An immersed boundary method for simulating vesicle dynamics in three dimensions
Introduction
A vesicle is a liquid droplet with a radius of about 10 μm enclosed by a phospholipid membrane suspended in an incompressible viscous fluid media. Such phospholipid membrane consists of two-layer tightly packed lipid molecules with hydrophilic heads facing the exterior and interior fluids while the hydrophobic tails hide in the middle. This bilayer membrane has the thickness about 6 nm and exhibits resistance against membrane dilation and bending. Therefore, it is quite natural to regard this membrane as an incompressible surface with mechanical functions determined by some energy functional [16]. Thus, the dynamics of vesicle in fluids can be determined by the membrane incompressibility, bending, and hydrodynamical forces. The study of vesicle dynamics in fluid flow has become an active research area in the communities of soft matter physics and computational fluid mechanics in the past years. For example, understanding of vesicle behaviors in fluid flows might lead to a better knowledge of red blood cells (RBCs) in blood simply because they both share similar mechanical behaviors [33]. Certainly, it has other practical applications such as a drug-delivery vehicle for cancer therapy [40] and a micro-reactor [13] for enzymatic mRNA synthesis in bioengineering.
The interaction between the vesicle and surrounding fluid makes the dynamics rich from physical point of view. For instance, a vesicle can undergo tank-treading, tumbling, or trembling motion under shear flow, see [6] and the references therein. For the past two decades, the vesicle dynamics in general flows (particularly in shear flow) have been extensively studied by experiments [22], [24], [6], theories [25], [32], [28], [11], and numerical simulations, see the detailed references below.
The numerical simulations of the vesicle problem not only involve a two-phase incompressible flow but also require to enforce an incompressibility constraint of the membrane surface, which makes the problem more challenging. The numerical methods for simulating vesicle problems in literature can be characterized by how the membrane surface is represented and how the fluid equations are solved. Based on this characterization, several methods have been developed such as boundary integral method [26], [43], [44], [3], [45], [50], [12], level set method [29], [37], [27], [30], [8], phase field method [9], [2], [30], [1], particle collision method [34], immersed interface method [21], [41], and immersed boundary method or front-tracking method [18], [20], [48], [17]; just to name a few recent ones. In all of these numerical methods, how to impose the membrane incompressibility constraint is an important issue. The surface tension in vesicle problems, which has a different physical meaning from that in general two-phase flow problems, is unknown a priori and in fact acts like Lagrange multiplier to enforce the local incompressibility along the surface. This is exactly the same role played by the pressure to enforce the fluid incompressibility in Navier–Stokes equations.
There are two different approaches to enforce the local incompressibility constraint in literature. The first one needs to discretize the whole equations first (regardless of using boundary integral, finite element, or finite difference method) and then to solve the discretized equations simultaneously for the tension and fluid variables. This approach can be explicit or semi-implicit depending on how we treat the tension force computations. There usually exists a trade-off between the time-step stability and efficiency in those algorithms simply because iterative procedures are needed. Most of the boundary integral method [45], [3], [50] or level set method [37], [27] fall into this category. Another approach, which was used in our previous 3D axisymmetric case [17], is called a penalty idea. Instead of keeping the vesicle membrane locally incompressible, the penalty idea makes the vesicle surface patch nearly incompressible by introducing a modified elastic tension energy. This approach replaces the unknown tension by a spring-like tension depending on the surface configuration so that we can avoid solving the whole system to obtain the variable tension, which significantly simplifies the numerical algorithm. In this paper, we extend our previous immersed boundary (IB) method for simulating incompressible vesicles in 3D axisymmetric Navier–Stokes flows [17] to general three dimensions. We shall show that the new elastic force derived from the modified tension energy has exactly the same mathematical form as the original elastic force except for the different definitions of the tension. We validate this approach by performing several numerical tests in our simulations.
The rest of the paper are organized as follows. In Section 2, we present the governing equations for the vesicle problem under the immersed boundary formulation. We also provide some notions in classical differential geometry that will be used to compute the geometrical quantities of the vesicle surface mathematically and numerically. Then we introduce our approach for a nearly incompressible vesicle surface and the modified energy. The detailed numerical algorithm is described in Section 3; we first explain how to evaluate the mean curvature vector and bending force terms on a triangulated surface, then outline the complete time-stepping scheme for the algorithm, and finally discuss how to maintain the surface mesh quality during the simulations. A series of numerical tests to validate our present algorithm is given in Section 4 which is followed by conclusion and future work in Section 5.
