scholarly journals The Human Meniscus Behaves as a Functionally Graded Fractional Porous Medium under Confined Compression Conditions

2021 ◽  
Vol 11 (20) ◽  
pp. 9405
Author(s):  
Raphaël Bulle ◽  
Gioacchino Alotta ◽  
Gregorio Marchiori ◽  
Matteo Berni ◽  
Nicola F. Lopomo ◽  
...  
Keyword(s):  
Porous Medium ◽  
Darcy’S Law ◽  
Time Derivative ◽  
Applied Pressure ◽  
Darcy's Law ◽  
Functionally Graded ◽  
Compression Tests ◽  
Confined Compression ◽  
Fluid Flux ◽  
Poroelastic Model

In this study, we observe that the poromechanical parameters in human meniscus vary spatially throughout the tissue. The response is anisotropic and the porosity is functionally graded. To draw these conclusions, we measured the anisotropic permeability and the “aggregate modulus” of the tissue, i.e., the stiffness of the material at equilibrium, after the interstitial fluid has ceased flowing. We estimated those parameters within the central portion of the meniscus in three directions (i.e., vertical, radial and circumferential) by fitting an enhanced model on stress relation confined compression tests. We noticed that a classical biphasic model was not sufficient to reproduce the observed experimental behaviour. We propose a poroelastic model based on the assumption that the fluid flow inside the human meniscus is described by a fractional porous medium equation analogous to Darcy’s law, which involves fractional operators. The fluid flux is then time-dependent for a constant applied pressure gradient (in contrast with the classical Darcy’s law, which describes a time independent fluid flux relation). We show that a fractional poroelastic model is well-suited to describe the flow within the meniscus and to identify the associated parameters (i.e., the order of the time derivative and the permeability). The results indicate that mean values of λβ,β in the central body are λβ=5.5443×10−10m4Ns1−β, β=0.0434, while, in the posterior and anterior regions, are λβ=2.851×10−10m4Ns1−β, β=0.0326 and λβ=1.2636×10−10m4Ns1−β, β=0.0232, respectively. Furthermore, numerical simulations show that the fluid flux diffusion is facilitated in the central part of the meniscus and hindered in the posterior and anterior regions.

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

Gas Separation Using a Membrane

2014 ◽  
Author(s):  
Nawaf Alkhamis ◽  
Ali Anqi ◽  
Dennis E. Oztekin ◽  
Abdulmohsen Alsaiari ◽  
Alparslan Oztekin
Keyword(s):  
Porous Medium ◽  
Natural Gas ◽  
Gas Separation ◽  
Stokes Equations ◽  
Darcy’S Law ◽  
Circular Cross Section ◽  
Darcy's Law ◽  
Mass Separation ◽  
Functional Surface ◽  
Wide Range

Gas-gas separation, to purify natural gas, is simulated using a membrane supported by a porous medium. Removing acidic gasses from the natural gas is gaining attention recently. Computational fluid dynamics simulations are conducted for asymmetric multi-component fluid flows in a channel. The flow system consists of a circular cross-section channel bounded by a porous layer which supports the membrane wall. The Navier-Stokes equations model the flow in the channel, while the flow in the porous medium is modeled by both the Darcy’s law and the extended Darcy’s law. Mass transport equations, including mass diffusion of mixtures of two gasses (CO2 and CH4), are employed to determine the concentration distribution. The membrane will be modeled as a functional surface; where the flux of each component will be determined based on the local partial pressure of each species, composition, and permeability and selectivity of the membrane. The effect of the porous medium on the membrane performance will be determined for a wide range of Reynolds number. The performance of the system will be measured by maximum mass separation with minimal frictional losses.

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 The Structure of Filter Curve and Methods of Its Approximation. Part 1. Darcy’s Law and the Lower Limit of Its Application. Filtration of Liquids in Microporous Media

2021 ◽  
Vol 27 (1) ◽  
pp. 086-094
Author(s):  
N. A. Merentsov ◽  
◽  
V. A. Balashov ◽  
A. B. Golovanchikov ◽  
M. V. Topilin ◽  
...  
Keyword(s):  
Porous Medium ◽  
Boundary Layers ◽  
Lower Limit ◽  
Electric Potential ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Low Permeability ◽  
Filtration Flow ◽  
Potential Gradients ◽  
Microporous Media

The description of the lower limit of the application of Darcy’s law is given, which is due to the influence of a number of anomalous factors that arise during the filtration flow of liquids through low-permeability finely dispersed media. The influence of such factors as the action of the forces of intermolecular interaction is considered; boundary layers and surface wettability; concentration and electric potential gradients; the presence of impurities in the liquid; gas saturation and vaporization; changes in the structure of the porous medium, separately or in the aggregate, leading to a violation of Darcy’s law.

