scholarly journals Flow of Stokesian fluid through a cellular medium and thermal effects

2014 ◽  
Vol 62 (2) ◽  
pp. 321-329
Author(s):  
R. Wojnar
Keyword(s):  
Porous Medium ◽  
Elastic Medium ◽  
Thermal Effects ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Compressible Fluids ◽  
Scale Analysis ◽  
Flow Through ◽  
Cellular Medium

Abstract The thermal effects of a stationary Stokesian flow through an elastic micro-porous medium are compared with the entropy produced by Darcy’s flow. A micro-cellular elastic medium is considered as an approximation of the elastic porous medium. It is shown that after asymptotic two-scale analysis these two approaches, one analytical, starting from Stoke’s equation and the second phenomenological, starting from Darcy’s law give the same result. The incompressible and linearly compressible fluids are considered, and it is shown that in micro-porous systems the seepage of both types of fluids is described by the same equations.

Numerical modelling of nonlinear effects in laminar flow through a porous medium

1988 ◽  
Vol 190 ◽  
pp. 393-407 ◽  
Author(s):  
O. Coulaud ◽  
P. Morel ◽  
J. P. Caltagirone
Keyword(s):  
Porous Medium ◽  
Pressure Drop ◽  
Nonlinear Term ◽  
Numerical Solutions ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Quadratic Term ◽  
Inertial Effects ◽  
Operator Splitting Methods ◽  
Flow Through

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.

scholarly journals Heuristic derivation of Brinkman's seepage equation

2017 ◽  
Vol 4 (20) ◽  
pp. 359-374
Author(s):  
Ryszard Wojnar
Keyword(s):  
Porous Medium ◽  
Viscous Fluid ◽  
Stokes Equation ◽  
Approximation Method ◽  
Stokes Flow ◽  
Darcy’S Law ◽  
Flow Equation ◽  
Darcy's Law ◽  
Porous Solid ◽  
Flow Through

Brinkman’s law is describing the seepage of viscous fluid through a porous medium and is more acurate than the classical Darcy’s law. Namely, Brinkman’s law permits to conform the flow through a porous medium to the free Stokes’ flow. However, Brinkman’s law, similarly as Schro¨dinger’s equation was only devined. Fluid in its motion through a porous solid is interacting at every point with the walls of pores, but the interactions of the fluid particles inside pores are different than the interactions at the walls, and are described by Stokes’ equation. Here, we arrive at Brinkman’s law from Stokes’ flow equation making use of successive iterations, in type of Born’s approximation method, and using Darcy’s law as a zero-th approximation.

THE INHOMOGENEOUS DAM PROBLEM WITH LINEAR DARCY’S LAW AND DIRICHLET BOUNDARY CONDITIONS

1996 ◽  
Vol 06 (08) ◽  
pp. 1051-1077 ◽  
Author(s):  
A. LYAGHFOURI
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Dirichlet Boundary Conditions ◽  
Flow Through

In this paper we study a fluid flow through a porous medium with Dirichlet boundary conditions and a general permeability. We establish the continuity of the free boundary and the uniqueness of the S3-connected solution.

Incorporation of Interfacial Areas in Models of Two-Phase Flow

1999 ◽  
Author(s):  
William G. Gray ◽  
Michael A. Celia
Keyword(s):  
Porous Media ◽  
Hydraulic Conductivity ◽  
Single Phase ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Hydraulic Gradient ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Α Phase ◽  
Flow Through

The mathematical study of flow in porous media is typically based on the 1856 empirical result of Henri Darcy. This result, known as Darcy’s law, states that the velocity of a single-phase flow through a porous medium is proportional to the hydraulic gradient. The publication of Darcy’s work has been referred to as “the birth of groundwater hydrology as a quantitative science” (Freeze and Cherry, 1979). Although Darcy’s original equation was found to be valid for slow, steady, one-dimensional, single-phase flow through a homogeneous and isotropic sand, it has been applied in the succeeding 140 years to complex transient flows that involve multiple phases in heterogeneous media. To attain this generality, a modification has been made to the original formula, such that the constant of proportionality between flow and hydraulic gradient is allowed to be a spatially varying function of the system properties. The extended version of Darcy’s law is expressed in the following form: qα=-Kα . Jα (2.1) where qα is the volumetric flow rate per unit area vector of the α-phase fluid, Kα is the hydraulic conductivity tensor of the α-phase and is a function of the viscosity and saturation of the α-phase and of the solid matrix, and Jα is the vector hydraulic gradient that drives the flow. The quantities Jα and Kα account for pressure and gravitational effects as well as the interactions that occur between adjacent phases. Although this generalization is occasionally criticized for its shortcomings, equation (2.1) is considered today to be a fundamental principle in analysis of porous media flows (e.g., McWhorter and Sunada, 1977). If, indeed, Darcy’s experimental result is the birth of quantitative hydrology, a need still remains to build quantitative analysis of porous media flow on a strong theoretical foundation. The problem of unsaturated flow of water has been attacked using experimental and theoretical tools since the early part of this century. Sposito (1986) attributes the beginnings of the study of soil water flow as a subdiscipline of physics to the fundamental work of Buckingham (1907), which uses a saturation-dependent hydraulic conductivity and a capillary potential for the hydraulic gradient.

scholarly journals On the Transition From Creeping to Inertial Flow in Arrays of Cylinders

2010 ◽  
Author(s):  
K. Yazdchi ◽  
S. Srivastava ◽  
S. Luding
Keyword(s):  
Porous Media ◽  
Darcy’S Law ◽  
Computational Study ◽  
Darcy's Law ◽  
Natural Processes ◽  
Inertial Flow ◽  
Flow Through ◽  
High Reynolds ◽  
An Array Of Cylinders ◽  
Element Method

Many important natural processes involving flow through porous media are characterized by large filtration velocity. Therefore, it is important to know when the transition from viscous to the inertial flow regime actually occurs in order to obtain accurate models for these processes. In this paper, a detailed computational study of laminar and inertial, incompressible, Newtonian fluid flow across an array of cylinders is presented. Due to the non-linear contribution of inertia to the transport of momentum at the pore scale, we observe a typical departure from Darcy’s law at sufficiently high Reynolds number (Re). Our numerical results show that the weak inertia correction to Darcy’s law is not a square or a cubic term in velocity, as it is in the Forchheimer equation. Best fitted functions for the macroscopic properties of porous media in terms of microstructure and porosity are derived and comparisons are made to the Ergun and Forchheimer relations to examine their relevance in the given porosity and Re range. The results from this study can be used for verification and validation of more advanced models for particle fluid interaction and for the coupling of the discrete element method (DEM) with finite element method (FEM).

On steady convection in a porous medium

1972 ◽  
Vol 54 (1) ◽  
pp. 153-161 ◽  
Author(s):  
Enok Palm ◽  
Jan Erik Weber ◽  
Oddmund Kvernvold
Keyword(s):  
Porous Medium ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Abrupt Change ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Sixth Order ◽  
Steady Convection ◽  
Good Agreement

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.

Darcy's Law for a Heterogeneous Porous Medium

2001 ◽  
Vol 4 (2) ◽  
pp. 14 ◽  
Author(s):  
F. D. Moura Neto ◽  
S. T. Melo
Keyword(s):  
Porous Medium ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Heterogeneous Porous Medium
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