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Laminar source flow between parallel porous disks with equal suction and injection

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Abstract

An analysis is presented for laminar source flow between parallel stationary porous disks with suction at one of the disks and equal injection at the other. The solution is in the form of an infinite series expansion about the solution at infinite radius, and is valid for all suction and injection rates. Expressions for the velocity, pressure, and shear stress are presented and the effect of the cross flow is discussed.

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Abbreviations

a :

distance between disks

A, B, ..., J :

functions of R w only

F:

static pressure

p :

dimensionless static pressure, p(a 2/ρν 2)

Q :

volumetric flow rate of the source

r:

radial coordinate

r :

dimensionless radial coordinate, r/a

R:

radial coordinate of a point in the flow region

R :

dimensionless radial coordinate of a point in the flow region, R

Re :

source Reynolds number, Q/2πνa

R w :

wall Reynolds number, Va/ν

\(\dot Re\) :

reduced Reynolds number, Re/r 2

\(\dot Re_c\) :

critical Reynolds number

ū :

velocity component in radial direction

u :

dimensionless velocity component in radial direction, ūa/ν

〈ū〉:

average radial velocity, Q/2πa

〈u〉:

dimensionless average radial velocity, Re/r

\(\mathop u\limits^*\) :

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

\(\bar v\) :

velocity component in axial direction

v :

dimensionless velocity component in axial direction, v ν

V :

magnitude of suction or injection velocity

z :

axial coordinate

z :

dimensionless axial coordinate, z a

μ :

viscosity

ρ :

density

ν :

kinematic viscosity, μ/ρ

\(\bar \tau 0\) :

shear stress at lower disk

\(\bar \tau 1\) :

shear stress at upper disk

τ 0 :

dimensionless shear stress at lower disk, \(\frac{{\bar \tau 0}}{{\left( {\mu \left\langle {\bar u} \right\rangle /a} \right)}}\)

τ 1 :

dimensionless shear stress at upper disk, \(\frac{{\bar \tau 1}}{{\left( {\mu \left\langle {\bar u} \right\rangle /a} \right)}}\)

ψ :

dimensionless stream function

References

  1. Peube, J. L., J. de Mécanique, 2 (1963) 377.

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  2. Savage, S. B., Trans. ASME J. of Appl. Mechanics, 31 (1964) 594.

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  3. Elkouh, A. F., Appl. Sci. Res. 21 (1969) 284.

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  4. Khan, Mohd. Abdul Aleem, J. de Mécanique, 7 (1968) 575.

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Authors and Affiliations

  1. Mechanical Engineering Dept., Marquette University, Milwaukee, Wisconsin, U.S.A.

    A. F. Elkouh

Authors
  1. A. F. Elkouh
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Elkouh, A.F. Laminar source flow between parallel porous disks with equal suction and injection. Appl. Sci. Res. 23, 431–445 (1971). https://doi.org/10.1007/BF00413217

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  • DOI: https://doi.org/10.1007/BF00413217

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