Use of STAR CD for viscoplastic flow

Theoretical study of the problem

II.1 General theory

        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.

II.2 Behavior's law

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

and if one considers yo such as , two cases are possible:

if yo>h, then : the difference of pressure is not strong enough to put the fluid in movement.
if yo<h, then two cases are possible:

 | y | < yo then 
                 | y | > yo then 

        Moreover, in the case one considers,  so when | y | < yo, which means u(y) is constant. This constant will be    determinate later. When | y | > yo , 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 = yo  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, 
and in this case,