Applied Scientific Research

, Volume 57, Issue 2, pp 133–149 | Cite as

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

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Abstract

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.

Key words

yield stress Casson fluid porous medium 

Nomenclature

h

step length in radial direction

I0

modified Bessel function of first kind order zero

Il

modified Bessel function of first kind of order one

k(r)

permeability function =\(\bar k(\bar r)/R^2 = k_0 f(r)\)

k0

permeability factor

f

radial variation of permeability (=1 or (1−r)/r)

N

number of sub-interval in radial direction

p

dimensionless pressure

P0

characteristic pressure gradient

Q

dimensionless flow rate

R

radius of the tube

r

dimensionless radial distance =\(\bar r/R\)

rj

jth radial nodal point

rp

dimensionless plug flow radius

u

dimensionless axial velocity =\(\bar u/u_c \)

uc

characteristic velocity = −R 2 P 0/2μ∞

up

dimensionless plug flow velocity

Greek symbols

ɛ1

error of tolerance for τ

ɛ2

error of tolerance forr p

μ∞

Newtonian viscosity (viscosity at high shear rate)

τ

dimensionless shear stress = τ/τɛc

τc

characteristic shear stress = μ∞ u c /R

τy

dimensionless yield stress

Subscripts

+

shear flow region,r p ≤r≤1

−

plug flow region, 0≤r≤r p

Overscore

−

‘overbar’; represents all dimensional quantities

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Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyNew DelhiIndia

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