In this thesis, the applications of the proposed immersed interface method based on finite volume formulation are presented. The merit of the present scheme is that the required jump conditions are pressure and first order derivative of velocity. Despite the simplicity of the jump conditions, the second second order accuracy of the scheme is still retained. This is demonstrated by predicting the discontinuous Poisson equations, where the second order accuracy in maximum errors is obtained for the cases investigated. The scheme is further applied to the moving elastic interface problems, where the rigid surface is modeled using the immersed boundary method. To ensure the stable solution of the elastic problem, Fourier filtering is adopted to smooth the elastic boundary at each time step after the interface are moved and the cubic spline is used to redistribute the Lagrangian markers along the interface. The constricted channel with elastic membrane demonstrates the capability of modeling elastic interface and rigid boundary flow. Also, developments of the membrane in passing through the constricted channel at different membrane surface tension and diameters are explored.

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