TY - JOUR

T1 - On the drag-out problem in liquid film theory

AU - Benilov, Eugene S.

AU - Zubkov, V. S.

PY - 2008

Y1 - 2008

N2 - We consider an infinite plate being withdrawn (at an angle α to the horizontal, with a constant velocity U) from an infinite pool of viscous liquid.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/ρg sin α)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.

AB - We consider an infinite plate being withdrawn (at an angle α to the horizontal, with a constant velocity U) from an infinite pool of viscous liquid.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/ρg sin α)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.

UR - http://www.scopus.com/inward/record.url?scp=57149118686&partnerID=8YFLogxK

U2 - 10.1017/S002211200800431X

DO - 10.1017/S002211200800431X

M3 - Article

AN - SCOPUS:57149118686

SN - 0022-1120

VL - 617

SP - 283

EP - 299

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -