Stability of Streaming in an Electrified Maxwell Fluid Sheet Influenced by a Vertical Periodic Field in the Absence of Surface Charges

Journal of Colloid and Interface Science 229, 29–52 (2000) doi:10.1006/jcis.2000.6855, available online at http://www.idealibrary.com on

Stability of Streaming in an Electrified Maxwell Fluid Sheet Influenced by a Vertical Periodic Field in the Absence of Surface Charges Yusry O. El-Dib∗,1 and R. T. Matoog† ∗ Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt; and †Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah, Saudi Arabia Received April 22, 1999; accepted March 23, 2000

electrohydrodynamics of a generalized surface wave by linearizing the electrohydrodynamic equations of motion and boundary conditions and the associated electric, velocity, pressure, and surface deformation fields (1, 2). Consistent with the first-order linear approximation, he used perturbation theory to derive the surface wave solutions for the electric and velocity potentials and the surface deformations for different wave modes. Following Melcher and Schwartz (1) and applying the electrohydrodynamic capillary wave theory derived by Miskovsky et al. (3) to the planar model including viscosity, the dispersion relation was obtained within the harmonic approximation from the set of electrohydrodynamic equations, surface coupling, and other boundary conditions (3). Recently, Mohamed et al. (4–7) and El-Dib (8) have applied Melcher’s and Miskovsky’s model to the planar model including viscoelasticity of the Maxwell and Kelvin types. When different layers of a stratified fluid are in horizontal motion, we get another type of instability. The instability of the plane interface between two superposed fluids with a relative horizontal velocity is called Kelvin–Helmholtz instability. In the Kelvin–Helmholtz model, the effect of streaming is destabilizing in the linear sense (Chandrasekhar [9]). Lyon (10) added the effect of compressibility and applied electric field, but neglected the surface tension. Drazin (11), Nayfeh and Saric (12), and Weissman (13) have studied the nonlinear development of the Kelvin–Helmholtz instability. The study of the nonlinear electrohydrodynamic Kelvin–Helmholtz instability of an interface separating two semi-infinite dielectric inviscid fluids influenced by a normal electric field in the presence of mass and heat transfer has been discussed by Elshehawey (14), Mohamed and Elshehawey (15), and Mohamed et al. (16). In one of the most recent works on this subject the effect of periodic body forces is discussed. The effect of a time-dependent acceleration in the presence of a tangential magnetic field on the nonlinear stability of Kelvin–Helmholtz waves has been discussed by El-Dib (17). The object of this work is to provide an extension to the model Mohamed et al. (4, 7) in order to discuss the influence of an unsteady electric force on the interfacial stability of a streaming fluid sheet of Maxwell type. The influence of periodic forces on the stability of flow is a relatively new topic in the theory of hydrodynamic stability.

The problem of electroviscoelastic Kelvin–Helmholtz waves of Maxwellian fluids under the influence of a vertical periodic electric field is studied in the absence of surface charges. The system is composed of a streaming dielectric fluid sheet of finite thickness embedded between two different streaming semi-infinite dielectric fluids. Due to the streaming flow and the influence of a periodic force, a mathematical simplification is considered. The weak viscoelastic effects are taken into account so that their contributions are demonstrated in the boundary conditions. The approximate equations of motion are solved in the absence of viscoelastic effects. The solutions of the linearized equations of motion and boundary conditions lead to two simultaneous Mathieu equations of damping terms having complex coefficients. Symmetric or antisymmetric deformation that relaxes the coupled Mathieu equations and yields a single Mathieu equation is considered. Stability criteria are discussed and numerical estimation shows that the increase in the sheet thickness plays a destabilizing effect in the presence or in the absence of the field frequency as well as the field intensity. In the absence of the field frequency the velocity ratio between the upper fluid velocity and the sheet velocity has a destabilizing influence, while that between the velocity of the lower fluid and the velocity of the sheet has a stabilizing influence. Moreover, the viscosity ratios have a damping influence while the elasticity ratios have a destabilizing influence. Furthermore, a range of general deformations of the surface deflections is studied. Moreover, the stability behavior for the resonance cases is studied and discussed. The coupled Mathieu equations are analyzed by the multiple scale method. The numerical examination for stability yields some changes in the stability behavior. The fluid sheet thickness plays a stabilizing role in the presence of a constant field while the damping role is observed for the resonance case. Similar results are found for both the stratified velocities and the stratified relaxation times. The dual role of the stratified viscosities is observed in the presence or the absence of the field frequency. °C 2000 Academic Press Key Words: electrohydrodynamics; Maxwell viscoelastic fluid; Kelvin–Helmohltz; periodic force; interfacial stability.

1. INTRODUCTION

A surface wave is essentially a two-dimensional deformation of an interface separating two media. Melcher studied the 1

To whom correspondence should be addressed. 29

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EL-DIB AND MATOOG

The mathematics is more 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. However, the investigation of a viscous Raleigh–Taylor fluid leads to a transcendental dispersion relation (see Chandasekhar [9]). If the viscous or viscoelastic problem is influenced by periodic force, a differential equation with a transcendental differential operator will result, which causes a difficulty in the mathematical problem (see Mohamed et al. [18]). Because of the complexity of the mathematical problem, a perturbation technique was used by El-Dib (19) to modify the dispersion differential equation. He carried out the stability analysis of a viscous fluid by perturbing viscous flow about inviscid flow. Mohamed et al. (4, 5) have carried out the stability examination of an electroviscoelastic Maxwell fluid or Kelvin fluid by perturbing the non-Newtonian fluid about the viscous fluid. The motion of a viscoelastic fluid that is bounded by a deformable interface constitutes a difficulty. The shape of this interface is not known in advance, but must be determined as part of the solution. The goal is to develop methods of solution that are not restricted to infinitesimal deformations from some obvious shape. In general, however, viscoelastic flow problems involving two fluids are exceedingly difficult to solve even when the shapes of the interface are assumed to be known. On the other hand, the inclusion of external periodic forces leads to more difficulty. Another difficulty arises when the two viscous fluids are in relative tangential motion. The Kelvin–Helmholtz model for viscous flow or for viscoelastic flow represents an ill-posed problem. This is because the tangential velocity could have a nonzero jump. In order to modify the Kelvin–Helmholtz waves for viscoelastic flow we restrict ourselves to the case of weak viscoelastic effects. It is assumed that the viscous effects appear at the interface and gradually decrease so that they are negligible in the bulk (20). Normally, the assumption in physicochemical hydrodynamics is that viscous or viscoelastic forces dominate the inertia terms. Much work on instabilities in viscous films with space charges and varying dielectric constants (with practical connections to foam stability, flotation processes, detergent cleansing, and the stability of biomembranes) has been performed by researchers including Jain and Ruckenstein (21), Invanov et al. (22), Maldarelli and Jain (23), Bisch (24), Sanfeld and Steinchen (25), Gallez et al. (26) Wendel et al. (27), and Mohamed et al. (18). However, for horizontal constant velocities, it can be excluded that solutions exist. The assumption of weak viscoelastic forces seems to be very urgent. However, we confine the analysis here to consider weak viscous or viscoelastic effects. The effects are believed to be significant only within a vertical surface layer so that the motions elsewhere in the fluid sheet may reasonably be assumed to be irrotational. In the case of pure viscous flow, El-Dib (28) has successfully applied this technique to viscous fluid column stability, in the manner of Feng and Beard (20).

The breakup mechanism for a low-speed circular liquid jet and for a moving thin liquid sheet has several distinctly different features (Reitz [29], Lefebvre [30]): (i) For the jet, the dominate type of disturbance which induces instability and eventually lead to its breakup into droplets is axisymmetrical, whereas it is antisymmetrical for the sheet. (ii) For the jet, the principal sources of instability are the capillary forces for Rayleigh instability. The effect of surface tension induces instability and results in drop formation at a negligible relative velocity. For the sheet, surface tension always opposes the onset and development of instability. (iii) For the jet, a velocity difference between two fluid phases contributes to, but is not necessary for, breakup, whereas it is required for the breakup of the sheet. In view of the above differences, a theoretical investigation has been conducted on the instability of a moving thin liquid sheet of uniform thickness, including the effect of low viscoelasticity for dielectric fluids. Another motivation for the present work is that the moving thin fluid sheet is a reasonably good approximation to sheet breakup processes in hollow cone sprays, which has numerous practical applications. We are mainly interested in the sheet breakup processes, which eventually lead to drop formation. Thus, the present study focuses primarily on the growth of disturbance waves through instability and does not yield any prediction on the sizes of the subsequently formed drops. 2. FORMULATION OF THE PROBLEM

The derivation of the dynamical system for viscoelastic fluids of Maxwellian type is presented in this section. Consider a horizontal fluid sheet of a finite thickness 2a embedded in two semi-infinite fluids (Fig. 1). Both of the fluids are incompressible, dielectric, and isotropic. There are no volume charges in the layers. In addition there are no free surface charges at the

FIG. 1. Sketch of the system under consideration.

31

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

interfaces between the layers. The fluids have a viscoelastic nature described by the Maxwellian constitutive relation (4) µ

µ ¶ ¶ ∂v j ∂ ∂vi ∂ τi j + λ + vk + τi j = µ , ∂t ∂ xk ∂x j ∂ xi

[1]

where τi j is the stress deviator of the stress tensor σi j , µ is the coefficient of viscosity, λ is the Maxwell relaxation time, and v is the fluid velocity vector. The axis y = 0 is taken to be the middle plane of the sheet. The effect of gravity is taken in the negative y-direction. The fluid interfaces are parallel and the flow in each phase is everywhere parallel to each other. Figure 1 represents a sketch of the system under consideration where the y-axis is taken vertically upward and the x-axis is taken horizontally to be at the middle sheet. E 0(1) cos ωt, E 0(2) cos ωt, and E 0(3) cos ωt are periodic vertical electric fields in the fluid layers. ρ (1) , ρ (2) , and ρ (3) are the fluid densities. ε (1) , ε(2) , and ε (3) are the fluid dielectric constants. µ(1) , µ(2) , and µ(3) are the viscosity coefficients. λ(1) , λ(2) , and λ(3) are the Maxwell relaxation times. The fluids are streaming at constant velocities v0(1) , v0(2) , and v0(3) along the positive x-direction. The superscripts (1), (2), and (3) refer to quantities in the upper fluid, plane sheet, and lower fluid, respectively, g is the gravitational acceleration, and ξ1 and ξ2 are the deflection of the interfaces. Two forces must be accounted for by the stress tensor σi j . One is the surface force, which results from the Newtonian effect and the elastic contribution of the Maxwell type as given by µ

σivis j

µ

∂ +v·∇ = −Pδi j + µ 1 + λ ∂t

¶¶−1 µ

¶ ∂v j ∂vi + , [2] ∂ x j ∂ xi

where P is the hydrostatic pressure and ∇ is the gradient operator. The other is the body force, which is caused by the electrical forces as given by the equation µ ¶ 1 ∂ε 1 F = ρf E − E · E∇ε + ∇ E · E ρ , 2 2 ∂ρ

combined free charge and polarization force densities, 1 2 σiele j = εE i E j − εE δi j 2 ele σi j = σivis j + σi j ,

[4] [5]

where the superscript vis or ele on the stress tensor σi j represents the viscoelastic or the electric stress tensor (31, 32), respectively. 3. EQUATIONS OF MOTION

Electrohydrodynamics includes the part of fluid mechanics concerned with electrical force effects. In formulating Maxwell relations for the system, we assume that the quasi-static approximation is valid for the problem (32). With a quasi-static model, it is recognized that relevant time rates of change are sufficiently low so that contributions due to a particular dynamic process may be ignored. The theory electrified fluids is concerned with phenomena in which electric energy storage greatly exceeds magnetic energy storage, and where the propagation times of electromagnetic waves are short times compared with those of interest to us. In the presence of the volume charge density in the bulk of the fluid the electric potential should be determined by the Poisson equation, in the quasi-static approximation. Without injection of free charges directly into the bulk of the system the initial free charges at each point in the bulk decay to zero after relaxation. At this stage the electrified potential should be determined by the Laplace equation (31). Thus under the quasi-electrostatic approximation Maxwell’s electric equations are reduced to ¢ ¡ ∇ · ε (r ) E(r ) = 0;

r = 1, 2, 3

∇ × E(r ) = 0.

[6] [7]

The potential φ has to satisfy Laplace’s equation, [3]

where ρf is the free charge density, ε is the dielectric constant, and ρ is the mass density of the fluid (31). The first term, called the Coulomb force, is the force per unit volume on a medium containing free electric charge. It is the strongest electro-fluid dynamic term and usually dominates when the dc electric field is present. The second term, called the dielectric force, is caused by the force exerted on a dielectric fluid by a nonuniform electric field. It is usually weaker than the Coulomb force and only dominates when an electric field of high enough frequency is imposed. The third term, called the electrostriction term, being the gradient of a scalar, is treated as a modification to the compressible fluid pressure and should be omitted from this analysis because incompressible flow is presented here (31). Manipulation of Eq. [3] that incorporates the irrotational nature of the electric field intensity shows the stress tensor representation of

∇ 2 φ (r ) = 0.

