Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries

Abstract

Motivated by models for thin films coating cylinders in two physical cases proposed in Kerchman (J Fluid Mech 290:131–166, 1994) and Kerchman and Frenkel (Theor Comput Fluid Dyn 6:235–254, 1994), we analyze the dynamics of corresponding thin-film models. The models are governed by nonlinear, fourth-order, degenerate, parabolic PDEs. We prove, given positive and suitably regular initial data, the existence of weak solutions in all length scales of the cylinder, where all solutions are only local in time. We also prove that given a length constraint on the cylinder, long time and global in time weak solutions exist. This analytical result is motivated by numerical work on related models of Reed Ogrosky (Modeling liquid film flow inside a vertical tube, Ph.D. thesis, The University of North Carolina at Chapel Hill, 2013) in conjunction with the works (Camassa et al. in Phys Rev E 86(6):066305, 2012; Physica D Nonlinear Phenom 333:254–265, 2016; J Fluid Mech 745:682–715, 2014; J Fluid Mech 825:1056–1090, 2017).