Benard-Marangoni convection and instability in a layered fluid system

Abstract

Buoyancy-driven instability has been well established since the pioneering work of Rayleigh, who showed that convection, called Bénard convection, occurs only when the Rayleigh number, which is a ratio of buoyancy force to the dissipative force, exceeds a critical value. On the other hand, Pearson showed that surface-tension-gradient effects can also cause convection, usually called Marangoni convection, when the Marangoni number, which is a ratio of thermocapillary force to the dissipative force, exceeds a critical value. Inhomogeneities in solidified electronic materials have a very undesirable effect on the material’s electronic properties. Therefore, in the processing of high-quality electronic materials, stringent control of electronic melt stoichiometry is required during the solidification process. One source for inhomogeneities in the solidified crystal is thermal convective flow in the liquid melt. Thus, the control of melt convection affords control of crystalline structure. Therefore, it is necessary to develop methods and mathematical models to suppress thermal convection caused by buoyancy and surface tension forces. A reduction in the convection due to buoyancy is achieved in a microgravity environment. The rotation of a liquid column to confine thermocapillary or Marangoni convection near the surface is another technique to suppress convection. Favorable results of reduced convection are also reported using an applied magnetic field. Another alternative method has been proposed, whereby the liquid metal layer is encapsulated by another immiscible and noncorrosive liquid layer to suppress any thermocapillary convection that may develop in space processing. This method has several other advantages, namely (i) one can enhance or reduce the convection as needed by selecting a liquid encapsulant with appropriate physicochemical properties and (ii) a fluid-fluid system will not develop thermal stress problems at the phase boundary as in solid encapsulation. Motivated by this, several workers have studied the thermocapillary convection in two-immiscible liquid layers with flat free surface and flat liquid-liquid interface with differential heating applied parallel or normal to the interface. This thesis is devoted to the study of Bénard-Marangoni convection and instability in two-layered immiscible viscous liquid systems or a single-layer system with or without magnetic field in finite or infinite domains. The thesis is divided into seven chapters, with Chapter 1 giving the general introduction. Chapters 2, 3, and 4 deal with convective flows, whereas Chapters 5 and 6 with convective instability in two-layered fluid systems. The last chapter deals with the linear stability of a single fluid in a three-dimensional container. The literature connected with thermal convection in bounded and unbounded domains is surveyed in Chapter 1, and the general equations governing the flow for the problems discussed in the thesis, with the required appropriate assumptions, are presented. Further, the physical interpretations and derivations of the boundary conditions at the fluid-fluid interface, the free surface, and the rigid boundaries are explained. In Chapter 2, steady thermocapillary convection is investigated in a system of two superposed layers of immiscible, incompressible, viscous liquids with a curved free surface and a flat liquid-liquid interface in a configuration similar to that of an encapsulated crystal growth. The layers are bounded on the sides by isothermal vertical walls maintained at different constant temperatures. An analytical solution is obtained for infinite layers under lubrication approximation aspect ratio A (ratio of length of the cavity to height of the cavity) ? 0. There exist four different flow regimes under zero gravity condition depending on the values of A, the ratio of the temperature coefficient of the interfacial tension to that of the surface tension. The solutions for the core flow, temperature, and the free surface are determined for a finite domain by using matched expansion with the aspect ratio A (small) as a parameter correct to O(A²). In obtaining the solutions, we have used either fixed lines or fixed angles at the contact between the free surface and side walls. Streamlines for the flow near the vertical boundary are determined numerically using finite-difference over-relaxation method with suitable relaxation parameter and matching it with the analytical solution for the stream functions in the core region. In general, the free surface is not flat, and the conditions under which the free surface becomes flat are presented. It is observed that the strength of the vortices near the walls, the free surface height, and bending of the isotherms are closely connected and depend on the value of A. Further, for certain value of A = 0.5, it is observed that there is no flow in lower layer called halt condition as it halts the fluid motion. The results for a single layer can be obtained by taking (i) the viscosity of the lower layer very large tending to infinity or (ii) the width of the lower layer tending to zero or (iii) by taking the fluid properties to be the same for both the fluids. We have obtained the analytical results for a single layer by taking the viscosity of the lower layer tending to infinity. Analysis in the previous chapter for the steady thermocapillary convection of two superposed layers of immiscible incompressible, viscous liquids with deformed free surface and flat liquid-liquid interface in a configuration similar to that of an encapsulated crystal growth has been extended to the case with both interface and free surface being deformed in Chapter 3. The analytical solutions are presented for the stream function and temperature correct to O(A²). The shapes of free surface and interface are determined for different parameters and the effect of these parameters on streamline structure is also analyzed using fixed contact lines or fixed contact angles. In the absence of appropriate numerical techniques to deal with the problem in which the shapes of interface and free surface are not known a priori, the solutions for streamlines, temperature, interface and free surface shapes presented here give some insight of the flow structure. The structure of the streamlines and the isotherms for deformable interface and free surface qualitatively remains the same as that for flat interface and free surface. It is observed that for a given fixed angles of contact, the shape determined for the free surface is convex and the corresponding shape for the interface is either convex or concave depending on the value of A. The critical parameters which make the interface and free surface flat are determined. The assumption of flat