An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane Zhijun Tan a , D.V. Le a , Zhilin Li b , K.M. Lim a,c , B.C. Khoo a,c, * a Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore b Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA c Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore article info Article history: Received 27 January 2008 Received in revised form 18 August 2008 Accepted 19 August 2008 Available online 30 August 2008 Keywords: Incompressible viscous flows Piecewise constant viscosity Singular force Immersed interface method Projection method Front tracking method abstract This paper presents an implementation of the second-order accurate immersed interface method to simulate the motion of the flexible elastic membrane immersed in two viscous incompressible fluids with different viscosities, which further develops the work reported in Tan et al. [Z.-J. Tan, D.V. Le, K.M. Lim, B.C. Khoo, An Immersed Interface Method for the Incompressible Navier–Stokes Equations with Discontinuous Viscosity Across the Interface, submitted for publication] focussing mainly on the fixed interface problems. In this work, we introduce the velocity components at the membrane as two augmented unknown interface variables to decouple the originally coupled jump conditions for the velocity and pressure. Three forms of augmented equation are derived to determine the augmented variables to satisfy the continuous condition of the velocity. The velocity at the membrane, which determine the motion of the membrane, is then solved by the GMRES iterative method. The forces calculated from the configuration of the flexible elastic membrane and the augmented variables are interpolated using cubic splines and applied to the fluid through the jump conditions. The position of the flexible elastic membrane is updated implicitly using a quasi-Newton method (BFGS) within each time step. The Navier–Stokes equations are solved on a staggered Cartesian grid using a second order accurate projection method with the incorporation of spatial and temporal jump conditions. In addition, we also show that the inclusion of the temporal jump contributions has non-negligible effect on the simulation results when the grids are crossed by the membrane. Using the above method, we assess the effect of different viscosities on the flow solution and membrane motion. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction Many problems of fluid mechanics involve the interaction of a viscous incompressible fluid and a moving elastic mem- brane (called fluid-membrane interactions). One can consider the membrane as a part of the fluid which exerts forces to the surrounding viscous fluid, while also moving with the velocity of adjacent fluid particles. The mathematical formulation and numerical method for this type of problems was first introduced by Peskin in what we now commonly called the immersed boundary method to simulate the blood flow in the heart and through heart valves [25,30]; the method has been used for many other applications particularly in bio-fluid dynamics. Examples include the deformation of red blood cell in a shear 0021-9991/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2008.08.013 * Corresponding author. Address: Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore. Tel.: +65 65162889; fax: +65 67791459. E-mail addresses: smatz@nus.edu.sg (Z. Tan), smaldv@nus.edu.sg (D.V. Le), zhilin@math.ncsu.edu (Z. Li), mpelimkm@nus.edu.sg (K.M. Lim), mpekbc@nus.edu.sg (B.C. Khoo). Journal of Computational Physics 227 (2008) 9955–9983 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp