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On the drag-out problem in liquid film theory

Published online by Cambridge University Press:  25 December 2008

E. S. BENILOV  and
V. S. ZUBKOV
E. S. BENILOV*
Affiliation:
Department of Mathematics, University of Limerick, Ireland
V. S. ZUBKOV
Affiliation:
Department of Mathematics, University of Limerick, Ireland
*
†Email address for correspondence: eugene.benilov@ul.ie
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Abstract

We consider an infinite plate being withdrawn (at an angle α to the horizontal, with a constant velocity U) from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin (C. R. Dokl. Acad. Sci. URSS, vol. 39, 1943, p. 13.) conjectured that the ‘load’ l, i.e. the thickness of the liquid film clinging to the plate, is l=(μU/ρgsinα)1/2, where ρ and μ are the liquid's density and viscosity, and g is the acceleration due to gravity.

In the present work, the above formula is derived from the Stokes equations in the limit of small slopes of the plate (without this assumption, the formula is invalid). It is shown that the problem has infinitely many steady solutions, all of which are stable – but only one of these corresponds to Derjaguin's formula. This particular steady solution can only be singled out by matching it to a self-similar solution describing the non-steady part of the film between the pool and the film's ‘tip’.

Even though the near-pool region where the steady state has been established expands with time, the upper, non-steady part of the film (with its thickness decreasing towards the tip) expands faster and, thus, occupies a larger portion of the plate. As a result, the mean thickness of the film is 1.5 times smaller than the load.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 617 , 25 December 2008 , pp. 283 - 299
Copyright
Copyright © Cambridge University Press 2008

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References

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