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.

scholarly journals Darcy’s law for yield stress fluid flowing through a porous medium

2013 ◽  
Vol 195 ◽  
pp. 57-66 ◽  
Author(s):  
T. Chevalier ◽  
C. Chevalier ◽  
X. Clain ◽  
J.C. Dupla ◽  
J. Canou ◽  
...  
Keyword(s):  
Porous Medium ◽  
Yield Stress ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Yield Stress Fluid

A Unified Theory for Stable and Unstable Miscible Displacement

1963 ◽  
Vol 3 (03) ◽  
pp. 205-213 ◽  
Author(s):  
R.L. Perrine
Keyword(s):  
Porous Medium ◽  
Turbulent Flow ◽  
Darcy’S Law ◽  
Flow Behavior ◽  
Viscous Fingering ◽  
Darcy's Law ◽  
Miscible Displacement ◽  
Fluid Viscosity ◽  
Flow Conditions ◽  
Unstable Displacement

Introduction Many experimental studies of miscible displacement in porous media have been conducted with prediction of reservoir displacement efficiency as the ultimate objective. Most such studies have utilized lower displacing than displaced fluid viscosity, scaled to potential reservoir fluid pairs. Theoretical approaches have been largely limited to unit viscosity ratios, however, in spite of the necessity for an understanding of the mechanism of the unstable, adverse viscosity ratio displacement process. An obvious reason is the difficulty of describing in mathematical form the viscous fingering characteristic of these conditions. Observation of experiments conducted with dyed fluids in transparent systems suggests an analogy between unstable miscible displacement and turbulent flow in a pipe. In both cases there are fluctuations around a simpler, mean flow behavior. An important difference is that flow behavior of interest in the porous medium is entirely transient, contained within a transition zone between displacing and displaced fluids. Transient behavior complicates description in that coefficients in the equations become variables rather than constants. In the study reported here, the analogy with turbulent flow has been used in creating unstable miscible displacement as a quasi-turbulent displacement process. The purpose has been to derive, even if restricted to an idealized conceptual model, a unified theoretical relationship applicable to both stable and unstable displacement. A relationship meeting these specifications up to moderately adverse viscosity ratios, such as 17:1, is presented. One fluctuation parameter in the theory and dispersion coefficients are obtained by empirical means. The idealized theory is compared with experimental results. UNSTABLE MISCIBLE DISPLACEMENT AS QUASI-TURBULENT DARCY FLOW The miscible displacement of one fluid by another within a porous medium is considered in this study. Flow conditions are such that Darcy's law is applicable. It is further assumed that, by the stability criterion proposed by Perrine, initial flow conditions lie well within the regime of instability. Thus substantial viscous fingering is assured.We wish to show how this flow regime can be represented as quasi-turbulent. That is, the Reynolds number for the established flow conditions is below that for inertial or turbulent flow within a porous medium, and lies within the regime for which Darcy's law is valid. Yet flow behavior can be described as the combination of some relatively simple average result, and characteristic fluctuations that are superimposed on the simpler behavior. Stated another way, flow behavior includes the movement of layers of fluids with different velocities past or over one another. Such descriptions are characteristic of turbulent pipe flow. The fluctuations in the present case are a direct consequence of the local viscous fingering which accompanies the unstable displacement process. Should displacement become stable, fluctuations would die out. It is of particular importance to note that fluctuations such as these can interact in a way that contributes to material transport by the basic flow. The basic transport equations required for engineering calculations must be modified to reflect this fact. SPEJ P. 205^

Natural convection in porous media

1985 ◽  
Vol 150 ◽  
pp. 89-119 ◽  
Author(s):  
V. Prasad ◽  
F. A. Kulacki ◽  
M. Keyhani
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Porous Medium ◽  
Rayleigh Number ◽  
Effective Thermal Conductivity ◽  
Darcy’S Law ◽  
Saturated Porous Medium ◽  
Darcy's Law ◽  
Experimental Results ◽  
Rayleigh Numbers

Experimental results on free convection in a vertical annulus filled with a saturated porous medium are reported for height-to-gap ratios of 1.46, 1 and 0.545, and radius ratio of 5.338. In these experiments, the inner and outer walls are maintained at constant temperatures. The use of several fluid–solid combinations indicates a divergence in the Nusselt-number–Rayleigh-number relation, as also reported by previous investigators for horizontal layers and vertical cavities. The reason for this divergence is the use of the stagnant thermal conductivity of the fluid-filled solid matrix. A simple model is presented to obtain an effective thermal conductivity as a function of the convective state, and thereby eliminate the aforementioned divergence. A reasonable agreement between experimentally and theoretically determined Nusselt numbers is then achieved for the present and previous experimental results. It is thus concluded that a unique relationship exists between the Nusselt and Rayleigh numbers unless Darcy's law is inapplicable. The factors that influence the breakdown of Darcian behaviour are characterized and their effects on heat-transfer rates are explained. It is observed that, once the relation between the Nusselt and Rayleigh numbers branches out from that obtained via the mathematical formulation based on Darcy's law, its slope approaches that for a fluid-filled enclosure of the same geometry when the Rayleigh number is large enough. An iterative scheme is also presented for estimation of effective thermal conductivity of a saturated porous medium by using the existing results for overall heat transfer.

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