[8]

The equation that governs the behavior of a viscoelastic Maxwell fluid is ¸ · ∂v + (v · ∇)v ρ ∂t · µ ¶¸−1 ∂ ∇ 2 v, [9] +v · ∇ = −∇ P − ρge y + µ 1 + λ ∂t associated with the continuity equation ∇ · v = 0,

[10]

where ex and e y are the unit vectors in the x- and y-directions. Due to the intricacy of the problem for the generalization to the Kelvin–Helmholtz flow of non-Newtonian dielectric fluids, we confine the analysis to weak viscoelastic effects. The introduction of the weak viscous effects is customary in the case of a

32

EL-DIB AND MATOOG

Newtonian fluid (27, 28) and should be understood in a manner similar to a viscoelastic problem, which makes the problem formulation much easier to handle, since periodic force and relative motions are included. If the viscous forces are very small in comparison with nonviscous forces, then viscoelasticty produces only a thin, weak vertical layer at the surface of separation, while the motion remains irrotational throughout the bulk of the fluids. Thus the derivations in this problem deal completely with potential flow so that the complicated manipulation of the boundary-layer equations for the weak vertical flow can be avoided. Since the field equation governing the irrotational flow is the Laplace equation, modifying the boundary conditions at the surface should be an acceptable means of including the small viscoelastic effects. The nonzero irrotational tangential stress near the surface drags a thin vertical layer along, making a modification to the velocity field. In the present theory, viscoelastic effects are extremely weak and they can be formulated from the normal stress boundary condition. The present calculation is expected to lead to a good first approximation due to this treatment. There are no electrical volume force density terms here because ε is constant in a fluid phase and there is no volume charge in the bulk of the fluids. Thus electrical forces act only on interfaces. Their contribution passes through the normal component stress term in the boundary conditions at the surface of separation (1–3, 31). Therefore, and in view of the weak viscoelastic effects approximation considered here, the governing equation for the bulk of a fluid phase is ¸ · ∂v + (v · ∇)v = −∇ P − ρge y . [11] ρ ∂t Since the configuration must satisfy the above equation of motion in the equilibrium state, the equilibrium solution is P0(r ) = −ρ (r ) gy + c(r ) ,

[12]

where c is the constant of integration. The system of Eqs. [10] and [11] with [6]–[8] will be referred to as the tilde system. The solution of this system can be facilitated by defining a stream function, ψ, which automatically satisfies [10] and removes the explicit dependence of the pressure in Eq. [11], where v(r ) =

∂ψ ∂ψ ex − ey , ∂y ∂x (r )

(r )

[13]

which should be substituted into the above system and an equivalent boundary value problem to solve for ψ and will be obtained later. 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 dis-

turbances are assumed to be two-dimensional, that is, uniform across the width of the fluid sheet. In addition, it is assumed that the fluid interface has surface tensions T j( j + 1) , where j represents the upper fluid, and that there are no surfactants present there. The amplitude of waves that form on the fluid sheet is assumed to be small; thus, wave solutions can be obtained by assuming linear perturbations about the uniform flow solution, discussed in the previous section as E(r ) = −E 0(r ) cos ωte y − ∇φ1(r ) v(r ) = v(r0 ) +

∂ψ1(r ) ∂ψ1(r ) ex − ey ∂y ∂x

P (r ) = P0(r ) + P1(r ) ,

[14] [15] [16]

where ψ1 , φ1 , and P1 are the increments in the stream function ψ, the electric potential φ, and the pressure P, respectively. These expansions are introduced into the governing equations and boundary conditions, and terms quadratic or higher in the perturbed quantities are ignored; that is, a linear approximation is considered (9). To test the stability of the present problem, the interface between the two fluids will be assumed to be perturbed about its equilibrium location and to cause a displacement of the material particles of the fluid system. This displacement may be described by the equation ξ j (x, t) = γ j (t) exp(ikx);

j = 1, 2,

[17]

where k is the √ wavenumber, which is assumed to be real and positive, i = −1, and γ j (t) are arbitrary functions of time t which determine the behavior of the amplitude of the disturbance of the interfaces. The deformation in the interfaces y = ±a is due to the perturbation about the equilibrium values for all the other variables. According to linear perturbation theory (9), the unit vector normal to the interfaces can be derived as nj = −

∂ξ j ex + e y ; ∂x

j = 1, 2.

[18]

The equations of motion and the boundary conditions will be solved for these perturbations under the assumption that the perturbations are small; that is, all equations and boundary conditions will be linearized in the perturbation quantities. Actually, the linearized equations governing the perturbation quantities are readily found to satisfy Laplace’s equations: µ

¶ ∂ ∂ + v0(r ) ∇ 2 ψ1(r ) (x, y, t) = 0 ∂t ∂x

[19]

∇ 2 φ1(r ) (x, y, t) = 0.

[20]

As a result of the perturbation of ψ1 , φ1 and in view of [17] one

33

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

may assume the following dependence in the three fluid regions, ψ1(r ) = ψˆ (r1 ) (t, y) exp(ikx); φ1(r )

=

φˆ (r1 ) (t,

r = 1, 2, 3

phases of the system is continuous at the dividing surface. This implies that

[21]

y) exp(ikx),

¡ ¢ n J˙ · v( j) − v( j+1) = 0;

[22]

where it is necessary to determine the unknown eigenfunctions φˆ (r1 ) and ψˆ (r1 ) . The operator (∂/∂t + ikv0(r ) ) is included in the unknown eigenfunction ψˆ (r1 ) . These eigenfunctions for electric potential φ and stream function ψ die away exponentially with distance from the interfaces. However, if we substitute into Eqs. [19] and [20], the solutions of the resulting differential equations are as given in the Appendix. Note that the implication of the inertia term has been included in the process of deriving the pressure P1(r ) as shown in the Appendix. 5. BOUNDARY CONDITIONS

( j),( j+1)

∂ξ j ∂ψ =− 1 ∂t ∂x

= 0;

j = 1, 2.

[23]

To verify this fact the continuity of the normal component of electric displacement at the surfaces of separation is applicable, which requires that ( j)

ε( j)

( j+1)

∂φ1 ∂φ − ε( j+1) 1 ∂y ∂y

= 0;

y = (−1) j+1 a.

[24]

In addition, the jump in the tangential component of the electric field is zero across the interfaces. This requires that ( j)

( j+1)

= 0;

y = (−1) j+1 a, j = 1, 2. [25] 5.2. Kinematic Boundary Conditions The kinematic boundary conditions at the interfaces of the system are as follows: (i) The first kinematic relation follows from the assumption that the normal component of the velocity vector in each of the

.

[27]

(i) The normal component of the stress tensor is discontinuous by the amount of the surface tension T j( j+1) . Thus, the balance at the dividing surfaces gives ¡¡

¢ ¢ ρ ( j) − ρ ( j+1) g − k 2 T j( j+1) ξ j à ! ( j) ( j+1) ( j) ( j) ∂φ1 ( j+1) ( j+1) ∂φ1 + ε E0 E0 −ε cos ωt ∂y ∂y

( j+1)

P1

( j)

− P1 +

¶¶−1 2 ( j) µ µ ∂ ∂ ψ1 ( j) + v0 ik − 2µ( j) 1 + λ( j) ∂t ∂ x∂ y ¶¶−1 µ µ ( j+1) ( j+1) ( j+1) ∂ ik + v0 1+λ + 2µ ∂t ( j+1)

×

∂ 2 ψ1 ∂ x∂ y

= 0;

y = (−1) j+1 a,

[28]

where T j( j+1) is the surface tension through 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 stresses due to perturbations along the surfaces requires that à ε

( j)

( j) ( j) ∂φ1 E0

∂x

−ε

( j+1)

( j+1) ( j+1) ∂φ1 E0

∂x

! cos ωt

! ¶¶−1 à 2 ( j) ( j) ∂ 2 ψ1 ∂ ψ1 ∂ ( j) − + ikv0 1+λ +µ ∂t ∂ y2 ∂x2 µ µ ¶¶−1 ∂ ( j+1) + ikv0 − µ( j+1) 1 + λ( j+1) ∂t ! à ( j+1) ( j+1) ∂ 2 ψ1 ∂ 2 ψ1 × − ∂ y2 ∂x2 µ

( j)

∂ξ j ¡ ( j) ∂φ ∂φ ( j+1) ¢ cos ωt + 1 − 1 E0 − E0 ∂x ∂x ∂x

∂x

At the fluid interfaces, two sets of conditions determine the shape of the surfaces:

Because of wave propagation attention is restricted to the vertical electric field where no surface charges are present on the interfaces. Thus it is understood that the equilibrium of electric field has satisfied the relation (30) j+1

( j),( j+1) ∂ξ j

− v0

5.3. Components of the Stress Tensor Conditions

5.1. Maxwell’s Electric Conditions

( j)

[26]

(ii) The conditions that the interfaces y = ±a + ξ j (x, t) are moving with the fluids lead to

Solution of the equations of motion cited earlier is accomplished by utilizing convenient boundary conditions. At the boundary between fluids, the fluids and the electrical stresses must be balanced. The components of these stresses consist of the hydrodynamic, pressure, viscoelastic, surface tension, and electrical stresses. The electrical stresses result from the dielectric forces (29, 30).

ε ( j) E 0 − ε ( j+1) E 0

y = (−1) j+1 a.

+

µ

( j)

∂ξ j ¡ ( j) ( j)2 ( j+1)2 ¢ ε E 0 − ε ( j+1) E 0 cos2 ωt = 0; ∂x y = (−1) j+1 a. [29]

34

EL-DIB AND MATOOG

6. DERIVATION OF THE CHARACTERISTIC EQUATIONS

The goal of this section is to determine the boundary-value problem. It constitutes a homogeneous system of equations and boundary conditions, cited above to explain the factors governing the surface waves propagation. Because there are no surface charges on the unperturbed interfaces y = ±a, condition [23] will be taken into account in deriving the characteristic equations. The electric conditions which have to be satisfied at y = ±a are condition [24] and condition [25]. With these boundary conditions, the solution of the Laplace equation [20] yields the distribution of the potential field φ1 in the three layers as given in the Appendix. Further, the solution of the Laplace equation [19] in view of the boundary conditions [26] and [27] gives the stream function ψ1 and thus the distribution of the pressure P1 in the three layers as presented in the Appendix. It is known for inviscid fluids that in the absence of surface charges conditions [25] and [29] are equivalent, but this is not so in the presence of viscosity or viscoelasticity. Therefore, there exists a contribution for condition [29] that gives the following conditions:

conditions [30] allows us to eliminate the imaginary part of the variables γ j from Eqs. [31] and [32]. In Section 9 conditions [30] are used to eliminate the second derivative of [31] and [32]. 7. CHARACTERISTIC EQUATIONS IN THE CASE OF SYMMETRIC MODES OR ANTISYMMETRIC MODES

The solution to the coupled Mathieu equations [31] and [32] can be facilitated by defining the symmetric and antisymmetric deformation of the surface deflections ξ1 and ξ2 . Therefore, the variables ξ1 and ξ2 may be related by ξ2 = J ξ1 ,

where J = 1 refers to the antisymmetric deformation, while J = −1 indicates the symmetric deformation. To solve the characteristic equations [31] and [32] in view of symmetric or antisymmetric deformation we use the definition [33] keeping in mind the use of condition [30], which allows us to eliminate the imaginary part of γ j . The algebra is lengthy but straightforward. Finally we obtain

¡

¡¡ ¢ ¢ µ( j+1) λ( j+1) − µ( j) λ( j) γ¨ j + µ( j) − µ( j+1) ¡ ( j+1) ( j) ¢¢ − λ( j) µ( j) v0 γ˙ j + 2ik λ( j+1) µ( j+1) v0 ¡ ¡ ¡ ( j) ( j+1) ¢ ( j)2 + ik µ( j) v0 − µ( j+1) v0 + k 2 λ( j) v0 µ( j) ( j+1)2 ( j+1) ¢¢ − λ( j+1) v0 µ j = 1, 2. [30] γ j = 0;

Substituting the solutions for φ1(r ) , ψ1(r ) , and P1(r ) into the normal stress condition [28] gives the coupled equations in γ1 and γ2 ¡ ¢ 2 γ¨1 + ( f 11 + il11 )γ˙1 + s11 + i h 11 + E 0(2) k11 cos2 ωt γ1 ¡ ¢ 2 + ( f 22 + il22 )γ˙2 + s22 + i h 22 + E 0(2) k22 cos2 ωt γ2 = 0,

¡ ¢ 2 γ¨ + (a1 + ib1 )γ˙ + c1 + d1 E 0(2) cos2 ωt γ = 0 ¡ ¢ 2 γ¨ + (a2 + ib2 )γ˙ + c2 + d2 E 0(2) cos2 ωt γ = 0,

[32] which are two coupled Mathieu equations with damping terms and having complex coefficients. These coefficients are defined in the Appendix. These equations are used to control the stability behavior for the fluid sheet having viscoelastic interfaces. The boundary-value problem, Eqs. [30], [31], and [32], constitutes a homogeneous system of equations and boundary conditions, that is, a linear eigenvalue problem to determine the surface elevation amplitudes γ1 and γ2 as functions of t for given physical, geometrical, and flow properties. The combination between the system of [31] and [32] and the system of [30] will lead to simplified characteristic equations. Two different treatments are discussed. In Section 7, the use of

[34] [35]

where the constant coefficients a, b, c, and d are given in the Appendix. The result, given by Eqs. [34] and [35], allows a boundary-value problem to be obtained totally in terms of one unknown variable γ . It is easy to show that the above system of Eqs. [34] and [35] can be reduced to a single Mathieu equation with or without a damped term. Eliminating b1 and b2 from Eqs. [34] and [35] yields a damped Mathieu equation in the form ¡ ¢ 2 dγ d 2γ + δ˜ + q E 0(2) cos2 ωt γ = 0, + α˜ 0 2 dt dt

[31]

¡ ¢ 2 γ¨2 + ( f 12 + il12 )γ˙2 + s12 + i h 12 + E 0(2) k12 cos2 ωt γ2 ¡ ¢ 2 + ( f 21 + il21 )γ˙1 + s21 + i h 21 + E 0(2) k21 cos2 ωt γ1 = 0,

[33]

[36]

where α˜ 0 =

a 1 b2 − a 2 b1 b2 − b 1

c1 b2 − c2 b1 δ˜ = b2 − b1 q=

d1 b2 − d2 b1 . b2 − b 1

In the case of a static field the damped Mathieu equation [36] reduces to ¡ 2¢ dγ d 2γ + δ˜ + q E 0(2) γ = 0, + α˜ 0 2 dt dt

[37]

which is a linear differential equation with constant coefficients and can be satisfied by a growth rate solution having the form

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

γ = exp(σ0 + iω0 )t. The stability examination depends on the sign of σ0 (the real part). Both σ0 and ω0 are real constants, and they satisfy the following equations: ¡ 2¢ σ02 − ω02 + α˜ 0 σ0 + δ˜ + q E 0(2) = 0

[38]

2σ0 + α˜ 0 = 0.

