3c0 (7.15) are the transition curves separating the stable from unstable regions. According to the Floquet theory [5], the region bounded by the two branches of the transition curves E1∗ and E2∗ is unstable. The area outside these curves is a stable one. The width of the unstable region is represented by (E1∗ − E2∗ ). The increase of this width refers to the destabilizing influence, while its decrease represents a stabilizing effect. In what follows, numerical calculations are made for the stability condition (7.13). The results of calculations for the
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transition curves (7.14) are displayed in Figs. 4 and 5 to indicate the effect of the mass and heat transfer parameters and influence of the field frequency, respectively. The graphs (2)2
are displayed in the plane (E0 − k) for the same system as considered in Fig. 2 with some changes. In Fig. 4, the 2 plane (E0(2) − k) is displayed to indicate the influence of the mass and heat transfer parameter α23 . The plane is partitioned by the transition curves E1∗ and E2∗ into stable and unstable regions. The stability picture is drawn at a single value for α12 and various values of α23 , as shown in the figure. It is found that an increase of the parameter α23 increases the stability regions. Therefore, in contrast to the case of the uniform electric field, the increase of the parameter α23 increases the stabilizing influence of the normal periodic field. In Fig. 5, the stability picture is due to the presence of the field frequency ω. It is found that as the periodicity increases, the width of the unstable region bounded by (E1∗ − E2∗ ) decreases, which shows a stabilizing influence of the frequency of the normal field intensity.
8. Analysis of the marginal state The single governing Eq. (7.4) for all symmetric and antisymmetric modes is used to discuss the stability picture for the surface wave propagation in the absence of the damped term a0∗ , which is the so-called the marginal state. This case may occur when the primary streaming velocities are the same as in the inviscid flow, considering also the absence of the mass and heat parameters. The marginal state may occur mathematically by eliminating a1 and a2 between Eqs. (7.2) and (7.3) and using the transformation i a2 b1 − a1 b2 t γ (t) = Γ (t) exp − (8.1) 2 a2 − a1 to obtain the normal form 2 d 2Γ + p + qE0(2) cos2 ωt Γ = 0, (8.2) dt 2 which has been extensively studied. This equation, which is a linear differential equation with periodic coefficients, is known as Mathieu’s equation. Equations similar to this appear in many problems in applied mathematics such as stability of a transverse column subject to a periodic longitudinal load, stability of periodic solutions of a nonlinear conservative system, electromagnetic wave propagation in a medium with periodic structure, and the excitation of certain electrical systems. Equation (8.2) may be used to discuss the stability near the marginal state in the presence of all physical parameters acted on the problem at hand. The new constants p and q are defined as a 2 c1 − a 1 c2 1 a 2 b 1 − a 1 b 2 2 + and p= a2 − a1 4 a2 − a1 a2 d1 − a1 d2 , q= (8.3) a2 − a1
The stability analysis may be classified into two cases: first, stability behavior in the presence of a uniform normal electric field in the absence of the field frequency, and second, stability examination in the influence of the periodic normal field. 8.1. Numerical simulation in the static case In the absence of the periodicity of the applied normal field ω → 0, wave propagation is excited using the electrocapillary technique. In this case, Eq. (8.2) reduces to 2 d 2Γ + p + qE0(2) Γ = 0. (8.4) 2 dt Equation (8.4) may be satisfied by the following solution: (2)2 Γ (t) = exp i p + qE0 t. (8.5)
Therefore, in the static case, the stability criterion reduces to 2
p + qE0(2) > 0,
(8.6)
which is automatically satisfied if p and q are positive. Otherwise, stability occurs when p (2)2 E0 > − , q > 0, (8.7) q or 2 p E0(2) < − , q < 0. (8.8) q Again, the uniform normal electric field intensity has a dual role in the stability picture. This role depends mainly on the sign of the constant q. To screen the contribution of the uniform normal electric field on the stability criteria, numerical calculations are made for the stability conditions (8.6). Only the case of antisymmetric interface deformation (J = 1) is considered in the following numerical calculations. The results for the calculations are displayed in Figs. 6–8 to indicate the (2)2
plane (E0 − k) with a transition curve as a function of all the physical parameters acting on the system. The examination of the influence of the sheet thickness and the mass and heat transfer parameters are the goal of the present calculations, which are displayed in these figures. In Fig. 6, the graph displays the natural stability curve (8.6) for various values of the sheet thickness a as shown in the figure. According to the physical parameters chosen in the figure, it is found that q > 0. Thus only the transition curve (8.7) is considered. It follows that the region above the natural curve is stable, while that below it is unstable. Here, the uniform normal electric field plays a stabilizing influence on the system under consideration. It is apparent from inspection of this graph that increasing sheet thickness increases the stabilizing influence of the normal field. A comparison between Figs. 3 and 6 shows that the sheet thickness plays a dual role in the stability criteria. The influence of mass and heat transfer parameters are displayed in Figs. 7 and 8.
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
197
8.2. Stability criterion in the oscillated case In the case the electro-capillary excitation was switched off, and the field was oscillated at the frequency ω, Eq. (8.2) may be rewritten in the form d 2Γ + (pˆ − 2qˆ cos 2ζ )Γ = 0, dζ 2
(8.9) (2)2
Fig. 6. Stability diagram for the same system as considered in Fig. 3.
Fig. 7. Stability diagram for the same system as considered in Fig. 6, except at fixed values of a = 0.5 cm, α23 = 1 g/cm3 s and for various values of α12 . The curves are drawn according to relation (8.7).
where ζ = ωt is used, and pˆ = (p + qE0 /2)/ω2 and qˆ = (2)2 −qE0 /4ω2 . Equation (8.9) is the canonical form of Mathieu differential equation. The properties of the Mathieu functions are well known [5]. The solutions of the Mathieu equation can be, under certain conditions, periodic and the system is then stable. The condition for the periodic Mathieu functions depends of the relation between the parameters pˆ and q. ˆ The p– ˆ qˆ plane is divided into stable and unstable regions bounded by the characteristic curves of Mathieu functions. The general solution of Eq. (8.9) is stable if the point (p, ˆ q) ˆ in the p– ˆ qˆ plane lies in the stable region; otherwise it is unstable. The stability condition reduces the problem to the bounded regions of the Mathieu functions, which gives the criterion [5] >(0) sin2 π pˆ 1/2 /2 1, (8.10) where >(0) is the infinite Hill’s determinate. In general, the analysis of inequality (8.10) is rather complicated due to the infinite Hill’s determinate. An approximate formula of stability, given by Morse and Feshbach [24], may be used for small values of q, ˆ which is a good approximation to high-frequency fields. According to this assumption the system is stable if 4
2
q 2 E0(2) + 16(ω2 − p)qE0(2) + 32p(ω2 − p) > 0.
(8.11)
In terms of the electric field E0(2) , this condition may be arranged in the form
Fig. 8. Stability diagram for the same system as considered in Fig. 3, except at fixed values of α23 = 1 g/cm3 s and for various values of α12 . The curves are drawn according to relations (8.7) and (8.8).
In Fig. 7, as before, only one transition curve is considered where q > 0. The graph shows various curves according to the variation of the parameter α12 . As α12 increases, the field shows a destabilizing influence. In Fig. 8, the graph includes the two cases q > 0 and q < 0. In the case when q > 0, stability is enhanced as α23 > α12 . In contrast, when q < 0 stability decreases. It follows that the parameter α23 plays a dual role in the stability criteria.
(2)2
E0
(2)2 − Eˆ 1 E0 − Eˆ 2 > 0,
(8.12)
where Eˆ 1 and Eˆ 2 are the stability boundaries, which are given by 8 3 Eˆ 1,2 = −(ω2 − p) ± (ω2 − p) ω2 − p . (8.13) q 2 Therefore, the stability criteria become (2)2
E0
> Eˆ 1
(2)2
or E0
> Eˆ 2 ,
(8.14)
provided that Eˆ 1 > Eˆ 2 , which are the transition curves separating the stable from unstable regions. From Floquet theory [5], the region bounded by the two branches of the transition curves Eˆ 1 and Eˆ 2 is unstable, while the region outside these curves is stable.
