An analysis for the momentum equation with unbounded pressure gradient: Numerical simulations

Abstract

We study the momentum equation with unbounded pressure gradient across the interior curve starting at a non-convex vertex. The horizontal directional vector U=(1,0)t on the L-shaped domain makes the inflow boundary disconnected. So, if the pressure function is integrated along the streamline, it must have a jump across the interior curve emanating from the vertex (0,0). Hence the pressure gradient is not well-defined. To handle this we construct a vector field which lifts the pressure jump value on the curve into the region. The precise structure of the solution is found by splitting from the solution the lifting vector field, the contact singularity, the corner singularity and the smoother part. The smoother one is shown to have the twice differentiability. The contact singularity is because the lifting vector show a non-smooth behavior at the contact point where the interface curve meets the boundary. The corner singularity is due to the non-convex vertex. Finally we give some numerical examples confirming the critical roles of each part in the decomposition.