An isochoric domain deformation method for computing steady free surface flows with conserved volumes

The domain deformation method has been applied successfully to steady state free surface flows where the volume of the flow domain is unknown [V.F. de Almeida, Gas–liquid counterflow through constricted passages, Ph.D. thesis, University of Minnesota, Minneapolis, MN 1995; P.A. Sackinger, P.R. Schunk, R.R. Rao, A Newton–Raphson pseudo-solid domain mapping technique for free and moving boundary problems: a finite element implementation, J. Comput. Phys. 125 (1996) 83–103; L.C. Musson, Two-layer slot coating, Ph.D. thesis, University of Minnesota, Minneapolis, MN 2001]; however, this method does not handle effectively problems where the volume of the flow domain is known a priori. This work extends the original domain deformation method to a new isochoric domain deformation method to account for the volume conservation. Like in the original domain deformation method, the unknown shape of the flow domain is mapped onto a reference domain by using the equations of an elastic pseudo-solid; the difference with the original method is that this pseudo-solid is considered incompressible. Because of the incompressibility, the pseudo-pressure of the mapping appears as a Lagrange multiplier in the equations, and it is determined only up to an arbitrary uniform datum. By analyzing the coupled fluid flow-mapping problem, we show that, in the finite-element setting, such pressure datum can be specified by replacing one continuity equation in the fluid domain.