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2 2αj (j +1) ν (2) × coth k(b − a) + ρ (2) v0(2) − ρ (2)
9. Conclusion In this study, we have presented the problem of electrohydrodynamic Kelvin–Helmholtz instability of a horizontal fluid sheet of finite thickness embedded between two finite fluid layers. The system has been analyzed including the effect of interfacial tension, mass and heat transfer parameters, and weak viscous effects. It is recognized that viscous effects are demonstrated in the boundary conditions. The stability analysis is based on a linear perturbation theory. Through this perturbation analysis, the solution of the boundary-value problem leads to two simultaneous differential equations of Mathieu types, which are used to control the stability of the fluid sheet. Only the symmetric and antisymmetric deformations for the surface deflections are considered. This constraint leads to a single Mathieu equation. The analytical results are numerically confirmed. The stability examination yields the following results: (1) In the presentation of the problem, the effect of mass and heat transfer is revealed through two parameters α12 and α23 . Thus, correlation of experimental data would be greatly facilitated by this simplification. (2) The sheet thickness, as well as mass and heat transfer parameters, has a dual role on the stability criteria. (3) The field frequency ω has a stabilizing influence, especially at small values of the wave number. (4) In contract to the case of a pure inviscid fluid, the uniform normal electric field intensity plays a dual role in the stability picture.
The coefficients appearing in Eqs. (6.2) and (6.3) are 1 f1j = ∗ αj (j +1) + 2k 2 µ(2j −1) ρ × ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka × coth k(b − a) + αj (j +1) + 2k 2 µ(2) × ρ (5−2j ) coth k(b − a) cosh 2ka + ρ (2) sinh 2ka , l1j =
ρ∗
× ρ
(2j −1)
ρ (2j −1) v0
sinh 2ka
k 2 (2) ( (2j −1) − (2) ) ρ ∗ ∗ (2j −1) × ρ (2) (2j −1) ( (2) − (5−2j ) ) + ( (2j −1) − (2) ) × (2) tanh k(b − a) sinh 2ka + (5−2j ) cosh 2ka × ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka ,
k1j = −
h1j =
f2j =
s2j =
k (2j −1) 2 (2j −1) v α + 2k µ j (j +1) 0 ρ∗ × ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka × coth k(b − a) + v0(2) αj (j +1) + 2k 2 µ(2) × ρ (5−2j ) coth k(b − a) cosh 2ka + ρ (2) sinh 2ka , coth k(b − a) (2) ρ αj (j +1) + 2k 2 µ(2j −1) ∗ ρ − ρ (2j −1) αj (j +1) + 2k 2 µ(2) , 2kρ (2)ρ (2j −1) coth k(b − a) (2j −1) v0 − v0(2) , ∗ ρ k 2 ρ (2) ρ (2j −1) coth k(b − a) ρ∗ (2j −1) ν ν (2) (2j −1)2 (2)2 × v0 − v0 + 2αj (j +1) (2j −1) − (2) ρ ρ −
k2j =
kρ (2) Σj (j +1) , ρ∗
k 2 (2) ( (2j −1) − (2) ) ρ ∗ ∗ (2j −1) × ρ (2j −1) (2j −1) ( (5−2j ) − (2) ) × coth k(b − a) sinh 2ka + ρ (2) (2) × ( (5−2j ) − (2j −1) ) cosh 2ka + ( (2) − (2j −1) ) × tanh k(b − a) sinh 2ka ,
× coth k(b − a) + ρ (2) v0(2)
× ρ (5−2j ) coth k(b − a) cosh 2ka + ρ (2) sinh 2ka ,
s1j
coth k(b − a) cosh 2ka + ρ
(2)
(5−2j ) k Σ coth k(b − a) sinh 2ka ρ j (j +1) ρ∗ + ρ (2) cosh 2ka ,
× ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka
k 2 (2j −1) (2j −1)2 2αj (j +1)ν (2j −1) =− ∗ ρ − v0 ρ ρ (2j −1) (5−2j ) × ρ coth k(b − a) sinh 2ka + ρ (2) cosh 2ka
(5−2j )
−
l2j =
Appendix A
2k
h2j =
k coth k(b − a) (2) (2j −1) ρ v0 αj (j +1) + 2k 2 µ(2j −1) ∗ ρ (2j −1) (2) −ρ v0 αj (j +1) + 2k 2 µ(2) ,
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
Σj (j +1) = (ρ (j ) − ρ (j +1) )g − k 2 Tj (j +1), and ρ ∗ = ρ (2) (ρ (1) + ρ (3) ) coth k(b − a) cosh 2ka + ρ (2)2 + ρ (1) ρ (3) coth2 k(b − a) sinh 2ka.
References [1] B.G. Newman, M.P. Paidoussis, J. Fluid Struc. 5 (1991) 443. [2] E.F. El-Shehawey, Y.O. El-Dib, A.A. Mohamed, Nuovo Cimento D 6 (4) (1985) 291. [3] A.A. Mohamed, E.F. El-Shehawey, Y.O. El-Dib, J. Chem. Phys. 85 (1) (1986) 445. [4] S. Parhi, G. Nath, Int. J. Eng. Sci. 29 (11) (1991) 1439. [5] N.W. McLachlan, Theory and Application of Mathieu Functions, Clarendon, Oxford, 1964. [6] W. Magnus, S. Winkler, Hill’s Equations, Wiley, New York, 1966. [7] A.H. Nayfeh, D.T. Mook, Non-Linear Oscillations, Wiley, New York, 1979.
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[8] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, Oxford, 1961. [9] A.A. Mohamed, E.F. El-Shehawey, Y.O. El-Dib, J. Colloid Interface Sci. 207 (1998) 54. [10] Y.O. El-Dib, R.T. Matoog, J. Colloid Interface Sci. 229 (2000) 29. [11] D.Y. Hsieh, Phys. Fluids 21 (5) (1978) 745. [12] D.Y. Hsieh, Phys. Fluids 22 (8) (1979) 1435. [13] G.M. Moatimid, Chaos Sol. Frac. 12 (2001) 1239. [14] G.M. Moatimid, Phys. Scripta 65 (2002) 490. [15] G.M. Moatimid, Int. J. Eng. Sci. 32 (3) (1994) 535. [16] G.M. Moatimid, J. Phys. D Appl. Phys. 27 (1994) 1398. [17] A.R.F. Elhefnawy, G.M. Moatimid, Phys. Scripta 50 (1994) 258. [18] C.S. Yih, Dynamics of Nonhomgeneous Fluids, Macmillan, New York, 1965. [19] H.H. Woodson, J.R. Melcher, Electromechanical Dynamics, Wiley, New York, 1968. [20] J.Q. Feng, K.V. Beard, J. Fluid Mech. 222 (1991) 417. [21] J.R. Melcher, Field Coupled Surface Waves, MIT Press, Cambridge, MA, 1963. [22] D.Y. Hsieh, Trans. ASME 94D (1972) 158. [23] D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations, Oxford Univ. Press, London, 1977. [24] P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I, McGraw–Hill, New York, 1953.
Stability conditions of an electrified miscible viscous fluid sheet Galal M. Moatimid Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt Received 6 February 2002; accepted 20 November 2002
Abstract The Kelvin–Helmholtz problem of viscous fluids under the influence of a normal periodic electric field in the absence of surface charges is studied. The system is composed of a streaming dielectric fluid sheet of finite thickness embedded between two different streaming finite dielectric fluids. The interfaces permit mass and heat transfer. Because of the complexity of the considered system, a mathematical simplification is adopted. The weak viscous effects are taken into account so that their contributions are incorporated into the boundary conditions. Therefore, the equations of motion are solved in the absence of viscous effects. The boundary value problem leads to two simultaneous Mathieu equations of damped terms having complex coefficients. The symmetric and antisymmetric deformations reduced the coupled Mathieu equations to a single Mathieu equation. The classical stability criterion is found to be substantially modified due to the effect of mass and heat transfer. The analytical results are numerically confirmed. It is found that the sheet thickness and mass and heat transfer parameters have a dual influence on the stability criteria. It is also found that the field frequency has a stabilizing influence especially at small values of the wave number. In contrast to the case of a pure inviscid fluid, it is found that the uniform normal electric field plays a dual role in the stability criteria. This role depends on the choice of the numerical values of the physical parameters of the system under consideration. 2003 Elsevier Science (USA). All rights reserved. Keywords: Electrohydrodynamic; Viscous fluids; Kelvin–Helmholtz; Periodic forces; Mass and heat transfer; Interfacial; Stability
1. Introduction The stability of thin films has received increasing interest due to their technological applications in many industries such as the petroleum industry. The fluid layer may be considered as a thin fluid sheet such as a bilayer lipid membrane. Studies on the stability of a fluid layer can contribute to a better understanding of colloidal and biological systems. Stability is used to rupture biological and artificial membranes and thin films which are involved in the mechanisms of a wide variety of biological and engineering processes and is important for practical applications. Typical examples are membrane fusion, membrane rupture and mechanical destruction, cell division and fusion, and colloid stability [1]. If the layer is stressed by an electric field, then the classical stability conditions will be considerably modified. El-Shehawey et al. [2] studied the linear electrohydrodynamic stability of inviscid fluids. The dielectric fluids are stressed by gravitational forces and a tangential periodic electric field. They found that this field has a stabilizing effect except at resonance points. Furthermore, the tangenE-mail address: [email protected].
