Onset of Darcy-Brinkman convection with a uniform internal heat source and vertical throughflow

The problem of thermal convection in a horizontal fluid saturated porous layer is examined, where the flow is governed by the Brinkman extension of Darcy's law. A uniform internal heat source and vertical throughflow are also considered. The linear and nonlinear stability analyses are performed in order to determine the stability characteristics of the system. The linear and nonlinear thresholds give good agreement in the absence of vertical throughflow. However, it is shown that there are potential regions of sub-critical instabilities for increasing values of internal heat source parameter Q, Peclet number Pe and Darcy number Da

Buoyancy Driven Convection Due to Thermal and Salinity Gradients in a Tilted Porous Slot

The stabilityof double diffusive convection driven by temperature and salinity gradients in a tilted porous slot saturated by porous media is investigated analytically using linear stability analysis. Here the boundaries are infinite and assumed to be free and isothermal, tilted at a small angle of inclination with respect to the horizontal. A sec - ond order perturbation method is employedin terms of small angle of inclination is used to determine the critical Rayleigh number and wave number at the critical point and is supported by numerical solution of the resulting differential equation at higher order of approximation for velocity, temperature and salinity. The method of solution and the principal results obtained are somewhat identical for horizontal boundaries and for slightly tilted slot as for as nature of flow and convective instability is concerned. But, however the expression shows a considerable deviation in the basic flow that imparts definite structure with the convection pattern under investigation

Stability Analyses for Porous Convection Including Second Sound Effects

We investigate various models of thermal convection in a fluid saturated porous medium of both Darcy and Brinkman types. The linear instability and global (unconditional) nonlinear stability thresholds are analysed. Analytical solutions and numerical solutions are obtained by employing the $D^2$ Chebyshev tau and compound matrix techniques, and we investigate the effect that the inertia term and other physical parameters have on the stability of the system. The thesis is split into two parts. In PartI we consider a coupled model of thermal convection in a fluid saturated porous material and theories of viscous fluid motion which allows heat to travel as a wave. This is discussed in the first three chapters. In Chapter 2 the instability mechanism is investigated in complete detail and it is shown that stationary convection is likely to prevail under normal terrestrial conditions, but if the thermal relaxation time is sufficiently large there is a possible parameter range which allows for oscillatory convection. However, the presence of the Guyer-Krumhansl terms has the effect of damping the oscillatory convection and returning the instability mechanism to one of stationary convection. In Chapter 3 the constitutive equation for the heat flux is governed by a couple of the Guyer-Krumhansl equations and the Cattaneo-Fox law. In particular, we study the effects of the Guyer-Krumhansl terms on oscillatory convection. It is found that for a certain range of the Guyer-Krumhansl coefficient stationary convection occurs while changing the range results in oscillatory convection. Numerical results quantify this effect. The thermal instability in a Brinkman porous medium incorporating fluid inertia for both free--free and fixed--fixed boundaries is considered in Chapter 4. We have incorporated the Cattaneo--Christov theory in the onstitutive equation for the heat flux. For fixed surfaces, the results are generated by using the $D^2$ Chebyshev tau method. The results reveal that employing the Cattaneo--Christov theory has a pronounced effect in determining the convection instability threshold. Part II concerns the effect of an anisotropic permeability on thermal instability in the modelling problems of thermal convection of Darcy type with and without the inclusion of an inertia term, which represented the last three chapters. In Chapter 5 we allow a non-zero inertia term and also allow the permeability to be an anisotropic tensor. For particular numerical results we consider the case when the vertical component of the permeability tensor is variable. Linear instability results are calculated numerically and it is proved that the nonlinear energy stability bound is the same as the linear one. We perform the linear instability and nonlinear stability analysis, in the case where the inertial term vanishes, to investigate the effect of anisotropy with rotation on the stability thresholds in Chapter 6, showing that the nonlinear critical Rayleigh numbers coincide with those of the linear analysis. The results reveal that the inclusion of the inertial term for this model can play an important role on the onset of convection in Chapter 7