[39]

The elimination of σ0 yields 2 α˜ 2 ω02 = δ˜ + q E 0(2) − 0 . 4

[40]

The assumption that ω0 is real imposes the following stability condition with the requirement that α˜ 0 is positive:

E 0(2)

2

2 α˜ 2 δ˜ + q E 0(2) − 0 > 0 4 ¢ 1 ¡ 2 α˜ − 4δ˜ ; > E˜ = q > 0. 4q 0

[41] [42]

In the presence of the field frequency ω the stability picture has changed dramatically. Grimshaw (33) has examined the Mathieu equation [36] by the perturbation technique. On the boundaries of the first unstable region in the Mathieu diagram there is a periodic solution of Eq. [36] with period 2π. The stability is thus described by 3q

2

4 2 E 0(2) + 16(δ˜ −ω2 )q E 0(2) +16(δ˜ − ω2 )2 +16ω2 α˜ 02

> 0, [43]

with the constraint that α˜ 0 is positive. In terms of the electric field E 0(2) the above stability condition can be arranged in the form ¡ (2)2 ¢¡ ¢ 2 E 0 − E˜ 1 E 0(2) − E˜ 2 > 0. [44] Thus the stability criterion is found to be 2

2

E 0(2) > E˜ 1 or E 0(2) < E˜ 2 ; where

E˜ 1 > E˜ 2 ,

q ¤ 4 £ E˜ 1,2 = −2(δ˜ − ω2 ) ± (δ˜ − ω2 )2 − 3ω2 α˜ 02 3q

[45]

[46]

are the transition curves separating the stable region from the unstable region. From Floquet theory (34) the region bounded by the two branches of the transition curves E˜ 1 and E˜ 2 is the unstable region. The area outside these curves is a stable region. The width of the unstable region is represented by ( E˜ 1 − E˜ 2 ). The increase of this width refers to the destabilizing influence, while its decrease represents a stabilizing role. To explain the influence of the parameter α˜ 0 on the stability we need to inspect the sign of the following relation: −24ω2 α˜ 0 ∂( E˜ 1 − E˜ 2 ) = q . ∂ α˜ 0 3q (δ˜ − ω2 )2 − 3ω2 α˜ 02

[47]

35

Since α˜ 0 is positive, the negative sign of ∂( E˜ 1 − E˜ 2 )/∂ α˜ 0 means that the area of the unstable region is decreased as α˜ 0 is increased, which shows the damping role of the parameter α˜ 0 . Inspection the definition of α˜ 0 shows that the viscosity parameter plays a stabilizing role in this case. Another conclusion can be observed ˜ α˜ 0 is positive, the unstable region is in the static case. Since ∂ E/∂ increased as α˜ 0 is increased. Therefore, a destabilizing influence for the viscosity parameter is found in the absence of ω. 8. MARGINAL STATE REPRESENTATION

The single governing equation [36] for all symmetric and antisymmetric modes is used to discuss the stability picture for the surface waves propagation since the coefficient α˜ 0 vanishes, giving the marginal state analysis. This can be satisfied in the case of matching velocities (v0(1) = v0(2) = v0(3) ) or in the inviscid case. On the other hand, eliminating a1 and a2 between Eqs. [34] and [35] and using the transformation · µ ¶¸ 1 a2 b1 − a1 b2 γ (t) = γ˜ (t) exp − i t 2 a2 − a1

[48]

yields an undamped Mathieu equation in the form ¡ ¢ 2 d 2 γ˜ + 1 + π E 0(2) cos2 ωt γ˜ = 0, dt 2

[49]

which can be used to discuss the stability near the marginal state even in the presence of both viscosity and velocity parameters. The constant coefficients 1 and π are given below: ¶ µ 1 a2 b1 − a1 b2 2 a 2 c1 − a 1 c2 + 1= a2 − a1 4 a2 − a1 π =

a 2 d1 − a 1 d2 . a2 − a1

Two scopes of the stability criteria are taken into account: the stability behavior in the absence of the field frequency and the influence of the presence of the field frequency on the stability examination. 8.1. Numerical Discussion for Stability Behavior Due to Electocapillarity Excitation For a static electric field as ω → 0, we present the wave propagation results obtained for a nonoscillating field, where waves were excited using the electrocapillarity technique. In this case Eq. [49] reduces to ¡ 2¢ d 2 γ˜ + 1 + π E 0(2) γ˜ = 0, dt 2

[50]

36

EL-DIB AND MATOOG

q 2 which can be satisfied by the solution γ˜ = exp[i 1 + E 0(2) π t]. Thus, the stability condition in the static case reduces to 2

1 + E 0(2) π > 0.

[51]

It is noted that for arbitrary field E 0(2) the stability is satisfied for 1 > 0 and π > 0. Otherwise stability occurs when 2

E 0(2) > E ∗

for

1<0

and π > 0,

[52]

for

1>0

and π < 0,

[53]

or 2

E 0(2) < E ∗ where

E∗ =

−1 . π

[54]

FIG. 2. Influence of the fluid sheet thickness on the stability where the marginal stability curves are plotted showing the calculations for [54]. Curves 1 to 5 indicate the increase of the parameter a from 0.3 to 1.5 with step 0.3.

It is easy to show that for negative values of E ∗ condition [52] is trivially satisfied and the field has no contribution for stability, while condition [53] is false and instability is present. Before we proceed to the numerical estimation, and in order to numerically assess the linear stability of the wave propagation on the interfaces, it is useful to investigate the transition curves which separate the stable region from the unstable region in a nondimensional form. To this end, the following dimensionless forms are introduced: the characteristic length l = v0(2) λ(2) , the characteristic time t = λ(2) , and the characteristic mass M = µ(2) λ(2)l. Other dimensionless quantities are given −2 −1 −1 by k = k ∗l −1 , ρ ( j) = ρ ∗( j) µ(2) v0(2) λ(2) , g = g ∗ v0(2) λ(2) , ω = −1 −1 ( j) ω∗ λ(2) , a = a ∗l, T j( j+1) = T j(∗ j+1) µ(2)lλ(2) , and E 0 = ∗( j)

−2

−1

E 0 µ(2) λ(2) v0(2) l. In this section, the goal is to determine the numerical profiles for the stability pictures in the static case for the field as well as for the oscillating case. There are two modes for the surface deformations included in the theoretical analysis: the symmetric deformation (J = −1) and the antisymmetric deformation (J = 1). The case of J = 1 is considered in our numerical estimation. In order to screen the examination of the constant field influence on the stability criteria, numerical calculations for the stability condition [51] are made. The results for the calculations are 2 displayed in Figs. 2, 3a, and 3b to indicate the plane (E 0(2) − k) with a transition curve as a function of the sheet thickness, the velocity ratio va = (v0(1) /v0(2) ), and the velocity ratio vb = (v0(3) /v0(2) ). The graphs displayed in this plane for a sample case are in a nondimensional form, where ρ (1) = 0.24, ρ (2) = 0.65, ρ (3) = 1.5, ε(1) = 3.3, ε(2) = 38, ε(3) = 17, T12 = 20, T23 = 12, µa = (µ(1) /µ(2) ) = 2.5, µb = (µ(3) /µ(2) ) = 1.5, va = 1.5, vb = 0.5, λa = (λ(1) /λ(2) ) = 1.6, λb = (λ(3) /λ(2) ) = 1, g = 340, and the halfthickness of the sheet a = 0.5, where the superposed ∗ has been omitted for simplicity.

(10)

(2)

FIG. 3. (a) Influence of the velocity ratio va = (v0 /v0 ) on the stability picture for the same system as in Fig. (2). Curves 1 to 6 refer to the parameter va = 0.5, 0.6, 0.7, 0.8, 0.9, and 1, respectively. (b) Influence of the variation of (3) (2) the velocity ratio vb = (v0 /v0 ) with fixed va = 1.5 on the stability diagram. The variation for vb = 0.5–1 with step 0.1 is represented by curves 1 to 6, respectively.

37

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

It is noted that the above specific values for the fluid densities ρ (r ) , r = 1, 2, 3, are chosen so that the system is statically stable in the absence of 2the electric field. The plane (E 0(2) − k) is partitioned by the transition curve [54] into two regions. The upper one is the unstable region and is labeled by the letter U . The lower one is the stable region and is labeled by the letter S. The introduction of the vertical field plays a destabilizing role, which is an early important phenomenon studied by Melcher (32) and by several other researchers for inviscid flow through the linear stability theory. Melcher (32) demonstrated that in the linear stability theory, the tangential field has a stabilizing effect (it increases the surface tension effect), while the vertical field has a destabilizing influence (it decreases the surface tension effect). Even if the fluids have a viscoelastic nature the vertical field still plays a destabilizing role, as demonstrated by El-Dib (8). He studied the effect of both vertical and tangential fields on surface wave propagation in the interface between two rheological fluids of Kelvin type. He demonstrated that the vertical field retards the stabilizing influence of both viscosity and elasticity parameters, whereas the presence of a tangential field suppresses the destabilizing influence of the stratified density. Regarding the contribution of the dielectric constant, Mohamed and Elshehawey (15, 35–37) discussed electric fields under the stability conditions in detail. They found that if a finite amplitude disturbance is stable then a small modulation to the wave is also stable. They reported that the tangential field plays a dual role in the stability criterion. The dielectric constants of the fluids affect the stability conditions. For example, for a range of relatively smaller values of the density ratio, the field is stabilizing or destabilizing according to whether the lower fluid has a larger or smaller dielectric constant than the upper one. As the electric field increases, a greater stabilizing influence appears for a band of values of the density ratio. The examination of the influence of the sheet thickness and the ratios of the velocities va and vb on the stability criteria for the surface wave proportions is the goal of the present calculations which are depicted in Figs. 2, 3a, and 3b, respectively. In Fig. 2, the graph displays the natural stability curves [54] for various values the sheet thickness. The curves marked by the numbers 1 to 5 refer to the variation values a = 0.3–1.5 with step size 0.3. It is apparent from inspection of this graph that increasing the sheet thickness increases the destabilizing effect of the vertical field. Thus, the increase of the sheet thickness has a destabilizing influence, especially for small values of the wavenumber k. It is observed that the increase of the wavenumber k has counteracted the effects of variation in a where for large values of k, the variation of a has no effect on the stability picture. On the other hand, the increase of k has the same effect as that of the increase of thickness. Further, the instability is 2 seen for arbitrary values of E 0(2) and for all thicknesses a when k has values larger than 8.1. This destabilizing influence of an increase in the sheet thickness has not been observed before for Newtonian fluids. In pure inviscid fluids, Mohamed et al. (38)

demonstrated that the increase of the thickness of the inviscid fluid layer has a stabilizing effect. Thus, we can see that the flow has undergone a dramatic change in its behavior due to the presence of the viscoelasticity factor. The influence of the velocity parameters va and vb on the stability picture is displayed in Figs. 3a and 3b, respectively. In the unstable region that is present due to the influence of the vertical fields, two different effect are found for the influence of the velocities va and vb . The variation of the velocity ratio va (=0.5, 0.6, 0.7, 0.8, 0.9, and 1) with fixed velocity ratio vb = 0.5 is displayed in Fig. 3a, which shows that the increase of va increases the unstable region. Thus, the increase of va increases the destabilizing influence of the electric field. The destabilizing influence of the Kelvin–Helmholtz waves has been demonstrated before by Chandrasekhar (9). The linear electrohydrodynamic Kelvin–Helmholtz instability was studied by Lyon (10). He showed that both the streaming and vertical fields have a strictly destabilizing influence. Another conclusion can be observed in Fig. 3b where the variation of the velocity vb is considered (vb = 0.5, 0.6, 0.7, 0.8, 0.9, and 1) with fixed va = 1.5. The increase of the velocity ratio vb increases the stable region. This effect increases as the wavenumber k is increased. Thus, the increase of vb retards the destabilizing effect of the field. A conclusion can be made here that the increase of the velocity of the uppermost fluid has a destabilizing effect while the increase of the velocity of the lower fluid has a stabilizing influence. 8.2. Numerical Illustration of Stability Behavior Due to Oscillation of the Field Next, the electrocapillary excitation was switched off, and the field was oscillated at the frequency ω. The ordinary Mathieu equation [49] has been studied extensively. The stability arises when the following inequality is satisfied: 4

2

E 0(2) π 2 + 16(ω2 − 1)E 0(2) π + 321(ω2 − 1) > 0. [55] This stability criterion reduces the problem of the bounded region of the Mathieu functions, which is discussed by Mclachlan (39). In terms of the electric field E 0(2) the above condition can be arranged in the form ¡

¢¡ ¢ 2 2 E 0(2) − E 1∗ E 0(2) − E 2∗ > 0,

[56]

where E 1∗ and E 2∗ are the stability boundaries, which are given by ∗ E 1,2

s µ µ ¶¶ 8 3 2 2 2 = −(ω − 1) ± (ω − 1) ω − 1 . [57] π 2

Condition [51] can be satisfied as 2

E 0(2) > E 1∗

or

2

E 0(2) < E 2∗ ;

E 1∗ > E 2∗ .

