Coriolis effect on thermal convection in a couple-stress fluid-saturated rotating rigid porous layer

Both linear and weakly nonlinear stability analyses are performed to study thermal convection in a rotating couple-stress fluid-saturated rigid porous layer. In the case of linear stability analysis, conditions for the occurrence of possible bifurcations are obtained. It is shown that Hopf bifurcation is possible due to Coriolis force, and it occurs at a lower value of the Rayleigh number at which the simple bifurcation occurs. In contrast to the nonrotating case, it is found that the couple-stress parameter plays a dual role in deciding the stability characteristics of the system, depending on the strength of rotation. Nonlinear stability analysis is carried out by constructing a set of coupled nonlinear ordinary differential equations using truncated representation of Fourier series. Sub-critical finite amplitude steady motions occur depending on the choice of physical parameters but at higher rotation rates oscillatory convection is found to be the preferred mode of instability. Besides, the stability of steady bifurcating equilibrium solution is discussed using modified perturbation theory. Heat transfer is calculated in terms of Nusselt number. Also, the transient behavior of the Nusselt number is investigated by solving the nonlinear differential equations numerically using the Runge–Kutta–Gill method. It is noted that increase in the value of Taylor number and the couple-stress parameter is to dampen the oscillations of Nusselt number and thereby to decrease the heat transfer

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

A Darcy-Brinkman Model for Penetrative Convection in LTNE

The aim of this paper is to investigate the onset of penetrative convection in a Darcy-Brinkmann porous medium under the hypothesis of local therma non-equilibrium. For the problem at stake, the strong form of the principle of exchange of stabilities has been proved, i.e. convective motions can occur only through a secondary stationary motion. We perform linear and nonlinear stability analyses of the basic state motion, with particular regard to the behaviour of the stability thresholds with respect to the relevant physical parameters characterizing the model. The Chebyshev-$\tau$ method and the shooting method are employed and implemented to solve the differential eigenvalue problems arising from linear and nonlinear analyses to determine critical Rayleigh numbers. Numerical simulations prove the stabilising effect of upper bounding plane temperature, Darcy's number and the interaction coefficient characterising the local thermal non-equilibrium regime

Stability Studies of Porous Media Including Surface Reactions

We investigate the onset of thermal convection in a number of porous models, with a focus on the influence of a boundary reaction. The models that we consider are: the Darcy porous model; the Darcy model with inclusion of the Soret effect and the Brinkman model, all with an exothermic surface reaction on the lower boundary. Numerical results are presented for each of these models and we show that the Darcy and Brinkman models with a surface reaction are structurally stable. Finally we derive stability results for a vertical porous channel that is in thermal non-equilibrium. In Chapter 2 we investigate how the parameters of an exothermic reaction on the lower boundary of a horizontal Darcy porous layer affect the linear instability boundary. We show that for low Lewis numbers stationary convection is dominant and for larger Lewis number oscillatory convection dominates. We use a non-linear analysis to and stability boundaries for this model in Chapter 3, showing how some of the reaction parameters affect this boundary. It is shown that the two boundaries do not coincide and there is a region in which sub-critical instabilities may occur. Structural stability on the reaction parameters is established for this model in Chapter 4. The impact of including the Soret effect on the stability of the Darcy model with a surface reaction on the lower boundary is considered in Chapter 5. When stationary convection dominates we find that increasing the Soret effect increases the critical Rayleigh number that defines the instability boundary. Chapter 6 discusses instabilities in a highly porous layer with an exothermic surface reaction on the lower boundary. The Brinkman model is used to take into account the impact of higher level derivatives of the fluid velocity. We show that this model is structurally stable on the parameters of the reaction in Chapter 7. Finally, in Chapter 8 we analytically derive two stability results for a vertical porous channel in thermal non-equilibrium. The first is that the model is stable for any initial data provided the Rayleigh number is below a given threshold. The second is that there is stability for any Rayleigh number given restrictions on the initial data

Stability of Triple Diffusive Convection in a Viscoelastic Fluid-Saturated Porous Layer

