MIXED CONVECTIVE FLOW IN VERTICAL CHANNEL FILLED WITH POROUS MEDIUM AND ITS STABILITY

Abstract

great deal of research activity in physics has recently been focused on transport in porous media because the theory of porous media underlines many unsolved problems in the engineering and applied sciences ranging from contaminant trans-port [36], paper manufacturing [47], geophysics and petroleum engineering [45], to marine science [23]. Most of the available studies are relevant to free convective heat transfer, whereas, convection due to the fact that fluid trapped in the pores of substance can be subjected to vaporization, condensation or to migration due to applied pressure gradients, is frequent. These are the inter-facial area of mixed convection which connects natural and forced convections. For example, convec-tion in permeable sediment due to hydrothermal vent is a combination of forced and natural convection. Because of the high pressure difference the hot and highly dense mixture of mineral water used to move in the upward direction through permeable sediment layer and leads to a forced convection, whereas, the temper-ature gradient of the sediment layer leads to density gradients in the fluid and in the presence of gravitational field, these variation in density induce varying body 11 forces in the fluid, which in turn, causes natural convection. This problem con-stitutes a very new research area, and theoretical investigation of it has been, in comparison, largely overlooked. In the present thesis, an attempt is made to understand the'mixed convection in porous medium by studying the flow dynamics and its transition behavior in vertical channel. Linear stability analysis is used to understand the stability char-acteristic of the fully developed base flow. In order to understand the fluid flow and heat transfer in a similar but in a confined domain the complete problem is solved by Spectral Element Method. The outline of the thesis is as follows. Chapter 1, is introductory and contains a brief outline of the fundamentals of flow in porous media, hydrodynamic stability theory, and a review of the liter-ature, mainly related to (i) possible models for mixed convection in porous me-dia, (ii) thermo-solutal convection and its stability, and (iii) natural convection in porous enclosure. In Chapter 2, the details of the linear theory of hydrodynamic stability that used in the thesis and mathematical formulation of Spectral Element method are given. In Chapter 3, the influence of different terms in the equation of motion in porous media is investigated while studying the stability of mixed convection due to external pressure gradient and buoyancy assisted force in a vertical channel. For this, a comparative study is made on the instability mechanism of the fluid flow 111 and heat transfer under four different models: (i) Darcy-Brinkman (MI), (ii) Darcy-Brinkman-Forchheimer (M2), (iii) Darcy-Brinkman-Wooding (M3), and (iv) Darcy-Brinkman-Forchheimer-Wooding (M4). It has been studied using linear theory of stability analysis and mainly gas and water as fluids. Details on the mechanism of the change of destabilizing characteristic of shear term Vii) of the momentum equation to stabilizing characteristic by changing fluid, media permeability, and flow strength; is given. Apart from these, the pattern similarity of the secondary flow of the basic flow under M2(M4) and M1(M3) models is also reported here. Chapter 4 concerns the stability of double-diffusive mixed convection in a ver-tical porous medium channel using Darcy-Brinkman-Wooding (M3) model. Since there are many controlling parameters in this problem, here a parametric study is carried out to isolate factors that are responsible for the instabilities of saline water through a permeable medium. As a consequence of it, the controlling parameters, Prandtl number (Pr) and Schimdt number (Sc) are fixed at 7 and 700 respectively. It has been reported that for certain range of media permeability the instability boundary curves are characterized by three different regimes: (i) Rayleigh-Taylor instability regime, (ii) non-linear variation regime, and (iii) linear variation regime. The details of the parametric analysis on instability boundary curve are also in-cluded. Similar to stability of cross-diffusive natural convection in vertical slot at critical state, 144], a linear relationship between logarithmic values of absolute heat source intensity ( I RaT I) and solute source intensity (Rac) is reported. Apart iv from these, it has been also mentioned that similar to single diffusive case, [11], for double-diffusive mixed convection there is a direct link between hyperbolic relation, -RaTDa = 2.467, and the Rayleigh-Taylor instability. In Chapter 5, a comprehensive numerical investigation on the natural convec-tion in a rectangular enclosure filled with hydro-dynamically anisotropic porous media is presented. The flow is due to sinusoidal temperature profile on the up-per wall and adiabatic conditions on the bottom as well as sidewalls of the cavity. Brinkman extended non-Darcy model, including material derivative, is considered here. The principal direction of the permeability tensor has been taken oblique to the gravity vector. The Spectral Element method has been adopted to solve nu-merically the governing conservative equations of mass, momentum, and energy, by using the vorticity-stream-function approach. Special attention is given to un-derstand the effect of anisotropic parameters on local heat transfer rate as well as on flow configurations. For small Rayleigh-Darcy number (RaDa) it is shown that the heat transfer is mainly due to conduction mode across the fluid layers, and convection started only for higher values of RaDa. Increasing the enclosure aspect ratio increases the fluid circulation and the depth of thermal penetration. Effect of Prandtl number in the range [0.5, 100] is negligible on the local heat transfer rate. Apart from these, it is also reported that for all values of RaDa considered here, the local heat transfer rate at the mid-plane of the upper wall is maximum when an-gles of orientation (0) of the media permeability tensor are 45° and 135°, whereas V the same is minimum at 0 = 0°, 90° and 180°. Chapter 6 contains an extension of the above problem by changing the bound-ary condition of the problem. The flow is induced by a non-uniform partial heating of the bottom wall and constant cooling of the vertical side walls along with an adi-abatic top wall. The results are presented in terms of isotherms, stream function and average Nusselt numbers (bottom heat transfer rate, Nub and side heat trans-fer rate, Nus). The present analysis shows that the parameters of anisotropy have significant influence on the flow behavior and heat transfer. The flow in the enclo-sure is governed by two different types of convective cells: rotating (i) clockwise, (ii) anticlockwise. Based on the value of media permeability as well as orientation angle, in anisotropic case, one of the cell will dominant on the other one. Both heat transfer rates and flow mechanism (streamlines as well as thermal lines) are symmetric about 0 equal to 45° and 135° in the domain [0°, 90°] and [90°, 180°] of ifp respectively. Finally, chapter 7 presents the summary and conclusions of this thesis, and the possible directions for future work.