[58]

38

EL-DIB AND MATOOG

From Floquet theory (34) the region bounded by the two branches of the transition curves E 1∗ and E 2∗ is the unstable region. The area outside these curves is a stable region. Despite this case, numerical calculations were made for the stability conditions [56]. The calculation results for the transition curves [57] are displayed in Figs. 4–8 to indicate the influence of the field frequency ω, the sheet thickness a, the velocity ratio va , the viscosity ratio µa , and the elasticity ratio λa , respectively. Due to the numerical parameter used here the calculation shows that the velocity ratio vb , the viscosity ratio µb , and the elasticity ratio λb play a very small role in the stability picture. The graphs 2 displayed in the plane (E 0(2) − k) for the same system are considered in Fig. 2 except that va = 1 and ω = 10. The influence of the presence of the field frequency ω on the stability criteria 2 shows that the plane (E 0(2) − k) is partitioned by the transition curves E 1∗ and E 2∗ into stable and unstable regions. These regions are functions of the field frequency ω, the sheet thickness a, the velocity ratio va , the viscosity ratio µa , and the elasticity ratio λa . The stability examinations are illustrated by three different numerical values for each parameter mentioned above. The transition curve labeled I refers to the lowest value for each parameter. The curve labeled II represents the case of the middle value. The curve marked III indicates the highest value that is chosen in the numerical estimation. 2 Inspection of the plane (E 0(2) − k) for the specific value ω = 10.5 as given in Fig. 4 shows that the two transition curves E 1∗ and E 2∗ bound an unstable region U . There are two stable regions S1 and S2 bounded by the curves E 1∗ and E 2∗ , respectively, where the S1 region lies under the curve E 1∗ and the S2 region lies in the area above the curve E 2∗ . These transition curves are labeled III in Fig. 4. This stability picture is due to the presence of the field frequency ω where the stable region S2 disappears in the absence of the field frequency that was shown in the previous

FIG. 4. Influence of the small variation of the field frequency ω. The curves labeled I refer to the case of ω = 9.5, II represents the case of ω = 10, while III denotes the case of ω = 10.5. The transition curves E 1∗ and E 2∗ indicate the relation [57] where the stability condition [55] is satisfied.

calculations as shown in Figs. 2, 3a, and 3b. Thus, the presence of ω bound the unstable region that occurs due to the influence of the vertical field intensity. This stability picture that occurs due to the influence of the oscillating vertical field is a function of the variation of ω. The influence of this variation ω = 9.5, 10, and 10.5 on the stability behavior is shown in Fig. 4. It appears from the inspection of Fig. 4 that as the field frequency ω is decreased, the transitions curve E 2∗ moves rapidly down and leads to an increase in the stable region S2 . On the other hand, the transition curve E 1∗ has moved up slightly and a very small increase in the stable region S1 is observed. This shifting of the transition curves E 1∗ and E 2∗ occurs until they contact at a point. The continued decrease in ω leads to the division of each of the curves E 1∗ and E 2∗ into two branches. Each set of branches contains an unstable region. However, the decrease in ω yields a decrease in the unstable region U so that it partitions into two smaller unstable regions U1 and U2 . A rapid decrease in U1 and U2 is observed as ω is decreased and a larger stable region composed of S1 and S2 is left. This behavior for the variation of the field frequency shows that the increase of the field frequency plays a destabilizing influence. In the linear theory of surface waves for non-Newtonian fluids of Maxwell type Mohamed et al. (7) demonstrated that the increase of the field frequency ω has a stabilizing effect. They found that in the absence of the elasticity parameter (i.e., pure viscous effects) the field frequency plays a destabilizing role. They showed that the field frequency plays against the kinematic viscosity in pure viscous fluids as well as in Maxwell fluids and behaves like the kinematic viscosity in Kelvin fluids. In small viscous fluids with the presence of surface charges the system exhibits a destabilizing influence as the field frequency is increased (19), while the stabilizing influence for ω is noted in the case of a periodic rotating inviscid fluid column (40). A dual role for the field frequency is observed in the nonlinear stability for inviscid fluids (41). Even in the linear stability for a small viscous fluid a dual effect the field frequency is observed in a rigidly rotating fluid column (27). The examination for the increase of the sheet thickness on the stability behavior in the presence of ω is displayed in Fig. 5. 2 The graph shown in the plane (E 0(2) − k) is for three different values of the parameter a (=0.4, 0.5, and 0.6). It is shown that increasing the sheet thickness leads to increases in the unstable regions U1 and U2 . The increase in the unstable region U2 is more rapid than that in the unstable region U1 . This increase in the unstable region continues until U1 and U2 are connected in one unstable region U . At this stage, the transition curves partition the stability diagram into stable region S1 , unstable region U, and stable region S2 . Thus, even in the presence of the field frequency ω the increase of fluid sheet thickness has a destabilizing influence. The influence of the velocity ratio va on the stability picture is displayed in Fig. 6. Three specific values of the parameter va are considered in this investigation: the case of va = 0.5 which is

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

39

FIG. 5. Influence of small variation in the sheet thickness a. I refers the case of a = 0.4, II denotes the case of a = 0.5, and III represents the case of a = 0.6. The graph represents the same system as in Fig. 4 except that ω has a fixed value ω = 10.

FIG. 7. Influence of a slight change in the viscosity ratio µa on the stability behavior. I, II, and III refer to µa = 2.3, 2.6, and 2.9, respectively, for va = 1, a = 0.5, and ω = 10.

represented by I, the case of va = 1 which is characterized by II, and the case of va = 1.5 which is represented by III. The graph shows that as va is increased both of the transition curves E 1∗ and E 2∗ move down and consequently the unstable regions U1 and U2 are shifted down, leading to a decrease in their width. This shows the stabilizing influence of increasing the velocity ratio va . Another conclusion was observed in Fig. 3a in the absence of the field periodicity ω. Thus, the velocity ratio va has changed its mechanisms in the presence of the frequency ω. Further, inspection of the stability diagram [6] reveals that the field intensity is a function of the parameter va . This is because the shifting in the unstable regions means that the critical value of the field intensity for stability has been affected by the increase of the parameter va . Large values of va should lead to small 2 values of E 0(2) , causing a bounded destabilizing influence.

The examination of the effect of increasing the viscosity ratio µa on the stability behavior is shown in Fig. 7 where three different values of µa (=2.3, 2.6, and 2.9) are considered. The graph shows that a small increase in the viscosity ratio µa leads to contraction of the unstable regions U1 and U2 . This behavior gives the stabilizing influence of µa . Thus, the viscosity ratio µa behaves as the velocity ratio va in the stability criteria in which the stabilizing influence is observed. The influence of the elasticity ratio λa plays another role in the stability picture as shown in Fig. 8, where a small increase in λa (=1.5, 1.6, and 1.7) yields a relatively large increase in the unstable region, especially in the region U2 associated with moving the transition curve E 2∗ up and leaving a large unstable area. The same destabilizing influence on the Maxwell relaxation time has been demonstrated

FIG. 6. Influence of the variation of the velocity ratio va on the stability picture. I refers to the case of va = 0.5, II represents the case of va = 1, and III refers to va = 1.5. In this graph a has the value 0.5 while ω has the value 10 and other parameters are held fixed as in Fig. 5.

FIG. 8. Influence of a slight increase in the elasticity ratio λa = 1.5, 1.6, and 1.7. These cases are represented by I, II, and III, respectively. The graph is for the same system considered in Fig. 5 except that a = 0.5, ω = 10, va = 1, and µa = 2.5.

40

EL-DIB AND MATOOG

before by Mohamed et al. (4) and by Saasen (42). The comparison between Figs. 4, 5, and 8 shows that the increase in the field frequency ω, the increase in the sheet thickness a, and the increase in the elasticity parameter λa have the same effect on the stability criteria. Their contribution has a destabilizing effect.

The above equations constitute a system having two degrees of freedom with periodic coefficients of a complex nature. There is no known analytic solution for this system. Therefore, we shall discuss its stability using the asymptotic expansion treatment and use the method of multiple scales to discuss the stability of the problem.

9. CHARACTERISTIC EQUATIONS WHERE THE SURFACE DEFLECTIONS ARE INDEPENDENT

9.1. Stability Analysis and Numerical Discussion in the Case of a Constant Field

The effect of general surface deformations on the onset of a periodic vertical electric field applied to the streaming fluid sheet is the main line adopted in the present section. Accordingly, the boundary-value problem which was described by the simultaneous Mathieu equations [31] and [32] needs treatment without any constraints on the surface deflections. Therefore, we shall examine this system with two degrees of freedom using the two differential equations of general surface deformations. The solutions and the properties of a single Mathieu equation with real coefficients are well known (39). In addition, there are many books (33, 43) which treat the damped Mathieu equation with real coefficients. El-Dib (40) has analyzed the single Mathieu equation with an imaginary parametric damped term. Further, the El-Dib (27) and Moatimed and El-Dib (44) additionally damped Mathieu equation included complex coefficients. They derived the stability criteria by using the multiple scale method. The same perturbation method allowed Mohamed et al. (45) to discuss two coupled Mathieu equations without damped terms and having real coefficients. There are no general analytical solutions available for the system of coupled Mathieu equations with damped terms having complex coefficients such as those described by [31] and [32]. Therefore, we shall discuss the stability of this system using an asymptotic expansion treatment. The method of using multiple scales enables us to discuss the stability of the problem (34). Since this section is concerned with the case of relaxing the relationship between γ1 and γ2 we may assume that the surface deflections ξ1 and ξ2 are neither symmetric nor antisymmetric. If we study the problem in the general case where ξ1 and ξ2 are independent, we may use [30] to eliminate the second derivative from the system [31] and [32]. The results are

Before we analyze the above system with periodic coefficients, we need to screen the examination of the constant field influence on the stability criteria. Hence, at the limiting case of ω → 0 in the above system the solutions may be written as

£ ¤ 2 γ˙1 + (a11 + ib11 ) + 4E 0(2) ( p11 + iq11 ) cos2 ωt γ1 £ ¤ 2 + (a12 + ib12 ) + 4E 0(2) ( p12 + iq12 ) cos2 ωt γ2 = 0 [59] and £ ¤ 2 γ˙2 + (a22 + ib22 ) + 4E 0(2) ( p22 + iq22 ) cos2 ωt γ2 £ ¤ 2 + (a21 + ib21 ) + 4E 0(2) ( p21 + iq21 ) cos2 ωt γ1 = 0, where the coefficients ai j , bi j , pi j , and qi j are as given in the Appendix.

£ ¤ 2 γ1 = − (a12 + ib12 ) + 4E 0(2) ( p12 + iq12 ) m 0 eÄt £ ¤ 2 γ1 = Ä + a11 + ib11 + 4E 0(2) ( p11 + iq11 ) m 0 eÄt ,

[60]

where m 0 is a complex constant of integration and Ä is the growth rate that is given by Ä2 + A0 Ä + B0 = 0,

[61]

which is a quadratic equation in Ä with complex coefficients where £ ¤ 2 A0 = a11 + a22 + 4E 0(2) ( p11 + p22 ) ¤ £ 2 + i b11 + b22 + 4E 0(2) (q11 + q22 ) £ ¤ 2 B0 = a11 + ib11 + 4E 0(2) ( p11 + iq11 ) ¤ £ 2 × a22 + ib22 + 4E 0(2) ( p22 + iq22 ) ¤ £ 2 − a12 + ib12 + 4E 0(2) ( p12 + iq12 ) ¤ £ 2 × a21 + ib21 + 4E 0(2) ( p21 + iq21 ) . For the polynomial with complex coefficients (46) the Hurwitz criterion for stability imposes the conditions. Re(A0 ) > 0 Re(A0 )Re(A0 B¯ 0 ) − (Im B0 )2 > 0,

[62]

where Re and Im represent the real and imaginary parts, respectively, and the bar refers to the complex conjugate. In terms of the electric field intensity, the above inequalities can be 2formulated in the form of the first order and fourth order in E 0(2) , respectively. This dependence has been borne out by numerical calculations. In numerical results detailed here the computed value of the 2 field intensity E 0(2) versus the wavenumber k is utilized. In the calculations given below all the physical parameters are sought

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

in the dimensionless form as defined in Section 8.1. The calculations are carried out for the same system as defined in Section 8.1. The stability examination will be made by fixing the value of all the physical parameters (the sheet thickness a, the field frequency ω, the velocity ratios va and vb , the viscosity ratios µa and µb , and the elasticity ratios λa and λb ) except for one parameter having varying values for comparison. First, we present the results in the stability analysis given in the absence of the field frequency (ω → 0) by drawing the transition curve represented by the equality of both relations [62]. Numerical calculations showed that the critical transition curve is that given by the second stability condition of [62]. This curve is displayed in the graphs in Figs. 9–12. Note that the system is stable in the absence of2the electric field. Figure 9 represents the variation of log (E 0(2) ) versus k for several values of the sheet thickness a (=0.03, 0.06, 0.09, 0.12, and 0.15) while the other parameters are held fixed. It is apparent that in the interval 0 < k < 1, there is a rise in the curve bounding a very narrow stable region which shows the destabilizing influence of increasing the wavenumber. This destabilizing influence of increasing k has been counteracted by increasing the thickness value. The stability examination shows that the constant vertical field has a destabilizing influence while the strong large values of the field have a stabilizing influence. This stabilizing influence is apparent as a function of the sheet thickness. The slight increase of a thin thickness increases the stabilizing influence on the field. This means that the variation of small thicknesses has a stabilizing influence. Similar conclusions were observed before by Mohamed et al. (47) and in addition, an increase in the stable region is observed for the present case. An opposite effect was observed in the case of antisymmetric deformation, which was discussed in Section 8.1. In Fig. 10 the graph shows the influence of both of the velocity ratios va and vb . This collection is for comparison. The curve carrying the symbol × refers to the case of va = 1 and vb = 0.5 while the other parameters are held fixed as in Fig. 9 with a fixed sheet thickness a = 0.1.

