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Arbitrary oscillatory Stokes flow past a porous sphere using Brinkman model

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Abstract

The present paper deals with the hydrodynamics of a porous sphere placed in an arbitrary oscillatory Stokes flow. Unsteady Stokes equation is used for the flow outside the porous sphere and Brinkman equation is used for the flow inside the porous sphere. Corresponding Faxén’s law for drag and torque is derived and compared with few existing results in some special cases. Examples like uniform flow, oscillatory shear flow and oscillating Stokeslet are discussed. Also, translational oscillation of a weakly permeable sphere is discussed.

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Abbreviations

a :

radius of the porous sphere [m]

k :

permeability of the porous sphere [m2]

r :

radial distance

v e :

oscillatory velocity external to the porous sphere [m/s]

p e :

oscillatory pressure external to the porous sphere [N/m2]

V e :

amplitude of the oscillatory velocity external to the porous sphere [m/s]

P e :

amplitude of the oscillatory pressure external to the porous sphere [N/m2]

V i :

velocity internal to the porous sphere [m/s]

P i :

pressure internal to the porous sphere [N/m2]

p 0 :

constant [N/m2]

U :

magnitude of the far field uniform velocity [m/s]

Da :

Darcy number

V 0 :

basic velocity [m/s]

V ∗ :

velocity due to the disturbance [m/s]

A, B :

scalars

f n :

modified spherical Bessel function of first kind

g n :

modified spherical Bessel function of second kind

S n , T n :

spherical harmonics

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

associated Legendre polynomial

\(|V_{\theta}^{i}|\) :

magnitude of the internal tangential velocity

θ :

inclination

φ :

azimuth angle

α :

slip coefficient

λ :

dimensionless parameter

ω :

frequency of oscillation [s−1]

ϖ :

dimensionless frequency of oscillation

ρ :

density of the fluid [kg/m3]

μ :

dynamic viscosity [kg/m−1/s−1]

ν :

kinematic viscosity [m2/s]

e :

external to the porous sphere

i :

internal to the porous sphere

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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

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

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Dedicated to Professor S.D. Nigam.

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Prakash, J., Raja Sekhar, G.P. Arbitrary oscillatory Stokes flow past a porous sphere using Brinkman model. Meccanica 47, 1079–1095 (2012). https://doi.org/10.1007/s11012-011-9494-1

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  • DOI: https://doi.org/10.1007/s11012-011-9494-1

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