Problem Formulation

We consider a fluid in a cavity, the unit square. The bottom wall (y=0) of the cavity is moved with a constant (horizontal) velocity. We want to determine a top velocity of the fluid so that the fluid is separated at the line y=0.4 in the cavity (the vertical velocity of the fluid is zero at y=0.4).

The fluid is modeled by the two dimensional incompressible steady state Navier Stokes equations. The fluid is described by its velocity vector u (the first component is the horizontal velocity, the second component denotes the vertical velocity) and by its pressure p. The vector b denotes given boundary velocities and g is the top velocity to be determined.
In the following, variables set in boldface are vector valued.

Mathematical Formulation