Thu, 02 Feb 2012

16:00 - 17:00
DH 1st floor SR

On advancing contact lines with a 180-degree contact angle

Eugene Benilov
(Limerick)
Abstract

This work builds on the foundation laid by Benney & Timson (1980), who

examined the flow near a contact line and showed that, if the contact

angle is 180 degrees, the usual contact-line singularity does not arise.

Their local analysis, however, does not allow one to determine the

velocity of the contact line and their expression for the shape of the

free boundary involves undetermined constants - for which they have been

severely criticised by Ngan & Dussan V. (1984). As a result, the ideas

of Benny & Timson (1980) have been largely forgotten.

The present work shows that the criticism of Ngan & Dussan V. (1984)

was, in fact, unjust. We consider a two-dimensional steady Couette flow

with a free boundary, for which the local analysis of Benney & Timson

(1980) can be complemented by an analysis of the global flow (provided

the slope of the free boundary is small, so the lubrication

approximation can be used). We show that the undetermined constants in

the solution of Benney & Timson (1980) can all be fixed by matching

their local solution to the global one. The latter also determines the

contact line's velocity, which we compute among other characteristics of

the global flow.

Thu, 22 Jan 2009

16:30 - 17:30
DH 1st floor SR

On the drag-out problem in liquid film theory

Eugene Benilov
(Limerick)
Abstract

We consider an infinite plate being withdrawn from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin (1943) conjectured that the 'load', i.e. the thickness of the liquid film clinging to the plate, is determined by a certain formula involving the liquid's density and viscosity, the plate's velocity and inclination angle, and the acceleration due to gravity.

In the present work, Deryagin's formula is derived from the Stokes equations in the limit of small slope 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.

The results obtained are extended to order-one inclinantion angles and the case where surface tension is present.

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