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# The velocity and temperature distributions in a liquid film

Part I

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## Summary

The steady state velocity distribution in a liquid film is investigated analytically. The liquid is assumed to appear on a flat surface due to transpiration through a porous medium or the melting of a solid material. The liquid is considered to be introduced into the film at the solid-liquid interface, in a direction normal to the interface, such that mass is continually being added to the film along the longitudinal path of flow. The flow, which is assumed to be laminar, occurs under the influence of gravity. Evaporation and gaseous boundary layer effects are not considered. Physical properties of the liquid, such as density and viscosity are considered constant.

The continuity and momentum equations are set up in integral form and the film thickness and velocity distribution are determined by four different approaches. These four approaches differ from one another by the simplifying assumptions made. These assumptions are made in such a way that each successive case takes different effects into account, and yields a solution based on a more complete analysis.

A comparison of the solutions shows that where the liquid film thickness is relatively small, the results of all four cases converge to the same expression. This tends to justify the use of certain simplifying assumptions, even though they are of such a nature that one might not initially regard them reasonable.

This is the first of a series of two papers. In the second paper, the temperature distribution is determined for a film flowing under the influence of gravity; in addition, the velocity and temperature distributions are determined for a film which is subjected to an externally applied surface shear stress.

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## Abbreviations

a, b, c, d :

coefficients in velocity profile polynomial

A :

dimensionless parameter, $$A = \left( {\frac{{\partial f}}{{\partial \eta }}} \right)_{\eta = 0} = \left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0}$$

B :

dimensionless parameter, $$\int\limits_0^1 {f(\eta ,x)d\eta = \frac{1}{{U\delta }}} \int\limits_0^\delta {u{\text{d}}y}$$

C :

dimensionless parameter, $$\int\limits_0^1 {f^2 (\eta ,x)d\eta = \frac{1}{{U^2 \delta }}} \int\limits_0^\delta {u^2 {\text{d}}y}$$

E :

dimensionless parameter E=C/B 2

f(x, η):

dimensionless velocity profile function

F :

term defined in Appendix D

g :

gravitational acceleration component in x-direction

h :

heat transfer coefficient at film surface

k :

thermal conductivity of liquid

M :

dimensionless parameter, M=A/B

R :

dimensionless parameter, R=V 0 δ/ν

S :

dimensionless parameter, S=1/2(g/V 0 ν)(δ 3/x)

u :

liquid velocity in x-direction

U :

surface velocity of liquid in x-direction

V 0 :

velocity of fluid in the y-direction at y=0, the feed-in velocity

x, y :

space coordinates

α, β, γ :

terms defined in Appendix D

δ :

film thickness

η :

dimensionless space coordinate, η=y/δ

ν :

kinematic viscosity

ø :

term defined by (25)

ω :

term defined by (42)

LL, UL :

lower limit, upper limit

NS :

no-slope or zero-slope

0:

refers to position y=0

δ:

refers to position y=δ

## References

1. 1)

Jakob, M., Heat Transfer. Vol. I, pp. 658–696, John Wiley, New York, 1949.

2. 2)

McAdams, W., Heat Transmission, pp. 325–343, McGraw-Hill, New York, 1954.

3. 3)

Chen, M. M., Trans. of ASME, J. of Heat Transfer, 83 (1961) 46–60.

4. 4)

Koh, J. C. Y., ASME Paper No. 60-WA-253, 1960.

5. 5)

Koh, J. C. Y., E. M. Sparrow and J. P. Hartnett, Int. J. of Heat and Mass Transfer 2 (1961) 69.

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Rohsenow, W. M., Trans. of ASME 78 (1956) 1645.

7. 7)

Sparrow, E. M. and J. L. Gregg, Trans. of ASME, J. of Heat Transfer 81 (1959) 291.

8. 8)

Eckert, E. R. G., Transpiration and Film Cooling, Heat Transfer (a symposium), Engineering Research Institute, University of Michigan, Ann Arbor, Mich, 1953.

9. 9)

Eckert, E. R. G. and R. J. Drake, Jr., Heat and Mass Transfer, pp. 301–315, McGraw-Hill, New York, 1959.

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Adams, M. C., American Rocket Society Journal 29 (1959) 625.

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Jakob, M., Heat Transfer, Vol. II, pp. 355–370, 394–415, John Wiley, New York, 1957.

12. 12)

Dukler, A. E. and O. P. Bergelin, Chem. Eng. Progress 48 (1952) 557.

13. 13)

McAdams, W. H., T. B. Drew and G. S. Bays, Jr., Trans. of ASME 62 (1940) 627.

14. 14)

Schlichting, H., Boundary Layer Theory, pp. 229–232, Pergamon Press, New York, 1955.

15. 15)

Mouradian, E. M., Velocity and Temperature Distributions in a Liquid Film, Ph. D. Thesis, Northwestern University, Evanston, Ill., Sept. 1961.

16. 16)

Ibid., pp. 156.

17. 17)

Schlichting, H., Op. cit. pp. 102–103, 116–117, 201–205.

18. 18)

Kamke, E., Differential Gleichungen, Losungmethoden und Losungen, Vol. I, pp. 24–25, Chelsea, New York, 1959.

19. 19)

Lu, C. S., The Velocity Distribution in a Liquid Film, M. S. Thesis, Georgia Institute of Technology, Atlanta, Ga., 1962.

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Mouradian, E.M., Sunderland, J.E. The velocity and temperature distributions in a liquid film. Appl. Sci. Res. 14, 431–452 (1965). https://doi.org/10.1007/BF00382265