FIG. 9. Stability chart for the variation of the sheet thickness a for the same system as in Fig. 2, where va = 1, vb = 0.5, µa = 2.5, µb = 0.5, λa = 1.6, and λb = 1. The graph indicates the second stability condition of [62].

41

FIG. 10. Stability diagram for the variation of the velocity ratios. The graph collects changes in both va and vb . The curve carrying the symbol × indicates the case of va = 1 and vb = 0.5. The curves labeled by I and II refer to the extensions in the value of va by 0.01 and 0.02, respectively, with fixed vb = 0.5. The curves labeled by III and IV represent the extensions in the value of vb by 0.03 and 0.06, respectively, with fixed va = 1.

This curve has divided the plane into an unstable region and a stable region. The unstable region lies below the transition curve while the stable region is presented in the upper part of the curve. In order to study the impact of each velocity ratio we shall fix one velocity ratio and change the other one to discuss the influence of one parameter on the stability behavior. The curve labeled I refers to va increased to 1.01, while the curve labeled II refers to va changed to 1.02. The value va = 1 is fixed and vb changed from 0.5 to 0.53 and then to 0.56. The results are represented by III and IV, respectively. It is apparent from the inspection of the graph for the case of 2 va = 1 and vb = 0.5 that the electric field E 0(2) has a nonlinear relation to the wavenumber k, and so the stability region has been largely affected by the increase in the wavenumber k. The consequence of the maximum and the minimum stability is observed. Large stability occurs in the interval 0 < k < 1. Maximum instability occurs in the interval of 1 < k < 2 and maximum stability is present in the interval 2.5 < k < 3.5. Then maximum instability is shown in the interval of 6 < k < 7. After that arbitrary stability occurs independent of the electric field. The stability picture has been affected by the increase of both va and vb . The increase of va produces a large stabilizing influence associated with small decrease in the maximum instability, which occurs in the interval of 1 < k < 2. At this stage, the increase of va has increased the stabilizing influence of the field, while the destabilizing influence has decreased. This behavior of the stability criteria is similar to that observed in the case of increasing sheet thickness a (see Fig. 9). It is noted that this stabilizing influence of increasing va has not been observed before in the case of the antisymmetric deformations as numerically discussed in Section 8.2. Another stability behavior with increasing velocity ratio vb with fixed va = 1 is observed. As vb is increased, two different effects are presented: a small stabilizing influence in

42

EL-DIB AND MATOOG

the interval 1 < k < 4 and a relatively large destabilizing influence for 4 < k < 10. The destabilizing efffect of an increase vb has increases as k is increased. This destabilizing influence has not been recorded in Section 8.2 in the case of a constant field. The small stabilizing effect has been observed before in the case of antisymmetric deformations. However, the dual role for increasing vb has been observed in the present study. The effect of increasing the viscosity ratio µa one time and increasing the viscosity ratio µb another time is examined in Fig. 11. The curve carrying the symbol × indicates the neutral case of µa = 1.5 and µb = 1.5 for fixed va = 1 and vb = 0.5. In addition, the other parameters are held fixed as in Fig. 2. As shown before in Figs. 9 and 10, a maximum and minimum stability occur as a function of the wavenumber k. These maximum and minimum stabilities have been affected by fixing µb to the value 1.5 and changing µa to 2 and then to 2.5. As µa increases the stability region decreases, associated with a small increase in its area, especially for large values of k because of the large destabilizing effect relative to small values of k and a stabilizing role relative to large values of k. We conclude that the dual effect phenomena are presented here. In order to examine the influence of changing µb on the stability behavior we fix µa to the value 1.5 and extend µb slightly to 1.55 and then to 1.6. A very narrow stabilizing influence relative to small values of k associated with a relatively large destabilizing influence for large wavenumber k is observed. This destabilizing influence increases as k increases. Again, the dual effect is observed for increasing µb , but in a manner opposite to the observed for increasing µa . It is noted that the influence of µa is different from the influence of µb . The influence of increasing λa one time and increasing λb another time is displayed in Fig. 12. The case of λa = 1.6 and λb = 1 has been indicated by the curve carrying the symbol ×.

FIG. 12. Influence of increases in both λa and λb . The neutral case is represented by the curve carrying the symbol × for λa = 1.6 and λb = 1, with the other parameters fixed. Increasing λa to 1.8 and then to 2 is represented by I and II, respectively, while increasing λb to 1.03 and then to 1.06 is denoted by III and IV, respectively.

Extending the value of λa to 1.8 and then to 2 is represented by I and II, respectively. Extending the value of λa with fixed λa is denoted by III and IV. This shows that the stabilizing influence for the vertical field increases as the stratified relaxation time λb increases. This shows the stabilizing role of the parameter λb . The increase of λb has a stabilizing influence. In dealing with the increase of the stratified relaxation time λa two effects are readily seen. As λa increases, a small increase in the stable region is observed. The stabilizing influence decreases as k increases until the destabilizing influence occurs. This destabilizing effect occurs for large k. Thus, we can conclude that the increase of λa has a dual effect. 9.2. Employing the Multiple Scales Method In order to assess the implications of the field periodicity for the theory in a specific case, we return to the description of the system [59] and analyze it in the presence of ω. To employ the method of multiple scale expansion [34], we shall introduce a 2 (2)2 ˜ 00 . dimensionless (smallness) parameter ε˜ such that E 0(2) = εE Further, we may use the time scales T0 and T1 , such that T0 = t ˜ as a fast time scale and a slow time scale, respecand T1 = εt, tively. The differential operator can now be expressed as the derivative expansions d = D0 + ε˜ D1 + ε˜ 2 D2 + · · · ; dt

Dn =

∂ . ∂ Tn

[63]

Also the dependent variables γ j can be expanded in the form ˜ = γ j0 (T0 , T1 ) + εγ ˜ j1 (T0 , T1 ) + · · · ; γ j (t, ε) FIG. 11. Influence of the viscosity ratios µa and µb . The curve carrying the symbol × refers to the case of µa = 1.5 and µb = 1.5. The curves marked by I and II refer to the increase of µa to 2 and then 2.5, respectively, with fixed µb . The curves represented by III and IV refer to the extension of µb to 1.55 and 1.6, respectively.

j = 1, 2. [64]

Substituting [63] and [64] into [59] and equating coefficients of like powers of ε˜ yields simpler equations, which can be solved successively. Two versions are of interest: the fast and the slow solutions.

43

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

The fast solution may be written as γ10 = −(a12 + ib12 )η j (T1 )eÄ j T0 + cc γ20 = (Ä j + a11 + ib11 )η j (T1 )eÄ j T0 + cc;

j = 1, 2,

[65]

where η j (T1 ) are unknown complex functions to be determined, cc represents the complex conjugate of the preceding terms, and ˜ j with real σ˜ j and $ ˜ j ) sathe growth rate Ä j (Ä j = σ˜ j + i $ (2)2 tisfy Eq. [61] as E 0 → 0 in both coefficients A0 and B0 . As mentioned before, the stability examination may be described by [62]. If these conditions are met, the instability will arise and shall be revealed in the higher-order perturbations. In analyzing the slow solution secular terms are imposed. These terms correspond to the fast solutions. The diminution of these terms produces the solvability condition. Here the solvability condition is divided into two cases. The first is valid in the nonresonance case in which the field frequency ω is away from both imaginary parts of (Ä j ) and their combinations. Otherwise the resonance case arises. However, the solvability condition in the nonresonance case is 2

(2) D1 η j + 2E 00 (α j + iβ j )η j = 0,

If η˜ 0 and ηˆ 0 are proportional to eνT1 , the coefficients matrix must vanish for nontrivial solution. This yields the following dispersion relation: ¡ ¢ (2)4 (2)2 (2)2 ν 2 +4α1 E 00 ν +ζ12 +4β1 E 00 ζ1 +3 α12 +β12 E 00 = 0. [71] An important feature of the waves is that the growth or decay is according to the sign of the real part of ν. In view of Hurwitz criterion (48), the stability arises when α1 > 0

[72]

¡ ¢ (2)4 (2)2 ζ1 + 3 α12 + β12 E 00 >0 ζ12 + 4β1 E 00

[73]

and

are satisfied. Condition [72] is satisfied in the nonresonance case, and the inequality [73] can be satisfied when (ζ1 − ζ˜1 )(ζ1 − ζ˜2 ) > 0,

[66]

where the real constants α j and β j are given in the Appendix. If the solution of Eq. [66] is proportional with exp[−(α j + iβ j )T1 ], the stability criterion depends on the parameter α j . The positive values of both α j give a stable case, while negative values of any α j lead to an unstable case. The resonance arises when the field frequency ω approaches the frequency ω˜ 1 or ω˜ 2 or their combinations. For a given frequency of the field oscillation only one mode is excited in the uniform sheet region. (i) The case of ω near ω˜ 1 . The selection of a particular field frequency ω is anticipated mathematically by introducing a detuning parameter ζ1 defined as ω = ω˜ 1 + εζ ˜ 1.

i.e., ζ1 > ζ˜1 and ζ1 < ζ˜2 ; where q ¤ (2)2 £ ˜ζ1,2 = E 00 −2β1 ± β12 − 3α12 .

ω − ω˜ 1 q ¢ −2β1 + β12 − 3α12

2

D 1 η1 +

(2)2 E 00 (α1

∗∗ =¡ E 0(2) = E 11

ω − ω˜ 1 q ¢, −2β1 − β12 − 3α12

2

+ iβ1 )(2η1 + η¯ 1 e

2iζ1 T1

) = 0.

[68]

The solution of this equation imposes a dispersion relation. This dispersion relation will be used to discuss the stability behavior in this resonance case. To obtain the solution of Eq. [68], we let η1 = (η˜ 0 + i ηˆ 0 )eiζ1 T1 ,

[69]

with real η˜ 0 and ηˆ 0 . Substituting from [69] into Eq. [68] and separating the real and imaginary parts we obtain ¡ ¢ (2)2 ¢ (2)2 + ζ1 ηˆ 0 = 0 η˜ 0 − β1 E 00 D1 + 3α1 E 00 ¡ ¢ ¡ (2)2 ¢ (2)2 + ζ1 η˜ 0 = 0. ηˆ 0 − 3β1 E 00 D1 + α1 E 00 ¡

[70]

[74]

These values indicate the critical values of the excitation waves, which are known as the transition values. Upon insertion of these values into [67] the transition curves will decrease in terms of the electric field intensity as

[67]

At this stage, the corresponding solvability condition is presented in the form

ζ˜1 > ζ˜2 ,

[75]

∗∗ =¡ E 0(2) = E 21

2

2

(2) where εE ˜ 00 = E 0(2) is used. ∗∗ ∗∗ The curves E 11 and E 21 represent the transition curves in (2)2 the plane (E 0 − k) which separate the stable region from the unstable one. According to Floquet’s theory (34), the region ∗∗ ∗∗ and E 21 is unstable, while bounded by the two branches of E 11 the area outside them is stable. It is clear that the two branches ∗∗ ∗∗ and E 21 have common fixed points known as the resonance E 11 ∗∗ ∗∗ and E 21 occurs points. The emergence of the two branches E 11 as ε˜ tends to zero in [67]. Thus, these resonance points will lie 2 on the k-axis in the plane (E 0(2) − k).

44

EL-DIB AND MATOOG

(ii) The case of ω approaching ω˜ 2 . This case is treated in the same manner as that of the above analysis by simply replacing 2 ω˜ 1 by ω˜ 2 . The transition curves in terms of E 0(2) are ω − ω˜ 2 q ¢, −2β2 + β22 − 3α22

2

∗∗ =¡ E 0(2) = E 12

2 E 0(2)

=

∗∗ E 22

[76]

ω − ω˜ 2 q =¡ ¢, −2β2 − β22 − 3α22

where α2 and β2 are functions of ω˜ 2 . (iii) The case of ω near (ω˜ 1 + ω˜ 2 )/2. We express the nearness of ω to (ω˜ 1 + ω˜ 2 )/2 by introducing the detuning parameter ζ2 defined by ω=

1 (ω˜ 1 + ω˜ 2 ) + εζ ˜ 2. 2

[77]

Consequently, the following solvability conditions are imposed, 2

2

(2) (2) (α1 + iβ1 )η1 + E 00 (α1∗ + iβ1∗ )η¯ 2 e2iζ2 T1 = 0, D1 η1 + 2E 00

[78] (2)2 (2)2 ∗ (α2 + iβ2 )η2 + E 00 (α2 + iβ2∗ )η¯ 1 e2iζ2 T1 D1 η2 + 2E 00

= 0,

where the bar represents the complex conjugate and the real coefficients α ∗j and β ∗j are as given in the Appendix. The solution of these coupled equations can be used to derive the stability conditions in the resonance case at hand. (iv) The case of ω near (ω˜ 1 − ω˜ 2 )/2. We express the nearness of ω to (ω˜ 1 − ω˜ 2 )/2 by introducing the detuning parameter ζ3 defined by ω=

1 ˜ 3. (ω˜ 1 − ω˜ 2 ) + εζ 2

stability characteristics are governed by Eq. [75] which requires specification of eight parameters: the sheet thickness a, the field frequency ω, the velocity ratios va and vb , the viscosity ratios µa and µb , and the elasticity ratios λa and λb . ∗∗ The numerical calculations for the transition curves E 11 and ∗∗ E 21 in the resonance case of ω near ω˜ 1 are displayed in Figs. 13– 18 for the same system described before in Fig. 9. In these graphs 2 E 0(2) is plotted versus the wavenumber k. The graphs emphasize the impact of the variation of the eight parameters mentioned above on the stability behavior in the resonance case. A numerical search was conducted to seek three consequent values for each parameter displayed in these graphs for comparison. The curves labeled by the symbol s refer to the first value (the low one), the curves marked by the symbol ∗ indicate the second value (the middle one), and the curve labeled with the symbol × represents the third value (the high one). The stable region involved in these graphs was decided by satisfying the inequalities [75], where S represents the stable region occurring in the nonresonance case and the U indicates the unstable resonance case. The instability is due to the balance between the field frequency ω and the disturbance frequency ω˜ 1 . If the eight parameters are held fixed as defined above except that the sheet thickness a varies from a = 0.05 to 0.06 and then 0.07, the results will be as shown in Fig. 13. As indicated above, the curves carrying the symbol s refer to the case of a = 0.05, the curves labeled by ∗ denote the case of a = 0.06, and the curves labeled by × represent the case of a = 0.07. It is noted that the exact resonance ω = ω˜ 1 occurs at a point known as a resonance point. For a specific sheet2 thickness two resonance points are presented in the plane (E 0(2) − k) and lie on the k-axis. For a = 0.05, the resonance points k1 and k2 lie at k1 = 1.655844 and k2 = 3.38243. As a is increased to 0.06, these points are shifted to k1 = 1.828446 and k2 = 2.88181. Thus, the distance between the resonance points decreases as the sheet thickness increases.