tial periodic field cannot stabilize a system which is unstable under the influence of a uniform electric field. Mohamed et al. [3] investigated the electrohydrodynamic stability of a fluid sheet at finite thickness. They found that when there are no surface charges present at the interfaces, the normal electric field showed a destabilizing influence. Also, in the presence of surfaces charges, the field is still destabilizing but its effect is partially shielded in some situations. Parhi and Nath [4] gave a new analytical criterion for the Rayleigh– Taylor instability of a three-layer viscous, stratified, incompressible steady flow, when the top and the bottom layers extend to infinity while the middle layer has a small thickness. They recovered the results of the corresponding problem for two fluid layers. The problem of parametric resonance arises in many branches of physics and engineering. One of the important problems is that of dynamic instability, which is the response of mechanical and elastic systems to time-varying loads, especially periodic loads. There are cases in which the introduction of a small vibration loading can stabilize a system which is statically unstable or destabilize a system which is statically stable. There are many books devoted to the analysis and applications of parametric excitation. McLachlan [5]
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00164-9
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
discussed the theory and applications of Mathieu functions, while Magnus and Winkler [6] discussed Hill’s equation and its engineering vibration problem. The treatment of the parametric excitation system having many degrees of freedom and distinct natural frequencies are usually operated on using the multiple time scales method as given by Nayfeh and Mook [7]. The influence of periodic forces on the stability of flow is a relatively new topic in the theory of hydrodynamic stability. The mathematical analysis is rather difficult, because the method of normal modes is not applicable. The linearized partial differential equations have coefficients which vary with time so that the exponential time dependence of the perturbation is not separable. The stability of a viscous Rayleigh–Taylor fluid results in a transcendental dispersion relation (see Chandrasekhar [8]). When different layers of a stratified fluid are in horizontal motion, another type of instability is obtained. The instability of the plane interface between two superposed fluids with a relative horizontal velocity is called the Kelvin–Helmholtz instability. In the Kelvin–Helmholtz model, the effect of streaming is destabilizing in the linear sense [8]. Mohamed et al. [9] studied the electroviscoelastic stability of a Kelvin fluid layer in the presence of field periodicity. They found that the surface elevations are governed by two transcendental coupled equations of Mathieu type. El-Dib and Matoog [10] investigated the stability of streaming in an electrified Maxwell fluid sheet influenced by a vertical periodic field in the absence of surface charges. Their calculations show that the increase in the sheet thickness plays a destabilizing effect in the presence or absence of the field frequency as well as being a function of the field intensity. A dual role of the stratified viscosities, in the presence or the absence of the field frequency, is observed in their analysis. These studies in interfacial instabilities are based on the assumption that the fluids are immiscible. Thus, there is no mass transfer across the interface. The immiscibility condition concerns the limited case of infinite latent heat. Ordinarily, since the latent heat is very large, it is a very good approximation to treat the fluids as immiscible when thermal effects are very small. However, when there is a strong temperature gradient in the fluid, thermal effects on the interfacial waves can be appreciable. Therefore, there is significant mass transfer across the interface, and in turn transfer of heat in the fluid has to be taken into consideration. The mechanism of heat and mass transfer across an interface (gas–liquid interface, say) is of great importance in numerous industrial and environmental processes. These include the design of many types of contacting equipment, e.g., boilers, condensers, evaporators, gas absorbers, pipelines, chemical reactors, nuclear reactors, and are relevant to other problems such as aeration of rivers and the sea. Hsieh [11] formulated the general problem of interfacial fluid flow with mass and heat transfer, which he applied to discuss the Kelvin– Helmholtz instability in the linear theory. He showed that the classical stability criterion is substantially modified due to the effect of mass and heat transfer. He [12] employed
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the multiple time scales analysis to investigate the nonlinear Rayleigh–Taylor stability with mass and heat transfer. He found that when the heat transfer rate is strong enough, the classically unstable system is stabilized by the nonlinear effect and the effect of mass and heat transfer across the interface. Moatimid [13] studied the problem of electrohydrodynamic instability of two superposed viscous, miscible, streaming fluids. His investigation includes the stability analysis of the presence of the periodic electric field as well the constant one. He found that the presence of surface charges plays a dual role in the stability criteria. He also found that the mass and heat transfer has a destabilizing effect whether the electric field is static or periodic. Recently, the stability of a basic flow of streaming magnetic fluids in the presence of an oblique periodic magnetic field has been investigated by Moatimid [14]. He found that the mass and heat transfer parameter has a destabilizing influence regardless of the field mechanism. He also found that the field frequency plays a dual role in the stability picture. The problem of hydrodynamic stability of two-phase fluids, acted upon by an external electric field in different situations in the presence of mass and heat transfer, separated by a single interface, has been extensively studied [15–17]. Moatimid [15,16] showed that, in contrast to the case of the uniform electric fields, the periodic one—frequently— shows the effect of mass and heat transfer parameter on the stability conditions. The analysis of Elhefnawy and Moatimid [17] indicates that the instability criterion is independent of mass and heat transfer parameters, but it is different from that in the same problem without mass and heat transfer. Their analysis showed the destabilizing effect of mass and heat transfer. To the best of my knowledge, no attempt has been made to examine—analytically—the hydrodynamic stability of three-phase fluids separated by double interfaces. Therefore, we have considered the Kelvin– Helmholtz instability problem with mass and heat transfer in a plane geometry. The model consists of three-phase fluids which are divided by double interfaces. The system is excited by an external periodic normal electric field distribution. Hsieh’s [11] simplified formulation of a single interface is extended to adopt double interfaces. The analysis is based on the linear theory of stability. Because of the complexity in solving the present boundary-value problem, only the symmetric and antisymmetric deformations of the surface deflections are considered. Therefore, we have divided the work into various sections for the clarity of presentation as follows: The formulation of the problem is presented in Section 2. The basic equations are introduced in Section 3. The perturbation equations are formulated in Section 4. Section 5 is devoted to presenting the boundary conditions, which include Maxwellian, kinematical, and stress tensor conditions. The derivation of the characteristic equations are presented in Section 6. The treatment of the symmetric and antisymmetric deformations of the surface deflections are introduced in Section 7, the analysis includes the case of a uniform normal electric field as well as a periodic one. Section 8 is de-
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voted to a discussion of the marginal state analysis. Finally, the outlines of the problem are introduced as a conclusion in Section 9.
2. Formulation of the problem The derivation of the dynamical system for a viscous fluid is presented in this section. We consider three fluid layers confined between two infinitely long horizontal parallel planes of thickness 2b. It has been shown by Yih [18] for stratified fluids that stability or instability of threedimensional disturbances can be determined from that of the corresponding two-dimensional disturbances. Hence, it is sufficient to consider only two-dimensional disturbances. Therefore, without any loss of generality, only Cartesian coordinates (x, y) are used with the y-axis being taken vertically. The z-axis is ignored. A horizontal fluid sheet of finite thickness 2a and density ρ (2) occupies the region −a < y < a. This layer is assumed to be an incompressible, viscous, and dielectric fluid. Its viscosity coefficient and dielectric constant are µ(2) and (2) , respectively. The middle plane of the fluid sheet is taken to be y = 0. This layer is embedded in two fluids, the upper fluid having density ρ (1) and bounded by the conducting plane y = b (b > a) which is raised to the periodic potential (φ0 cos ωt). The lower fluid with density ρ (3) is bounded from below by an earthed conducting plane y = −b. Both fluids are viscous and incompressible. The superscripts (1), (2), and (3) refer to quantities in the upper, middle, and lower fluid layers, respectively. Also, we consider that the primary flow state is given (j ) by three uniform velocities v0 (j = 1, 2, 3) along the positive x-direction. The temperatures at y = b, a, −a, and −b are taken as τ1 , τ2 , τ3 , and τ4 , in order. Gravity is taken to be in the negative y-direction. A sketch of the system under consideration is given in Fig. 1. As a result of the potential difference between the planes y = ±b, a normal electric field is produced in the three
regions. This field may be represented by (j )
E(j ) = −E0 cos ωtey
(j = 1, 2, 3),
(2.1)
where ω is the frequency of the applied normal electric field and ey is the unit vector along the y-direction. Consequently, the potential φ0 may be written as φ0 = a E0(1) − 2E0(2) + E0(3) − b E0(1) + E0(3) cos ωt. (2.2) The problem considered here is the stability of a dielectric miscible viscous fluid layer sandwiched between two dielectric miscible viscous fluids and stressed by a normal periodic electric field. The analysis of the present study focuses on the case where the interfaces are ideally conducting and have no surface charges. Two forces must be accounted for by the stress tensor σij : First, the surface force, which results from the Newtonian effect, is given by [8] ∂vj ∂vi (vis) + , σij = −P δij + µ (2.3) ∂xj ∂xi where P is the hydrostatic pressure and δij is the Kronecker delta. Second, the body force, which is caused by the electrical forces, is given by [19] 1 1 ∂ F = ρf E − E 2 ∇ + ∇ E 2 ρ (2.4) , 2 2 ∂ρ where ρf is the free charge density, is the dielectric constant, ρ is the mass density of the fluid, and ∇ is the gradient operator. The first term is the free charge force density, the second term is due to inhomogeneities in the dielectric, and the last one results from changes in the material density. This last term is called the electrostriction force density. Manipulation of Eq. (2.4), which incorporates the irrotational nature of the electric field intensity, shows that the stress tensor representation of combined free charge and polarization force densities is 1 ∂ (ele) , σij = Ei Ej − δij Ek Ek − ρ (2.5) 2 ∂ρ
Fig. 1. Schematic diagram of the system under consideration.
