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Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

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Summary

The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.

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Abbreviations

A :

cross-sectional area of duct

c :

mean value of random molecular speed

d :

diameter of tube

f p :

\({{h \cdot \frac{{dp}}{{dx}}} \mathord{\left/ {\vphantom {{h \cdot \frac{{dp}}{{dx}}} {\tfrac{1}{2}\rho u^2 }}} \right. \kern-\nulldelimiterspace} {\tfrac{1}{2}\rho u^2 }}\)

f t :

\({{\tfrac{1}{2}r_{\text{t}} \frac{{dp}}{{dx}}} \mathord{\left/ {\vphantom {{\tfrac{1}{2}r_{\text{t}} \frac{{dp}}{{dx}}} {\tfrac{1}{2}\rho u^2 }}} \right. \kern-\nulldelimiterspace} {\tfrac{1}{2}\rho u^2 }}\)

h :

half height of parallel plate channel

Kn :

Knudsen number

L :

molecular mean free path

n :

directional normal of solid boundary

p :

pressure

p 0 :

pressure at inlet

r :

radial co-ordinate

r t :

radius of tube

R :

non-dimensional radial co-ordinate

Re p :

4hU/ν

Re t :

2r t U/ν

s :

direction along solid boundary

T :

absolute temperature

u :

velocity in x direction

u*:

non-dimensional velocity

U :

bulk velocity = (1/A) ∫ A u dA

v :

velocity in y direction

x :

axial co-ordinate

x*:

stretched axial co-ordinate

X :

non-dimensional axial co-ordinate

X*:

non-dimensional stretched axial co-ordinate

Y :

non-dimensional channel co-ordinate

α :

eigenvalue in parallel plate channel

ε :

stretching factor

λ :

eigenvalue in tube

ρ :

density

ν :

kinematic viscosity

i :

index

p :

parallel plate

t :

tube

v :

velocity vector

∇ :

gradient operator

∇ 2 :

Laplacian operator

References

  1. Sparrow, E. M., T. S. Lundgren and S. H. Lin, Slip flow in the entrance region of a parallel plate channel. Proceedings of 1962 Heat Transfer and Fluid Mechanics Institute, Stanford Pg. 233.

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  3. Hanks, R. W., Phys. Fluids 6 (1963) 1645.

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  5. Patterson, G. N., Molecular flow of gases. Wiley, New York, 1956.

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  6. Kennard, E. H., Kinetic theory of gases. McGraw Hill, New York, 1938.

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  7. Lundgren, T. S., E. M. Sparrow and J. B. Starr, J. Basic Engg. 86 Series D (1964) 620.

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  8. Langhaar, H. L., J. Appl. Mech. 9 (1942) 55.

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  9. Dong, W. and L. A. Bromley, Vacuum flow of gases through channels with circular annular and rectangular cross-section. Trans 8th Vacuum Symp. Washington, Pergamon 1962.

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Authors and Affiliations

  1. Faculty of Technology, University of Manchester, England

    Alan Quarmby

Authors
  1. Alan Quarmby
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Quarmby, A. Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel. Appl. sci. Res. 15, 411–428 (1966). https://doi.org/10.1007/BF00411575

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  • DOI: https://doi.org/10.1007/BF00411575

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