interface and free surface is good for certain fluids with appropriate parameters. In Chapter 4, the effect of uniform magnetic field on thermal convection in a A shallow cavity, with differentially heated side walls, filled with two viscous, immiscible, incompressible and electrically conducting fluids in the presence of buoyancy force is studied. The fluid-fluid interface and the free surface are assumed to be flat and the driving forces for the flow are the thermocapillary and the buoyancy forces. Closed-form solutions, under thin layer approximation neglecting the side wall effects, are obtained for the stream function and temperature. In various limiting cases namely (i) absence of buoyancy force, (ii) absence of thermocapillary force and (iii) absence of magnetic fields, the solutions are obtained and they coincide with the existing results in the literature. The velocity is calculated and the resulting cell patterns are discussed for different values of A and Ha (Hartmann number). In absence of buoyancy forces, the results of this problem produce the results for infinite layers which include the effects of applied magnetic field. Here also there exist four different flow regimes depending on the values of A but with reduced convection compared to the non-magnetic case. It is observed that the halt condition decreases with an increase in Ha and for A < 0.5 it is possible to control the convection in the lower layer by a suitable choice of the magnetic field. The onset of steady Bénard-Marangoni convection in two horizontal liquid layers of electrically conducting immiscible, viscous, incompressible fluids subjected to a uniform vertical magnetic field and temperature gradient is analysed using a combination of analytical and numerical techniques in Chapter 5. The free surface is either deformable or non-deformable and the interface is always flat. The effect of the lower layer on the critical values of Rayleigh, Marangoni and wave numbers for the onset of steady convection is investigated. When the free surface is non-deformable the critical parameters for the onset of pure Marangoni convection are increased whereas the critical parameters for the onset of pure Bénard convection are decreased compared with the single layer model. Critical Marangoni and wave numbers are found to increase with an increase in A, the ratio of temperature coefficient of surface tension at the interface to that at the free surface, for pure Marangoni convection in the presence of magnetic field. All disturbances can be stabilized with sufficiently strong magnetic field when the free surface is non-deformable and insulated. If the free surface is allowed to deform and gravity waves are excluded then the layers are always unstable to disturbances with sufficiently small wave number with magnetic field. Inclusion of gravity waves has a stabilizing effect on certain disturbances of small wave number in the presence of weak or moderate magnetic field. The results for a single layer and for two layers are qualitatively similar when the free surface is deformable. It is observed that if the ratio of lower layer to the upper layer is greater than one then the critical value for stability of Marangoni convection increases whereas it decreases for Bénard convection. All results obtained in the present work agree with those obtained earlier in literature for a single layer in the limit of the height of the lower layer tending to zero. In Chapter 6, the effect of magnetic field and encapsulated layer on the onset of oscillatory Marangoni instability in two infinite horizontal immiscible, incompressible, viscous, electrically conducting liquid layers is investigated using normal mode analysis. The characteristic equation with complex coefficients is solved numerically by varying the frequency parameters until one gets a real Marangoni number as a solution. Oscillatory Marangoni instability is possible when the free surface is deformable and the interface is flat for certain negative Marangoni numbers (heated from top). When the free surface is flat and the interface is deformed there is no overstability in the absence of magnetic field. If both free surface and interface are deformed onset of overstability depends on the parameter A, the ratio of surface tensions at free surface and interface. Further, it is observed that if the lower layer height is more than that of the upper layer then the oscillatory convection is more stable. The corresponding single layer results with or without magnetic field are obtained by taking the fluid properties to be the same for both the fluids considered. The Bénard-Marangoni convection in a three-dimensional container with rigid lateral walls and prescribed heat flux at lower boundary is studied in Chapter 7. The upper surface of the single layer of incompressible, viscous fluid is assumed to be flat with temperature-dependent surface tension. A Galerkin-Tau method with odd and even trial functions satisfying all the essential boundary conditions except the natural boundary conditions at the free surface has been used to solve the problem. The critical Marangoni and Rayleigh numbers are determined for the onset of steady convection at the threshold as a function of aspect ratios a? (ratio of length to height) and a? (ratio of breadth to height) for the cases of Bénard-Marangoni, pure Marangoni and pure Bénard convections. Flow structures at the threshold are calculated for all the cases. The influence of Biot number on the critical parameter is also analyzed. The critical parameters for the heat flux prescribed case are higher than those for the case with prescribed temperature at the lower boundary. The following papers written based on the material embodied in this thesis: 1. Steady thermocapillary convection in two immiscible liquid layers with curved free surface (with Prof. A. Ramachandra Rao), Microgravity Sci. Technol. 8 (1995) 77-87. 2. Thermocapillary convection in two-layer liquid system with deformed interfaces (with Prof. A. Ramachandra Rao), Microgravity Sci. Technol. 8 (1995) 240-248. 3. Onset of steady Marangoni convection in two layers of electrically conducting fluids (with Prof. A. Ramachandra Rao), presented by P. C. Biswal in the international conference "Drop Tower Days 1996" held during July 8-11, 1996, University of Bremen, Germany. 4. The onset of steady Bénard-Marangoni convection in a two-layer system of conducting fluid in the presence of uniform magnetic field (with Prof. A. Ramachandra Rao). Revised version communicated to J. Engineering Mathematics. 5. Thermal convection in two immiscible liquid layers in the presence of a uniform magnetic field. In preparation. 6. The onset of oscillatory Marangoni convection in a two-layer system of conducting fluid in the presence of uniform magnetic field. In preparation.