This paper investigates the motion of a liquid blister, trapped between an elastic sheet and a rigid substrate. The blister is driven by a frictionless blade moving at a constant velocity, forcing a constant gap that causes fluid to bleed from the blister. The sheet adheres to the substrate ahead of the blister. The main goal of the study is to assess the magnitude and orientation of the force applied by the blade on the moving blister. The solution is constructed for the asymptotic case of a long blister. Thanks to a separation of scales, the asymptotic solution is obtained by matching the boundary layers at the front end and at the back end of the blister to an outer solution characterised by a uniform pressure in the bulk. Both boundary layers are formulated as travelling-wave equations for the gap between the sheet and the substrate. The formulation accounts for a moving fluid front, distinct from the separation edge, and for a tail with a gap tending to an a priori unknown value far behind the blister. Scaling of the governing equations indicates that the solution depends on two numbers: a dimensionless toughness and a scaled gap imposed by the moving blade. The key result concerns the dependence of the scaled force on the two numbers controlling the solution of the moving liquid blister. There are two asymptotic solutions: for small gaps at the blade, the force on the blade is dominated by viscous dissipation at the back end and only depends on aperture; for large gaps, the horizontal force only depends on toughness, a function of both fluid viscosity and energy of separation at the front end, whereas the vertical force depends on both and.
Bibliographical noteFunding Information:
Z.W. would like to acknowledge support from the China Scholarship Council and also from the MTS visiting Professorship in Geomechanics, which enabled the current research to be completed during visits to the University of Minnesota. Partial support of E.D. was provided by the T.W. Bennett Chair in Mining Engineering and Rock Mechanics. The authors are grateful for the critical comments and insight provided by three anonymous reviewers. They are particularly indebted to one reviewer who identified a conceptual mistake in the original manuscript, thus saving them from future embarrassment.
© 2021 Cambridge University Press. All rights reserved.
- boundary-layer structure
- lubrication theory