Advertisement

Applied Scientific Research

, Volume 32, Issue 2, pp 149–166 | Cite as

Squeezing flows of Newtonian liquid films an analysis including fluid inertia

  • R. J. Grimm
Article

Abstract

A theoretical study is made of the flow behavior of thin Newtonian liquid films being “squeezed” between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations.

Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.

Keywords

Reynolds Number Aspect Ratio Numerical Calculation Flow Behavior Liquid Film 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

a

half of the distance, or gap, between the two plates

a0

the value of a at time t=0

adot

da/dt

ä

d2a/dt2

\(\dddot a\)

d3a/dt3

ai

components of a contravariant acceleration vector

f

unknown function of z0 and t defined in (6)

fi

function defined in (9) f1=r0g(z0, t) f2=θ0f3=f(z0, t)

F

force applied to the plates

g

unknown function of z0 and t defined in (6)

g′

∂g/∂z 0

h

grid dimension in the z0 direction (see Fig. 5)

\(\left\{ \begin{gathered} i \hfill \\ jk \hfill \\ \end{gathered} \right\}\)

Christoffel symbol

i, j, k, l

indices

k

grid dimension in the t direction (see Fig. 5)

r

radial coordinate direction defined in Fig. 1

r0

radial convected coordinate

R

radius of the circular plates

t

time

vr

fluid velocity in the r direction

vz

fluid velocity in the z direction

vθ

fluid velocity in the θ direction

xi

cylindrical coordinate x1=r x2=θ x3=z

z

vertical coordinate direction defined in Fig. 1

z0

vertical convected coordinate

θ

tangential coordinate direction

θ0

tangential convected coordinate

μ

viscosity

ν

kinematic viscosity, μ/ρ

ξi

convected coordinate ξ1=r0 ξ20 ξ3=z0

ρ

density

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Kauzlarich, J. J., ASLE Trans. 15 (1972) 37.Google Scholar
  2. [2]
    Fontana, E. H., American Ceramic Society Bulletin 49 (1970) 594.Google Scholar
  3. [3]
    Scott, J. R., Trans. I.R.I. 29 (1953) 175.Google Scholar
  4. [4]
    Leider, P. J. and R. B. Bird, I&EC Fund. 13 (1974) 336; also Leider, P. J., I&EC Fund. 13 (1974) 342.Google Scholar
  5. [5]
    Brindley, G., J. M. Davies, and K. Walters, Elastico-Viscous Squeeze Films Part I, to be published in J.N.N.F.M.Google Scholar
  6. [6]
    Ishizawa, S., Bulletin of JSME 9 (1966) 533.Google Scholar
  7. [7]
    Stefan, J., Akad. Wiss. Math. Natur., Wien 69 (1874) 713.Google Scholar
  8. [8]
    Cameron, A., The Principles of Lubrication, Longmans, Green and Co., London, 1966, pp. 389–392.Google Scholar
  9. [9]
    Kuzma, D. C., Appl. Sci. Res. 18 (1967) 15.Google Scholar
  10. [10]
    Van Dyke, M., Perturbation Methods in Fluid Mechanics, Academic Press, London and New York, 1964, 35.Google Scholar
  11. [11]
    Terrill, R. M., J. Lubric. Tech. 91 (1969) 126.Google Scholar
  12. [12]
    Jones, A. F. and S. D. R. Wilson, J. Lubric. Tech. 97 (1975) 101.Google Scholar
  13. [13]
    Tichy, J. A. and W. O. Winer, J. Lubric. Tech. 92 (1970) 588.Google Scholar
  14. [14]
    Kramer, J. M., Appl. Sci. Res. 30 (1974) 1.Google Scholar
  15. [15]
    Lodge, A. S., Body Tensor Fields in Continuum Mechanics, Academic Press, London and New York, 1974.Google Scholar
  16. [16]
    Gill, S., Proc. Cambridge Phil. Soc. 47 (1951) 96.Google Scholar
  17. [17]
    Greenspan, D., Discrete Numerical Methods in Physics and Engineering, Academic Press, London and New York, 1974, pp. 32–43.Google Scholar

Copyright information

© Martinus Nijhoff 1976

Authors and Affiliations

  • R. J. Grimm
    • 1
  1. 1.Dept. of Chem. Eng. and Rheology Res. CenterUniv. of WisconsinMadisonUSA

Personalised recommendations