Boundary and inertia effects on the stability of natural convection in a vertical layer of an anisotropic Lapwood�Brinkman porous medium

The stability of natural convection in a fluid-saturated vertical anisotropic porous layer is investigated. The vertical rigid walls of the porous layer are maintained at different constant temperatures, and anisotropy in both permeability and thermal diffusivity is considered. The flow in the porous medium is described by the Lapwood�Brinkman model, and the stability of the basic flow is analysed numerically using Chebyshev collocation method. The presence of inertia is to inflict instability on the system and in the absence of which the system is always found to be stable. The mechanical and thermal anisotropies exhibit opposing contributions on the stability characteristics of the system. The mode of instability is interdependent on the values of Prandtl number and thermal anisotropy parameter, while it remains unaltered with the mechanical anisotropy parameter. The effect of increasing Prandtl and Darcy numbers shows a destabilizing effect on the system. Besides, simulations of secondary flow and energy spectrum have been analysed for various values of physical parameters at the critical state. © 2017, Springer-Verlag Wien

The Onset of Stationary and Oscillatory Convection in a Horizontal Porous Layer Saturated with Viscoelastic Liquid Heated and Soluted From Below: Effect of Anisotropy

The onset of double diffusive stationary and oscillatory convection in a viscoelastic Oldroyd type fluid saturated in an anisotropic porous layer heated and soluted from below is studied. The flow is governed by the extended Darcy model for Oldroyd fluid. Stability analysis based on the method of perturbations of infinitesimal amplitude is performed using the normal mode technique. The analysis examines the effect of the Darcy Rayleigh number, the solutal Darcy the Rayleigh number, the relaxation time, the retardation time and the Lewis number. Important conclusions include the destabilizing effect of the relaxation time, the Darcy Rayleigh number and the Lewis number and the stabilizing effect of the solutal Darcy Rayleigh number, the retardation time and anisotropy parameter. Some of the results are generalization of the previous findings for isotropic porous medium

Linear and Weak Nonlinear Double Diffusive Convection in a Viscoelastic Fluid Saturated Anisotropic Porous Medium with Internal Heat Source

This paper deals linear and weak nonlinear stability analysis of double-diffusive convection in an anisotropic porous layer with internal heat source saturated by viscoelastic fluid. For linear stability analysis we use normal mode technique and obtained the expression for oscillatory thermal Rayleigh number which is used to plot neutral stability curve for oscillatory case. For nonlinear analysis truncated representation of Fourier series upto two terms is used. The system of time dependent nonlinear equation is solved numerically and plot the curve for heat transfer and mass transfer with respect to time for different parameters. Effect of thermal anisotropy parameter, mechanical anisotropy parameter, relaxation parameter, retardation parameter, internal heat source parameter, solute Rayleigh number, diffusivity ratio, Darcy-Prandtl number on the onset of convection, heat and mass transfers have been discussed. We also draw the stream lines, isotherms, isohalines at different times

