An asymptotic analysis of small holes in thin fluid layers

Abstract

In this paper we obtain the description of axisymmetric equilibrium holes in thin fluid layers lying on a horizontal substrate under the influence of surface tension and gravity effects in the asymptotic limit when the radius of the hole is small. For values of the contact angle between the fluid and the substrate not equal to π we demonstrate that James' (J. Fluid Mech. 63, 657–664 (1974)) solution for the meniscus surrounding a narrow cylindrical rod dipped into a bath of fluid also provides the correct asymptotic solution to the present problem. In the case when the contact angle is equal to π we obtain the asymptotic solution for the first time. In both cases we obtain asymptotic expressions for the radius of the hole at the substrate and the thickness of the layer far from the hole. The correctness of these expressions is confirmed by comparison with numerical solutions to the full problem. In the light of the present study we are able to highlight shortcomings in previous studies and, in particular, show that their predictions for the thickness of the layer are correct only at leading order in the limit of small holes.