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where ∂σij . Fj = ∂xi
(2.6)
The resultant stress tensor acting on the system is then given by σij = σij(vis) + σij(ele) .
(2.7)
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during perturbation [19]. A number of simplifications of Maxwell’s equations are appropriate to the description of electrodynamic phenomena. The fluids under consideration are poor conductors compared with fluids such as mercury and other liquid metals and so induced magnetic effects are very small. In electrodynamics, it is generally assumed that the quasi-static approximation is valid [21]. Thus, the electrical equations are
3. Basic equations
∇ ∧ E = 0,
(3.6)
The equation that governs the behavior of the viscous fluid is [8] ∂v + (v · ∇)v = −∇P + µ∇ 2 v − ρgey , ρ (3.1) ∂t
∇ · E = 0.
(3.7)
associated with incompressibility condition ∇ · v = 0,
(3.2)
where v = (u, v, 0) is the fluid velocity. Because of the intricacy of the problem for the generalization to the Kelvin–Helmholtz flow of a viscous dielectric fluid, we confine the analysis to weak viscous effects [20]. If the viscous forces are very small in comparison with nonviscous ones, then viscosity produces only a thin weak vertical layer at the surfaces of separation, while the motion remains irrotational throughout the bulk of the fluids. Thus, the derivation in this problem deals completely with potential flow so that the complicated manipulation of the boundarylayer equations for the weak vortical flow can be avoided. In the present analysis, viscous effects are extremely weak and they can be incorporated into the boundary conditions. Therefore, and in view of the weak viscous effects considered here, the governing equation for the bulk of a fluid phases becomes ∂v + (v · ∇v) = −∇P − ρgey . ρ (3.3) ∂t Since the configuration must satisfy this equation of motion in the equilibrium state, the equilibrium solution then becomes (j )
P0
= ρ (j ) gy + C (j ) (t)
(j = 1, 2, 3),
(3.4)
where C (j ) (t) is an arbitrary time-dependent function of time. From the continuity of the normal stresses at the interfaces, one gets C
(j +1)
(j +1)
(j +1)
(t) − C (t) = (−) (ρ − ρ 2 1 (j ) (j )2 (j +1) + E0 − (j +1) E0 cos2 ωt 2 (j = 1, 2).
(j )
(j )
)ga
(3.5)
No volume charges are assumed to be present in the bulk of the fluids. Also, because of the continuity of the electric field, no surface charges are present at the interface in the equilibrium state and the electric field will, therefore, vanish
Therefore, one can construct an electrostatic potential φ (j ) (x, y, t) such that (j )
E(j ) = E0 cos ωt ey − ∇φ (j )
(j = 1, 2, 3).
(3.8)
Therefore, the electrostatic potential satisfies the Laplace equation ∇ 2 φ (j ) = 0 (j = 1, 2, 3).
(3.9)
To facilitate the stability analysis, a simplified formulation of the interfacial flow problems with mass and heat transfer has been employed by Hsieh [11,22]. In his formulation, the effect of mass and heat transfer on the dynamics of the system is expressed through an interfacial condition which makes use of the equilibrium temperature distribution of the system. In this work, we extend Hsieh’s [11] simplified formulation to include the double-interface problem. Therefore, the influence of mass and heat transfer is considered only through the interfacial boundary conditions.
4. Perturbation equations In what follows, the equations used to assess the linear stability of the uniform flow equations are presented; to this end, the same coordinate system as illustrated in Fig. 1 is used. All disturbances are assumed to be two dimensional. In addition, it is assumed that the fluid interfaces have surface tension coefficient Tj (j +1) , where the label j refers to the upper fluid, and that there are no surfactants present there. The amplitude of waves formed on the fluid sheet is assumed to be small. For a small departure from the equilibrium state, every physical perturbed quantity may be expressed in the form F1 (x, y; t) = f1 (y, t)eikx ,
(4.1)
where k is the wave number and is assumed to be real and positive, and F1 stands for each of φ1 , P1 , and v1 . These, expansions are introduced into the governing equations and the relevant boundary conditions. The linearized terms in these perturbed quantities are only maintained in view of the linear stability theory [8]. To perform a linear stability analysis of the present problem, the interfaces between the three fluids will be
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assumed to be perturbed about their equilibrium locations to cause displacements of the material particles of the fluid system. Consider the effect of small wave disturbances to the interfaces y = ±a, propagating in the positive x-direction. Assuming that the surface deflections are given by y = (−)j +1 a + ξj
(j = 1, 2),
(4.2)
5.1. Maxwell’s electric conditions Because of wave propagation, attention is restricted to the normal electric field where no surface charges are present on the interfaces. It follows that the equilibrium electric field intensity satisfies the relation [21] (j +1)
(j ) E0 − (j +1) E0 (j )
where ξj = γj (t)eikx ,
(4.3)
γ1 (t) and γ2 (t) are arbitrary time-dependent functions which determine the behavior of the amplitude of the disturbances on the interfaces. These functions are determined after applying the appropriate boundary conditions. According to linear perturbation theory [8], the unit vectors normal to the interfaces are given by ∂ξj ex + ey (j = 1, 2), nj = − (4.4) ∂x where ex is the unit vector along the x-direction. The equations of motion and the relevant boundary conditions will be solved for these perturbations under the assumption that the perturbations are small; that is, all equations will be linearized in the perturbation quantities. Actually, the equations governing the perturbation quantities are readily found to satisfy Laplace’s equations (j )
(4.5)
(j )
(4.6)
∇ 2 P1 (x, y; t) = 0, ∇ 2 φ1 (x, y; t) = 0 (j = 1, 2, 3).
As a result of the perturbations P1 , v1 , and φ1 and in view of the dependence given by Eq. (4.1), we have the solutions in the three fluid layers as ρ (j ) d (j ) (j ) + ikv0 P1 = − k dt (j ) × C (t) cosh ky + D (j ) (t) sinh ky eikx , (4.7) (j ) v1 = C (j ) (t) sinh ky + D (j ) (t) cosh ky eikx ,
(4.8)
(j ) φ1 = A(j ) (t) cosh ky + B (j ) (t) sinh ky eikx (j = 1, 2, 3), A(j ) (t),
C (j ) (t),
(j )
(j )
y = (−)j +1 a
= 0,
(j = 1, 2).
(5.2)
In addition, the jump in the tangential component of the electric field is zero across the interfaces. This requires (j ) (j +1) ∂φ ∂φ ∂ξi (j ) (j +1) E0 − E0 = 0, cos ωt + 1 − 1 ∂x ∂x ∂x y = (−)j +1 a (j = 1, 2).
(5.3)
The electric part is not completely specified until the electrical properties of the external electrodes are given. In an experimental apparatus, these electrodes provide the electric potential difference that induces surface charges on the interfaces. It is, therefore, reasonable to assume that they are perfectly conducting so as to form an equipotential. Then the boundary conditions on the electric field at y = ±b, as a consequence of the fact that the electric field is zero inside the conductor, are the same as condition (5.3). Therefore, at the rigid boundaries, one gets (j )
∂φ1 = 0, ∂x
y = (−)j +1 b
(j = 1, 2).
(5.4)
Applying the boundary conditions (5.2), (5.3), and (5.4) on the general solution (4.9), one gets (1) E0 (2) ( − (1) ) ∗ × (2) tanh k(b − a) sinh 2ka + (3) cosh 2ka γ1 sinh k(b − y) cos ωt eikx , + ( (3) − (2) ) (1)γ2 cosh k(b − a) (5.5)
φ1 = −
where and are arbitrary functions of time which are to be determined by making use of the appropriate boundary conditions.
