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consumption at the interface
Acta Astronautica 123 (2016) 137–144
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Nonlinear traveling waves in a two-layer system with heat release/ consumption at the interface Ilya B. Simanovskii a,n, Antonio Viviani b, Frank Dubois c, Jean-Claude Legros c a
Department of Mathematics, Technion – Israel, Institute of Technology, 32000 Haifa, Israel Seconda Universita di Napoli (SUN), Dipartimento di Ingegneria Aerospaziale e Meccanica (DIAM), via Roma 29, 81031 Aversa, Italy c Universite Libre de Bruxelles, Service de Chimie Physique EP, CP165-62, 50 Av. F.D. Roosevelt 1050, Brussels, Belgium b
art ic l e i nf o
a b s t r a c t
Article history: Received 21 December 2015 Accepted 20 February 2016 Available online 17 March 2016
The influence of an interfacial heat release and heat consumption on nonlinear convective flows, developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system with periodic boundary conditions, is investigated. Regimes of traveling waves and modulated traveling waves have been obtained. It is found that rather intensive heat sinks at the interface can lead to the change of the direction of the waves' propagation. & 2016 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Interface Instabilities Two-layer system
1. Introduction It is known that two-layer liquid systems are subject to numerous instabilities (for a review, see [1,2]). Several classes of instabilities have been found by means of the linear stability theory for purely thermocapillary flows [3–6] and for buoyant-thermocapillary flows [7–10]. For the most typical kind of instability, hydrothermal instability, the appearance of oblique waves moving upstream has been predicted by the theory and justified in experiments [11–13]. However, two-dimensional waves moving downstream have also been observed in experiments [14]. The change of the direction of waves propagation can be caused by the influence of buoyancy [7]. Most of the investigations have been fulfilled for a sole liquid layer with a free surface, i.e., in the framework of the one-layer approach. Recently, Madruga et al. [15,16] studied the linear stability of two superposed horizontal liquid layers bounded by two solid planes and subjected to a horizontal temperature gradient. The analysis has revealed a variety of instability modes. The nonlinear wavy convective regimes in two-layer systems have been described in [17]. In the investigations of convection in two-layer systems, it is typically assumed that the normal components of the heat flux are equal on both sides of the interface. However, there are various physical phenomena which are characterized by a heat release or heat consumption at the interface. Specifically, the interfacial heat release accompanies an interfacial chemical reaction (see, e.g., n
Corresponding author. E-mail address: [email protected] (I.B. Simanovskii).
http://dx.doi.org/10.1016/j.actaastro.2016.02.026 0094-5765/& 2016 IAA. Published by Elsevier Ltd. All rights reserved.
[18]) and the evaporation [19]. The interfacial heating can be generated, e.g., by an infrared light source. The infrared absorption bands of water and silicone fluids are essentially different [20], therefore the light frequency can be chosen in a way that one of the fluids is transparent, while the characteristic length of the light absorption in another liquid is short. Another possibility of the interfacial heating may be realized by the use of the ultra-violet radiation: the process of photolysis of hydrogen peroxide H2 O2 due to the radiation with a wavelength lower than 400 nm, leads to the appearance of OH_radicals, accompanying by the reaction between silicone fluids and OH_radicals with an interfacial heat extraction [21]. It is known that the presence of a constant, spatially uniform heat release or heat consumption at the interface can lead to the appearance of an oscillatory instability [22,23]. Oscillations in [22,23] have been obtained in a two-layer system heated from below. Nonlinear convective regimes in a two-layer system filling a closed cavity with heat release at the interface have been studied in [24]. The system was heated from the lateral wall. Let us note that the theoretical predictions obtained for flows in closed cavities cannot be automatically applied for the infinite layers. For the observation of waves in a closed cavity a global instability is needed, while in the case of periodic boundary conditions one observes waves generated by a convective instability of a parallel flow [25]. Also, it should be taken into account, that in the presence of rigid lateral walls the basic flow is not parallel – the lateral walls act as a stationary finite-amplitude perturbation that can produce steady multicellular flow in the part of the cavity and in the whole cavity [25]. In the present paper, the influence of the interfacial heat release and heat consumption on nonlinear convective regimes,
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developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system with periodic boundary conditions, has been investigated. Specific regimes of traveling waves and modulated traveling waves have been obtained. It is found that rather intensive heat sinks at the interface can lead to the change of the direction of the waves' propagation. The paper is organized as follows. In Section 2, the mathematical formulation of the problem in the two-layer system is presented. Numerical simulations of the finite-amplitude convective regimes are considered in Section 3. Section 4 contains some concluding remarks.
→ Here, vm = (vmx , vmy, vmz ) is the velocity vector, Tm is the temperature and pm is the pressure in the m-th fluid; → γ is the unit vector directed upwards; b1 = c1 = d1 = e1 = 1; b2 = 1/β , c2 = 1/ν, d2 = 1/χ , e2 = ρ ; G = gβ1Aa14 /ν12 is the Grashof number and P = ν1/χ1 is the Prandtl number for the liquid in layer 1. The conditions on the rigid horizontal boundaries are:
→ v1 = 0;
z = 1:
→ v2 = 0;
z = − a:
2.1. Equations and boundary conditions We consider a system of two horizontal layers of immiscible viscous fluids with different physical properties (see Fig. 1). The variables referring to the top layer are marked by subscript 1, and the variables referring to the bottom layer are marked by subscript 2. The system is bounded from above and from below by two rigid plates, z = a1 and z = − a2. A constant temperature gradient is imposed in the direction of the axis x: T1 (x, y, a1, t ) =T2 (x, y, − a2, t ) = − Ax + const, A > 0. A constant heat release of the rate Q0 (Q0 may be positive or negative) is set on the interface. It is assumed that the interfacial tension s decreases linearly with an increase of the temperature: σ = σ0 − αT , where α > 0. Let us introduce the following notation:
ν = ν1/ν2,
η = η1/η2,
χ = χ1 /χ2 ,
β = β1/β2 ,
a = a2/a1.
κ = κ1/κ 2,
Here ρm, νm, ηm, κm, χm, βm and am are, respectively, density, kinematic and dynamic viscosity, heat conductivity, thermal diffusivity, thermal expansion coefficient and the thickness of the m-th layer (m = 1, 2). As the units of length, time, velocity, pressure and temperature we choose a1, a12/ν1, ν1/a1, ρ1ν12/a12 and Aa1, respectively. The nonlinear equations of convection in the framework of the Boussinesq approximation for both fluids have the following form (see [1]):
→ ∇· vm = 0.
