A short note on Navier-Stokes flows with an incompressible interface and its approximations

In biological applications, a cell membrane consisting of a lipid bilayer usually behaves as fluid-like interface with surface incompressibility. Here we consider a mathematical formulation for an incompressible interface immersed in Navier-Stokes flows and study the mathematical and physical features for this incompressible interface. The model formulation introduces an unknown tension which acts as a Lagrange's multiplier to enforce such surface incompressibility. In this note, we show that the spreading operator of the tension and the surface divergence operator of the velocity are skew-adjoint with each other which indicates physically that the tension does not do extra work to the fluid under the condition of surface incompressibility. In order to avoid solving the unknown tension to enforce the surface incompressibility, we adopt a nearly surface incompressible approach (or penalty approach) by introducing two different modified elastic tensions which can be used efficiently in practical numerical simulations. Furthermore, we show that the resultant modified elastic forces have the same mathematical form as the original one derived from the unknown tension.