[79]

At this stage, solvability conditions yield the equations 2

2

(2) (2) (α1 + iβ1 )η1 + E 00 (α˜ 1 + i β˜ 1 )η2 e2iζ3 T1 = 0, D1 η1 + 2E 00

[80] (2)2 (2)2 (α2 + iβ2 )η2 + E 00 (α˜ 2 + i β˜ 2 )η1 e−2iζ3 T1 D1 η2 + 2E 00

= 0,

which give the stability conditions in this resonance case. 9.3. Numerical Illustration of the Resonance Cases In graphing the stability picture at resonance, numerical computations are made for one of the resonance cases discussed above. The resonance case of the field frequency ω approaching one of the disturbance frequencies (ω˜ 1 ) is carried out. The

FIG. 13. Stability diagram for the resonance case of ω near ω˜ 1 . The stability boundaries are given for three different cases for the sheet thickness a. Curves marked with s refer to the case of a = 0.05, curves labeled by the symbol ∗ indicate the case of a = 0.06, and the curve carrying the symbol × represents ∗∗ and E ∗∗ the case of a = 0.07. The calculations indicate the transition curves E 11 21 as defined by [89].

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

Consequently, a stable region S0 between these resonance points decreases in its width and its height as a increases. The significance of these observations is that though the sheet thickness is increased the unstable resonance region is contracted in association with the contraction in the stable region S0 , while the stable regions S are extended. This presented two roles in the stability criteria for increasing the sheet thickness at the resonance case: a stabilizing influence on nonresonance stable regions and a destabilizing influence on the stable region S0 , which lies between the two resonance regions. This phenomenon is known as a dual effect on stability criteria, which has not been observed before for the thickness influence on the linear stability. The continuous increase of the sheet thickness leads to a decrease in the distance between k1 and k2 until they become one point. After that the unstable resonance region leaves the k-axis. Further increasing the thickness a further decreases the unstable region associated with moving it up. At this stage, the effect of a damping role are presented. The effect of the damping role, which is caused by the sheet thickness, has not been observed before in the case of antisymmetric modes numerically discussed in Section 8.2, while the stabilizing influence of increasing the sheet thickness has been recorded in the case the of absence of the field frequency (Fig. 9). Examination of the influence of the field frequency ω on the stability picture for the fluid sheet at the resonance case of2 ω near ω˜ 1 is illustrated in Fig. 14. The graph in the plane (E 0(2) − k) is sketched for a fixed value of the sheet thickness (a = 0.05) and the other parameters are held fixed as defined in the previous graphs. Variations of the field frequency ω are considered where the curves with the symbol s refer to ω = 7, the curves with the symbol ∗ represent the case ω = 8.5, and the curves with the symbol × indicate the case ω = 10. Inspection of the stability diagram reveals that there is an unstable region U bounded by the transition curve for the case of

FIG. 14. Stability diagram showing the influence of the field frequency ω at the resonance case for the same system as in Fig. 9 except that a has a fixed value a = 0.05. Curves marked with s refer to the case of ω = 7, curves labeled by the symbol ∗ indicate the case of ω = 8.5, and curves carrying the symbol × represent the case of ω = 10.

45

ω = 7. As ω is increased, this unstable region is increased where the transition curve has moved down until it contacts the k-axis at a point known as a resonance point. At this point maximum instability occurs due to the increase of ω. We conclude that the increase of ω has a destabilizing influence. Another mechanism can be observed as ω continues increasing its value. The continuous increase of the frequency ω divides the resonance point into two different points k1 = 1.83654 and k2 = 2.99243 at the specific value of ω = 8.5. As ω is increased to the value ω = 10, k1 is found at 1.655844 while k2 shifts to 3.38243. Consequently, a stable region S0 has been placed between the resonance points k1 and k2 and bounded by one of the branches of the transition ∗∗ ∗∗ and E 21 . This stable region is increased in both width curves E 11 and height as ω is increased. The increase of the stable region S0 yields a contraction in the unstable region U . Therefore, there is another conclusion for increasing ω: the stabilizing influence is presented. However, the phenomena of the dual effect of ω are presented in this investigation. Moreover, further decreases in the field frequency ω have a damping influence on the stability criteria. It is noted that the phenomenon of the dual effect of the field frequency ω has not been observed before in inviscid or viscous fluids but it is observed for a rigidly rotating fluid, column with weak viscosity under the influence of a periodic magnetic field (27). Mohamed et al. (18) observed the effect of damping on the field frequency in a non-Newtonian fluid of Kelvin type. A dual role for the field frequency occurs in the nonlinear stability for inviscid fluids (41). The study of the influence of a periodic acceleration in nonlinear stability analysis (49, 50) shows that the acceleration frequency plays the same dual role played by the field frequency. The influences of the velocity ratios va and vb are displayed in Figs. 15a and 15b, respectively. The stability diagrams that are shown in these graphs represent the same system as in the previous graphs where both ω and a have fixed values of ω = 10 and a = 0.05. In both Figs. 15a and 15b the transition curves are distinguished for the variations of va and vb as follows: the curves labeled by s refer to va = 1.0 and vb = 0.5, the curves marked by ∗ represent va = 1.01 and vb = 0.52, and the curves carrying × indicate va = 1.02 and vb = 0.54. It is seen from these graphs that slight variations of both va and vb introduce a large effect in the stability criteria, especially for va rather than vb . A slight increase in both va and vb yields a decrease in the stable region S0 associated with further increasing in the nonresonance stable region. At this stage, there are two roles for stability examination: a destabilizing influence for the stable region embedded in between the unstable resonance regions and a stabilizing influence for the nonresonance stable region. Again, the phenomenon for the dual role is found for increasing both va and vb . For slighter increases in va and vb the two resonance points are collected in one point and then leave the k-axis. However, the damping influence of increasing va and vb is apparent. This damping influence is more rapid for va than for vb . The stabilizing role of

46

EL-DIB AND MATOOG

µb . The decrease in the viscosity ratio µa has the same effect as slightly increasing the viscosity ratio µb . Slightly increasing µb plays the role of the velocities va and vb , while decreasing the viscosity ratio µa plays the same role the field frequency has played. However, the stabilizing role of increasing µb and decreasing µa is observed here. The destabilizing influence of increasing µa has not been observed in the case of antisymmetric deformation in the presence of the field frequency ω. But this destabilizing influence is observed in the absence of the field frequency ω as shown in Fig. 11. Numerical calculations for the stability boundaries as a function of both the elasticity ratio λa and λb are illustrated in Figs. 17a and 17b, respectively. The stability picture has been affected by slightly increasing λb rather than increasing the relaxation time ratio λa . The inspection of these graphs shows that both λa and λb behave as va and vb in the stability examination. The stabilizing influence for both ratios λa and λb has not been observed before in the case of antisymmetric

FIG. 15. (a) Influence of the velocity ratio va on the stability criteria for the resonance case. Slight variations of va are considered where va = 1.0 is represented by the curves carrying the symbol s, va = 1.01 is denoted by the curves with the symbol ∗, and va = 1.02 is indicated by the curves with the symbol ×. (b) Influence of the velocity ratio vb on the stability examination at the resonance case. The stability boundaries of the symbol s refer to vb = 0.5, the curves with the symbol ∗ represent the case of vb = 0.52, while the case of vb = 0.54 is distinguished by the symbol ×.

the velocity ratios is in agreement with those effects occurring for the antisymmetric deformations as studied in Section 8.2 and with the numerical study for the case of the absence of ω (Fig. 10). In Figs. 16a and 16b we repeat the same diagrams as illustrated in Figs. 15a and 15b with a change in the values of the viscosity fixed ratios µa and µb while the values of va and vb are held 2 as va = 1.0 and vb = 1.5. The graphs in the plane (E 0(2) − k) are sketched for three different values of both µa and µb . In Fig. 16a the case of µa = 1.5 is represented by the symbol s, the case of µa = 2 is indicated by the symbol ∗, while the case of µa = 2.5 is denoted by the symbol ×. Similarly in Fig. 16b the variations of µb from 1.5 to 1.55 and then 1.6 are represented by the symbols s, ∗, and ×, respectively. It is apparent from the comparison between the graph of Fig. 16a and the graph of Fig. 16b that the variation of the viscosity ratio µa plays against the slight variation of the viscosity ratio

FIG. 16. (a) Variation of the viscosity ratio µa where µb is held fixed at 0.5. The curves carrying the symbol s refer to µa = 1.5, the symbol ∗ represents the case of µa = 2, while the symbol × indicates the case of µa = 2.5. (b) Variation of the viscosity ratio µb where µa is held fixed at 2.5. The curves labeled by the symbol s refer to µb = 1.5, the curves marked by ∗ refer to µb = 1.55, while the curves with × represent the case of µb = 1.6.

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

examination, while in the case of the constant field this destabilizing influence is observed as presented in Fig. 12. However, a dramatic change occurs at the resonance case in the effect of the sheet thickness a, the velocity ratios va , vb , the viscosity ratio µa , and the elasticity ratios λa and λb on the stability picture, in comparison with the numerical studies in the antisymmetric deformation, while the field frequency ω still exent a destabilizing influence at the resonance case. Also, the viscosity ratio µb still plays the same damping role in the resonance case. The transition from the resonance case of ω ≈ ω˜ 1 to the resonance case of ω ≈ ω˜ 2 is illustrated in Fig. 18. The graph shows both resonance cases. It is found that the resonance case of ω ≈ ω˜ 1 occurs at the field frequency having the value ω = 10, while the second resonance case of ω ≈ ω˜ 2 occurs at the frequency having the value ω = 30. One resonance point is presented at k = 2.618125 for the case ω ≈ ω˜ 2 corresponding to two resonance points for the case of ω ≈ ω˜ 1 located at k1 = 1.655844 and k1 = 3.38243.

FIG. 17. (a) Resonance region for the onset of the variation of the relaxation time ratio λa on the stability criteria. λa = 1.6 is represented by the symbol s, λa = 1.8 is indicated by the symbol ∗, while λa = 2 is denoted by the symbol ×. (b) Effect of slight increases in the relaxation time ratio λb on the stability in the resonance case. The variations of λb (1, 1.03, and 1.06) are represented by the symbols s, ∗, and ×, respectively.

47

FIG. 18. Stability boundaries for two resonance cases, ω ≈ ω˜ 1 and ω ≈ ω˜ 2 . The first resonance case occurs at ω = 10, while the second resonance case occurs at ω = 30.

