Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation

In a flow of two immiscible incompressible viscous fluids, jump discontinuities of flow quantities appear at the two-fluid interface. The im- mersed interface method can accurately and efficiently simulate the flow with- out smearing the sharp interface by incorporating necessary jump conditions into a numerical scheme. In this paper, we systematically derive the principal jump conditions for the velocity, the pressure, and their normal derivatives. 1. Introduction. The flow of two immiscible fluids is used in many technologi- cal applications, ranging from manufacturing to lubricated transport. The direct numerical simulations of two-fluid problems have a potentially huge domain for in- creased understanding (3, 4). Because of possible interface breakup/coalescence in a two-fluid flow, it is generally difficult and inefficient to simu late the dynamics of each fluid separately in its own domain using an interface-fitted grid method and couple the dynamics of the two fluids through interfacial conditions. Following Peskin's mathematical formulation in the immersed boundary method (9, 10), the two-fluid dynamics can be formulated in a single set of con- servation equations for the whole flow field (11). In particular, interfacial effects such as surface tension are included in the equations as a force term. This force term concentrates at the interface through the Dirac δ function, so it is called a sin- gular force. With this unified formulation, the two-fluid dynamics can be efficiently computed using a fixed grid, for example, a fixed Cartesian grid. Following Peskin's immersed boundary method, the Dirac δ function in the for- mulation can be approximated by a narrow-supported smooth function. This ap- proximation removes the force singularity and associated discontinuities, and thus allows for standard numerical schemes, but it has the drawbacks of interface smear- ing and low accuracy. The immersed interface method (6, 7, 8, 13, 14, 15) can overcome these drawbacks by directly incorporating jump conditions (caused by the singular force) into numerical schemes near an interface. With necessary jump conditions known, it can achieve second-order or higher accuracy. In this paper, the principal jump conditions for the velocity, the pressure, and their normal deriva- tives are derived for the immersed interface method to simulate two-fluid systems governed by the Navier-Stokes equations. In (5, 12), similar jump conditions are de- rived for single-fluid systems. In (2), principal pressure jump conditions are derived