Commun. Comput. Phys., 11 (2012), pp. 925-950.


An Immersed Interface Method for the Simulation of Inextensible Interfaces in Viscous Fluids

Zhijun Tan 1*, D. V. Le 2, K. M. Lim 3, B. C. Khoo 3

1 Guangdong Province Key Laboratory of Computational Science & School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China.
2 Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore.
3 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 20 January 2010; Accepted (in revised version) 4 May 2011
Available online 28 October 2011
doi:10.4208/cicp.200110.040511a

Abstract

In this paper, an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow. The tension is introduced as an augmented variable to satisfy the constraint of interface inextensibility, and the resulting augmented system is solved by the GMRES method. In this work, the arclength of the interface is locally and globally conserved as the enclosed region undergoes deformation. The forces at the interface are calculated from the configuration of the interface and the computed augmented variable, and then applied to the fluid through the related jump conditions. The governing equations are discretized on a MAC grid via a second-order finite difference scheme which incorporates jump contributions and solved by the conjugate gradient Uzawa-type method. The proposed method is applied to several examples including the deformation of a liquid capsule with inextensible interfaces in a shear flow. Numerical results reveal that both the area enclosed by interface and arclength of interface are conserved well simultaneously. These provide further evidence on the capability of the present method to simulate incompressible flows involving inextensible interfaces.

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

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Key words: Inextensible interface, Stokes flows, singular force, immersed interface method, CG-Uzawa method, front tracking.

*Corresponding author.
Email: tzhij@mail.sysu.edu.cn (Z.-J. Tan), ledv@ihpc.a-star.edu.sg (D. V. Le), mpelimkm@nus.edu.sg (K. M. Lim), mpekbc@nus.edu.sg (B. C. Khoo)
 

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