Zhijun Tan

An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane

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.

An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model

A phase field model which describes the motion of mixtures of two incompressible fluids is presented by Liu and Shen [C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D 179 (2003) 211-228]. The model is based on an energetic variational formulation. In this work, we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids. It is a coupled nonlinear system of Navier-Stokes equations and Allen-Cahn phase equation (phase-field equation) through an extra stress term and the transport term. The numerical strategy is based on the approach proposed by Li et al. [R. Li, T. Tang, P.-W. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys. 170 (2001) 562-588] to separate the mesh-moving and PDE evolution. In the PDE evolution part, the phase-field equation is numerically solved by a conservative scheme with a Lagrange multiplier, and the coupled incompressible Navier-Stokes equations are solved by the incremental pressure-correction projection scheme based on the semi-staggered grid method. In the mesh-moving part, the mesh points are iteratively redistributed by solving the Euler-Lagrange equations with a parameter-free monitor function. In each iteration, the pressure and the phase are updated on the resulting new grid by a conservative-interpolation formula, while the velocity is re-mapped in a non-conservative approach. A simple method for preserving divergence-free is obtained by projecting the velocity onto the divergence-free space after generating the new mesh at the last iterative step. Numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows.

Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

The deformation of a liquid capsule enclosed by a thin shell in a simple shear flow is studied numerically using an implicit immersed boundary method. We present a thin-shell model for computing the forces acting on the shell middle surface during the deformation within the framework of the Kirchhoff-Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. For hyperelastic materials, the plane-stress assumption is used to compute the hydrostatic pressure and the incompressibility condition yields the thickness strain component and the corresponding change in the thickness. The stresses developing over the cross-section of the shell are integrated over the thickness to yield the stress and moment resultants which are then used to compute the forces acting on the shell middle surface. The immersed boundary method is employed for calculating the hydrodynamics and fluid-structure interaction effects. The location of the thin shell is updated implicitly using the Newton-Krylov method. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical and oblate spheroidal capsules in the shear flow.

An Immersed Interface Method for the Incompressible Navier-Stokes Equations with Discontinuous Viscosity Across the Interface

Abstract SWe present an immersed interface algorithm forthe incompressible Navier Stokes equations. The interface isrepresented by cubic splines which are interpolated through aset of Lagrangian control points. The position of the controlpoints is implicitly updated using the uid velocity. The forcesthat the interface exerts on the uid are computed from theconstitutive relation of the interface and are applied to the uidthrough jumps in the pressure and jumps in the derivativesof pressure and velocity. A projection method is used to timeadvance the Navier-Stokes equations on a uniform cartesianmesh. The Poisson-like equations required for the implicit so-lution of the diffusive and pressure terms are solved using afast Fourier transform algorithm. The position of the interface isupdated implicitly using a quasi-Newton method (BFGS) withineach timestep. Several examples are presented to illustrate theexibility of the presented approach. I. I NTRODUCTION We consider an incompressible uid in a 2-dimensionaldomainthat contains a material interface (t). The Navier-Stokes equations are written as,

An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deformable interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains.

A numerical study of muco-ciliary transport under the condition of diseased cilia

Structural and functional disorders of pulmonary cilia may result from genetic disorders and acquired insults. A two-dimensional numerical model based on the immersed boundary method coupled with the projection method is used to study the flow physics of muco-ciliary transport of the human respiratory tract under various abnormalities of cilia. The effects of the cilia beat pattern (CBP), ciliary length, immotile cilia, beating amplitude and uncoordinated beating of cilia are investigated. As expected, the mucus velocity decreases as the beating amplitude reduces. The windscreen wiper motion and rigid planar motion, which are two abnormal CBPs owing to genetic disorders, greatly reduce or almost stop the mucus transport. If the ciliary length varies from its standard length, the mucus velocity would decrease. The mucus velocity decreases rather linearly if the number of uniformly distributed immotile cilia increases. The numerical results show that the mucus velocity would be further reduced marginally when the uniformly distributed immotile cilia are rearranged as a cluster of immotile cilia. Furthermore, if half of the cilia are immotile and uniformly distributed and motile cilia beat at reduced amplitude, the incoordination between the active motile cilia would not significantly affect the mucus velocity.

Adaptive variational curve smoothing based on level set method

This paper presents an adaptive method for variational curve smoothing based on level set implementation. A suitable cost functional is minimized via solving the derived Euler-Lagrangian equation, of which the discretization is conducted on unstructured triangular meshes by employing a simple and effective finite volume scheme. Through adaptive refinement of the mesh, the geometry features of the given curve can be well resolved in a cost-effective way. Various numerical experiments demonstrate the effectiveness and efficiency of the proposed approach.