(1)
η
∂v1x ∂v ηM ∂T1 = 2x + , ∂z ∂z P ∂x
∂v1y ∂v2y ηM ∂T2 = + ; ∂z ∂z P ∂y
(4)
the continuity of the velocity field:
→ → v1 = v2;
(5)
the continuity of the temperature field:
T1 = T2;
(6)
and the continuity of the heat flux normal components:
κ
GQ ∂T ∂T1 − 2 = −κ . ∂z ∂z G
(7)
Here M = αθa1/η1χ1 is the Marangoni number, which is the basic non-dimensional parameter characterizing the thermocapillary effect, and GQ = gβ1Q 0 a14 /ν12 κ1 is the modified Grashof number determined by the interfacial heat release. Let us note that in [17] GQ = 0 . The boundary-value problem (1)–(7) contains nine thermophysical (M, G, P, GQ, ν, η, κ, χ, β) and two geometrical (a, L) nondimensional parameters, where L = l/a1. 2.2. Nonlinear approach
vmx =
∂ψm , ∂z
vmz = −
∂ψm , ∂x
ϕm =
(m = 1, 2).
x
∂vmz ∂v − mx , ∂x ∂z
we can rewrite the boundary value problem (1)–(7) in terms of variables ϕm , ψm , and Tm (see [1]). The calculations have been performed in a finite region 0 ≤ x ≤ L , − a ≤ z ≤ 1 with periodic boundary conditions on the lateral walls:
ψm (x + L, z ) = ψm (x, z );
a2
η
Eliminating the pressure and defining the vorticity
z 1
z = 0:
In order to investigate the flow regimes generated by the convective instabilities, we perform nonlinear simulations of twodimensional flows ( vmy = 0 (m = 1, 2); the fields of physical variables do not depend on y). In this case, we can introduce the stream function ψ
→ ∂ vm → → → + ( vm·∇) vm = − em ∇pm + cm ∇2 vm + bm GTm → γ, ∂t ∂Tm → d + vm·∇Tm = m ∇2Tm, ∂t P
a1
(3)
T2 = T0 − x,
where T0 is constant. We assume that the interface is flat, and it is located at z¼ 0. The boundary conditions on the interface include relations for the tangential stresses:
2. Formulation of the problem
ρ = ρ1/ρ2 ,
(2)
T1 = T0 − x,
2 y
Fig. 1. Geometrical configuration of the two-layer system and coordinate axes.
ϕm (x + L, z ) = ϕm (x, z );
Tm (x + L, z )
= Tm (x, z ) − L; m = 1, 2.
(8)
The boundary value problem is integrated in time with some initial conditions for ψm and Tm (m = 1, 2) by means of a finitedifference method. Equations and boundary conditions are approximated on a uniform mesh using a second order approximation for the spatial coordinates. The nonlinear equations are solved
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using an explicit scheme on a rectangular uniform mesh 84 × 56 (L = 2.74) and 112 × 84 (L = 9). We checked the results on 112 × 84 meshes (L = 2.74) and 168 × 112 (L = 9). The relative changes of the stream function amplitudes for all the mesh sizes do not exceed 2%. The variation of vortices at the corners of the region is about 2.5%. The Poisson equation is solved by the iterative Liebman successive over-relaxation method on each time step. The accuracy of the solution is 10 5. The details of the numerical method can be found in the book by Simanovskii and Nepomnyashchy [1].
3. Numerical results 3.1. Short computational regions 3.1.1. The case GQ < 0 In the present section, we investigate the nonlinear convective regimes in the 47v2 silicone oil–water system with the following set of parameters: ν = 2.0; η = 1.7375; κ = 0.184; χ = 0.778; β = 5.66 ; P ¼25.7. This system was used in experiments carried out by Degen et al. [26]. To simulate the motions in a laterally infinite two-layer system, we use periodic boundary conditions (8) for L¼ 2.74. Let us take the ratio of the layers thicknesses a ¼1. Under conditions of the experiment, when the geometric configuration of the system is fixed while the temperature difference θ is changed, the Marangoni number M and the Grashof number G are proportional. We define the inverse dynamic Bond number
K=
Sl2 (t ) =
∫0
L /2
dx
139
0
∫−a dzψ2 (x, z, t ).
(11)
Though these variables lack a clear physical meaning, they are sufficient for a qualitative understanding of the spatial structure of the flow (location of positive and negative vortices), the intensity of vortices and the symmetry of the flow. The time evolution of quantities Slm(t), m ¼ 1,2, at G ¼44.9 for different values of GQ, is shown in Fig. 3. One can see that with an increase of |GQ |, the period of oscillations grows, i.e., the phase velocity of the wave decreases (cf., for example, lines 1a, 2a and 1c, 2c in Fig. 3). Now, let us take G ¼91.9 (K ¼0.025). For sufficiently small values of |GQ |, the periodic traveling wave, moving in the direction of the horizontal temperature gradient, is developed in the system. The streamlines and isotherms for the traveling wave at GQ = 0 are shown in Fig. 4. Let us note that in the case G ¼44.9 and GQ = 0 , the parallel flow appears in the system. At |GQ | > 375, the traveling wave disappears and the parallel flow takes place. At |GQ | > 1150, the parallel flow becomes unstable and the system makes a transition to the traveling wave, moving in the direction opposite to the direction of the horizontal temperature gradient. The streamlines and isotherms for the traveling wave are shown in Fig. 5. Let us note that the maximum values of stream functions in both layers ψmax, m = max ψm (x, z ) (m ¼ 1,2) are constant in time. With an increase of |GQ | (at |GQ | > 1190), the periodic traveling wave becomes
M α . = GP gβ1ρ1 a12
Let us take G ¼ 44.9 for the fixed value of K (K ¼ 0.025). Since the temperature gradient is imposed in the horizontal direction, for any small values of the Grashof number G ≠ 0 and the Marangoni number M ≠ 0, the mechanical equilibrium becomes impossible. According to the predictions of the linear theory, for any small values of M and G the boundary value problem has a solution ψm = ψm(0) (z ) , θm = θm(0) (z ), m ¼1,2, corresponding to a parallel flow. Exact expressions for the stream function and temperature profiles of the parallel flow are presented in [17]. Expressions for ψm(0) (z ) and θm(0) (z ) consist of two parts: the part proportional to G, that describes the contribution of the buoyancy convection, and the part proportional to M /P , that describes the contribution of the thermocapillary convection. Numerical results confirm the predictions of the linear theory – in the absence of the interfacial heat release GQ = 0 , the parallel flow takes place in the system. With an increase of |GQ |, the parallel flow becomes unstable and the traveling wave, moving from the left to the right-hand side, i.e., in the direction opposite to the direction of the horizontal temperature gradient, is developed:
ψm (x, z, t ) = ψm (x − ct , z ),
Tm (x, z, t ) = Tm (x − ct , z ),
(9)
where c is the phase velocity of the traveling wave. The streamlines and isotherms for the traveling wave at GQ ¼ 3000, are presented in Fig 2. The corresponding vortices in both layers rotate in the opposite direction (Fig. 2a) and isotherms in both layers are distorted in the opposite way (Fig. 2b). With an increase of |GQ |, the intensity of the motion grows. In order to describe the time evolution of the solution, we use the following integral variables:
Sl1 (t ) =
∫0
L /2
dx
∫0
1
dzψ1 (x, z, t ),
(10)
Fig. 2. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 44.9; K = 0.025; GQ = − 3000 ; L ¼ 2.74; a¼ 1; the wave moves from the left to the righthand side.
