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Numerical solution of the equations of motion for flow around objects in channels at low reynolds numbers

Summary

This paper describes the numerical integration of derived equations of motion for flow past an infinite set of parallel flat plates placed perpendicular to and between two infinite parallel planes. If the distance between the plates is greater than two channel depths, the problem can be reduced to two dimensions by assuming that the two major velocity components are parabolic in the channel depth direction. The solution of the derived equations has been compared with experimental data for Stokes flow and reasonable agreement was obtained. At Reynolds numbers greater than 10, the stability of the numerical method becomes increasingly troublesome and convergence of the numerical solution is very slow.

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Abbreviations

A, B, C, D :

parameters in (11), see text for definitions

h :

one-half the channel depth

n, j :

running indices for the finite difference network

P :

\(5\bar p/6\rho \bar u_0 ^2 \)

p :

pressure

R :

Reynolds number, 2ū 0 h/ν

U :

u/u 0

U i :

assumed value ofU for thei th iteration

Û i :

calculated value ofU for thei th iteration

U 0 :

U at the center line between plates

u m :

u at center line of channel atX → ∞

u 0 :

u m (1 −z2/h2), velocity atX → ∞

u, v, w :

velocities in thex, y andz directions, respectively

V :

υ/u 0

X :

x/h

Y :

y/h

Y 0 :

Y at the center line between plates

α :

constant, greater than zero and less than one

δ :

boundary-layer thickness, the distance in they-direction from the plate at whichU/U 0=0.95

ν :

kinematic viscosity

ρ :

density

References

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    Bruce, P. H., D. W. Peaceman and H. H. Rachford, Trans. AIME198, (1953) 79.

  3. 3)

    Hildebrand, F. B., Introduction to Numerical Analysis, McGraw-Hill, 1956.

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    Kuwabara, S., J. Phys. Soc. Japan13 (1958) 1516.

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    National Research Council Bulletin No. 84, p. 197, Dover Publications, 1956.

  6. 6)

    Schlichting, H., Boundary Layer Theory, McGraw-Hill, 1961.

  7. 7)

    Young, D. and L. Ehrlich, Boundary Problems in Differential Equations, University of Wisconsin Press, 1960.

  8. 8)

    Whitaker, S. and R. L. Pigford, Ind. Eng. Chem.52 (1960) 185.

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Author information

Correspondence to S. Whitaker.

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Whitaker, S., Wendel, M.M. Numerical solution of the equations of motion for flow around objects in channels at low reynolds numbers. Appl. sci. Res. 12, 91–104 (1964). https://doi.org/10.1007/BF03184753

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Keywords

  • Reynolds Number
  • Velocity Profile
  • Rectangular Channel
  • Stokes Flow
  • Channel Depth