Synchronous and Asynchronous Boundary Temperature Modulations of Bénard-Darcy Convection

A theoretical analysis of thermo-convective instability in a densely packed porous medium is carried out when the boundary temperatures vary with time in a sinusoidal manner. By performing a weakly non-linear stability analysis, the Nusselt number is obtained as a function of amplitude of convection which is governed by a non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. The paper succeeds in unifying the modulated Bénard–Darcy, Bénard–Rayleigh, Bénard–Brinkman and Bénard–Chandrasekhar convection problems and hence precludes the study of these individual problems in isolation. A new result that shows that asynchronous temperature modulation may be effectively used to either enhance or reduce heat transport by suitably adjusting the frequency and phase-difference of the modulated temperature is presented

Asymptotic approach to the generalized Brinkman’s equation with pressure dependent viscosity and drag coefficient

In this paper we investigate the fluid flow through a thin (or long) channel filled with a fluid saturated porous medium. We are motivated by some important applications of the porous medium flow in which the viscosity of fluids can change significantly with pressure. In view of that, we consider the generalized Brinkman’s equation which takes into account the exponential dependence of the viscosity and the drag coefficient on the pressure. We propose an approach using the concept of the transformed pressure combined with the asymptotic analysis with respect to the thickness of the channel. As a result, we derive the asymptotic solution in the explicit form and compare it with the solution of the standard Brinkman’s model with constant viscosity. To our knowledge, such analysis cannot be found in the existing literature and, thus, we believe that the provided result could improve the known engineering practice.Croatian Science FoundationFundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Economía y CompetitividadJunta de Andalucí

Marangoni Convection in a Fluid Saturated Porous Layer with a Prescribed Heat Flux at its Lower Boundary

The onset of Marangoni convection in a horizontal porous layer heated from below with a constant heat flux is investigated. The Brinkman model is used and the Darcy law is employed to describe the flow in the porous medium heated from below. We obtain for the first time the closed form analytical solution for the onset of steady Marangoni convection in a fluid saturated porous layer with a prescribed heat flux at its lower boundary. Besides, the asymptotic solution of the long-wavelength is also obtained using regular perturbation technique with wave number as a perturbation parameter. The Marangoni numbers are found to depend on the Darcy number and Biot number. Predictions for the onset of convection are studied in detail

Eddy Heat Conduction and Nonlinear Stability of a Darcy Lapwood System Analysed by the Finite Spectral Method

A finite Fourier transform is used to perform both linear and nonlinear stability analyses of a Darcy-Lapwood system of convective rolls. The method shows how many modes are unstable, the wave number instability band within each mode, the maximum growth rate (most critical) wave numbers on each mode, and the nonlinear growth rates for each amplitude as a function of the porous Rayleigh number. Single amplitude controls the nonlinear growth rates and thereby the physical flow rate and fluid velocity, on each mode. They are called the flak amplitudes. A discrete Fourier transform is used for numerical simulations and here frequency combinations appear that the traditional cut-off infinite transforms do not have. The discrete show a stationary solution in the weak instability phase, but when carried past 2 unstable modes they show fluctuating motion where all amplitudes except the flak may be zero on the average. This leads to a flak amplitude scaling process of the heat conduction, producing an eddy heat conduction coefficient where a Nu-RaL relationship is found. It fits better to experiments than previously found solutions but is lower than experiments

On the Onsetof Thermal Instability in a Low Prandtl Number Nanofluid Layer in a Porous Medium

Thermal instability in a low Prandtl number nanofluid in a porous medium is investigated by using Galerkin weighted residuals method for free-free boundaries. For porous medium, Brinkman-Darcy modelis applied. The model used for the nanofluid describes the effects of Brownian motion and thermophoresis. Linear stability theory based upon normal mode analysis is employed to find the expression for stationary and oscillatory convection. The effects of Prandtl- number, Darcy number, Lewis number and modified diffusivity ratio on the stationary convection are investigated both analytically and graphically. The results indicated that the Prandtl and Darcy numbers have a destabilizing effect while the Lewis number and modified diffusivity ratio have a stabilizing effect for the stationary convection

Thermal convection in a higher-gradient Navier–Stokes fluid

We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences

A thermal non-equilibrium model with Cattaneo effect for convection in a Brinkman porous layer

This paper aims to investigate the onset of thermal convection in a layer of fluid-saturated Brinkman porous medium taking into account fluid inertia and local thermal non-equilibrium (LTNE) between the solid and fluid phases with Cattaneo effect in the solid. A two-field model is used for the energy equations each representing the solid and fluid phases separately. The usual Fourier heat-transfer law is retained in the fluid phase while the solid phase is allowed to transfer heat via a Cattaneo heat flux theory. It is observed that the Cattaneo effect has a profound influence on the nature of convective instability. In contrast to the standard Brinkman convection with LTNE model, instability is found to occur through oscillatory convection depending on the value of solid thermal relaxation time parameter which in turn depends on other parametric values. The instability characteristics of the system are analyzed in detail for a wide range of parametric values including those for copper oxide and aluminium oxide solid skeletons.postprin

Asymptotic Approach to the Generalized Brinkman’s Equation with Pressure-Dependent Viscosity and Drag Coefficient

In this paper we investigate the fluid flow through a thin (or long) channel filled with a fluid saturated porous medium. We are motivated by some important applications of the porous medium flow in which the viscosity of fluids can change significantly with pressure. In view of that, we consider the generalized Brinkman’s equation which takes into account the exponential dependence of the viscosity and the drag coefficient on the pressure. We propose an approach using the concept of the transformed pressure combined with the asymptotic analysis with respect to the thickness of the channel. As a result, we derive the asymptotic solution in the explicit form and compare it with the solution of the standard Brinkman’s model with constant viscosity. To our knowledge, such analysis cannot be found in the existing literature and, thus, we believe that the provided result could improve the known engineering practice