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Fig. 3. The dependences of Sl, m on time (m = 1, 2) for the traveling wave at GQ = − 3000 (lines 1a, 2a); GQ = − 6000 (lines 1b, 2b); GQ = − 12, 000 (lines 1c, 2c); G ¼ 44.9; K = 0.025; L ¼ 2.74; a¼ 1; the wave moves from the left to the righthand side.
Fig. 5. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 91.9; K = 0.025; GQ = − 1190 ; L ¼ 2.74; a ¼1; the wave moves from the left to the righthand side.
Fig. 4. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 91.9; K = 0.025; GQ = 0 ; L ¼ 2.74; a¼ 1; the wave moves from the right to the left-hand side.
unstable and the modulated traveling wave, moving from the right to the left-hand side, appears in the system. The maximum values of stream functions ψmax, m , m ¼1,2, are not constant anymore but oscillate in a periodic way. The snapshots of streamlines for the modulated traveling wave at GQ = − 1455, are presented in Fig. 6. One can see that during an oscillatory process, the vortices change their shape and intensity. At |GQ | > 1500, another type of the periodic traveling wave is developed. The time evolution of quantities Slm(t), m¼1,2, at different values of GQ, is shown in Fig. 7. With an increase of |GQ |, the period of oscillations grows, i.e., the phase velocity of the wave decreases (cf. lines 1a, 2a and 1c, 2c in Fig. 7). The streamlines and isotherms for this wave at GQ = − 3000 are presented in Fig. 8. The wave moves from the left to the right-hand side and the “main” vortices fill, in practice, all the volume in the top and bottom layers (Fig. 8a). With an increase of |GQ |, the intensity of the motion grows in both layers. The maximum values of the stream function ψmax, m , m¼1,2, are constant in time. The dependences of the phase velocity of the traveling wave on |GQ | for different values of the Grashof number are shown in Fig. 9.
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Fig. 6. (a)–(d) A time sequence of snapshots of streamlines for the modulated traveling wave at G ¼ 91.9; K = 0.025; GQ = − 1455; L ¼ 2.74; a¼ 1; the wave moves from the left to the right-hand side.
Thus, rather intensive interfacial heat sinks can lead to the change of the direction of the waves' propagation (cf., for example, Figs. 4a and 5a; see, also, lines 2a, 2b and 3a, 3b in Fig. 9). The general diagram of regimes in the plane (|GQ | , G ) is presented in Fig. 10.
Fig. 7. The dependences of Sl, m on time (m = 1, 2) for the traveling wave at GQ = − 3000 (lines 1a, 2a); GQ = − 6000 (lines 1b, 2b); GQ = − 12, 000 (lines 1c, 2c); G ¼ 91.9; K = 0.025; L ¼ 2.74; a ¼1; the waves move from the left to the righthand side.
3.1.2. The case GQ > 0 Let us consider the influence of the interfacial heat sources (GQ > 0) on nonlinear regimes. We take G ¼ 44.9 and K ¼ 0.025. For sufficiently small values of GQ, the parallel flow takes place in the system. At GQ > 290, the parallel flow becomes unstable and the traveling wave, moving in the direction of the horizontal temperature gradient appears in the system. The streamlines and isotherms for the wave are shown in Fig. 11. One observes an essential asymmetry between the positive vortices, which occupy a large area in the top layer, and rather compact negative vortices localized near the upper rigid plate (Fig. 11a). With an increase of GQ , the intensity of the motion grows. The positive and negative vortices in the top layer become comparable in volume and intensity.
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Fig. 9. The dependence of the phase velocity |c| on |GQ | : line 1 (G ¼ 44.9); lines 2a, 2b (G ¼ 91.9); lines 3a, 3b (G ¼ 135.5); lines 1, 2b, 3b – the wave moves from the left to the right-hand side; lines 2a, 3a – the wave moves from the right to the left-hand side; K = 0.025; L ¼2.74; a ¼1.
Fig. 8. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 91.9; K = 0.025; GQ = − 3000 ; L ¼ 2.74; a¼ 1; the wave moves from the left to the righthand side.
3.2. Long computational regions 3.2.1. The case GQ < 0 Let us now consider the long computational regions (L ¼9). We fix G ¼22.4 (K ¼0.025). For sufficiently small values of |GQ |, the parallel flow takes place in the system. At |GQ | > 125, the parallel flow becomes unstable and the traveling wave, moving from the left to the right-hand side, is developed (see Fig. 12). The corresponding vortices in the top and bottom layers rotate in the opposite direction (Fig. 12a) and isotherms in both layers are distorted in the opposite way (Fig. 12b). The time evolution of quantities Slm(t), m ¼1,2, for different values of GQ, is shown in Fig. 13. With an increase of |GQ |, the period of oscillations grows, i.e., the phase velocity of the wave decreases (cf. lines 1a, 2a and 1c, 2c in Fig. 13). The situation is significantly changed at the larger values of the Grashof number. Let us fix G ¼300.8 (K ¼0.025). For sufficiently small values of |GQ |, the periodic traveling wave, moving in the direction of the horizontal temperature gradient, is developed in the system (Fig. 14). The positive vortices occupy a large area in the top layer and the small negative vortices are located near the upper rigid plate (Fig. 14a). Let us note that for G ¼22.4 and GQ = 0, we have observed a parallel flow. At |GQ | > 390, the traveling wave
Fig. 10. The general diagram of regimes in the plane (|GQ | , G ); K = 0.025; L ¼2.74; a¼ 1; ▵ – parallel flow, □ – the wave moves from the right to the left-hand size, n – the wave moves from the left to the right-hand size.