The solutions given in the previous section must satisfy certain boundary conditions through which the arbitrary functions A(j ) (t), B (j ) (t), C (j ) (t), and D (j ) (t) may be identical. For the problem under consideration, the appropriate boundary conditions may be listed as follows.
(5.1)
(j +1)
∂φ1 ∂φ − (j +1) 1 ∂y ∂y
D (j ) (t)
5. Boundary conditions and solutions of the perturbed functions
(j = 1, 2).
To verify this fact, the continuity of the normal component of electric displacement at the surfaces of separation is applicable, and this requires that
(1)
(4.9) B (j ) (t),
=0
(2)
φ1 =
(2) E0 (2) ( − (1)) (2) tanh k(b − a) cosh k(y + a) ∗ + (3) sinh k(y + a) γ1 − ( (3) − (2) ) (2) tanh k(b − a) cosh k(y − a) − (1) sinh k(y − a) γ2 cos ωt eikx , (5.6)
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(3) φ1
E0(3) (2) = − ∗ ( − (3) ) (2) × tanh k(b − a) sinh 2ka + (1) cosh 2ka γ2 sinh k(b + y) + ( (1) − (2)) (3) γ1 cos ωt eikx , cosh k(b − a) (5.7)
As indicated by Hsieh [11], we may expand his formula as χ1 (y) =
K (1) (τ2 − τ1 ) K (2) (τ3 − τ2 ) − , b−y a+y
(5.10)
χ2 (y) =
K (2) (τ3 − τ2 ) K (3) (τ4 − τ3 ) − . a−y b+y
(5.11)
where ∗ = (2) ( (1) + (3) ) tanh k(b − a) cosh 2ka + (2)2 tanh2 k(b − a) + (1) (3) sinh 2ka. 5.2. Kinematical conditions Hsieh [11] formulated the interfacial boundary conditions for a single interface. These conditions are necessary to obtain the unknowns coming from solutions of the hydrodynamic equations of motion. They include the conservation of mass and energy across the interface, accounting for interfacial transfer of mass due to heat transfer and latent heat effects. Hsieh [11] assumed that the amount of the released latent heat depends mainly on the instantaneous position of the interface. Hsieh’s [11] interfacial conditions, for a single interface, must be expanded to adopt the double interfaces in the problem under consideration. Therefore, let us define the function Sj (x, y; t) = y + (−)j a − ξj
(j = 1, 2),
(5.8)
where Sj (x, y; t) = 0 describes the wave like profile of the disturbed interfaces. The kinematical boundary conditions at the interfaces are as follows: (i) The interfacial conditions of energy transfer may be expressed as ∂Sj (j ) + v1 · ∇Sj = χj (ξj ) (j = 1, 2), Lj ρ (j ) (5.9) ∂t where Lj is the latent heat of transformation from the fluid of density ρ (j ) to the fluid of density ρ (j +1) , and χj (ξj ) is the perturbed heat flux at the interface used for evaporation or condensation. χj is a function of the instantaneous profile of the interface and is determined from the heat transfer relation at equilibrium. As shown by Hsieh [22], the heat fluxes have to be determined from equations governing the heat transfer in the fluids, thus completely coupling the dynamics and the thermal exchanges in the entire flow region. In this simplified version, the expression is that χj simply a function of ξj , and moreover, χj is to be determined from the heat exchange relations in the equilibrium state. Let us consider the heat fluxes in the positive y-direction in regions (3), (2), and (1) as given by K (3)(τ4 − τ3 )/(b − a), K (2) (τ3 − τ2 )/(2a), and K (1) (τ2 − τ1 )/(b − a), where K (1) , K (2) , and K (3) are the heat conductivities of the three-phase fluids, in order.
191
The perturbed heat flux at the interface results in evaporation or condensation at the troughs or crests of the interfacial wave, where the interface is located deep in the hotter vapor or cooler liquid, respectively. Hsieh [11] used a convenient way of representing the perturbed heat flux by a Taylor series approximation about the point of equilibrium with respect to interfacial displacements. Hsieh [11] argued that only the linear terms of the series should enter the linear stability analysis and, therefore, truncated the series beyond the second terms. Therefore, the perturbed heat flux χj (ξj ) may be expanded in the form χj (ξj ) = χj (−1)j +1 a + αj (j +1 )Lj ξj , (5.12) where αj (j +1) is the measure of the rate of interfacial heat and mass transfer across the interfaces. The basic state of heat flux to the interface that is conducted through the vapor is assumed to be equal to the heat conducted from the interface into the liquid. Therefore, at the basic state of the unperturbed interface, no heat is used for evaporation or condensation. It follows that χj ((−)j +1 a) represents the net heat flux from the interface into the fluid regions (j ) and (j + 1). Since it is an equilibrium state, we have χj (−)j +1 a = 0. (5.13) It follows that K (1) (τ2 − τ1 ) K (2) (τ3 − τ2 ) K (3)(τ4 − τ3 ) = = . b−a 2a b−a (5.14) Hence we set 1 G 1 + αj (j +1) = (5.15) (j = 1, 2). Lj b − a 2a G=
If liquid (3) is hotter than liquid (2) at the same time that liquid (2) is hotter than liquid (1) (i.e., τ4 > τ3 > τ2 > τ1 ), then G and Lj are simultaneously positive. On the other hand, if (τ4 < τ3 < τ2 < τ1 ), then G and Lj are both negative. Therefore, in both cases αj (j +1) is always positive. These parameters represent the contribution of heat and mass transfer to the problem. Thus, correlation of experimental data would be greatly facilitated by this simplification. (ii) The conservation of mass across the interfaces [22] leads to (j ) (j +1) (j ) ∂Sj (j +1) ∂Sj + v1 · ∇Sj = ρ + v1 · ∇Sj , ρ ∂t ∂t y = (−)j +1 a
(j = 1, 2).
(5.16)
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(iii) At the rigid boundaries, the normal velocities must vanish and so one gets (1)
y = b,
(5.17a)
(3)
y = −b.
(5.17b)
v1 = 0, v1 = 0,
Applying the foregoing boundary conditions (5.9), (5.16), and (5.17a), (5.17b) to the general solutions given by Eqs. (4.7) and (4.8), one gets the special solutions of the perturbed (j ) (j ) quantities v1 and P1 as follows: sinh k(b − y) d α12 + ikv0(1) + (1) γ1 eikx , (5.18) v1(1) = sinh k(b − a) dt ρ
d 1 α12 (2) (2) sinh k(y + a) + ikv0 + (2) γ1 v1 = sinh 2ka dt ρ
α23 d (2) + ikv0 + (2) γ2 eikx , − sinh k(y − a) dt ρ (5.19) (3) v1
(1) P1
P1(2)
sinh k(b + y) d α23 (3) + ikv0 + (3) γ2 eikx , = sinh k(b − a) dt ρ
(5.20)
ρ (1) d 2 α12 d (1) = + 2ikv0 + (1) k dt 2 dt ρ
cosh k(b − y) ikx α12 e , + ikv0(1) ikv0(1) + (1) γ1 sinh k(b − a) ρ (5.21) d2 ρ (2) α12 d (2) =− + 2ikv + 0 k sinh 2ka dt 2 ρ (2) dt
α12 (2) (2) + ikv0 ikv0 + (2) γ1 cosh k(y + a) ρ α23 d d2 (2) + 2ikv0 + (2) − dt dt 2 ρ
α23 (2) (2) + ikv0 ikv0 + (2) γ2 cosh k(y − a) eikx , ρ (5.22)
and (3) P1
ρ (3) d 2 α23 d (3) =− + 2ikv + 0 k dt 2 ρ (3) dt
α23 cosh k(b + y) ikx (3) (3) e . + ikv0 ikv0 + (3) γ2 sinh k(b − a) ρ (5.23)
5.3. Stress tensor conditions At the fluid interfaces, two sets of stress tensor conditions determine the shape of the surfaces as follows: (i) The normal component of the stress tensor is discontinuous by the amount of the surface tension Tj (j +1) . Therefore, the balance at the dividing surfaces gives (j +1) (j ) P1 − P1 + (ρ (j ) − ρ (j +1) )g − k 2 Tj (j +1) ξj (j ) (j +1) (j ) (j ) ∂φ1 (j +1) (j +1) ∂φ1 − cos ωt + E0 E0 ∂y ∂y (j ) (j +1) (j ) ∂v1 (j +1) ∂v1 +2 µ −µ = 0, ∂y ∂y y = (−)j +1 a
(j = 1, 2),
(5.24)
where Tj (j +1) is the surface tension coefficient of the surfaces separating fluid j from fluid j + 1. (ii) The tangential component of the stress tensor is continuous across the interfaces. The balance of the stresses due to perturbations along the interfaces requires that (j ) (j +1) (j ) (j ) ∂φ1 (j +1) (j +1) ∂φ1 E0 − E0 cos ωt ∂x ∂x ∂ξj (j ) (j )2 (j +1)2 + cos2 ωt E0 − (j +1) E0 ∂x 2 2 (j ) (j +1) (j ) ∂ 2 (j +1) ∂ 2 + k v1 − µ + k v1 = 0, +µ ∂y 2 ∂y 2 y = (−)j +1 a
(j = 1, 2).
