Commun. Comput. Phys., 10 (2011), pp. 1333-1362.

An Implementation of MAC Grid-Based IIM-Stokes Solver for Incompressible Two-Phase Flows

Zhijun Tan 1*, K. M. Lim 2, B. C. Khoo 2

1 Guangdong Province Key Laboratory of Computational Science and School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China.
2 Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore; and Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore.

Received 16 October 2009; Accepted (in revised version) 22 February 2011
Available online 2 September 2011


In this paper, a novel implementation of immersed interface method combined with Stokes solver on a MAC staggered grid for solving the steady two-fluid Stokes equations with interfaces. The velocity components along the interface are introduced as two augmented variables and the resulting augmented equation is then solved by the GMRES method. The augmented variables and/or the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines and are then applied to the fluid through the jump conditions. The Stokes equations are discretized on a staggered Cartesian grid via a second order finite difference method and solved by the conjugate gradient Uzawa-type method. The numerical results show that the overall scheme is second order accurate. The major advantages of the present IIM-Stokes solver are the efficiency and flexibility in terms of types of fluid flow and different boundary conditions. The proposed method avoids solution of the pressure Poisson equation, and comparisons are made to show the advantages of time savings by the present method. The generalized two-phase Stokes solver with correction terms has also been applied to incompressible two-phase Navier-Stokes flow.

AMS subject classifications: 65N06, 35R05, 65M12

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Key words: Incompressible two-phase Stokes equations, two-phase Navier-Stokes equations, discontinuous viscosity, singular force, immersed interface method, Uzawa method, front tracking method.

*Corresponding author.
Email: (Z.-J. Tan), (K. M. Lim), (B. C. Khoo)

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