becomes unstable and the parallel flow appears. At |GQ | > 1100, a system makes a transition to the periodic traveling wave moving from the left to the right-hand side. The streamlines and isotherms for this wave are shown in Fig. 15. The maximum values of the stream function in both layers ψmax, m, m = 1, 2, are constant in time. With an increase of |GQ | (at |GQ | > 1450), the periodic traveling wave becomes unstable and the modulated traveling wave, moving from the left to the right-hand side, appears in the system. The snapshots of the streamlines for the modulated traveling wave are presented in Fig. 16. During an oscillatory process, the vortices change their shape and intensity. The maximum values of stream
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Fig. 13. The dependences of Sl, m on time (m = 1, 2) for the traveling wave at GQ = − 900 (lines 1a, 2a); GQ = − 3000 (lines 1b, 2b); GQ = − 6000 (lines 1c, 2c); G ¼22.4; K = 0.025; L ¼9; a¼ 1; the waves move from the left to the right-hand side.
Fig. 14. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 300.8; K = 0.025; GQ = 0; L ¼ 9; a¼ 1; the wave moves from the right to the left-hand side.
Fig. 11. (a) Streamlines and (b) isotherms for the traveling wave at G ¼44.9; K = 0.025; GQ = 1000 ; L ¼2.74; a¼ 1; the wave moves from the right to the lefthand side.
Fig. 15. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 300.8; K = 0.025; GQ = − 1250 ; L ¼ 9; a¼ 1; the wave moves from the left to the righthand side.
layer) decreases in comparison with the wave presented in Fig. 15a. Fig. 12. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 22.4; K = 0.025; GQ = − 900 ; L ¼ 9; a ¼1; the wave moves from the left to the right-hand side.
functions ψmax, m , m ¼1,2, are not constant anymore but oscillate in a periodic way. At |GQ | > 1490, the system makes a transition to another type of the periodic wave; ψmax, m = const (m = 1, 2). The number of the vortices for this wave (five “main” vortices in each
3.2.2. The case GQ > 0 Let us consider the influence of the interfacial heat sources on nonlinear regimes. We take G ¼ 22.4 (K = 0.025). For sufficiently small values of GQ, the parallel flow appears in the system. At GQ > 450, the parallel flow becomes unstable and the traveling wave, moving in the direction of the horizontal temperature gradient is developed. An example of streamlines and isotherms for the
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regimes of traveling waves and modulated traveling waves have been obtained. It is found that the direction of the wave propagation depends on the intensity of the interfacial heat sinks. The general diagram of regimes has been constructed.
References
Fig. 16. (a)–(c) A time sequence of snapshots of streamlines for the modulated traveling wave at G ¼ 300.8; K = 0.025; GQ = − 1470 ; L ¼ 9; a¼ 1; the wave moves from the left to the right-hand side.
Fig. 17. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 22.4; K = 0.025; GQ = 500 ; L ¼ 9; a ¼1; the wave moves from the right to the left-hand side.
traveling wave is shown in Fig. 17. One can see that positive and negative vortices in the top layer are comparable in volume and intensity (Fig. 17a).
4. Conclusion The influence of heat release and heat consumption at the interface on nonlinear convective regimes, developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system, has been investigated. The periodic boundary conditions on the lateral walls have been considered. Analysis has been fulfilled in the framework of the two-layer approach. Transitions between the flows with different spacial structures have been studied. It is shown that the presence of heat release and heat consumption at the interface can lead to the development of oscillatory regimes in the system. Specifically,
[1] I.B. Simanovskii, A.A. Nepomnyashchy, Convective Instabilities in Systems with Interface, Gordon and Breach, London, 1993. [2] A.A. Nepomnyashchy, I.B. Simanovskii, J.C. Legros, Interfacial Convection in Multilayer Systems, second edition, Springer, New York, 2012. [3] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities, J. Fluid Mech. 132 (1983) 119. [4] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers. Part 2. Surface-wave instabilities, J. Fluid Mech. 132 (1983) 145. [5] M.K. Smith, Instability mechanisms in dynamic thermocapillary liquid layers, Phys. Fluids 29 (1986) 3182. [6] S.H. Davis, Thermocapillary instabilities, Annu. Rev. Fluid Mech. 19 (1987) 403. [7] P.M. Parmentier, V.C. Regnier, G. Lebon, Buoyant-thermocapillary instabilities in medium–Prandtl-number fluid layers subject to a horizontal temperature gradient, Int. J. Heat Mass Transfer 36 (1993) 2417. [8] J.F. Mercier, C. Normand, Buoyant-thermocapillary instabilities of different heated liquid layers, Phys. Fluids 8 (1996) 1433. [9] J. Priede, G. Gerbeth, Convective, absolute and global instabilities of thermocapillary-buoyancy convection in extended layers, Phys. Rev. E 56 (1997) 4187. [10] C. Perez-Garcia, S. Madruga, B. Echebarria, G. Lebon, J. Burguete, Hydrothermal waves and corotating rolls in laterally heated convection in simple liquids, J. Non-Equilib. Thermodyn. 29 (2004) 377. [11] R.J. Riley, G.P. Neitzel, Instability of thermocapillary-buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities, J. Fluid Mech. 359 (1998) 143. [12] M.A. Pelacho, A. Garcimartin, J. Burguete, Local Marangoni number at the onset of hydrothermal waves, Phys. Rev. E 62 (2000) 477. [13] J. Burguete, N. Mukolobwiez, F. Daviaud, N. Garnier, A. Chiffandel, Buoyantthermocapillary instabilities in extended liquid layer subjected to a horizontal temperature gradient, Phys. Fluids 13 (2001) 2773. [14] A.B. Ezersky, A. Garcimartin, H.L. Mancini, C. Perez-Garcia, Spatiotemporal structure of hydrothermal waves in Marangoni convection, Phys. Rev. E 48 (1993) 4414. [15] S. Madruga, C. Perez-Garcia, G. Lebon, Convective instabilities in two superposed horizontal liquid layers heated laterally, Phys. Rev. E 68 (2003) 041607. [16] S. Madruga, C. Perez-Garcia, G. Lebon, Instabilities in two-liquid layers subject to a horizontal temperature gradient, Theor. Comput. Fluid Dyn. 18 (2004) 277. [17] A. Nepomnyashchy, I. Simanovskii, Convective flows in a two-layer system with a temperature gradient along the interface, Phys. Fluids 18 (2006) 032105. [18] K. Eckert, A. Grahn, Plume and finger regimes driven by an exothermic interfacial reaction, Phys. Rev. Lett. 82 (1999) 4436. [19] K.D. Danov, V.N. Paunov, N. Alleborn, H. Raszillier, F. Durst, Stability of evaporating two-layered liquid film in the presence of surfactant. I. The equations of lubrication approximation, Chem. Eng. Sci. 53 (1998) 2809. [20] C.N.R. Rao, Chemical