In this paper, the jump conditions for the normal derivative of the pressure have been derived for two-phase Stokes (and Navier-Stokes) equations with discontinuous viscosity and singular sources in two and three dimensions. While different jump conditions for the pressure and the velocity can be found in the literature, the jump condition of the normal derivative of the pressure is new. The derivation is based on the idea of the immersed interface method [9, 8] that uses a fixed local coordinate system and the balance of forces along the interface that separates the two phases. The derivation process also provides a way to compute the jump conditions. The jump conditions for the pressure and the velocity are useful in developing accurate numerical methods for two-phase Stokes equations and Navier-Stokes equations.
"Pressure Jump Conditions for Stokes Equations with Discontinuous Viscosity in 2D and 3D." Methods Appl. Anal. 13 (2) 199 - 214, June 2006.