STABILITY OF NON-ISOTHERMAL POISEUILLE FLOW IN A VERTICAL CHANNEL

Abstract

This thesis reports a linear as well as nonlinear stability analysis of non-isothermal Poiseuille flow in a vertical channel. The flow is induced by external pressure gradient and buoyancy force due to maintenance of linearly varying wall temperature of the channel. The major part of the thesis is concerned with problems in porous medium, whereas, one chapter is concerned with viscous medium to understand the concept of nonlinear stability. In porous media, two different situations: local thermal equilibrium state (LTE) and local thermal non-equilibrium state (LTNE) are possible. The linear stability analysis of the above flow under LTE state is given by Bera et al. [10–12, 58], whereas, in this thesis same is analyzed under LTNE state. Apart from this, weakly nonlinear stability is analyzed for both LTE and LTNE states. The weakly nonlinear stability of Poiseuille flow in a vertical channel is developed on the basis of previous efforts of Stuart [110], Stuart & Stewartson [106] and Yao & Rogers [128]. This thesis is compiled in six chapters and the chapter wise description is given below. Chapter 1 is an introductory and contains some basic definitions and preliminaries of the flow in porous medium, brief description of hydrodynamic stability theory and work done by various authors in the field of linear and nonlinear stability analysis of Poiseuille flow. Chapter 2 addresses the mean flow characteristics of the above flow in a vertical channel filled with porous medium under LTNE state. The non-Darcy Brinkman-Forchheimerextended model is considered. The governing equations are solved numerically by spectral collocation method and analytically for a special case (form drag equal to zero). Special i ii attention is given to understand the effect of local thermal non-equilibrium parameters: inter-phase heat transfer coefficient (H) and porosity scaled thermal conductivity ratio (g ) on the rates of heat transfer (for solid and fluid phase) and fluid flow profiles, for both buoyancy assisted as well as opposed cases. A comprehensive investigation indicates that in the case of buoyancy assisted flow, for each Ra considered in this study and when g · 1, there exists a minimum value Ho of H such that the heat transfer rate of fluid (Nuf ) at the wall is an increasing function in [0, Ho]. Furthermore, Nuf decreases as H is increased beyond Ho. Similar results are also observed in buoyancy opposed case for very small values of g . The variation of velocity profile as a function of g indicates that change in velocity profile is sudden and abrupt. The velocity profile contains the point of inflection, which suggests a potential for the instability. The point of inflections in the velocity profile dies out on increasing the value of H, whereas, it reappears on increasing g . In Chapter 3, the influence of LTNE state on the linear stability mechanism of the above mean flow in a vertical channel filled with porous medium is examined. The Darcy- Brinkman-Forchheimer-Wooding model is used. The disturbance momentum and energy equations are solved by Galerkin method. To understand the impact of LTNE state on transition mechanism of mean flow, following controlling parameters related to (i) media permeability (Da), (ii) type of fluid (Pr), (iii) inter-phase heat transfer coefficient between solid and fluid phases of porous matrix (H) and (iv) conductivity ratio (g ) are considered. Apart from these, the rate of change of energy away from critical point and the energy budget spectrum at critical point are also examined. The pattern variation of the secondary flow of the basic flow under LTNE condition is also reported. The linear stability results show that, in general H stabilizes the mean flow and g destabilizes it. Stabilizing effect of H for fluid with low Prandtl number (Pr) becomes high when kinetic energy (KE) due to non-isothermal effects (Eb) is lost to mean flow. The energy loss to mean flow increases on increasing (decreasing) H (g ). The active role of g has been seen in the pattern variation of secondary flow as a function of Pr. Furthermore, depending on the magnitude of all the studied parameters, three types of instabilities namely, shear, mixed and buoyant are iii observed. In Chapter 4, a weakly nonlinear stability theory in terms of Landau equation is developed to analyze the nonlinear saturation of stably stratified non-isothermal Poiseuille flow in a vertical channel. The linear stability analysis of this flow is given by Chen & Chung [24]. The nonlinear results are presented with respect to fluids: mercury, gases, liquids and heavy oils. A perturbation solution method of [106] is used to find a linear relationship between D = Ra¡Rac and aci, where a, ci, Ra and Rac are the wavenumber, imaginary part of the eigenvalue, Rayleigh number and critical Rayleigh number respectively. The weakly nonlinear stability results predict only the supercritical instability, in agreement with another theoretical study of Chen & Chung [26] by direct numerical simulation. We find that the influence of nonlinear interaction among different superimposed waves on the heat transfer rate, real part of wavespeed and friction coefficient on the wall is similar for fluids as gases, liquids and heavy oils. Due to this nonlinear interaction the heat transfer rate increases beyond the critical Rayleigh number, whereas, the corresponding real part of wavespeed and friction coefficient decrease. However, in case of mercury influence of nonlinear interaction is complex and subtle. The amplitude analysis indicates that the equilibrium amplitude decreases on increasing the value of Reynolds number. We also establish a balance of kinetic energy for the fundamental disturbance that leads to amplitude equation. Analysis of nonlinear energy spectra for the disturbance also supports the supercritical instability at and beyond the critical point. Finally, we show the impact of nonlinear interaction of waves on the variation of pattern of secondary flow based on linear stability theory. Chapter 5 reports a theoretical development of finite amplitude instability in terms of Landau equation of non-isothermal Poiseuille flow in a vertical channel filled with porous medium under both LTE and LTNE states. The non-Darcy model is considered to describe the flow instability. The nonlinear results are presented with respect to fluid as air (Prandtl number (Pr) equal to 0.7). We establish a perturbation series solution with the help of Stewartson & Stuart [106] to find the growth rate of the most unstable wave. The finite iv amplitude analysis predicts only the supercritical bifurcation for both LTE and LTNE situations. The equilibrium amplitude of the most unstable disturbance wave increases on increasing the value of Rayleigh number beyond the bifurcation point. The influence of nonlinear interaction among different superimposed waves on the heat transfer rate, friction coefficient and pattern variation dies out for relatively large value of inter-phase heat transfer coefficient (H). The heat transfer rate in time space is almost constant under both situations. Finally, Chapter 6 presents the summary and concluding remarks of this thesis and the possible directions of the future scope.