Applications of Infrared Spectroscopy, Academic, New York, 1963. [21] S. Esplugas, J. Gimienez, S. Contreras, E. Pascual, M. Rodriguez, Comparison of different advanced oxidation processes for phenol degradation, Water Res. 36 (2002) 1034. [22] A.A. Nepomnyashchy, I.B. Simanovskii, Influence of thermocapillary effect and interfacial heat release on convective oscillations in a two-layer system, Phys. Fluids 16 (2004) 1127. [23] A.Yu. Gilev, A.A. Nepomnyashchy, I.B. Simanovskii, Thermogravitational convection in a two-layer system with heat release at the interface, Fluid Dyn. 25 (1990) 148. [24] I.B. Simanovskii, A. Viviani, F. Dubois, J.C. Legros, Influence of an interfacial heat release on nonlinear convective flows in a laterally heated two-layer system, Eur. J. Mech. B/Fluids 48 (2014) 49. [25] J. Priede, G. Gerbeth, Convective, absolute and global instabilities of thermocapillary-buoyancy convection in extended layers, Phys. Rev. E 56 (1997) 4187. [26] M.M. Degen, P.W. Colovas, C.D. Andereck, Time-dependent patterns in the two-layer Rayleigh-Bénard system, Phys. Rev. E 57 (1998) 6647.
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Acta Astronautica journal homepage: www.elsevier.com/locate/aa
Nonlinear traveling waves in a two-layer system with heat release/ consumption at the interface Ilya B. Simanovskii a,n, Antonio Viviani b, Frank Dubois c, Jean-Claude Legros c a
Department of Mathematics, Technion – Israel, Institute of Technology, 32000 Haifa, Israel Seconda Universita di Napoli (SUN), Dipartimento di Ingegneria Aerospaziale e Meccanica (DIAM), via Roma 29, 81031 Aversa, Italy c Universite Libre de Bruxelles, Service de Chimie Physique EP, CP165-62, 50 Av. F.D. Roosevelt 1050, Brussels, Belgium b
art ic l e i nf o
a b s t r a c t
Article history: Received 21 December 2015 Accepted 20 February 2016 Available online 17 March 2016
The influence of an interfacial heat release and heat consumption on nonlinear convective flows, developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system with periodic boundary conditions, is investigated. Regimes of traveling waves and modulated traveling waves have been obtained. It is found that rather intensive heat sinks at the interface can lead to the change of the direction of the waves' propagation. & 2016 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Interface Instabilities Two-layer system
1. Introduction It is known that two-layer liquid systems are subject to numerous instabilities (for a review, see [1,2]). Several classes of instabilities have been found by means of the linear stability theory for purely thermocapillary flows [3–6] and for buoyant-thermocapillary flows [7–10]. For the most typical kind of instability, hydrothermal instability, the appearance of oblique waves moving upstream has been predicted by the theory and justified in experiments [11–13]. However, two-dimensional waves moving downstream have also been observed in experiments [14]. The change of the direction of waves propagation can be caused by the influence of buoyancy [7]. Most of the investigations have been fulfilled for a sole liquid layer with a free surface, i.e., in the framework of the one-layer approach. Recently, Madruga et al. [15,16] studied the linear stability of two superposed horizontal liquid layers bounded by two solid planes and subjected to a horizontal temperature gradient. The analysis has revealed a variety of instability modes. The nonlinear wavy convective regimes in two-layer systems have been described in [17]. In the investigations of convection in two-layer systems, it is typically assumed that the normal components of the heat flux are equal on both sides of the interface. However, there are various physical phenomena which are characterized by a heat release or heat consumption at the interface. Specifically, the interfacial heat release accompanies an interfacial chemical reaction (see, e.g., n
Corresponding author. E-mail address: [email protected] (I.B. Simanovskii).
http://dx.doi.org/10.1016/j.actaastro.2016.02.026 0094-5765/& 2016 IAA. Published by Elsevier Ltd. All rights reserved.
[18]) and the evaporation [19]. The interfacial heating can be generated, e.g., by an infrared light source. The infrared absorption bands of water and silicone fluids are essentially different [20], therefore the light frequency can be chosen in a way that one of the fluids is transparent, while the characteristic length of the light absorption in another liquid is short. Another possibility of the interfacial heating may be realized by the use of the ultra-violet radiation: the process of photolysis of hydrogen peroxide H2 O2 due to the radiation with a wavelength lower than 400 nm, leads to the appearance of OH_radicals, accompanying by the reaction between silicone fluids and OH_radicals with an interfacial heat extraction [21]. It is known that the presence of a constant, spatially uniform heat release or heat consumption at the interface can lead to the appearance of an oscillatory instability [22,23]. Oscillations in [22,23] have been obtained in a two-layer system heated from below. Nonlinear convective regimes in a two-layer system filling a closed cavity with heat release at the interface have been studied in [24]. The system was heated from the lateral wall. Let us note that the theoretical predictions obtained for flows in closed cavities cannot be automatically applied for the infinite layers. For the observation of waves in a closed cavity a global instability is needed, while in the case of periodic boundary conditions one observes waves generated by a convective instability of a parallel flow [25]. Also, it should be taken into account, that in the presence of rigid lateral walls the basic flow is not parallel – the lateral walls act as a stationary finite-amplitude perturbation that can produce steady multicellular flow in the part of the cavity and in the whole cavity [25]. In the present paper, the influence of the interfacial heat release and heat consumption on nonlinear convective regimes,
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developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system with periodic boundary conditions, has been investigated. Specific regimes of traveling waves and modulated traveling waves have been obtained. It is found that rather intensive heat sinks at the interface can lead to the change of the direction of the waves' propagation. The paper is organized as follows. In Section 2, the mathematical formulation of the problem in the two-layer system is presented. Numerical simulations of the finite-amplitude convective regimes are considered in Section 3. Section 4 contains some concluding remarks.
