Shuwang Li

Dynamics of multicomponent vesicles in a viscous fluid

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibil-ity of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.

A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell

In this paper, we present a time and space rescaling scheme for the computation of moving interface problems. The idea is to map time-space such that the interfaces can evolve exponentially fast in the new time scale while the area/volume enclosed by the interface remains unchanged. The rescaling scheme significantly reduces the computation time (especially for slow growth), and enables one to accurately simulate the very long-time dynamics of moving interfaces. We then implement this scheme in a Hele-Shaw problem, examine the dynamics for a number of different injection fluxes, and present the largest and most pronounced viscous fingering simulations to date.

A Boundary Integral Method for Computing the Dynamics of an Epitaxial Island

In this paper, we present a boundary integral method for computing the quasi-steady evolution of an epitaxial island. The problem consists of an adatom diffusion equation (with desorption) on terrace and a kinetic boundary condition at the step (island boundary). The normal velocity for step motion is determined by a two-sided flux. The integral formulation of the problem involves both double- and single-layer potentials due to the kinetic boundary condition. Numerical tests on a growing/shrinking circular or a slightly perturbed circular island are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient, and spectrally accurate in space. Nonlinear simulations for the growth of perturbed circular islands show that sharp tips and flat edges will form during growth instead of the usual dense branching morphology seen throughout physical and biological systems driven out of equilibrium. In particular, Bales-Zangwill instability is manifested in the form of wave-like fronts (meandering instability) around the tip regions. The numerical techniques presented here can be applied generally to a class of free/moving boundary problems in physical and biological science.

Nonlinear studies of tumor morphological stability using a two-fluid flow model

We consider the nonlinear dynamics of an avascular tumor at the tissue scale using a two-fluid flow Stokes model, where the viscosity of the tumor and host microenvironment may be different. The viscosities reflect the combined properties of cell and extracellular matrix mixtures. We perform a linear morphological stability analysis of the tumors, and we investigate the role of nonlinearity using boundary-integral simulations in two dimensions. The tumor is non-necrotic, although cell death may occur through apoptosis. We demonstrate that tumor evolution is regulated by a reduced set of nondimensional parameters that characterize apoptosis, cell–cell/cell-extracellular matrix adhesion, vascularization and the ratio of tumor and host viscosities. A novel reformulation of the equations enables the use of standard boundary integral techniques to solve the equations numerically. Nonlinear simulation results are consistent with linear predictions for nearly circular tumors. As perturbations develop and grow, the linear and nonlinear results deviate and linear theory tends to underpredict the growth of perturbations. Simulations reveal two basic types of tumor shapes, depending on the viscosities of the tumor and microenvironment. When the tumor is more viscous than its environment, the tumors tend to develop invasive fingers and a branched-like structure. As the relative ratio of the tumor and host viscosities decreases, the tumors tend to grow with a more compact shape and develop complex invaginations of healthy regions that may become encapsulated in the tumor interior. Although our model utilizes a simplified description of the tumor and host biomechanics, our results are consistent with experiments in a variety of tumor types that suggest that there is a positive correlation between tumor stiffness and tumor aggressiveness.

Convergence of boundary integral method for a free boundary system

Abstract Boundary integral method has been implemented successfully in practice for simulating problems with free boundaries. Though the method produces accurate and efficient numerical results, its convergence study is usually limited to numerical demonstrations by successively reducing time step and increasing resolution for a test problem. In this paper, we present a rigorous convergence and error analysis of the boundary integral method for a free boundary system. We focus our study on a nonlinear tumor growth problem. The boundary integral formulation yields a Fredholm type integral equation with moving boundaries. We show that in two dimensions, the convergence of the scheme in the L ∞ norm has first order accuracy on the time direction and Δ θ α on the spatial direction.

Computation of a Shrinking Interface in a Hele-Shaw Cell

In this paper, we present an adaptive rescaling method for computing a shrinking interface in a Hele-Shaw cell with a time increasing gap width b(t). We focus our study on a one-phase interior Hele...

Nonlinear simulations of elastic fingering in a Hele-Shaw cell

This work is motivated by the recent experiments of two reacting fluids in a Hele-Shaw cell (Podgorski etźal., 2007) and associated linear stability analysis of a curvature weakening model (He etźal., 2012). Unlike the classical Hele-Shaw problem posed for moving interfaces with surface tension, the curvature weakening model is concerned with a newly-produced gel-like phase that stiffens the interface, thus the interface is modeled as an elastic membrane with curvature dependent rigidity that reflects geometrically induced breaking of intermolecular bonds. Here we are interested in exploring the long-time interface dynamics in the nonlinear regime. We perform simulations using a spectrally accurate boundary integral method, together with a rescaling scheme to dramatically speed up the intrinsically slow evolution of the interface. We find curvature weakening inhibits tip-splitting and promotes side-branching morphology. At long times, numerical results reveal that there exist nonlinear, stable, self-similarly evolving morphologies.