Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation

In this paper, we systematically derive jump conditions for the immersed interface method [ SIAM J. Numer. Anal., 31 ( 1994), pp. 1019 - 1044; SIAM J. Sci. Comput., 18 ( 1997), pp. 709 - 735] to simulate three-dimensional incompressible viscous flows subject to moving surfaces. The surfaces are represented as singular forces in the Navier - Stokes equations, which give rise to discontinuities of flow quantities. The principal jump conditions across a closed surface of the velocity, the pressure, and their normal derivatives have been derived by Lai and Li [ Appl. Math. Lett., 14 ( 2001), pp. 149 - 154]. In this paper, we first extend their derivation to generalized surface parametrization. Starting from the principal jump conditions, we then derive the jump conditions of all first-, second-, and third- order spatial derivatives of the velocity and the pressure. We also derive the jump conditions of first- and second- order temporal derivatives of the velocity. Using these jump conditions, the immersed interface method is applicable to the simulation of three-dimensional incompressible viscous flows subject to moving surfaces, where near the surfaces the first- and second- order spatial derivatives of the velocity and the pressure can be discretized with, respectively, third- and second- order accuracy, and the first-order temporal derivatives of the velocity can be discretized with second- order accuracy.