The triple diffusive convection in an Oldroyd-B fluid-saturated porous layer is investigated by performing linear and weakly nonlinear stability analyses. The condition for the onset of stationary and oscillatory is derived analytically. Contrary to the observed phenomenon in Newtonian fluids, the presence of viscoelasticity of the fluid is to degenerate the quasiperiodic bifurcation from the steady quiescent state. Under certain conditions, it is found that disconnected closed convex oscillatory neutral curves occur, indicating the requirement of three critical values of the thermal Darcy-Rayleigh number to identify the linear instability criteria instead of the usual single value, which is a novel result enunciated from the present study for an Oldroyd-B fluid saturating a porous medium. The similarities and differences of linear instability characteristics of Oldroyd-B, Maxwell, and Newtonian fluids are also highlighted. The stability of oscillatory finite amplitude convection is discussed by deriving a cubic Landau equation, and the convective heat and mass transfer are analyzed for different values of physical parameters

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

Abstract 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

Parallel simulation of coupled flow and geomechanics in porous media

textIn this research we consider developing a reservoir simulator capable of simulating complex coupled poromechanical processes on massively parallel computers. A variety of problems arising from petroleum and environmental engineering inherently necessitate the understanding of interactions between fluid flow and solid mechanics. Examples in petroleum engineering include reservoir compaction, wellbore collapse, sand production, and hydraulic fracturing. In environmental engineering, surface subsidence, carbon sequestration, and waste disposal are also coupled poromechanical processes. These economically and environmentally important problems motivate the active pursuit of robust, efficient, and accurate simulation tools for coupled poromechanical problems. Three coupling approaches are currently employed in the reservoir simulation community to solve the poromechanics system, namely, the fully implicit coupling (FIM), the explicit coupling, and the iterative coupling. The choice of the coupling scheme significantly affects the efficiency of the simulator and the accuracy of the solution. We adopt the fixed-stress iterative coupling scheme to solve the coupled system due to its advantages over the other two. Unlike the explicit coupling, the fixed-stress split has been theoretically proven to converge to the FIM for linear poroelasticity model. In addition, it is more efficient and easier to implement than the FIM. Our computational results indicate that this approach is also valid for multiphase flow. We discretize the quasi-static linear elasticity model for geomechanics in space using the continuous Galerkin (CG) finite element method (FEM) on general hexahedral grids. Fluid flow models are discretized by locally mass conservative schemes, specifically, the mixed finite element method (MFE) for the equation of state compositional flow on Cartesian grids and the multipoint flux mixed finite element method (MFMFE) for the single phase and two-phase flows on general hexahedral grids. While both the MFE and the MFMFE generate cell-centered stencils for pressure, the MFMFE has advantages in handling full tensor permeabilities and general geometry and boundary conditions. The MFMFE also obtains accurate fluxes at cell interfaces. These characteristics enable the simulation of more practical problems. For many reservoir simulation applications, for instance, the carbon sequestration simulation, we need to account for thermal effects on the compositional flow phase behavior and the solid structure stress evolution. We explicitly couple the poromechanics equations to a simplified energy conservation equation. A time-split scheme is used to solve heat convection and conduction successively. For the convection equation, a higher order Godunov method is employed to capture the sharp temperature front; for the conduction equation, the MFE is utilized. Simulations of coupled poromechanical or thermoporomechanical processes in field scales with high resolution usually require parallel computing capabilities. The flow models, the geomechanics model, and the thermodynamics model are modularized in the Integrated Parallel Accurate Reservoir Simulator (IPARS) which has been developed at the Center for Subsurface Modeling at the University of Texas at Austin. The IPARS framework handles structured (logically rectangular) grids and was originally designed for element-based data communication, such as the pressure data in the flow models. To parallelize the node-based geomechanics model, we enhance the capabilities of the IPARS framework for node-based data communication. Because the geomechanics linear system is more costly to solve than those of flow and thermodynamics models, the performance of linear solvers for the geomechanics model largely dictates the speed and scalability of the coupled simulator. We use the generalized minimal residual (GMRES) solver with the BoomerAMG preconditioner from the hypre library and the geometric multigrid (GMG) solver from the UG4 software toolbox to solve the geomechanics linear system. Additionally, the multilevel k-way mesh partitioning algorithm from METIS is used to generate high quality mesh partitioning to improve solver performance. Numerical examples of coupled poromechanics and thermoporomechanics simulations are presented to show the capabilities of the coupled simulator in solving practical problems accurately and efficiently. These examples include a real carbon sequestration field case with stress-dependent permeability, a synthetic thermoporoelastic reservoir simulation, poroelasticity simulations on highly distorted hexahedral grids, and parallel scalability tests on a massively parallel computer.Engineering Mechanic

On thermal convective instability in rotating fluids.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.Abstract available on the PDF