Academic literature on the topic 'Darcy-Lapwood-Brinkman equation'

This article involves the study and analysis of the fully developed flow of a magnetorheological fluid through a non-isotropic porous medium under the effect of an external, uniform, and transversal magnetic field. Permeability is taken as an exponential distribution function of the transverse direction. The Darcy-Brinkman-Lapwood-Lorentz equation for the fluid flow in porous media has been used and solved under non-slip boundary conditions by Modified Homotopy Perturbation Method and the results validated by the Numerical Shooting Method. Finally, the analysis of results is made of the influence on the velocity, volumetric flow, and wall shear stress.

Circular Couette flow between porous cylinders is treated in this paper. Exact solutions of Navier-Stokes equations for the flow in the gap between cylinders and of Darcy-Brinkman-Lapwood equations for the flow in porous rings are found analytically by matching velocities and stresses on the porous boundaries, without making any previous assumptions concerning the slip velocities. In the special case in which the inner cylinder is fully porous and stationary, and the outer cylinder is fully rigid and rotating, the torque exerted by the rotation of the outer cylinder on the inner one is found, and it is shown how the effective viscosity of the liquid can be determined in possible experiments.

An improved numerical study on mixed convection from a heated vertical plate embedded in a Newtonian fluid saturated sparsely packed porous medium is undertaken by considering the variation of permeability, porosity and thermal conductivity. The boundary layer flow in the porous medium is governed by Lapwood – Forchheimer – Brinkman extended Darcy model. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge – Kutta – Fehlberg integration scheme to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. Further, the obtained results under the limiting conditions were found to be in good agreement with the existing ones.

The onset of thermomagnetic convection in a ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated for a variety of velocity and temperature boundary conditions. The Brinkman–Lapwood extended Darcy equation, with fluid viscosity different from effective viscosity, is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-ferromagnetic, while the upper boundary is considered to be either rigid-ferromagnetic or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective-radiative exchange at the upper boundary, which encompasses fixed temperature and heat flux as particular cases. The resulting eigenvalue problem is solved using the Galerkin technique and also by using regular perturbation technique when both boundaries are insulated to temperature perturbations. It is found that the increase in the Biot number and the viscosity ratio, and the decrease in the magnetic as well as in the Darcy number is to delay the onset of ferroconvection. Besides, the nonlinearity of fluid magnetization has no effect on the onset of convection in the case of fixed heat flux boundary conditions.

