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 firstand 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 firstand second-order spatial derivatives of the velocity and the pressure can be discretized with, respectively, thirdand second-order accuracy, and the first-order temporal derivatives of the velocity can be discretized with second-order accuracy.