Nonlinear Gravity Waves in a Thin Sheet of Viscous Fluid

摘要

A nonlinear theory of long gravity waves is developed for a highly viscous fluid of small depth. The expansion scheme of Lin and Clark for inviscid shallow waters is used, and discussions are then made for three different cases: a = O(E), O(E2), and O(E3), where a is the dimension less amplitude and E is the dimensionless depth. In the first case a new partial differential equation is obtained which involves a nonlinear diffusion term. In the second case the governing equation is shown to be of Burgers' type. In all three cases permanent waves are treated explicitly. A variety of wave forms is found in the third case when a = O(E3): monoclinal and polyclinal waves over an inclined bottom, as well as solitary and cnoidal waves on a vertical wall. Surface tension is not considered. (Author)