→ Here, vm = (vmx , vmy, vmz ) is the velocity vector, Tm is the temperature and pm is the pressure in the m-th fluid; → γ is the unit vector directed upwards; b1 = c1 = d1 = e1 = 1; b2 = 1/β , c2 = 1/ν, d2 = 1/χ , e2 = ρ ; G = gβ1Aa14 /ν12 is the Grashof number and P = ν1/χ1 is the Prandtl number for the liquid in layer 1. The conditions on the rigid horizontal boundaries are:
→ v1 = 0;
z = 1:
→ v2 = 0;
z = − a:
2.1. Equations and boundary conditions We consider a system of two horizontal layers of immiscible viscous fluids with different physical properties (see Fig. 1). The variables referring to the top layer are marked by subscript 1, and the variables referring to the bottom layer are marked by subscript 2. The system is bounded from above and from below by two rigid plates, z = a1 and z = − a2. A constant temperature gradient is imposed in the direction of the axis x: T1 (x, y, a1, t ) =T2 (x, y, − a2, t ) = − Ax + const, A > 0. A constant heat release of the rate Q0 (Q0 may be positive or negative) is set on the interface. It is assumed that the interfacial tension s decreases linearly with an increase of the temperature: σ = σ0 − αT , where α > 0. Let us introduce the following notation:
ν = ν1/ν2,
η = η1/η2,
χ = χ1 /χ2 ,
β = β1/β2 ,
a = a2/a1.
κ = κ1/κ 2,
Here ρm, νm, ηm, κm, χm, βm and am are, respectively, density, kinematic and dynamic viscosity, heat conductivity, thermal diffusivity, thermal expansion coefficient and the thickness of the m-th layer (m = 1, 2). As the units of length, time, velocity, pressure and temperature we choose a1, a12/ν1, ν1/a1, ρ1ν12/a12 and Aa1, respectively. The nonlinear equations of convection in the framework of the Boussinesq approximation for both fluids have the following form (see [1]):
→ ∇· vm = 0.
(1)
η
∂v1x ∂v ηM ∂T1 = 2x + , ∂z ∂z P ∂x
∂v1y ∂v2y ηM ∂T2 = + ; ∂z ∂z P ∂y
(4)
the continuity of the velocity field:
→ → v1 = v2;
(5)
the continuity of the temperature field:
T1 = T2;
(6)
and the continuity of the heat flux normal components:
κ
GQ ∂T ∂T1 − 2 = −κ . ∂z ∂z G
(7)
Here M = αθa1/η1χ1 is the Marangoni number, which is the basic non-dimensional parameter characterizing the thermocapillary effect, and GQ = gβ1Q 0 a14 /ν12 κ1 is the modified Grashof number determined by the interfacial heat release. Let us note that in [17] GQ = 0 . The boundary-value problem (1)–(7) contains nine thermophysical (M, G, P, GQ, ν, η, κ, χ, β) and two geometrical (a, L) nondimensional parameters, where L = l/a1. 2.2. Nonlinear approach
vmx =
∂ψm , ∂z
vmz = −
∂ψm , ∂x
ϕm =
(m = 1, 2).
x
∂vmz ∂v − mx , ∂x ∂z
we can rewrite the boundary value problem (1)–(7) in terms of variables ϕm , ψm , and Tm (see [1]). The calculations have been performed in a finite region 0 ≤ x ≤ L , − a ≤ z ≤ 1 with periodic boundary conditions on the lateral walls:
ψm (x + L, z ) = ψm (x, z );
a2
η
Eliminating the pressure and defining the vorticity
z 1
z = 0:
In order to investigate the flow regimes generated by the convective instabilities, we perform nonlinear simulations of twodimensional flows ( vmy = 0 (m = 1, 2); the fields of physical variables do not depend on y). In this case, we can introduce the stream function ψ
→ ∂ vm → → → + ( vm·∇) vm = − em ∇pm + cm ∇2 vm + bm GTm → γ, ∂t ∂Tm → d + vm·∇Tm = m ∇2Tm, ∂t P
a1
(3)
T2 = T0 − x,
where T0 is constant. We assume that the interface is flat, and it is located at z¼ 0. The boundary conditions on the interface include relations for the tangential stresses:
2. Formulation of the problem
ρ = ρ1/ρ2 ,
(2)
T1 = T0 − x,
2 y
Fig. 1. Geometrical configuration of the two-layer system and coordinate axes.
ϕm (x + L, z ) = ϕm (x, z );
Tm (x + L, z )
= Tm (x, z ) − L; m = 1, 2.
(8)
The boundary value problem is integrated in time with some initial conditions for ψm and Tm (m = 1, 2) by means of a finitedifference method. Equations and boundary conditions are approximated on a uniform mesh using a second order approximation for the spatial coordinates. The nonlinear equations are solved
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using an explicit scheme on a rectangular uniform mesh 84 × 56 (L = 2.74) and 112 × 84 (L = 9). We checked the results on 112 × 84 meshes (L = 2.74) and 168 × 112 (L = 9). The relative changes of the stream function amplitudes for all the mesh sizes do not exceed 2%. The variation of vortices at the corners of the region is about 2.5%. The Poisson equation is solved by the iterative Liebman successive over-relaxation method on each time step. The accuracy of the solution is 10 5. The details of the numerical method can be found in the book by Simanovskii and Nepomnyashchy [1].
3. Numerical results 3.1. Short computational regions 3.1.1. The case GQ < 0 In the present section, we investigate the nonlinear convective regimes in the 47v2 silicone oil–water system with the following set of parameters: ν = 2.0; η = 1.7375; κ = 0.184; χ = 0.778; β = 5.66 ; P ¼25.7. This system was used in experiments carried out by Degen et al. [26]. To simulate the motions in a laterally infinite two-layer system, we use periodic boundary conditions (8) for L¼ 2.74. Let us take the ratio of the layers thicknesses a ¼1. Under conditions of the experiment, when the geometric configuration of the system is fixed while the temperature difference θ is changed, the Marangoni number M and the Grashof number G are proportional. We define the inverse dynamic Bond number
K=
Sl2 (t ) =
∫0
L /2
dx
139
0
∫−a dzψ2 (x, z, t ).
