Eshelbian force on a steadily moving liquid blister

This paper describes the dynamics of a liquid blister forced to advance between a thin elastic sheet and a rigid substrate, by the dual action of a piston and a flat frictionless sleeve at the receding end. Compared to the removal of a viscous blister by sliding a frictionless blade (Wang and Detournay, 2021), the present problem is a steady-state one due to the absence of bleeding at the back end. We seek to obtain a travelling-wave solution, in particular the dependence of the external driving force on the blister velocity and other parameters characterizing this problem, such as the fluid viscosity, the elastic properties of the sheet, and the interface toughness. The peculiarity of this problem lays in the Eshelbian (rather than Newtonian) nature of the horizontal driving force applied by the sleeve on the elastic sheet. The Eshelbian nature of this horizontal force is then discussed and alternative expressions of this force are derived from both variational and energy balance considerations. Scaling of the governing equations indicates that the solution depends on three numbers: namely, dimensionless toughness K, residual gap W and length γ_f of the fluid-filled part of the blister, a proxy for the volume of the fluid. We use the method of matched asymptotic expansions to predict the horizontal force on a long blister in both the viscosity- and toughness-dominated asymptotic regimes in the back end boundary layer. The numerical solution of the finite blister is then compared with the asymptotic solutions. The key result concerns the dependence of the scaled horizontal force on the three numbers controlling the solution of the moving liquid blister.