Keywords: asymptotic analysis, shallow waters with viscosity, Euler equations, Navier-Stokes equations.

In this paper, we study the Euler and Navier-Stokes equations in a domain with small depth in order to obtain shallow water models. With this aim, we introduce a small adimensional parameter related to the depth, and then we use asymptotic analysis to study what happens when becomes small.

Usually, when used asymptotics to analyze fluids, they are used in the original domain (see, for example, [1] and [2]), that in this case depends on parameter and time , or the surface is supposed to be constant (see, for example, [3]). We, however, shall use the asymptotic technique in the same way as in [4], [5] and related works, that is, we do a change of variable to a reference domain independent of the parameter and the time. This change of variable is applied to each function and equation of both models (Euler and Navier-Stokes) and to the initial and boundary conditions too.

We suppose that the solutions of both problems on the reference domain allow an expansion in powers of . We replace this expansion into the equations obtained, after the change of variable. Next step is to identify terms multiplied by the same power of . Then, equaling the coefficients of every power of to zero, we obtain a series of equations that are used to determine the first terms of the expansions. Finally, we do the change of variable back to the original domain.

In this way we obtain two models for small that, without making a priori assumptions about velocity or pressure behavior, give us in the case of Euler equations a shallow water model in which the vorticity equations are taken into account due to the fact that we consider non conservative external forces (Coriolis) acting on the fluid. In this model we present expressions for the horizontal components of the velocity in which the dependency on (vertical coordinate) is explicit. If we take as starting point Navier-Stokes, we obtain a shallow water model including a new diffusion term. In both cases the pressure expression is quite different from the classic shallow waters model. We have also obtained a non zero vertical velocity which takes into account the effects of a non constant bottom.

As example, next we present the shallow waters model obtained from the Navier-Stokes equations:

where , and do not depend on .

1

Gerbeau, J.-F., Perthame, B., "Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation", Discrete and Continuous Dynamical Systems-Series B, vol. 1, no. 1, pp. 89-102, 2001. doi:10.3934/dcdsb.2001.1.89

2

Zeytounian, R. K., "Modélisation asymptotique en mécanique des fluides newtoniens", Springer-Verlag, Berlin, 1994.

3

Azérad, P., Guillén, F., "Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics", Siam J. Math. Anal., vol. 33, no. 4, pp. 847-859, 2001.

4

Ciarlet, P.G., "Mathematical Elasticity. Volume II: Theory of Plates", North-Holland, Amsterdam, 1997. doi:10.1137/S0036141000375962

5

Trabucho, L., Viaño, J.M., "Mathematical modelling of rods", in: P. G. Ciarlet, J.-L. Lions (Eds.), Handbook of Numerical Analysis, Vol. IV, North-Holland, Amsterdam, pp. 487-974, 1996.

6

Rodríguez, J.M., Taboada-Vázquez, R., "Un modelo de aguas someras obtenido a partir de las ecuaciones de Euler usando la técnica de desarrollos asintóticos", in "Proceedins of the XVIII C.E.D.Y.A./VIII C.M.A.", Universitat Rovira i Virgili, 2003.

7

Rodríguez, J.M., Taboada-Vázquez, R., "Un modelo de aguas someras con viscosidad obtenido usando la técnica de desarrollos asintóticos", in "Proceedins of the XVIII C.E.D.Y.A./VIII C.M.A", Universitat Rovira i Virgili, 2003.

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