(11)
Though these variables lack a clear physical meaning, they are sufficient for a qualitative understanding of the spatial structure of the flow (location of positive and negative vortices), the intensity of vortices and the symmetry of the flow. The time evolution of quantities Slm(t), m ¼ 1,2, at G ¼44.9 for different values of GQ, is shown in Fig. 3. One can see that with an increase of |GQ |, the period of oscillations grows, i.e., the phase velocity of the wave decreases (cf., for example, lines 1a, 2a and 1c, 2c in Fig. 3). Now, let us take G ¼91.9 (K ¼0.025). For sufficiently small values of |GQ |, the periodic traveling wave, moving in the direction of the horizontal temperature gradient, is developed in the system. The streamlines and isotherms for the traveling wave at GQ = 0 are shown in Fig. 4. Let us note that in the case G ¼44.9 and GQ = 0 , the parallel flow appears in the system. At |GQ | > 375, the traveling wave disappears and the parallel flow takes place. At |GQ | > 1150, the parallel flow becomes unstable and the system makes a transition to the traveling wave, moving in the direction opposite to the direction of the horizontal temperature gradient. The streamlines and isotherms for the traveling wave are shown in Fig. 5. Let us note that the maximum values of stream functions in both layers ψmax, m = max ψm (x, z ) (m ¼ 1,2) are constant in time. With an increase of |GQ | (at |GQ | > 1190), the periodic traveling wave becomes
M α . = GP gβ1ρ1 a12
Let us take G ¼ 44.9 for the fixed value of K (K ¼ 0.025). Since the temperature gradient is imposed in the horizontal direction, for any small values of the Grashof number G ≠ 0 and the Marangoni number M ≠ 0, the mechanical equilibrium becomes impossible. According to the predictions of the linear theory, for any small values of M and G the boundary value problem has a solution ψm = ψm(0) (z ) , θm = θm(0) (z ), m ¼1,2, corresponding to a parallel flow. Exact expressions for the stream function and temperature profiles of the parallel flow are presented in [17]. Expressions for ψm(0) (z ) and θm(0) (z ) consist of two parts: the part proportional to G, that describes the contribution of the buoyancy convection, and the part proportional to M /P , that describes the contribution of the thermocapillary convection. Numerical results confirm the predictions of the linear theory – in the absence of the interfacial heat release GQ = 0 , the parallel flow takes place in the system. With an increase of |GQ |, the parallel flow becomes unstable and the traveling wave, moving from the left to the right-hand side, i.e., in the direction opposite to the direction of the horizontal temperature gradient, is developed:
ψm (x, z, t ) = ψm (x − ct , z ),
Tm (x, z, t ) = Tm (x − ct , z ),
(9)
where c is the phase velocity of the traveling wave. The streamlines and isotherms for the traveling wave at GQ ¼ 3000, are presented in Fig 2. The corresponding vortices in both layers rotate in the opposite direction (Fig. 2a) and isotherms in both layers are distorted in the opposite way (Fig. 2b). With an increase of |GQ |, the intensity of the motion grows. In order to describe the time evolution of the solution, we use the following integral variables:
Sl1 (t ) =
∫0
L /2
dx
∫0
1
dzψ1 (x, z, t ),
(10)
Fig. 2. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 44.9; K = 0.025; GQ = − 3000 ; L ¼ 2.74; a¼ 1; the wave moves from the left to the righthand side.
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Fig. 3. The dependences of Sl, m on time (m = 1, 2) for the traveling wave at GQ = − 3000 (lines 1a, 2a); GQ = − 6000 (lines 1b, 2b); GQ = − 12, 000 (lines 1c, 2c); G ¼ 44.9; K = 0.025; L ¼ 2.74; a¼ 1; the wave moves from the left to the righthand side.
Fig. 5. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 91.9; K = 0.025; GQ = − 1190 ; L ¼ 2.74; a ¼1; the wave moves from the left to the righthand side.
Fig. 4. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 91.9; K = 0.025; GQ = 0 ; L ¼ 2.74; a¼ 1; the wave moves from the right to the left-hand side.
unstable and the modulated traveling wave, moving from the right to the left-hand side, appears in the system. The maximum values of stream functions ψmax, m , m ¼1,2, are not constant anymore but oscillate in a periodic way. The snapshots of streamlines for the modulated traveling wave at GQ = − 1455, are presented in Fig. 6. One can see that during an oscillatory process, the vortices change their shape and intensity. At |GQ | > 1500, another type of the periodic traveling wave is developed. The time evolution of quantities Slm(t), m¼1,2, at different values of GQ, is shown in Fig. 7. With an increase of |GQ |, the period of oscillations grows, i.e., the phase velocity of the wave decreases (cf. lines 1a, 2a and 1c, 2c in Fig. 7). The streamlines and isotherms for this wave at GQ = − 3000 are presented in Fig. 8. The wave moves from the left to the right-hand side and the “main” vortices fill, in practice, all the volume in the top and bottom layers (Fig. 8a). With an increase of |GQ |, the intensity of the motion grows in both layers. The maximum values of the stream function ψmax, m , m¼1,2, are constant in time. The dependences of the phase velocity of the traveling wave on |GQ | for different values of the Grashof number are shown in Fig. 9.
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141
Fig. 6. (a)–(d) A time sequence of snapshots of streamlines for the modulated traveling wave at G ¼ 91.9; K = 0.025; GQ = − 1455; L ¼ 2.74; a¼ 1; the wave moves from the left to the right-hand side.
Thus, rather intensive interfacial heat sinks can lead to the change of the direction of the waves' propagation (cf., for example, Figs. 4a and 5a; see, also, lines 2a, 2b and 3a, 3b in Fig. 9). The general diagram of regimes in the plane (|GQ | , G ) is presented in Fig. 10.
Fig. 7. The dependences of Sl, m on time (m = 1, 2) for the traveling wave at GQ = − 3000 (lines 1a, 2a); GQ = − 6000 (lines 1b, 2b); GQ = − 12, 000 (lines 1c, 2c); G ¼ 91.9; K = 0.025; L ¼ 2.74; a ¼1; the waves move from the left to the righthand side.