Section snippets
Equations of motion
We consider a single incompressible vesicle suspended in a three-dimensional domain Ω filled with viscous incompressible Navier–Stokes fluid. For the IB formulation in which the fluid-related quantities are represented in Eulerian manner while the vesicle-related ones are in Lagrangian manner, the governing equations can be written as follows. Here, we assume
Numerical scheme
In this section, we describe our numerical scheme and some implementation details for solving the equations of motion presented in the previous section. Before to proceed, we perform a non-dimensionalization on those governing equations first. We use the effective radius of vesicle as the scaling length scale, where A and V are the surface area and the enclosed volume of the vesicle, respectively. The characteristic time scale can be chosen depending on different flow
Numerical results
We perform a series of numerical tests for three-dimensional vesicle simulations in fluid flows. We begin by checking the rate of convergence of the developed scheme for the computation of the mean curvature, bending force, fluid variables, and vesicle configuration. Then, we choose different stiffness coefficient to study its effect on the nearly incompressibility by comparing local and global surface areas. As applications, we simulate the dynamics of a suspended vesicle in quiescent,
Conclusion and future work
We have proposed an immersed boundary method to simulate the fully three-dimensional vesicle dynamics in various flows. We relax the incompressibility constraint of a vesicle membrane by using a spring-like tension to keep the local surface area nearly incompressible. The new elastic force is derived from a modified tension energy which has exactly the same form as the original one. The vesicle surface is discretized on a triangular mesh, and the elastic tension and bending force are calculated
Acknowledgements
The work of M.-C. Lai is supported in part by Ministry of Science and Technology of Taiwan under research grant NSC-104-2115-M-009-014-MY3 and NCTS. Y. Kim was supported by National Research Foundation of Korea Grant (2015R1A2A2A01005420).
References (51)
- et al.
Diffuse interface models of locally inextensible vesicles in a viscous fluid
J. Comput. Phys.
(2014) - et al.
3D vesicle dynamics simulations with a linearly triangulated surface
J. Comput. Phys.
(2011) - et al.
A phase field approach in the numerical study of the elastic bending energy for vesicle membranes
J. Comput. Phys.
(2004) - et al.
3D numerical simulations of vesicle and inextensible capsule dynamics
J. Comput. Phys.
(2014) - et al.
An overview of projection methods for incompressible flows
Comput. Methods Appl. Mech. Eng.
(2006) - et al.
An immersed boundary method for simulating the dynamics of three-dimensional axisymmetric vesicles in Navier–Stokes flows
J. Comput. Phys.
(2014) - et al.
Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method
J. Comput. Phys.
(2010) - et al.
Numerical simulations of three-dimensional foam by the immersed boundary method
J. Comput. Phys.
(2014) - et al.
Computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods
J. Comput. Phys.
(2014) - et al.
Applications of level set methods in computational biophysics
Math. Comput. Model.
(2009)
Comparison between advected-field and level-set methods in the study of vesicle dynamics
Physica D
A level set projection model of lipid vesicles in general flows
J. Comput. Phys.
Strain energy function of red blood cell membranes
Biophys. J.
A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D
J. Comput. Phys.
A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows
J. Comput. Phys.
A fast algorithm for simulating vesicle flows in three dimensions
J. Comput. Phys.
Phase-field approach to three-dimensional vesicle dynamics
Phys. Rev. E
High-order algorithms for Laplace–Beltrami operators and geometric invariants over curved surfaces
J. Sci. Comput.
Shape diagram of vesicles in Poiseuille flow
Phys. Rev. Lett.
Dynamics of a vesicle in general flow
Proc. Natl. Acad. Sci. USA
Differential Geometry of Curves and Surfaces
Simulation of vesicle using level set method solved by high order finite element
Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers I: numerical method and results
SIAM J. Sci. Comput.
Analytical progress in the theory of vesicles under linear flow
Phys. Rev. E
Giant vesicles as microreactors for enzymatic mRNA synthesis
ChemBioChem
Cited by (30)
A semi-implicit semi-Lagrangian method for simulating immersed boundary motion under high inertia and elasticity
2023, Applied Mathematics and ComputationNumerical shape optimization of the Canham-Helfrich-Evans bending energy
2023, Journal of Computational PhysicsA stable and accurate immersed boundary method for simulating vesicle dynamics via spherical harmonics
2022, Journal of Computational PhysicsCitation Excerpt :A study on vesicle deformation under Poiseuille flow is also carried out. For simulating quasi-spherical vesicles in unsteady Navier-Stokes flow, we have improved our prior 3D immersed boundary (IB) method using triangulation in nearly incompressible interface approach [31]. In this paper, four improvements are made; namely, the spherical harmonic representation of vesicle interface, the logarithmic form of approximate elastic tension, the divergence form of discrete elastic tension force, and the modified approach of nearly incompressible interface coupled with mesh control.
Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach
2022, Computer Methods in Applied Mechanics and EngineeringThe smooth forcing extension method: A high-order technique for solving elliptic equations on complex domains
2021, Journal of Computational PhysicsAn immersed boundary projection method for incompressible interface simulations in 3D flows
2021, Journal of Computational Physics