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Consistent models with three equations for thin films and openchannel flows
Gael Richard (IMT Toulouse)
Consistent 1D models are derived with an asymptotic method in two different cases, viscous thin films and turbulent shallowwater flows, paying special attention to the mathematical structure of the equations. Besides the fluid depth and the average velocity, the deviation of the velocity to its average value is taken into account by a third variable called enstrophy, which acts as the entropy of the hyperbolic part of the equations. The third equation needed to close the problem is provided by the workenergy theorem averaged over the depth. The equations have the structure of the Euler equations of compressible fluids with relaxation and, in the case of thin films, diffusion and dispersive capillary terms. In the case of turbulent flows, two dimensionless systems are used, a shallowwater scaling in an external layer and a viscous scaling in an internal layer. A matching procedure is used in a buffer layer to connect both solutions.

