MCIP D

Use of STAR CD for viscoplastic flow

**Theoretical study of the problem**

For the following study, some assumptions have been made.

The fluid is
incompressible

The flow is
isothermal, stationary and established

The pressure
in the flow only depends on the coordinate

The continuity equation gives for an incompressible fluid:

which gives, with the assumptions one has:

Therefore, v is constant and if one considers the boundary conditions, v=0.

For the movement equation, it has to be noticed that:

so there is no convection. Therefore, the equations system one has is :

In the first equation, the first term depends on y and the second depends on x. This means that each term are constant and one has:

where Delta P is the difference of pressure between the inlet and the outlet of the channel and L is the length of the channel. An integration of the previous equation leads to . Indeed, there is no constant because of the symmetry by y = 0 plan. From the two last equations of the previous system, one obtains and .

It is to notice
that one has not yet used the behavior law of the fluid. These results
are therefore right for each type of fluid.

Now, one will use the behavior's law. For the Bingahm fluid, this law is described by following system:

and if one considers y_{o }such as ,
two cases are possible:

if y_{o}>h,
then : the
difference of pressure is not strong enough to put the fluid in movement.

if y_{o}<h,
then two cases are possible:

| y | > y

Moreover, in
the case one considers,
so when | y | < y_{o} , ,
which means u(y) is constant. This constant will be determinate
later. When | y | > y_{o }, one has, with the definition of gamma,
, which gives u(y) by integration and consideration of the boundary conditions
(u is nil for y = -h and y = +h). Finally, the expression of the speed
is given by

the joining of the two velocity profiles in y = y_{o}
gives the value of the constant previously seen.

Finally, one
can resume the situation with the following cases:

if
then u = 0. The fluid doesn't move.

if ,
then two cases are possible:

and in this case,