(5.25)
6. The characteristic equations The aim in this section is to complete the boundary-value problem. It consists of (1) a homogenous set of equations of motion, (2) a collection of appropriate boundary conditions, and (3) corresponding consistent solutions. All these components explain the time-dependent functions governing surface wave propagation. Equations that determine the surface deflections are called the characteristic equations. Because there are no surface charges on the unperturbed interfaces y = ±a, condition (5.1) will be taken into account in driving the characteristic equations. For inviscid fluids it is known that, in the absence of surface charges, conditions (5.2) and (5.25) are equivalent, but this is not so in the presence of viscosity. Therefore, there exists a contribution for condition (5.25), given by dγj + αj (j +1) (ν (j ) − ν (j +1) ) dt (j ) (j +1) γj = 0 (j = 1, 2), + ik µ(j ) v0 − µ(j +1) v0
(µ(j ) − µ(j +1) )
(6.1) where ν (j ) = µ(j ) /ρ (j ) (the kinematic viscosity) is used.
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199 (j )
(j )
(j )
Substituting the solutions φ1 , v1 , and P1 (j = 1, 2, 3) into the normal stress condition (5.24), after straightforward calculations, one obtains the coupled equations d 2 γ1 dγ1 + (f11 + il11 ) 2 dt dt
(2)2 + s11 + k11E0 cos2 ωt + ih11 γ1 dγ2 + (f22 + il22 ) dt
2 + s22 + k22E0(2) cos2 ωt + ih22 γ2 = 0,
(6.2)
dγ2 d 2 γ2 + (f12 + il12 ) 2 dt dt
(2)2 + s12 + k12E0 cos2 ωt + ih12 γ2 dγ1 + (f21 + il21 ) dt
(2)2 + s21 + k21E0 cos2 ωt + ih21 γ1 = 0,
which are two coupled Mathieu equations having damping terms and complex coefficients. These coefficients are defined in Appendix A. By making use of these equations, the stability behavior of the fluid sheet is controlled. The aim of this linear eigenvalue problem is to determine the surface elevation amplitudes γ1 and γ2 as time-dependent functions for the given physical geometrical and flow properties. The combination of the system of equations (6.1), (6.2), and (6.3) will lead to a simplification of the characteristic equations.
7. The symmetric and antisymmetric analysis The solution of the coupled Mathieu equations (6.2) and (6.3) can be simplified by considering the symmetric and antisymmetric deformations of the surface deflections ξ1 and ξ2 . Therefore, the variables ξ1 and ξ2 may be connected by ξ2 = J ξ1 = ξ,
where the constant coefficients a’s, b’s, c’s, and d’s are given by 1 mf1j + (µ(2j −1) − µ(2) ) aj = m (5−2j ) × µ(5−2j )v0 − µ(2)v0(2) h1j + J mf2(3−j ) + (µ(2) − µ(5−2j ) ) (2j −1) × µ(2)v0(2) − µ(2j −1)v0 h2(3−j ) , bj = l1j + J l2(3−j ), cj =
(6.3)
(7.1)
where J = 1 refers to the antisymmetric deformation, while J = −1 defines the symmetric one. In solving the characteristic equations (6.2) and (6.3), in view of symmetric and antisymmetric deformations, we use definition (7.1), keeping in mind the condition (6.1), which allows us to eliminate the imaginary part of γj . This procedure leads to
193
1 ms1j + αj (j +1) (ν (2j −1) − ν (2) ) m (5−2j ) (2) × µ(5−2j ) v0 − µ(2) v0 h1j + J ms2(3−j ) + α(3−j )(4−j )(ν (2) − ν (5−2j ) ) (2j −1) (2) h2(3−j ) , × µ(2) v0 − µ(2j −1) v0
and dj = k1j + J k2(3−j ), with (2) (1) (3) (2) m = k µ(2)v0 − µ(1) v0 µ(3) v0 − µ(2) v0 . Therefore, Eqs. (7.2) and (7.3) allow the boundary-value problem to be obtained entirely in terms of the unknown time-dependent function γ . It is easy to show that this system of equations, (7.2) and (7.3), may be reduced to a single Mathieu equation with a real damped term. The elimination of the constants b1 and b2 from these equations yields the damped Mathieu equation dγ ∗ d 2γ (2)2 + b0 + c0∗ E0 cos2 ωt γ = 0, + a0∗ 2 dt dt where a0∗ , b0∗ , and c0∗ are given by a1 b2 − a2 b1 , b2 − b1 d1 b2 − d2 b1 . c0∗ = b2 − b1
a0∗ =
b0∗ =
c1 b 2 − c2 b 1 , b2 − b1
(7.4)
and
The following analysis may be classified to include the case of a uniform normal electric field as well as the periodic one. 7.1. Stability analysis in the presence of a uniform electric field
d 2γ dγ (2)2 + c1 + d1 E0 cos2 ωt γ = 0, (7.2) + (a1 + ib1 ) 2 dt dt
For a uniform normal electric field, the periodicity of the field will be absent. Therefore, wave propagation is excited by using the electro-capillarity technique. The damped Mathieu equation (7.4) then becomes
2 d 2γ dγ + c2 + d2 E0(2) cos2 ωt γ = 0, (7.3) + (a2 + ib2 ) 2 dt dt
2 dγ ∗ d 2γ + b0 + c0∗ E0(2) γ = 0, + a0∗ 2 dt dt
(7.5)
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which is a linear differential equation with constant coefficients. It can be satisfied by a growth rate solution having the form γ (t) = exp(σ0 + iω0 )t,
(7.6)
where σ0 and ω0 are assumed to be real constants. This assumption imposes the following stability conditions: (2)2
E0
>
1 ∗2 a0 − 4b0∗ , ∗ 4c0
c0∗ > 0,
(7.7)
<
1 ∗2 a0 − 4b0∗ , ∗ 4c0
c0∗ < 0.
(7.8)
or (2)2
E0
In contrast to the linear stability theory of a pure inviscid fluid, it is found that the normal electric field intensity plays a dual role on the stability criteria. This role depends on the sign of the constant c0∗ . The goal, in what follows, is to determine a numerical calculation for the stability picture in the case of a uniform normal electric field. The theoretical analysis includes the two modes of surface deformation: the symmetric deformation (J = −1) and the antisymmetric one (J = 1). Since the calculations are too extensive, the following numerical calculations considered only the antisymmetric deformation (J = 1) as an example. In spite of this, Fig. 2b includes the case of a symmetric mode (J = −1). To screen the examination of the uniform normal electric field on the stability criteria, numerical calculations for the stability conditions (7.7) and (7.8) are made. The results are 2 displayed in Figs. 2a, 2b, and 3 to indicate the (E0(2) − k) plane. The sample chosen in these figures makes the system statically stable in the absence of the electric field. The plane 2 (E0(2) − k) is partitioned by the transition curve (7.8) into two regions. The calculations show that the constant c0∗ is negative. Therefore, the upper region of the transition curve (7.8) is unstable while the lower one is stable. The upper unstable region is labeled by the letter U , and the lower stable one is referred to by the letter S. The presence of the uniform electric field plays a destabilizing role, which is an early phenomena through the linear stability theory found by Melcher [21] and by several other researchers for inviscid flow. The same sense is obtained by El-Dib and Matoog [10]. Melcher [21] demonstrated that in the linear stability theory, the tangential field has a stabilizing effect (it increases the surface tension effect), while the normal field has a destabilizing influence (it decreases the surface tension effect). The examination of the influence of the mass and heat transfer parameters and the sheet thickness on the stability criteria for surface wave propagation is the aim of the present calculations. These calculations are depicted in Figs. 2 and 3, (2)2
respectively. In Fig. 2a, the plane (E0 − k) displays the natural stability curve (7.8) at the fixed value α12 = 1 g/cm3 s for various values of the parameter α23 as shown
(a)
(b) (2)2
− k) for a system having Fig. 2. (a) The stability diagram (E0 ρ (1) = 0.24 g/cm3 , ρ (2) = 0.65 g/cm3 , ρ (3) = 1.3 g/cm3 , a = 0.1 cm, (1) (2) (3) b = 1 cm, v0 = 5 cm/s, v0 = 10 cm/s, v0 = 15 cm/s, (1) = 3.3, (2) (3) = 30, = 18, T12 = 29 dynes/cm, T23 = 35 dynes/cm, µ(1) = 2.5 g/cm s, µ(2) = 4 g/cm s, µ(3) = 6 g/cm s, g = 980.665 cm/s2 , J = 1, α12 = 1 g/cm3 s, and different values of α23 . The curves are drawn according to relation (7.8). (b) Stability diagram for the same system as considered in Fig. 2a, but in the case of a symmetric deformation (J = −1).
