Systematic Derivation of Jump Conditions for the Immersed Interface Method in Three-Dimensional Flow Simulation

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Abstract
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.