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Stokes flow of an assemblage of porous particles: stress jump condition

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Abstract

The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (β), volume fraction (ε), and effective viscosity.

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Abbreviations

a :

Radius of the porous sphere

b :

Radius of fluid envelope

k :

Permeability of the porous sphere

V f :

Velocity inside fluid envelope

P f :

Pressure inside fluid envelope

V p :

Velocity inside porous sphere

P p :

Pressure inside porous sphere

p 0 :

Constant

U imp :

Imposed velocity at the cell boundary

U :

Magnitude of imposed velocity

V 0 :

Basic velocity

A f, A p :

Scalars

A imp, B :

Scalars

f n :

Modified spherical Bessel function of first kind

S n :

Spherical harmonics

\({P_{n}^{m}}\) :

Associated legendre polynomial

Da :

Darcy number

Da*:

Modified Darcy number

T f, T p :

Stresses in fluid and porous region

λ:

Dimensionless parameter

λ*:

Dimensionless parameter

δ :

Viscosity ratio

ε :

Volume fraction

ρ :

Density of the fluid

μ f :

Dynamic viscosity

μ eff :

Effective viscosity

β :

Stress jump coefficient

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721 302, India

    Jai Prakash & G. P. Raja Sekhar

  2. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Mirela Kohr

Authors
  1. Jai Prakash
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  2. G. P. Raja Sekhar
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  3. Mirela Kohr
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Correspondence to G. P. Raja Sekhar.

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Prakash, J., Raja Sekhar, G.P. & Kohr, M. Stokes flow of an assemblage of porous particles: stress jump condition. Z. Angew. Math. Phys. 62, 1027–1046 (2011). https://doi.org/10.1007/s00033-011-0123-6

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  • DOI: https://doi.org/10.1007/s00033-011-0123-6

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