Convection with Chemical Reaction, and Waves in Double Porosity Materials

We consider two cases of solid skeleton of porous materials: fixed skeleton saturated with fluid in motion and deformed skeleton. In the first case, we study a problem involving the onset of thermosolutal convection in a fluid saturated porous media when the solute concentration is subject to a chemical reaction in which the solubility of the dissolved mineral is a function of temperature, particularly the effect of a reaction rate on the stability of the systems. We consider the Darcy model, the Brinkman model, and the Darcy model with anisotropic permeability and thermal diffusivity. Moreover, in all models the systems are subjected to heat on the lower boundary and salt on the upper or lower boundary. In chapter 2 we show that the solutions to the Darcy and the Brinkman thermosolutal convection depend continuously on the reaction term when the chemical equilibrium is a linear function in temperature by establishing a priori bounds. While in chapter 3 we show continuous dependence of the solution to the Brinkman thermosolutal convection on reaction using a priori bounds for the solution when the chemical equilibrium function is an arbitrary function of temperature. In chapter 4 we investigate the effect of the reaction terms on the onset of stability in a Darcy type porous medium using the energy method. We use the D^2 Chebyshev Tau technique to solve the associated system of equations and the corresponding boundary conditions. We obtain the energy stability boundaries for different values of the reaction terms and compare them with the linear instability boundaries obtained by Pritchard & Richardson(2007). We find that the two boundaries do not coincide when there is reaction and a region of potential sub-critical instability occur. In chapter 5 we use the energy method to obtain the non-linear stability boundaries for thermosolutal convection porous medium of a Brinkman type with reaction. We implement the compound matrix technique to solve the associated system of equations with the corresponding boundary conditions. We compare the non-linear stability boundaries for different values of the reaction terms and the Brinkman coefficient with the relevant linear instability boundaries obtained by Wang & Tan(2009). Our investigation shows that a region of potential sub-critical instability may appear as we increase the reaction rate. We study the effect of the mechanical anisotropy parameter and the thermal anisotropy parameter on the stability of a Darcy reactive thermosolutal porous medium in chapter 6 using the energy method. Particularly, we restrict consideration to horizontal isotropy in mechanical and thermal properties of the porous skeleton. We find that the anisotropic permeability has opposite effect to that of the thermal anisotropy parameter on the stability on the system. In the second case, deformed solid skeleton, we study wave motion in elastic materials of double porosity structure. In chapter 7 we derive the amplitude and describe the behaviour of a one-dimensional acceleration wave based on an internal strain energy function. The overall situation is complicated as a wave moves in a three-dimensional body, therefore in chapter 8 we investigate the propagation of an acceleration wave in three-dimensional fully non-linear model

Stability of mixed convection in an anisotropic vertical porous channel

This paper addresses the stability of mixed convective buoyancy assisted flow due to external pressure gradient and buoyancy force in a vertical fluid saturated porous channel with linearly varying wall temperature. The porous medium is assumed to be both hydrodynamically and thermally anisotropic. Two different types of temperature perturbations, (i) zero temperature and (ii) zero heat flux, have been considered to study the effect of anisotropic permeability and thermal diffusivity on the flow stability. The stability analysis indicated that the least stable mode is two-dimensional. Furthermore, the results show that for the same Reynolds number, the fully developed base flow is highly unstable (stable) for high (low) permeable porous media as well as for a porous medium with small (large) thermal diffusivity ratio. Depending on the magnitude of all parameters studied, three types of instabilities (shear, thermal, and mixed instability) occurred. The transition of instability from one type to another took place smoothly, except when the permeability ratio exceeded 6. Based on the value of the permeability ratio, the flow in an anisotropic medium for a specific Reynolds number may be either more or less stable than the flow in an isotropic medium. In addition, the fully developed flow is more stable for relatively small values of the modified Darcy number than for larger values. The effect of Brinkman as well as Forchheimer terms are negligible for the set of other parameters studied here. In contrast to a pure viscous fluid or an isotropic porous medium, which are characterized by unicellular convective cells, in anisotropic porous media convective cells may be unicellular or bicellular. The stability analysis of mixed convection in channels filled either with a viscous fluid or with an isotropic saturated porous medium may be obtained as special cases of the general study presented here. (C) 2002 American Institute of Physics

Effect of anisotropy on the onset of convection in rotating bi-disperse Brinkman porous media

AbstractThermal convection in a horizontally isotropic bi-disperse porous medium (BDPM) uniformly heated from below is analysed. The combined effects of uniform vertical rotation and Brinkman law on the stability of the steady state of the momentum equations in a BDPM are investigated. Linear and nonlinear stability analysis of the conduction solution is performed, and the coincidence between linear instability and nonlinear stability thresholds in the$$L^2$$L2-norm is obtained