140
I. AN EXISTENCE THEOREM FOR A SHALLOW WATER PROB·
LEM
1.1. Introduction
In the first and second chapter, we work with a two dimensional model and then to
simplify the notations, we denote by n the fluid surface domain nx, "( its boundary (which
is "(x) and n the exterior normal unit to n on "(.
In all that follows, we note Q the cylinder ]0, T[ x n with boundary 8Q. We denote
by "(- (resp. 7+) the part of the boundary where the flow enters (resp. goes out or is zero).
Then, we define :
~ = ]0, T[ x ,,(, ~- = ]0, T[ x "(- and ~+ = ]0, T[ x "(+
We have seen the three dimensional equations of the model in the introduction.
To obtain the Shallow water problem, we made an important hypothesis, there is no
stratification effect in a Shallow water model; and then we have a null buoyancy.
The depth averaged motion is described in terms of the mean velocity denoted by
u(x) (where x = (XJ,X2) is a point of the surface n) defined by:
where h is the total depth i.e. h = H + E. Let u' be the deviation from the vertical mean,
with 1€ u' dx3 = O.
-H
Integrating the three dimensional momentum equation over depth, we obtain:
1
Ut + divR + "2\lu 2 + curlua(u) + g\!E + Du lulR2 =1
Where R is the Reynolds stress tensor that results from the non linear interactions of the
fluctuations products and g the acceleration of gravity. To simplify the mathematical study,
we're going to take g = 1. Another simplification for the composition of this paper is done.
Indeed we consider the variable h instead of E that leads to a new term (g\lH) which can
be put in the second part of the equation.
We can simply write a good approximation of the Reynolds stress tensor:
div R = -A.6.u
where A is the eddy viscosity.
The integration of the term a~3 (0 :~) gives the term I modelling the wind effect and
a term D u lulR2 modelling the shear effect at the bottom.
I. AN EXISTENCE THEOREM FOR A SHALLOW WATER PROB·
LEM
1.1. Introduction
In the first and second chapter, we work with a two dimensional model and then to
simplify the notations, we denote by n the fluid surface domain nx, "( its boundary (which
is "(x) and n the exterior normal unit to n on "(.
In all that follows, we note Q the cylinder ]0, T[ x n with boundary 8Q. We denote
by "(- (resp. 7+) the part of the boundary where the flow enters (resp. goes out or is zero).
Then, we define :
~ = ]0, T[ x ,,(, ~- = ]0, T[ x "(- and ~+ = ]0, T[ x "(+
We have seen the three dimensional equations of the model in the introduction.
To obtain the Shallow water problem, we made an important hypothesis, there is no
stratification effect in a Shallow water model; and then we have a null buoyancy.
The depth averaged motion is described in terms of the mean velocity denoted by
u(x) (where x = (XJ,X2) is a point of the surface n) defined by:
where h is the total depth i.e. h = H + E. Let u' be the deviation from the vertical mean,
with 1€ u' dx3 = O.
-H
Integrating the three dimensional momentum equation over depth, we obtain:
1
Ut + divR + "2\lu 2 + curlua(u) + g\!E + Du lulR2 =1
Where R is the Reynolds stress tensor that results from the non linear interactions of the
fluctuations products and g the acceleration of gravity. To simplify the mathematical study,
we're going to take g = 1. Another simplification for the composition of this paper is done.
Indeed we consider the variable h instead of E that leads to a new term (g\lH) which can
be put in the second part of the equation.
We can simply write a good approximation of the Reynolds stress tensor:
div R = -A.6.u
where A is the eddy viscosity.
The integration of the term a~3 (0 :~) gives the term I modelling the wind effect and
a term D u lulR2 modelling the shear effect at the bottom.