10. CONCLUSION

The gravitational stability of a fluid sheet of finite thickness impeded between two semi-infinite fluids under relative shearing motion (Kelvin–Helmholtz) has been analyzed including the effect of interfacial tensions and the effect of a vertical periodic electric field in the absence of free surface charges and weak viscoelastic effects of Maxwell type. With weak viscoelasticity, it is recognized that viscoelastic effects are included in the boundary conditions of normal and tangential stress tensor balance, so that the field equation governing the fluid motion is the Laplace equation. The dynamics of the problem are described by the simultaneous solution of three field equations: the equations of motion governing the Maxwellian type, the continuity equation, and Maxwell’s electric equations. From the electric point of view all media are generally considered to be homogeneous, isotropic liquids with some dielectric. The mechanism of Maxwell polarization is taken into consideration and the resulting Maxwell stress is calculated. A zero volumetric density of free charges is assumed. Additional effects related to changes in the dielectric preemptively (due to the deformation) as well as to the magnetic component of stress are neglected. We have also considered temporal stability (configuration). The stability analysis is based on linear perturbation theory. Through linear perturbation analysis, the solution of the system leads to two simultaneous differential equations of Mathieu type, which are used to control the stability of the fluid sheet motion. The symmetric and antisymmetric deformations for the surface deflections are considered and analyzed. This constraint leads to a single Mathieu equation governing both the symmetric and antisymmetric cases. Numerical estimations are made where the physical parameters are put in the dimensionless form. Some stability diagrams are plotted and discussed. The stability examination yields the following results: (i) Small values of the vertical electric field intensity have a stabilizing effect. The destabilizing influence appears suddenly

48

EL-DIB AND MATOOG

as the field intensity is increased. The instability occurs for short wave perturbations (k → ∞). (ii) The presence of a constant electric field has a destabilizing influence. A more destabilizing influence occurs as the sheet thickness increases. The destabilizing influence for the sheet thickness is very clear for long wave (k → 0) perturbations. The influence of the increase of the fluid sheet still plays a destabilizing role, even if the electric field is oscillating. (iii) The motion (va ) of the upper fluid (1) relative to the lower fluid (2) shows that the increase of this ratio increases the destabilizing influence of the constant electric field. On the other hand, the motion (vb ) of the lower fluid (3) relative to the upper fluid (2) indicates that the increase of this ratio retards the destabilizing influence of the constant electric field, especially for large values of the wavenumber k. (iv) The stability picture that shows the influence of oscillating the vertical electric field is found to be a function of the variation of the field periodicity ω, the velocity ratio va , the viscosity ratio µa , the elasticity ratio λa , and the sheet thickness a. The destabilizing behavior is maintained for increasing sheet thickness, field frequency, and elasticity ratio. Also, the stabilizing behavior is noticed for both the viscosity ratio µa and the velocity ratio va . It is noted that the influence of the velocity ratio va follows a change in its mechanisms in the presence of the field frequency ω. Furthermore, we present in this calculation an analysis undertaken principally by clarifying the coupling between the surface deflections. The general case where the surface deflections are assumed to be independent is discussed. The use of the tangential stress condition yields two simplified coupled Mathieu equations of first order. The method of multiple scales is used in order to derive stability criteria in resonance and nonresonance cases. Numerical estimates are made where the physical parameters are put in the dimensionless form. Numerical discussion of the stability of the system under consideration is given for arbitrary relations between the two surface deflections ξ1 and ξ2 . The underlying operator of the linear problem is neither symmetric nor antisymmetric. Some stability diagrams are plotted and discussed. Numerical examination of stability yields some dramatic changes in the stability behavior of the fluid sheet, in contrast to the results obtained in the case of antisymmetric deformation, which are discussed numerically. Thus, for general deformation, the fluid sheet thickness plays a stabilizing role in the presence of constant field while a damping effect is observed in the resonance case. Similar results are found for both the stratified velocities va and vb and the stratified relaxation times λa and λb . The dual effect of the stratified viscosities µa and µb is observed in the presence or the absence of the field frequency, where the influence of µa differs from the influence of µb . Their effects are opposite. In contrast to the influence of the field frequency, the dual role is observed in the resonance case.

APPENDIX

The linear formulates for the distribution of the filed potential φ1 , the stream function ψ1 and the pressure P1 in the three layers are µ ¡ ¢ ¢ ¢ 1 ¡¡ (3) (1) ε − ε (2) e3ka − ε (3) − ε (2) e−ka φ1 (x, y, t) = 2ε ∗ ¡ ¢ ¢ 1 ¡¡ × ε(2) − ε (1) E 0(2) γ1 + ∗ ε(1) − ε (2) eka 2ε ¡ (1) ¢ ¢¡ (3) ¢ (2) ka − ε +ε e ε − ε (2) E 0(2) γ2 ¶ ¡ ¢ − γ1 E 0(1) − E 0(2) eka × cos ωteikx−ky ;

y > a,

·

φ1(2) (x, y, t) =

¢¡ ¢ 1 ¡¡ (3) ε + ε (2) ε(2) − ε (1) E 0(2) γ1 eka ∗ 2ε ¡ ¢¡ ¢ ¢ + ε(1) − ε (2) ε(3) − ε (2) E 0(2) γ2 e−ka eikx+ky ¢¡ ¢ 1 ¡¡ (3) ε − ε (2) ε(2) − ε (1) E 0(2) e−ka γ1 ∗ 2ε ¸ ¡ (1) ¢¡ ¢ ¢ + ε + ε (2) ε(3) − ε (2) E 0(2) γ2 eka eikx−ky −

× cos ωt; · φ1(3) (x, y, t) =

|y| < a,

¡ ¢ ¢ ¢ 1 ¡¡ (3) ε + ε (2) eka − ε (3) − ε (2) eka ∗ 2ε

¡ ¢ ¢ 1 ¡¡ × ε(2) − ε (1) E 0(2) γ1 + ∗ ε(1) − ε (2) e−ka 2ε ¡ ¢ ¢¡ ¢ − ε(1) + ε (2) e3ka ε(3) − ε (2) E 0(2) γ2 ¸ ¡ (2) (3) ¢ ka + γ2 E 0 − E 0 e eikx+ky cos ωt; y < −a, ¡ ¡ ¢ 2¢ ε ∗ = ε(1) ε(3) + ε (2) sinh 2ka + ε (2) ε(1) + ε (3) cosh 2ka. µ ψ1(1) (x, y, t) = −i

¶ γ˙1 + v0(1) iγ1 eka+ikx−ky ; k

ψ1(2) (x, y, t) · ¡ ¢ 1 γ˙1 − γ˙2 + v0(2) ik(γ1 − γ2 ) =i 4k sinh ka

y > a,

¸ ¡ ¢ 1 γ˙1 + γ˙2 + v0(2) ik(γ1 + γ2 ) eikx−ky 4k cosh ka · ¡ ¢ 1 γ˙1 + γ˙2 + v0(2) ik(γ1 + γ2 ) +i 4k cosh ka ¸ ¢ ikx+ky ¡ 1 (2) ; |y| < a, γ˙1 − γ˙2 + v0 ik(γ1 − γ2 ) e + 4k sinh ka −

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

ψ1(3) (x, y, t) ¶ µ γ˙2 (3) + v0 iγ2 eka+ikx+ky ; =i k

¢¤ ¡ + µ(2) λ(2) v0(2) ρ (5−2 j) cosh 2ka + ρ (2) sinh 2ka 8k 5 £ (2 j−1) (2 j−1) (2 j−1) ¡ (2) (2) λ v0 µ µ λ cosh 2ka ρ˜ ¢ ¡ + µ(5−2 j) λ(5−2 j) sinh 2ka + µ(2) λ(2) v0(2) µ(2) λ(2) sinh 2ka ¢¤ + µ(5−2 j) λ(5−2 j) cosh 2ka ,

y < −a,

+

P1(1) (x, y, t) =

¢ 2 ρ (1) ¡ γ¨1 + 2v0(1) ik γ˙1 − v0(1) k 2 γ1 eka+ikx−ky ; k

y > a, l2 j =

P1(2) (x, y, t) · ¡ ρ (2) 1 =− γ¨1 − γ¨2 + 2v0(2) ik(γ˙1 − γ˙2 ) k 4 sinh ka ¢ 2 − v0(2) k 2 (γ1 − γ2 ) (eky + e−ky )eikx + 2

¢

¡ 1 γ¨1 + γ¨2 4 cosh ka

y < −a.

¢ 2k 2 £ (2 j−1) ¡ (5−2 j) µ sinh 2ka + ρ (2) cosh 2ka ρ ρ˜ ¢¤ ¡ + µ(2) ρ (5−2 j) cosh 2ka + ρ (2) sinh 2ka 4k 4 £ (2 j−1) ¡ (2) (2) µ λ cosh 2ka µ ρ˜ ¢ ¡ + µ(5−2 j) λ(5−2 j) sinh 2ka + µ(2) µ(2) λ(2) sinh 2ka ¢¤ + µ(5−2 j) λ(5−2 j) cosh 2ka ,

−

f2 j =

¤ 2k 2 £ (2 j−1) (2) µ ρ − µ(2) ρ (2 j−1) ρ˜ −

l1 j =

¢¤ 8k 5 £ (2 j−1) (2 j−1) (2) (2) ¡ (2 j−1) µ λ µ λ v0 − v0(2) , ρ˜

¢ k 2 £ (2) (2)2 ¡ (5−2 j) cosh 2ka + ρ (2) sinh 2ka ρ v0 ρ ρ˜ ¢¤ (2 j−1)2 ¡ (5−2 j) sinh 2ka + ρ (2) cosh 2ka ρ + ρ (2 j−1) v0

s1 j = −

The coefficients that appear in Eqs. [31] and [32] are f1 j =

¢ 4k 3 £ (2 j−1) (2) (2) ¡ (2 j−1) µ λ v0 − v0(2) ρ ρ˜ ¡ (2 j−1) ¢¤ + ρ (2) µ(2 j−1) λ(2 j−1) v0 − v0(2) +

P1(3) (x, y, t) ¢ 2 ρ (3) ¡ γ¨2 + 2ikv0(3) γ˙2 − v0(3) k 2 γ2 eikx+ka+ky ; k

¢¤ 2k £ (2) (2 j−1) ¡ (2 j−1) − v0(2) ρ ρ v0 ρ˜ −

+ 2v0(2) ik(γ˙1 + γ˙2 ) − v0(2) k 2 (γ1 + γ2 ) ¸ ky −ky ikx ; |y| < a, × (e − e )e

=−

49

¢¤ 4k 4 £ (2) (2 j−1) ¡ (2) µ µ λ − λ(2 j−1) , ρ˜

¢ 2k £ (2) (2) ¡ (5−2 j) ρ v0 ρ cosh 2ka + ρ (2) sinh 2ka ρ˜ ¢¤ (2 j−1) ¡ (5−2 j) + ρ (2 j−1) v0 sinh 2ka + ρ (2) cosh 2ka ρ 4k 3 £ (2) (2) ¡ (2) (2) ρ v0 µ λ sinh 2ka ρ˜ ¢ + µ(5−2 j) λ(5−2 j) cosh 2ka (2 j−1) ¡ (2) (2) + ρ (2 j−1) v0 µ λ cosh 2ka ¢ (2 j−1) + µ(5−2 j) λ(5−2 j) sinh 2ka + µ(2 j−1) λ(2 j−1) v0 ¡ ¢ × ρ (5−2 j) sinh 2ka + ρ (2) cosh 2ka

+

¡ ¢ ¢¡ k £¡ 2 k T j( j+1) − ρ ( j) − ρ ( j+1) g ρ (5−2 j) sinh 2ka ρ˜

¡ ¢ ¢ 2k 3 £¡ 2 k T j( j+1) − ρ ( j) − ρ ( j+1) g ρ˜ ¢¤ ¡ (2) (2) × µ λ cosh 2ka + µ(5−2 j) λ(5−2 j) sinh 2ka

+ ρ (2) cosh 2ka

¢¤

−

2k 4 £ (2) (2)2 ¡ (2) (2) ρ v0 µ λ sinh 2ka ρ˜ ¢ + µ(5−2 j) λ(5−2 j) cosh 2ka (2 j−1)2 ¡ (2) (2) µ λ cosh 2ka + ρ (2 j−1) v0 ¢ + µ(5−2 j) λ(5−2 j) sinh 2ka ¢ (2 j−1)2 ¡ (5−2 j) sinh 2ka + ρ (2) cosh 2ka ρ + µ(2 j−1) λ(2 j−1) v0 ¢¤ 2¡ + µ(2) λ(2) v0(2) ρ (5−2 j) cosh 2ka + ρ (2) sinh 2ka

+

4k 6 £ (2 j−1) (2 j−1) (2 j−1)2 ¡ (2) (2) µ λ v0 µ λ cosh 2ka ρ˜ ¢ 2¡ + µ(5−2 j) λ(5−2 j) sinh 2ka + µ(2) λ(2) v0(2) µ(2) λ(2) sinh 2ka ¢¤ + µ(5−2 j) λ(5−2 j) cosh 2ka ,

−

−

s2 j =

k 2 £ (2) (2 j−1) ¡ (2)2 kρ (2) £ 2 (2 j−1)2 ¢¤ ρ ρ k T j( j+1) v0 − v0 + ρ˜ ρ˜ ¡ ¢ ¤ 2k 3 µ(2) λ(2) £ 2 k T j( j+1) − (ρ ( j) − ρ ( j) − ρ ( j+1) g − ρ˜ ¢ ¤ 2k 4 £ (2 j−1) (2) (2) ¡ (2)2 (2 j−1)2 ¢ ρ − ρ ( j+1) g − µ λ v0 − v0 ρ˜

50

EL-DIB AND MATOOG

£ ¡ ¢ + 4k 4 µ(2) λ(2) µ(1) λ(1) + µ(3) λ(3) cosh 2ka ¡ ¢ ¤ 2 2 + µ(2) λ(2) + µ(1) µ(3) λ(1) λ(3) sinh 2ka .

