XXIII ICTAM, 19-24 August 2012, Beijing, China THREE-DIMENSIONAL SIMULATION OF A RED BLOOD CELL IN SHEAR FLOW Wei-Xi Huang * a) , Cheong Bong Chang ** & Hyung Jin Sung ** * Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China ** Department of Mechanical Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Korea Summary We present an improved penalty immersed boundary (pIB) method for 3D simulation of the fluid-membrane interaction. The membrane deformation takes full account of the bending and twisting effects as well as the stretching and shearing effects. Hence, the method of subdivision surfaces is adopted to generate the mesh of membrane and the corresponding shape functions, which are required to be C 1 continuous. Using the proposed pIB method, the deformation of a biconcave circular disk as a model of the red blood cell in a linear shear flow is studied in detail. In our simulation, the swinging motion is observed due to the shape memory effect. By decreasing the dimensionless shear rate or increasing the reduced bending modulus, the swinging motion is transited into the tumbling motion. INTRODUCTION Deformation of a liquid-filled capsule enclosed by an elastic membrane due to external shear flow is an important problem in fundamental research as well as in bioengineering applications. An example of widespread concern is the red blood cell (RBC) with its membrane composed of a lipid bilayer and a cytoskeleton, which behaves like a two- dimensional flexible but nearly area-conserved material [1]. The deformability of the RBC has a significant effect on the physiological function of the RBC and the rheology of the whole blood [2]. Thus understanding of the dynamics of the RBC is indispensable in cellular biology and hemodynamics, and may help us diagnose blood diseases. Experimental observations on the RBC in shear flow have revealed rich dynamics of the system, e.g. the swing motion and the tumbling motion [3, 4]. Theoretical studies have also been performed to qualitatively describe the tumbling-to- swinging mode transition observed in experiments using simplified models [5]. However, to fully resolve the dynamics of capsules with complex geometries and finite deformations where theoretical prediction is difficult if not impossible, numerical simulations have to be performed. NUMERICAL METHOD To solve the fluid-flexible body coupled system, we adopt the improved pIB method proposed in our previous study [6]. The immersed boundary is assumed to be composed of two parts: massive material points and massless material points, which are assumed to be linked closely by a stiff spring with damping. The massive material points are subjected to the elastic force of solid deformation but do not interact with the fluid directly, while the massless material points interact with the fluid by moving with the local fluid velocity. In this way, we can make use of the governing equations for fluid and solid motions, which are linked through the Lagrangian and Eulerian momentum forcings [6]. In the framework of the pIB method, the flow solver and the solid solver can be developed separately by different methods. In the present study, the fractional step method is adopted to solve the incompressible fluid motion on a staggered Cartesian grid, while the FE method is developed to simulate the membrane deformation using an unstructured triangular mesh. Since the bending force term of the RBC membrane takes the fourth-order partial differential form, the discretized surface is required to be C 1 continuous in order to ensure the convergence of the finite- element solutions. A method to solve the C 1 continuity difficulty has been found by using the subdivision surface, which represents the limit of a recursive iteration of surface node positions. The corresponding formulation for geometric interpolation of the subdivision surface can be used as the shape functions in the FE method for plate and shell computations [7]. In the present study, we apply the subdivision FE method in conjunction with the pIB method for 3D simulation of the fluid-membrane interaction. Details of the problem formulation and the computational procedure can be found in [8]. RESULTS AND DISCUSSION In this section, we study the deformation of the RBC in a linear shear flow by using the proposed method. The fluid domain size is 10×10×10 scaled by the equivalent volumetric radius a eq in the x-, y- and z-directions, with the origin located at the center of the domain, which coincides with the center of the RBC. Dirichlet conditions ( , u yv w 0 γ = = =  ) are applied at the top and bottom boundaries, while periodic conditions are adopted in the x- and z- directions. The geometry of the RBC is taken to be a biconcave circular disk as shown in Fig.1 [9]. According to the grid convergence test, we use a resolution of 8192 elements for the RBC mesh and 192×192×192 grids for the fluid domain. Moreover, the computational time step is 0.0001 t Δ = and the Reynolds number based on a eq is set as Re=0.025. a) Corresponding author. Email: hwx@tsinghua.edu.cn