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Laminar source flow between parallel porous disks

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Abstract

An analysis is presented for laminar source flow between infinite parallel porous disks. The solution is in the form of a perturbation from the creeping flow solution. Expressions for the velocity, pressure, and shear stress are obtained and compared with the results based on the assumption of creeping flow.

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Abbreviations

a :

half distance between disks

\(\bar r\) :

radial coordinate

r :

dimensionless radial coordinate, \(\bar r\)/a

\(\bar z\) :

axial coordinate

z :

dimensionless axial coordinate, \(\bar z\)/a

\(\bar R\) :

radial coordinate of a point in the flow

R :

dimensionless radial coordinate of a point in the flow, \(\bar R\)/a

ū :

velocity component in radial direction

u :

=ūa/ν, dimensionless velocity component in radial direction

\(\bar v\) :

velocity component in axial direction

v :

=\(\bar v\)a/ν}, dimensionless velocity component in axial direction

\(\bar p\) :

static pressure

p :

=\(\bar p\)(a 2/ρν 2), dimensionless static pressure

\(\begin{array}{*{20}c} {\text{*}} \\ p \\ \end{array}\) :

=p(r, z)−p(R, z), dimensionless pressure drop

V :

magnitude of suction or injection velocity

Q :

volumetric flow rate of the source

Re :

source Reynolds number, Q/4πνa

\(\begin{array}{*{20}c} {\text{*}} \\ {Re} \\ \end{array}\) :

reduced Reynolds number, Re/r 2

\(\begin{array}{*{20}c} {\text{*}} \\ {Re_{\text{c}} } \\ \end{array}\) :

critical Reynolds number

R w :

wall Reynolds number, Va/ν

μ :

viscosity

ρ :

density

ν :

=μ/ρ, kinematic viscosity

\(\bar \tau _0\) :

shear stress at upper disk

τ 0 :

=\(\bar \tau _0\)(a 2/ρν 2), dimensionless shear stress at upper disk

\(\begin{array}{*{20}c} {\text{*}} \\ {\tau _{\text{0}} } \\ \end{array}\) :

shear stress ratio, τ 0/(τ 0)inertialess

〈u〉:

=\(\int\limits_0^1 {u{\text{ d}}z}\), dimensionless average radial velocity

\(\begin{array}{*{20}c} {\text{*}} \\ u \\ \end{array}\) :

u/〈u〉, ratio of radial velocity to average radial velocity

ψ :

dimensionless stream function

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Authors and Affiliations

  1. Mechanical Engineering Dept., Marquette University, Milwaukee, Wisc., USA

    A. F. Elkouh

Authors
  1. A. F. Elkouh
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Elkouh, A.F. Laminar source flow between parallel porous disks. Appl. sci. Res. 21, 284–302 (1969). https://doi.org/10.1007/BF00411613

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  • DOI: https://doi.org/10.1007/BF00411613

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