in the figure. It is apparent from inspection of this graph that increasing the values of the parameter α23 increases the stabilizing influence of the normal electric field. Figure 2b represents the case of a symmetric deformation (J = −1). The numerical calculations show that the constant c0∗ is also negative. The destabilizing influence of the mass and heat transfer parameter α23 is also seen in Fig. 2a. A comparison between Fig. 2a and Fig. 2b shows that the instability of the system is enhanced under the influence of the symmetric deformations. The influence of the sheet thickness on the stability picture is displayed in Fig. 3. The calculations on this figure are made for the same values of the parameters α12 and α23 (= 1g/cm3 s). It is apparent from inspection of this graph that increasing the sheet thickness increases the destabilizing effect of the normal field. Thus the increase of sheet thickness has a destabilizing influence, which has not been observed before for Newtonian fluids. In pure inviscid fluids, El-Shehawey et al. [2] demonstrated that the increase of the inviscid fluid layer has a stabilizing influence. Therefore, one can see that the flow has undergone a dramatic change in its behavior due to the presence of the
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
Fig. 3. Stability diagram for the same system as considered in Fig. 2a, except α12 = α23 = 1 g/cm3 s and for different values of a. The curves are drawn according to relation (7.8).
195
Fig. 4. Stability diagram for the same system as considered in Fig. 1, but when ω = 5 Hz. The figure indicates the transition curves (7.14).
viscous factor. The same result is also found in the presence of the viscoelastic factor [10]. 7.2. Stability behavior due to the periodicity of the field In the presence of the periodicity of the normal electric field, the stability picture has changed dramatically. In this case, Eq. (7.4) may be written as dγ d 2γ + (b0 + c0 cos τ )γ = 0, + a0 dτ 2 dτ where τ = 2ωt is used, and a0 = a0∗ /2ω, (2)2
c0 = c0∗ E0
b0 =
b0∗
2 + c0∗ E0(2) /2 4ω2 ,
(7.9) Fig. 5. Stability diagram for the same system as considered in Fig. 4, except α12 = 1 g/cm3 s, α23 = 2 g/cm3 s and for different values of ω. The figure indicates the transition curves (7.14).
and
/8ω2 .
Consider a0 > 0 and b0 > 0. If we write γ (τ ) = e−a0 τ/2 η(τ ),
(7.10)
Eq. (7.9) is transformed to d 2η + (d0 + c0 cos τ )η = 0, dτ 2
(2)
In terms of the electric field E0 , conditions (7.13) may be rewritten in the form (7.11)
which is the standard form the Mathieu’s equation, with d0 = b0 − a02 /4.
periodic 4π . Jordan and Smith [23] deduced the stability criterion (b0 − 1/4)2 − c02 − a02 4 > 0. (7.13)
(7.12)
Jordan and Smith [23] try to construct part of a diagram of the stable regions for Eq. (7.9). Certainly, whenever d0 and c0 have values such that only bounded solutions of Eq. (7.11) exist, then only bounded solutions of Eq. (7.9) exist, by Eq. (7.11). Also, by Eq. (7.10), even some exponential growth in η is allowed, with γ still remaining bounded. Therefore, the regions of Eq. (7.9) will be the extensions of those for Mathieu’s equation. Jordan and Smith [23] have examined particularly the first unstable region for Eq. (7.11), which occurs near d0 = 1/4 or near b0 = (1/4)(1 + a02 ). On the boundaries of this unstable region, there are periodic solutions of Eq. (7.11) of
(2)2
E0
> E1∗
(2)2
or E0
< E2∗ ,
(7.14)
provided that E1∗ > E2∗ , where 2 4 ∗ E1,2 = ∗ −2 b0∗ − ω2 ± b0∗ − ω2 − 3a0∗2 ω2 3c0 (7.15) are the transition curves separating the stable from unstable regions. According to the Floquet theory [5], the region bounded by the two branches of the transition curves E1∗ and E2∗ is unstable. The area outside these curves is a stable one. The width of the unstable region is represented by (E1∗ − E2∗ ). The increase of this width refers to the destabilizing influence, while its decrease represents a stabilizing effect. In what follows, numerical calculations are made for the stability condition (7.13). The results of calculations for the
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transition curves (7.14) are displayed in Figs. 4 and 5 to indicate the effect of the mass and heat transfer parameters and influence of the field frequency, respectively. The graphs (2)2
are displayed in the plane (E0 − k) for the same system as considered in Fig. 2 with some changes. In Fig. 4, the 2 plane (E0(2) − k) is displayed to indicate the influence of the mass and heat transfer parameter α23 . The plane is partitioned by the transition curves E1∗ and E2∗ into stable and unstable regions. The stability picture is drawn at a single value for α12 and various values of α23 , as shown in the figure. It is found that an increase of the parameter α23 increases the stability regions. Therefore, in contrast to the case of the uniform electric field, the increase of the parameter α23 increases the stabilizing influence of the normal periodic field. In Fig. 5, the stability picture is due to the presence of the field frequency ω. It is found that as the periodicity increases, the width of the unstable region bounded by (E1∗ − E2∗ ) decreases, which shows a stabilizing influence of the frequency of the normal field intensity.
8. Analysis of the marginal state The single governing Eq. (7.4) for all symmetric and antisymmetric modes is used to discuss the stability picture for the surface wave propagation in the absence of the damped term a0∗ , which is the so-called the marginal state. This case may occur when the primary streaming velocities are the same as in the inviscid flow, considering also the absence of the mass and heat parameters. The marginal state may occur mathematically by eliminating a1 and a2 between Eqs. (7.2) and (7.3) and using the transformation i a2 b1 − a1 b2 t γ (t) = Γ (t) exp − (8.1) 2 a2 − a1 to obtain the normal form 2 d 2Γ + p + qE0(2) cos2 ωt Γ = 0, (8.2) dt 2 which has been extensively studied. This equation, which is a linear differential equation with periodic coefficients, is known as Mathieu’s equation. Equations similar to this appear in many problems in applied mathematics such as stability of a transverse column subject to a periodic longitudinal load, stability of periodic solutions of a nonlinear conservative system, electromagnetic wave propagation in a medium with periodic structure, and the excitation of certain electrical systems. Equation (8.2) may be used to discuss the stability near the marginal state in the presence of all physical parameters acted on the problem at hand. The new constants p and q are defined as a 2 c1 − a 1 c2 1 a 2 b 1 − a 1 b 2 2 + and p= a2 − a1 4 a2 − a1 a2 d1 − a1 d2 , q= (8.3) a2 − a1
The stability analysis may be classified into two cases: first, stability behavior in the presence of a uniform normal electric field in the absence of the field frequency, and second, stability examination in the influence of the periodic normal field. 8.1. Numerical simulation in the static case In the absence of the periodicity of the applied normal field ω → 0, wave propagation is excited using the electrocapillary technique. In this case, Eq. (8.2) reduces to 2 d 2Γ + p + qE0(2) Γ = 0. (8.4) 2 dt Equation (8.4) may be satisfied by the following solution: (2)2 Γ (t) = exp i p + qE0 t. (8.5)
Therefore, in the static case, the stability criterion reduces to 2
p + qE0(2) > 0,
(8.6)
which is automatically satisfied if p and q are positive. Otherwise, stability occurs when p (2)2 E0 > − , q > 0, (8.7) q or 2 p E0(2) < − , q < 0. (8.8) q Again, the uniform normal electric field intensity has a dual role in the stability picture. This role depends mainly on the sign of the constant q. To screen the contribution of the uniform normal electric field on the stability criteria, numerical calculations are made for the stability conditions (8.6). Only the case of antisymmetric interface deformation (J = 1) is considered in the following numerical calculations. The results for the calculations are displayed in Figs. 6–8 to indicate the (2)2
plane (E0 − k) with a transition curve as a function of all the physical parameters acting on the system. The examination of the influence of the sheet thickness and the mass and heat transfer parameters are the goal of the present calculations, which are displayed in these figures. In Fig. 6, the graph displays the natural stability curve (8.6) for various values of the sheet thickness a as shown in the figure. According to the physical parameters chosen in the figure, it is found that q > 0. Thus only the transition curve (8.7) is considered. It follows that the region above the natural curve is stable, while that below it is unstable. Here, the uniform normal electric field plays a stabilizing influence on the system under consideration. It is apparent from inspection of this graph that increasing sheet thickness increases the stabilizing influence of the normal field. A comparison between Figs. 3 and 6 shows that the sheet thickness plays a dual role in the stability criteria. The influence of mass and heat transfer parameters are displayed in Figs. 7 and 8.