¡ 2 (2 j−1)2 ¢¤ + ρ (2) µ(2 j−1) λ(2 j−1) v0(2) − v0 +

4k 6 £ (2) (2) (2 j−1) (2 j−1) ¡ (2)2 (2 j−1)2 ¢¤ λ v0 − v0 , µ λ µ ρ˜

The values of the coefficients of Eqs. [34] and [35] are h1 j

¢ 2k 3 £ (2 j−1) (2 j−1) ¡ (5−2 j) µ = v0 sinh 2ka + ρ (2) cosh 2ka ρ ρ˜ ¢¤ ¡ + µ(2) v0(2) ρ (5−2 j) cosh 2ka + ρ (2) sinh 2ka

aj =

4k 5 £ (2 j−1) (2 j−1) ¡ (2) (2) µ v0 µ λ cosh 2ka ρ˜ ¢ ¡ + µ(5−2 j) λ(5−2 j) sinh 2ka + µ(2) v0(2) µ(2) λ(2) sinh 2ka ¢¤ + µ(5−2 j) λ(5−2 j) cosh 2ka , −

h2 j =

¤ 2k 3 £ (2) (2 j−1) (2 j−1) v0 − ρ (2 j−1) µ(2) v0(2) ρ µ ρ˜ −

¢¤ 4k 5 £ (2) (2 j−1) ¡ (2) (2 j−1) − λ(2 j−1) v0(2) , λ v0 µ µ ρ˜

¢¡ ¢ −k 2 ε(2) £ (2) (2 j−1) ¡ (2) ρ ε ε − ε (5−2 j) ε(2 j−1) − ε (2) ∗ (2 j−1) ρε ˜ ε ¡ ¢ ¢2 ¡ + ε (2 j−1) − ε (2) ε(5−2 j) cosh 2ka + ε (2) sinh 2ka

k1 j =

¢¤ ¡ 2k 4 ε(2) × ρ (5−2 j) sinh 2ka + ρ (2) cosh 2ka + ∗ (2 j−1) ρε ˜ ε £ (2) (2) (2 j−1) ¡ (2) ¢¡ (2 j−1) ¢ (5−2 j) × µ λ ε − ε (5−2 j) ε −ε ε ¢ ¢2 ¡ ¡ + ε (2 j−1) − ε (2) ε(5−2 j) cosh 2ka + ε(2) sinh 2ka ¢¤ ¡ × µ(5−2 j) λ(5−2 j) sinh 2ka + µ(2) λ(2) cosh 2ka , k2 j =

¢¡ k 2 ε(2) £ (2 j−1) (2 j−1) ¡ (2 j−1) ε − ε (2) ε(5−2 j) ρ ε ρε ˜ ∗ ε(2 j−1) ¢ ¡ ¢£¡ − ε(2) sinh 2ka + ρ (2) ε(2) ε(2 j−1) − ε (2) ε(5−2 j) ¢ ¡ ¢ ¤¤ − ε(2 j−1) cosh 2ka + ε(2) − ε (2 j−1) sinh 2ka ¢¡ 2k 4 ε(2) £ (2 j−1) (2 j−1) (2 j−1) ¡ (2 j−1) λ ε − ε (2) ε(5−2 j) µ ε ∗ (2 j−1) ρε ˜ ε ¢ ¡ ¢£¡ − ε(2) sinh 2ka + µ(2) λ(2) ε(2) ε(2 j−1) − ε (2) ε(5−2 j) ¢ ¡ ¢ ¤¤ − ε(2 j−1) cosh 2ka + ε(2) − ε (2 j−1) sinh 2ka ,

−

ρ˜ = ρ

bj =

¡ (2)

¢ (3)

¡

2

¢ (3)

ρ (1) + ρ cosh 2ka + ρ (2) + ρ (1) ρ sinh 2ka ¢ £ ¡ − 2k 2 ρ (2) µ(2) λ(2) sinh 2ka + µ(3) λ(3) cosh 2ka ¢ ¡ + ρ (1) µ(2) λ(2) cosh 2ka + µ(3) λ(3) sinh 2ka ¢ ¡ + ρ (3) µ(1) λ(1) sinh 2ka + µ(2) λ(2) cosh 2ka ¢¤ ¡ + ρ (2) µ(1) λ(1) cosh 2ka + µ(2) λ(2) sinh 2ka

cj =

¡ 1 £¡ (2 j−1) ¢¡ (5−2 j) (5−2 j) v0 f 1 j k µ(2) v0(2) − µ(2 j−1) v0 µ mj ¢ ¡ ¢¡ (5−2 j) − µ(2) v0(2) + h 1 j µ(2 j−1) − µ(2) µ(5−2 j) v0 ¡ ¢ (2 j−1) ¢ − µ(2) v0(2) + J f 2(3− j) k µ(2) v0(2) − µ(2 j−1) v0 ¡ ¢ ¡ ¢ (5−2 j) − µ(2) v0(2) + J h 2(3− j) µ(2) − µ(5−2 j) × µ(5−2 j) v0 ¡ (2 j−1) ¢¤ × µ(2) v0(2) − µ(2 j−1) v0 , ¡ 1 £ (2 j−1) ¢¡ (5−2 j) (5−2 j) v0 l1 j k µ(2) v0(2) − µ(2 j−1) v0 µ mj ¡ ¢ (2 j−1) ¢ − µ(2) v0(2) + 2h 1 j k µ(2) λ(2) v0(2) − µ(2 j−1) λ(2 j−1) v0 ¡ ¡ ¢ (5−2 j) − µ(2) v0(2) + Jl2(3− j) k µ(2) v0(2) × µ(5−2 j) v0 ¢ (2 j−1) ¢¡ (5−2 j) (5−2 j) − µ(2 j−1) v0 v0 − µ(2) v0(2) µ ¡ (5−2 j) + 2J h 2(3− j) k µ(5−2 j) λ(5−2 j) v0 ¢¡ (2 j−1) ¢¤ − µ(2) λ(2) v0(2) µ(2) v0(2) − µ(2 j−1) v0 , ¡ 1 £ (2 j−1) ¢¡ (5−2 j) (5−2 j) v0 s1 j k µ(2) v0(2) − µ(2 j−1) v0 µ mj ¢ ¡ (2 j−1)2 − µ(2) v0(2) + h 1 j k 2 µ(2 j−1) λ(2 j−1) v0 ¢ 2 ¢¡ (5−2 j) − µ(2) λ(2) v0(2) µ(5−2 j) v0 − µ(2) v0(2) ¡¡ ¡ (2 j−1) ¢ + J s2(3− j) k µ(2) v0(2) − µ(2) v0(2) − µ(2 j−1) v0 ¢ ¡ ¡ 2 (5−2 j) − µ(2) v0(2) + J h 2(3− j) k 2 µ(2) λ(2) v0(2) × µ(5−2 j) v0 (2 j−1)2 ¢¡ (2) (2) (2 j−1) ¢¤ − µ(2 j−1) λ(2 j−1) v0 µ v0 − µ(2 j−1) v0 ,

dj =

1 £¡ ¡ (2) (2) (2 j−1) ¢¡ (5−2 j) (5−2 j) v0 k µ v0 − µ(2 j−1) v0 µ mj ¢¢¡ ¢¤ − µ(2) v0(2) k1 j + J k2(3− j) ,

¡ ¢ (2 j−1) ¢¡ (5−2 j) (5−2 j) m j = k µ(2) v0(2) − µ(2 j−1) v0 v0 − µ(2) v0(2) µ ¡ ¢¡ (5−2 j) + h 1 j µ(2) λ(2) − µ(2 j−1) λ(2 j−1) µ(5−2 j) v0 ¢ ¡ ¢ − µ(2) v0(2) + J h 2(3− j) µ(5−2 j) λ(5−2 j) − µ(2) λ(2) ¡ (2 j−1) ¢¤ × µ(2) v0(2) − µ(2 j−1) v0 . The formulas for the coefficients which are used in Eq. [59]

STREAMING IN AN ELECTRIFIED MAXWELL FLUID SHEET

are ajj =

q j(3− j) =

£ ©¡ ¢¡ ¢ 1 a00 s1 j + <∗j f 1(3− j) + <3− j 2 − b00 ¡ ¢¡ ¢ ª − h 1 j + ℵ∗j l j(3− j) + ℵ3− j − s2 j f 2(3− j) + h 2 j l2(3− j) ©¡ ¢¡ ¢ ¡ ¢¡ + b00 s1 j + <∗j l1(3− j) + ℵ3− j + h 1 j + ℵ∗j f 1(3− j) ¢ ª¤ + <3− j − s2 j l2(3− j) − h 2 j f 2(3− j) , 2 a00

bjj =

pjj =

qjj =

a j(3− j)

− f 22 f 21 + l21l22 },

¡ £ © ¢ 1 a00 (s1 j + < j ) l1(3− j) + ℵ3− j 2 − b00 ¡ ¢¡ ¢ + h 1 j + ℵ∗j f 1(3− j) + <3− j − s2 j l2(3− j) ª ©¡ ¢¡ ¢ − h 2 j f 2(3− j) + b00 s1 j + <∗j f 1(3− j) + <3− j ¡ ¢¡ ¢ + h 1 j + ℵ∗j l1(3− j) + ℵ3− j ª¤ − s2 j f 2(3− j) + h 2 j l2(3− j) ,

b00 = {( f 11 + <1 )(l12 + ℵ2 ) + (l11 + ℵ1 )( f 12 + <2 ) − l22 f 21 − l21 f 22 }, µ( j+1) − µ( j) , µ( j+1) λ( j+1) − µ( j) λ( j) ¡ ( j+1)2 ( j)2 ¢ − µ( j) λ( j) v0 k 2 µ( j+1) λ( j+1) v0 ∗ ,
£ © ¡ ¢ ª 1 a00 k1 j f 1(3− j) + <3− j − f 2(3− j) k2 j 2 − b00 © ¢ ª¤ + b00 k1 j (l1(3− j) + ℵ3− j − k2 j l2(3− j) ,

¡ ( j) ( j+1) ¢ 2k µ( j) λ( j) v0 − µ( j+1) λ( j+1) v0 , ℵj = µ( j+1) λ( j+1) − µ( j) λ( j) ¡ ( j+1) ( j) ¢ k µ( j+1) v0 − µ( j) v0 ∗ . ℵj = µ( j+1) λ( j+1) − µ( j) λ( j)

2 a00

£ © ¡ ¢ 1 = 2 a00 s2(3− j) f 1(3− j) + <3− j 2 a00 − b00 ¡ ¢ ¡ − h 2(3− j) l1(3− j) + ℵ3− j − f 2(3− j) s1(3− j) ¢ ¡ ¢ª + <∗3− j + l2(3− j) h 1(3− j) + ℵ∗3− j © ¡ ¢ ¡ + b00 h 2(3− j) f 1(3− j) + <3− j + s2(3− j) l1(3− j) ¢ ¡ ¢ + ℵ3− j − f 2(3− j) h 1(3− j) + ℵ∗3− j ¡ ¢ª¤ − l2(3− j) s1(3− j) + <∗3− j ,

b j(3− j) =

£ © ¡ ¢ 1 a00 h 2(3− j) f 1(3− j) + <3− j 2 − b00 ¡ ¢ ¡ + s2(3− j) l1(3− j) + ℵ3− j − f 2(3− j) h 1(3− j) ¢ ¡ ¢ª + ℵ∗3− j − l2(3− j) s1(3− j) + <∗3− j © ¡ ¢ ¡ − b00 s2(3− j) f 1(3− j) + <3− j − h 2(3− j) l1(3− j) ¢ ¡ ¢ + ℵ3− j − f 2(3− j) s1(3− j) + <∗3− j ¡ ¢ª¤ + l2(3− j) h 1(3− j) + ℵ∗3− j ,

The calculations of the coefficients of Eqs. [66] and [79] are αj =

1 {[σ˜ j p11 + a22 p11 − ω˜ j q11 − b22 q11 − σ˜ j p22 − a11 p22 b∗∗ + q22 ω˜ j + b11 q22 + a12 p21 − b12 q21 − p12 a21 + b21 q12 ][2σ˜ j + a11 + a22 ] + [ω˜ j p11 + b22 p11 + σ˜ j q11 + q11 a22 − ω˜ j p22 − b11 p22 − q22 σ˜ j − q22 a11 + b12 p21 + q21 a12 − p12 b21 − q12 a21 ][2ω˜ j + b11 + b22 ]},

βj =

£ © ¡ ¢ 1 a00 k2(3− j) f 1(3− j) + <3− j 2 − b00 ª © ¡ − f 2(3− j) k1(3− j) + b00 k2(3− j) l1(3− j) ¢ ª¤ + ℵ3− j − l2(3− j) k1(3− j) , 2 a00

1 {[ω˜ j p11 + b22 p11 + σ˜ j q11 + q11 a22 − ω˜ j p22 b∗∗ − b11 p22 − q22 σ˜ j − q22 a11 + b12 p21 + q21 a12 − p12 b21 − q12 a21 ][2σ˜ j + a11 + a22 ] − [σ˜ j p11 + a22 p11 − ω˜ j q11

2 a00

p j(3− j) =

£ © ¡ ¢ 1 a00 k2(3− j) l1(3− j) + ℵ3− j 2 − b00 ª © ¡ − l2(3− j) k1(3− j) − b00 k2(3− j) f 1(3− j) ¢ ª¤ + <3− j − f 2(3− j) k1(3− j) , 2 a00

a00 = {( f 11 + <1 )( f 12 + <2 ) − (l11 + ℵ1 )(l12 + ℵ2 )

2 a00

£ © ¡ ¢ ª 1 a00 k1 j l1(3− j) + ℵ3− j − l2(3− j) k2 j 2 2 a00 − b00 © ¡ ¢ ª¤ − b00 k1 j f 1(3− j) + <3− j − k2 j f 2(3− j) ,

51

− b22 q11 − σ˜ j p22 − a11 p22 + q22 ω˜ j + b11 q22 + a12 p21 − b12 q21 − p12 a21 + b21 q12 ][2ω˜ j + b11 + b22 ]}, b∗∗ = [2σ˜ j + a11 + a22 ]2 + [2ω˜ j + b11 + b22 ]2 , α ∗j =

1 {[ p11 σ˜ j + q11 ω˜ j + a22 p11 + q11 b22 + b11 q22 b∗∗ − p22 σ˜ 3− j + ω˜ 3− j q22 − a11 p22 + p21 a12 − p12 a21 ][2σ˜ j + a11 + a22 ] + [−q11 σ˜ j + ω˜ j p11 − q11 a22 + b22 p11 + q22 σ˜ 3− j + p22 ω˜ 3− j + q22 a11 + b11 p22 − a12 q21 + a21 q12 ][2ω˜ j + b11 + b22 ]},

52 β ∗j =

EL-DIB AND MATOOG

1 {[−q11 σ˜ j + ω˜ j p11 − q11 a22 + b22 p11 + q22 σ˜ 3− j b∗∗ + p22 ω˜ 3− j + q22 a11 + b11 p22 − a12 q21 + a21 q12 ][2σ˜ j + a11 + a22 ] − [ p11 σ˜ j + q11 ω˜ j + a22 p11 + q11 b22 + b11 q22 − p22 σ˜ 3− j + ω˜ 3− j q22 − a11 p22 + p21 a12 − p12 a21 ][2ω˜ j + b11 + b22 ]}.

23. 24. 25. 26. 27. 28. 29. 30. 31.

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