Computational and Mathematical Methods in Medicine

Volume 2016, Article ID 7981386, 9 pages

http://dx.doi.org/10.1155/2016/7981386

## Deformation of a Capsule in a Power-Law Shear Flow

School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2600, Australia

Received 28 April 2016; Revised 10 July 2016; Accepted 11 August 2016

Academic Editor: Giuseppe Pontrelli

Copyright © 2016 Fang-Bao Tian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An immersed boundary-lattice Boltzmann method is developed for fluid-structure interactions involving non-Newtonian fluids (e.g., power-law fluid). In this method, the flexible structure (e.g., capsule) dynamics and the fluid dynamics are coupled by using the immersed boundary method. The incompressible viscous power-law fluid motion is obtained by solving the lattice Boltzmann equation. The non-Newtonian rheology is achieved by using a shear rate-dependant relaxation time in the lattice Boltzmann method. The non-Newtonian flow solver is then validated by considering a power-law flow in a straight channel which is one of the benchmark problems to validate an in-house solver. The numerical results present a good agreement with the analytical solutions for various values of power-law index. Finally, we apply this method to study the deformation of a capsule in a power-law shear flow by varying the Reynolds number from 0.025 to 0.1, dimensionless shear rate from 0.004 to 0.1, and power-law index from 0.2 to 1.8. It is found that the deformation of the capsule increases with the power-law index for different Reynolds numbers and nondimensional shear rates. In addition, the Reynolds number does not have significant effect on the capsule deformation in the flow regime considered. Moreover, the power-law index effect is stronger for larger dimensionless shear rate compared to smaller values.

#### 1. Introduction

Flow induced deformation of a capsule consisting of a membrane enclosing an internal medium such as a gel or a liquid is an important problem in fundamental research as well as bioengineering applications. For example, a capsule in shear flow is a fundamental process that is related to erythrocytes (or red blood cells), leukocytes (or white blood cells), and platelets in blood flow [1–6]. Deformation is essential for red blood cells to perform their physiological functions in the circulation of capillary blood vessels and thus affects the rheology of the blood [6–8]. The deformations of white blood cells and red blood cells can, respectively, affect the immune response and the oxygen load release [9, 10]. The synthetic microcapsules with polymerized interfaces are designed for drug delivery, cosmetic production, and other technical usages [11, 12]. Therefore, great effort has been made to study this problem (e.g., [1, 4, 6, 8, 10, 12–14]).

Both experimental and numerical methods have been conducted to observe capsule behaviors and the relevant underneath fluid-structure interaction physics. Schmid-Schönbein and Wells [15] and Goldsmith [16] observed that red blood cells tumble like rigid particles at low shear rates while they deform to a steady configuration and direction after which the membrane rotates around the internal liquid (tank-treading movement) at high shear rates. Later, Goldsmith and Marlow [17] and Keller and Skalak [18] found that the viscosity ratio between the liquids inside and outside the cell may also affect the type of behaviors. A higher viscosity inside would cause unsteady tumbling-rotating motion, while a smaller viscosity inside would lead to the tank-treading movement with a stationary shape. These phenomena were captured by Xu et al. [14]. More recently, Dupire et al. [19] reported rolling motion in addition to other behaviors. A hysteresis cycle and two transient dynamics driven by the shear rate (i.e., an intermittent regime during the “tank-treading-to-flipping” transition and a Frisbee-like “spinning” regime during the “rolling-to-tank-treading” transition) were highlighted.

There are several numerical methods that have been used to study capsule dynamics. Examples are the boundary element method (e.g., [20]), arbitrary Lagrangian-Euler method (e.g., [21–23]), immersed finite element method (e.g., [24]), and immersed boundary method (IBM) (e.g., [12–14, 25–34]). Specifically, Zhou and Pozrikidis [20] studied the transient and large deformation of capsules with position-dependent membrane tension. Choi and Kim [21] simulated the motion of red blood cells freely suspended in shear flow to investigate the nature of pairwise interception of red blood cells using a fluid-particle interaction method based on the arbitrary Lagrangian–Eulerian method. Villone et al. [22, 23] studied the effect of the non-Newtonian fluid on flexible particle deformation and migration in shear and channel flows by using the arbitrary Lagrangian–Eulerian method. The Navier–Stokes equations and cell-cell interaction were coupled in the framework of the immersed finite element method and mesh-free method by Y. Liu and W. K. Liu [24] to model complex blood flows with deformable red blood cells within micro and capillary vessels in three dimensions. The transient deformation of a liquid-filled elastic capsule in simple shear flow was studied by Sui et al. [1, 4, 35, 36]. The fluid inertia on the dynamics of deformable particles has been studied by Krüger et al. [32] and Kaoui and Harting [34]. More recently, optical force based separation of particles/capsules was simulated by Chang et al. [37–39]. Still, as far as known to us, the existing numerical simulations seldom consider the non-Newtonian rheology effects on the capsule behaviors, while blood and most fluids involved in biomedical engineering are non-Newtonian fluids [6, 8, 40, 41].

Following the work by Sui et al. [1] and Xu et al. [14], we develop an immersed boundary-lattice Boltzmann method (IB-LBM) to study the non-Newtonian effects on the deformation of a capsule in a shear flow. As a typical rheology, the power-law fluid is used. In this method, the capsule dynamics and the fluid dynamics are coupled by using the IBM, and the incompressible viscous power-law fluid motion is acquired by solving the lattice Boltzmann equation (LBE).

The rest of this paper is organized as follows. Section 2 briefly introduces the governing equations of the fluid and solid structures and describes the numerical approach. Section 3 presents the numerical results. Final conclusions are given in Section 4.

#### 2. Mathematical Formulation and Numerical Method

##### 2.1. Physical Model and Mathematical Formulation

In this work, a two-dimensional liquid capsule enclosed by an elastic membrane and immersed in an incompressible non-Newtonian fluid is considered, as illustrated in Figure 1 where is the arch length coordinate, denotes the surface normal that points into the outer fluid, denotes the tangent unit vector that points to the increasing arc length, and is the velocity applied at both top and bottom walls to form a simple shear flow. The incompressible non-Newtonian fluid dynamics is achieved by using LBM [42, 43]. Great effort has been made in using LBM to solve the complex flows (see several reviews [42–44] for the effort). Many publications have presented the details of LBM; thus we just provide a brief description in this paper and discuss the extension for non-Newtonian fluids. The details of LBM and its applications are referred to the references provided. Using the IB-LBM, the lattice Boltzmann equation (LBE) that governs the viscous flow dynamics and incorporates the traction jump across the interface due to the elastic membrane is written as [1, 14, 42, 43, 45, 46]where is the distribution function for particles with velocity at position and time , is the size of the time step, is the equilibrium distribution function, represents the dimensionless relaxation time, is the term representing the body force effect on the distribution function, are the weights, is the velocity of the fluid, is the speed of sound defined by with being grid spacing, is the body force acting on the fluid, is the Lagrangian force density on the fluid by the elastic boundary, is the position vector on the membrane, and is Dirac’s delta function.