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The inertial draining of a thin fluid layer between parallel plates with a constant normal force. Part 1. Analytic solutions; inviscid and small-but finite-Reynolds-number limits

Published online by Cambridge University Press:  20 April 2006

S. Weinbaum ,
C. J. Lawrence  and
Y. Kuang
S. Weinbaum
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, New York, New York 10031
C. J. Lawrence
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, New York, New York 10031
Y. Kuang
Affiliation:
Department of Mechanical Engineering, The City College of The City University of New York, New York, New York 10031
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Abstract

The draining of thin fluid layers between rigid or deformable surfaces has been extensively studied in the limit of thin films where inertial effects are of negligible importance. In the present investigation, which is in two parts, we shall examine the inertial draining of a thin fluid layer between planar parallel surfaces under the action of a constant normal force. This is a simple model for dropping a sheet of paper or a book on a table or applying a piston to a microchip. The novelty of the problem is that we shall consider both the inertia of the object and that of the fluid for all Reynolds numbers where the flow remains laminar. In Part 1 of the study we shall derive a simplified Navier–Stokes equation for the general case which contains the dynamic equation for the motion of the object. Solutions will be presented for the time-dependent motion of the object and the intervening fluid in the gap for all ratios of object to fluid inertia in the limit of infinite Reynolds number and for small Reynolds numbers (Re < 10) in the limit where the time rate of change of momentum of the object is small compared with that of the fluid in the gap. In Part 2, we shall examine the limit where time-dependent boundary layers develop along the top and bottom surfaces in response to the time-varying core flow and also present a new exact Navier–Stokes solution for a time-dependent double-axisymmetric stagnation-point flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 156 , July 1985 , pp. 463 - 477
Copyright
© 1985 Cambridge University Press

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