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Transport in Porous Media

, Volume 114, Issue 1, pp 49–64 | Cite as

A Semi-analytical Solution for Start-Up Flow in an Annulus Partially Filled with Porous Material

  • Basant K. Jha
  • J. O. OdengleEmail author
Article

Abstract

This study investigates the start-up flow in a horizontal annulus partially filled with clear fluid and partially with a fluid-saturated porous material (composite channel) as a result of the sudden application of constant pressure gradient in horizontal direction. Momentum transfer in the porous medium is simulated using the Brinkman-extended Darcy model. The fluid and porous regions are linked together by equating their velocities and incorporating the shear stress jump conditions at the interface. With the application of Laplace transform technique, the governing partial differential equations are transformed into ordinary differential equations which are solved in the Laplace domain to obtain exact solutions. Inversion of the exact solution in Laplace domain to time domain is done using the Riemann-sum approximation approach. Validation of the Riemann-sum approximation method is achieved by comparing numerical values obtained with those of exact solution obtained for steady state flow and transient solution obtained by implicit finite difference method. At large values of time, transient solution obtained using Riemann-sum approximation approach and implicit finite difference method coincide with closed form solution obtain exactly for steady state flow showing excellent agreement between these methods. Line graphs are used to discuss the effect of the different flow parameters involved in the flow formation.

Keywords

Riemann-sum approximation Annulus Composite channel Start-up flow 

List of symbols

a

Radius of outer cylinder

b

Radius of inner cylinder

d

Non-dimensional thickness of clear fluid

\({d}'\)

Dimensional thickness of clear fluid

Da

Darcy number

\(I_0\)

Zeroth-order modified Bessel function of the first kind

\(I_1\)

First-order modified Bessel function of the first kind

k

Permeability of the porous medium

\(K_0\)

Zeroth-order modified Bessel function of the second kind

\(K_1\)

First-order modified Bessel function of the second kind

\({r}'\)

Dimensional radial coordinate

R

Non-dimensional radial coordinate

t

Time in non-dimensional form

\({t}'\)

Time in dimensional form

u

Fluid velocity in non-dimensional form

\({u}'\)

Fluid velocity in dimensional form

\(U_0\)

Reference velocity

Greek symbols

\(\beta \)

Adjustable coefficient in the stress jump condition

\(\gamma \)

Ratio of viscosity

\(\lambda \)

Radius ratio

\(\nu _\mathrm{eff}\)

Effective kinematic viscosity of the porous medium

\(\nu \)

Kinematic viscosity of the fluid

\(\rho \)

Density

Subscripts

f

Fluid region

i

Interface between clear fluid and porous region

p

Porous region

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsAhmadu Bello UniversityZariaNigeria
  2. 2.Institute of Computing & ICTAhmadu Bello UniversityZariaNigeria

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