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
197
8.2. Stability criterion in the oscillated case In the case the electro-capillary excitation was switched off, and the field was oscillated at the frequency ω, Eq. (8.2) may be rewritten in the form d 2Γ + (pˆ − 2qˆ cos 2ζ )Γ = 0, dζ 2
(8.9) (2)2
Fig. 6. Stability diagram for the same system as considered in Fig. 3.
Fig. 7. Stability diagram for the same system as considered in Fig. 6, except at fixed values of a = 0.5 cm, α23 = 1 g/cm3 s and for various values of α12 . The curves are drawn according to relation (8.7).
where ζ = ωt is used, and pˆ = (p + qE0 /2)/ω2 and qˆ = (2)2 −qE0 /4ω2 . Equation (8.9) is the canonical form of Mathieu differential equation. The properties of the Mathieu functions are well known [5]. The solutions of the Mathieu equation can be, under certain conditions, periodic and the system is then stable. The condition for the periodic Mathieu functions depends of the relation between the parameters pˆ and q. ˆ The p– ˆ qˆ plane is divided into stable and unstable regions bounded by the characteristic curves of Mathieu functions. The general solution of Eq. (8.9) is stable if the point (p, ˆ q) ˆ in the p– ˆ qˆ plane lies in the stable region; otherwise it is unstable. The stability condition reduces the problem to the bounded regions of the Mathieu functions, which gives the criterion [5] >(0) sin2 π pˆ 1/2 /2 1, (8.10) where >(0) is the infinite Hill’s determinate. In general, the analysis of inequality (8.10) is rather complicated due to the infinite Hill’s determinate. An approximate formula of stability, given by Morse and Feshbach [24], may be used for small values of q, ˆ which is a good approximation to high-frequency fields. According to this assumption the system is stable if 4
2
q 2 E0(2) + 16(ω2 − p)qE0(2) + 32p(ω2 − p) > 0.
(8.11)
In terms of the electric field E0(2) , this condition may be arranged in the form
Fig. 8. Stability diagram for the same system as considered in Fig. 3, except at fixed values of α23 = 1 g/cm3 s and for various values of α12 . The curves are drawn according to relations (8.7) and (8.8).
In Fig. 7, as before, only one transition curve is considered where q > 0. The graph shows various curves according to the variation of the parameter α12 . As α12 increases, the field shows a destabilizing influence. In Fig. 8, the graph includes the two cases q > 0 and q < 0. In the case when q > 0, stability is enhanced as α23 > α12 . In contrast, when q < 0 stability decreases. It follows that the parameter α23 plays a dual role in the stability criteria.
(2)2
E0
(2)2 − Eˆ 1 E0 − Eˆ 2 > 0,
(8.12)
where Eˆ 1 and Eˆ 2 are the stability boundaries, which are given by 8 3 Eˆ 1,2 = −(ω2 − p) ± (ω2 − p) ω2 − p . (8.13) q 2 Therefore, the stability criteria become (2)2
E0
> Eˆ 1
(2)2
or E0
> Eˆ 2 ,
(8.14)
provided that Eˆ 1 > Eˆ 2 , which are the transition curves separating the stable from unstable regions. From Floquet theory [5], the region bounded by the two branches of the transition curves Eˆ 1 and Eˆ 2 is unstable, while the region outside these curves is stable.
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2 2αj (j +1) ν (2) × coth k(b − a) + ρ (2) v0(2) − ρ (2)
9. Conclusion In this study, we have presented the problem of electrohydrodynamic Kelvin–Helmholtz instability of a horizontal fluid sheet of finite thickness embedded between two finite fluid layers. The system has been analyzed including the effect of interfacial tension, mass and heat transfer parameters, and weak viscous effects. It is recognized that viscous effects are demonstrated in the boundary conditions. The stability analysis is based on a linear perturbation theory. Through this perturbation analysis, the solution of the boundary-value problem leads to two simultaneous differential equations of Mathieu types, which are used to control the stability of the fluid sheet. Only the symmetric and antisymmetric deformations for the surface deflections are considered. This constraint leads to a single Mathieu equation. The analytical results are numerically confirmed. The stability examination yields the following results: (1) In the presentation of the problem, the effect of mass and heat transfer is revealed through two parameters α12 and α23 . Thus, correlation of experimental data would be greatly facilitated by this simplification. (2) The sheet thickness, as well as mass and heat transfer parameters, has a dual role on the stability criteria. (3) The field frequency ω has a stabilizing influence, especially at small values of the wave number. (4) In contract to the case of a pure inviscid fluid, the uniform normal electric field intensity plays a dual role in the stability picture.
The coefficients appearing in Eqs. (6.2) and (6.3) are 1 f1j = ∗ αj (j +1) + 2k 2 µ(2j −1) ρ × ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka × coth k(b − a) + αj (j +1) + 2k 2 µ(2) × ρ (5−2j ) coth k(b − a) cosh 2ka + ρ (2) sinh 2ka , l1j =
ρ∗
× ρ
(2j −1)
ρ (2j −1) v0
sinh 2ka
k 2 (2) ( (2j −1) − (2) ) ρ ∗ ∗ (2j −1) × ρ (2) (2j −1) ( (2) − (5−2j ) ) + ( (2j −1) − (2) ) × (2) tanh k(b − a) sinh 2ka + (5−2j ) cosh 2ka × ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka ,
k1j = −
h1j =
f2j =
s2j =
k (2j −1) 2 (2j −1) v α + 2k µ j (j +1) 0 ρ∗ × ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka × coth k(b − a) + v0(2) αj (j +1) + 2k 2 µ(2) × ρ (5−2j ) coth k(b − a) cosh 2ka + ρ (2) sinh 2ka , coth k(b − a) (2) ρ αj (j +1) + 2k 2 µ(2j −1) ∗ ρ − ρ (2j −1) αj (j +1) + 2k 2 µ(2) , 2kρ (2)ρ (2j −1) coth k(b − a) (2j −1) v0 − v0(2) , ∗ ρ k 2 ρ (2) ρ (2j −1) coth k(b − a) ρ∗ (2j −1) ν ν (2) (2j −1)2 (2)2 × v0 − v0 + 2αj (j +1) (2j −1) − (2) ρ ρ −
k2j =
kρ (2) Σj (j +1) , ρ∗
k 2 (2) ( (2j −1) − (2) ) ρ ∗ ∗ (2j −1) × ρ (2j −1) (2j −1) ( (5−2j ) − (2) ) × coth k(b − a) sinh 2ka + ρ (2) (2) × ( (5−2j ) − (2j −1) ) cosh 2ka + ( (2) − (2j −1) ) × tanh k(b − a) sinh 2ka ,
× coth k(b − a) + ρ (2) v0(2)
× ρ (5−2j ) coth k(b − a) cosh 2ka + ρ (2) sinh 2ka ,
s1j
coth k(b − a) cosh 2ka + ρ
(2)
(5−2j ) k Σ coth k(b − a) sinh 2ka ρ j (j +1) ρ∗ + ρ (2) cosh 2ka ,
× ρ (5−2j ) coth k(b − a) sinh 2ka + ρ (2) cosh 2ka
k 2 (2j −1) (2j −1)2 2αj (j +1)ν (2j −1) =− ∗ ρ − v0 ρ ρ (2j −1) (5−2j ) × ρ coth k(b − a) sinh 2ka + ρ (2) cosh 2ka
(5−2j )
−
l2j =
Appendix A
2k
h2j =
k coth k(b − a) (2) (2j −1) ρ v0 αj (j +1) + 2k 2 µ(2j −1) ∗ ρ (2j −1) (2) −ρ v0 αj (j +1) + 2k 2 µ(2) ,
G.M. Moatimid / Journal of Colloid and Interface Science 259 (2003) 186–199
Σj (j +1) = (ρ (j ) − ρ (j +1) )g − k 2 Tj (j +1), and ρ ∗ = ρ (2) (ρ (1) + ρ (3) ) coth k(b − a) cosh 2ka + ρ (2)2 + ρ (1) ρ (3) coth2 k(b − a) sinh 2ka.
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