Le problème de la cavité poreuse carrée est largement utilisé comme cas de référence courant pour les problèmes de Convection Naturelle (CN) en milieux poreux. Il peut être utilisé pour plusieurs applications numériques, théoriques et pratiques. Par ailleurs, toutes les solutions de haute précision existantes dans la littérature scientifique sont développées dans des conditions de régime permanent. Cependant, il est bien connu que les processus de CN dans les milieux poreux se produisent naturellement dans un régime dépendant du temps, car les conditions aux limites peuvent être variables dans le temps. Pour surmonter cette difficulté, la solution en régime permanent est souvent simulée comme une solution transitoire qui évolue jusqu'à atteindre l'état d'équilibre. Ces régimes dépendant du temps sont très efficaces pour détecter les effets des variations de paramètres sur le processus physique de CN, en particulier pour les sujets d'intérêt de cette thèse: la variation du niveau d'inclinaison du domaine et la prise en compte des variations de température de la paroi chaude dans le temps. À cet effet, trois objectifs sont identifiés dans cette thèse: 1. Développer une solution de convection naturelle en fonction du temps dans des milieux poreux en utilisant le Modèle Darcy en deux modes: transitoire et instable. 2. Étudier le comportement en fonction du temps de la convection naturelle dans des milieux poreux ayant le niveau d'inclinaison du domaine comme paramètre variable dans deux modes: transitoire et instable. 3. Développer une solution de convection naturelle en fonction du temps dans des milieux poreux en utilisant le Modèle Darcy-Lapwood-Brinkman en deux modes: transitoire et instable. Pour ce faire, du fait de la grande précision dans les domaines simplement connectés, une méthode spectrale de résidus pondérés de type Galerkin est choisie pour développer une solution au problème de CN dans une cavité carrée poreuse. L’application de la procédure de Fourier-Galerkin (FG), deux configurations traitant des régimes instables sont considérées où chaque solution est dérivée pour une large gamme des nombres de Rayleigh (Ra) avec d'autres conditions spéciales. Ce travail de thèse est subdivisé en cinq chapitres. Dans le premier chapitre, nous avons présenté un aperçu physique du processus de convection naturelle en milieux poreux. Dans le deuxième chapitre, le développement mathématique des équations, la méthode de résolution et la procédure de résolution sont décrits en détails. Dans le chapitre trois, la première étude de cas de cette thèse, la solution dépendante du temps de la convection naturelle dans une cavité carrée remplie de milieux poreux saturé utilisant le modèle de Darcy est développé. Dans le chapitre quatre, le problème de variation temporelle de Darcy-Lapwood- Brinkman de CN dans une enceinte poreuse saturée carrée est étudié. Dans le chapitre cinq, les solutions dépendant du temps sont développées pour le problème de convection naturelle utilisant la loi de Darcy dans une cavité poreuse inclinée et considéré comme une étude complète sur les effets de l'inclinaison du domaine sur le processus physique du problème de convection libre. Pour tous les cas, les régimes transitoires et instables sont considérés
The problem of the porous square cavity is extensively used as a common benchmark case for Natural convection (NC) problem in porous media. It can be used for several numerical, theoretical, and practical purposes. All the existing high accurate solutions are developed under steady-state conditions. However, it is well known that the processes of NC in porous media occurs naturally in a time-dependent procedure, as boundary conditions can be variable in time. Also, the convergence of the steady-state solution is known to be difficult. To overcome this difficulty, the steady-state solution is often simulated as a transient solution that evolves until reaching the steady-state condition. These time-dependent modes are very efficient to detect the effects of the parameter variations on the physical process of NC, especially for the subject of interest in this thesis: the domain inclination level and hot wall temperature variation in time. For this purpose, three goals are identified in this Thesis: 1. Developing a time-dependent solution of natural convection in porous media using the Darcy model in two modes: Transient and unsteady. 2. Investigating the time-dependent behavior of natural convection in porous media having the domain inclination level as a variable parameter in two modes: Transient and unsteady. 3. Developing a time-dependent solution of natural convection in porous media using the Darcy-Lapwood-Brinkman model in two modes: Transient and unsteady. To do so, according to the high accuracy in the simply connected domains, one of the Galerkin spectral weighted residual method is chosen to develop a space-time dependent solution for NC problem in a square porous cavity. Applying the Fourier-Galerkin (FG) procedure, two configurations dealing with transient and unsteady regimes are considered where each solution is derived for a wide range of Rayleigh numbers with other special conditions. This work of thesis is explained in details as five chapters.The NC physical process with the time-dependent variations is described in the transient mode to reach the steady-state solution and for the unsteady mode during a one period using periodic sinusoidal boundary conditions on the cavity hot wall. Finally, the work of this thesis is described in details in five chapters; while the sixth and last chapter is devoted to the summary and conclusion.The results in this thesis work provide a set of high-accurate data that are published in three papers to be used for testing numerical codes of heat transfer in time-dependent configurations

Abstract By developing our proper CFD code under Fortran, the performances of a Stirling engine are studied in unsteady laminar regime and closely linked to the properties of its regenerator. However, it is responsible about the maximum part of losses in the Stirling engine. These losses depend on geometric and physical properties of the material constituting the regenerator. Thus, finding the suitable regenerator material that generates the greatest heat exchange and the lowest pressure drop is a good solution to reduce sources of irreversibility and ameliorate the global performances of the Stirling engine. The aim of this paper is to describe oxillatory flow and heat transfer inside porous regenerator materials and to determine the most suitable regenerator material. Brinkman-Forchheimer-Lapwood extended Darcy model is assumed to simulate momentum transfer within the porous regenerator. And the oscillatory flow is described by the Navier-Stockes compressible equations. The local thermal equilibrium of the gas and the matrix is taken into account for the modelling of the porous regenerator. The governing equations with the appropriate boundary conditions are solved by the control volume based finite element method (CVFEM). A numerical code on the software Fortran is elaborated to evaluate flow and heat transfer characteristics inside regenerator. Results showed that the fluid flow and heat transfer between the compression and expansion phases were varied significantly. It was shown that the superior comprehensive performance of the regenerator makes it possible to improve the performance of Stirling engines.