3.1.2. The case GQ > 0 Let us consider the influence of the interfacial heat sources (GQ > 0) on nonlinear regimes. We take G ¼ 44.9 and K ¼ 0.025. For sufficiently small values of GQ, the parallel flow takes place in the system. At GQ > 290, the parallel flow becomes unstable and the traveling wave, moving in the direction of the horizontal temperature gradient appears in the system. The streamlines and isotherms for the wave are shown in Fig. 11. One observes an essential asymmetry between the positive vortices, which occupy a large area in the top layer, and rather compact negative vortices localized near the upper rigid plate (Fig. 11a). With an increase of GQ , the intensity of the motion grows. The positive and negative vortices in the top layer become comparable in volume and intensity.
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Fig. 9. The dependence of the phase velocity |c| on |GQ | : line 1 (G ¼ 44.9); lines 2a, 2b (G ¼ 91.9); lines 3a, 3b (G ¼ 135.5); lines 1, 2b, 3b – the wave moves from the left to the right-hand side; lines 2a, 3a – the wave moves from the right to the left-hand side; K = 0.025; L ¼2.74; a ¼1.
Fig. 8. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 91.9; K = 0.025; GQ = − 3000 ; L ¼ 2.74; a¼ 1; the wave moves from the left to the righthand side.
3.2. Long computational regions 3.2.1. The case GQ < 0 Let us now consider the long computational regions (L ¼9). We fix G ¼22.4 (K ¼0.025). For sufficiently small values of |GQ |, the parallel flow takes place in the system. At |GQ | > 125, the parallel flow becomes unstable and the traveling wave, moving from the left to the right-hand side, is developed (see Fig. 12). The corresponding vortices in the top and bottom layers rotate in the opposite direction (Fig. 12a) and isotherms in both layers are distorted in the opposite way (Fig. 12b). The time evolution of quantities Slm(t), m ¼1,2, for different values of GQ, is shown in Fig. 13. With an increase of |GQ |, the period of oscillations grows, i.e., the phase velocity of the wave decreases (cf. lines 1a, 2a and 1c, 2c in Fig. 13). The situation is significantly changed at the larger values of the Grashof number. Let us fix G ¼300.8 (K ¼0.025). For sufficiently small values of |GQ |, the periodic traveling wave, moving in the direction of the horizontal temperature gradient, is developed in the system (Fig. 14). The positive vortices occupy a large area in the top layer and the small negative vortices are located near the upper rigid plate (Fig. 14a). Let us note that for G ¼22.4 and GQ = 0, we have observed a parallel flow. At |GQ | > 390, the traveling wave
Fig. 10. The general diagram of regimes in the plane (|GQ | , G ); K = 0.025; L ¼2.74; a¼ 1; ▵ – parallel flow, □ – the wave moves from the right to the left-hand size, n – the wave moves from the left to the right-hand size.
becomes unstable and the parallel flow appears. At |GQ | > 1100, a system makes a transition to the periodic traveling wave moving from the left to the right-hand side. The streamlines and isotherms for this wave are shown in Fig. 15. The maximum values of the stream function in both layers ψmax, m, m = 1, 2, are constant in time. With an increase of |GQ | (at |GQ | > 1450), the periodic traveling wave becomes unstable and the modulated traveling wave, moving from the left to the right-hand side, appears in the system. The snapshots of the streamlines for the modulated traveling wave are presented in Fig. 16. During an oscillatory process, the vortices change their shape and intensity. The maximum values of stream
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143
Fig. 13. The dependences of Sl, m on time (m = 1, 2) for the traveling wave at GQ = − 900 (lines 1a, 2a); GQ = − 3000 (lines 1b, 2b); GQ = − 6000 (lines 1c, 2c); G ¼22.4; K = 0.025; L ¼9; a¼ 1; the waves move from the left to the right-hand side.
Fig. 14. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 300.8; K = 0.025; GQ = 0; L ¼ 9; a¼ 1; the wave moves from the right to the left-hand side.
Fig. 11. (a) Streamlines and (b) isotherms for the traveling wave at G ¼44.9; K = 0.025; GQ = 1000 ; L ¼2.74; a¼ 1; the wave moves from the right to the lefthand side.
Fig. 15. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 300.8; K = 0.025; GQ = − 1250 ; L ¼ 9; a¼ 1; the wave moves from the left to the righthand side.
layer) decreases in comparison with the wave presented in Fig. 15a. Fig. 12. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 22.4; K = 0.025; GQ = − 900 ; L ¼ 9; a ¼1; the wave moves from the left to the right-hand side.
functions ψmax, m , m ¼1,2, are not constant anymore but oscillate in a periodic way. At |GQ | > 1490, the system makes a transition to another type of the periodic wave; ψmax, m = const (m = 1, 2). The number of the vortices for this wave (five “main” vortices in each
3.2.2. The case GQ > 0 Let us consider the influence of the interfacial heat sources on nonlinear regimes. We take G ¼ 22.4 (K = 0.025). For sufficiently small values of GQ, the parallel flow appears in the system. At GQ > 450, the parallel flow becomes unstable and the traveling wave, moving in the direction of the horizontal temperature gradient is developed. An example of streamlines and isotherms for the
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regimes of traveling waves and modulated traveling waves have been obtained. It is found that the direction of the wave propagation depends on the intensity of the interfacial heat sinks. The general diagram of regimes has been constructed.
References
Fig. 16. (a)–(c) A time sequence of snapshots of streamlines for the modulated traveling wave at G ¼ 300.8; K = 0.025; GQ = − 1470 ; L ¼ 9; a¼ 1; the wave moves from the left to the right-hand side.
Fig. 17. (a) Streamlines and (b) isotherms for the traveling wave at G ¼ 22.4; K = 0.025; GQ = 500 ; L ¼ 9; a ¼1; the wave moves from the right to the left-hand side.
traveling wave is shown in Fig. 17. One can see that positive and negative vortices in the top layer are comparable in volume and intensity (Fig. 17a).
4. Conclusion The influence of heat release and heat consumption at the interface on nonlinear convective regimes, developed under the joint action of buoyant and thermocapillary effects in a laterally heated two-layer system, has been investigated. The periodic boundary conditions on the lateral walls have been considered. Analysis has been fulfilled in the framework of the two-layer approach. Transitions between the flows with different spacial structures have been studied. It is shown that the presence of heat release and heat consumption at the interface can lead to the development of oscillatory regimes in the system. Specifically,
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