Mathematical Justification of a Shallow Water Model

The shallow water equations are widely used to model the flow of a thin layer of fluid submitted to gravity forces. They are usually formally derived from the full incompressible Navier-Stokes equations with free surface under the modeling hypothesis that the pressure is hydrostatic, the flow is laminar, gradually varied and the characteristic fluid height is small relative to the characteristics flow length. This paper deals with the mathematical justification of such asymptotic process assuming a non zero surface tension coefficient and some constraints on the data. We also discuss relation between lubrication models and shallow water systems with no surface tension coefficient necessity.

Keywords

Navier-Stokes, shallow water, lubrication models, thin domain, free surface, asymptotic analysis, Sobolev spaces

2010 Mathematics Subject Classification

35Q30, 35R35, 76A20, 76B45, 76D08

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Published 1 January 2007