Numerical Simulation Of Red Blood Cell Motion And Deformation Based On Level-set Method

The red blood cells (RBCs) are important components in blood because of theircrucial role in oxygen and nutrients transport. A normal human RBC has a biconcaveshape, as the membrane is made of a bilayer of phospholipid molecules. The cellmembrane is highly deformable so that RBCs can pass through capillaries of smallerdiameter than themselves with large deformation, then they return to the initial shape.The production of many blood diseases is associated with the decreased deformationof the red blood cell. The research of the motion and deformation of the RBCs isimportant to reveal the mechanism of blood diseases.In this paper, a new numerical model of RBC motion and deformation invascular has been developed. The RBC has been simplified as a mass ofincompressible fluid surrounded by an elastic membrane which immersed in anotherincompressible fluid. Its internal fluid (hemoglobin) has a viscosity about5to10times than that of the suspending fluid (plasma). The level set method is used tocapture elastic interfaces, and the stretching information on the interface can berecovered from the level set function in incompressible flows. As in the immersedboundary method, the elastic force is derived from an energy principle and appears asa singular forcing term in the Navier-Stokes equations. In our Eulerian approach, wethus have to express this volume approximation of a Dirac mass with support on theinterface. We can obtain the parameters of the flow field by solving Navier-Stokesequations.A projection method on staggered grids is introduced to solve the Navier-Stokesequations. Level set equation is solved using the fifth order WENO scheme and thethird order TVD Runge-Kutta scheme for capturing the motion interface. The testcases show the elastic characteristic of a pressurized membrane with variable densityand viscosity properties. Different Reynolds numbers and different elastic modulesare introduced to study the deformation of the membrane. The conclusion is that themethod we have presented satisfies mass and energy conservation without the leakingproblems in classical immersed boundary method. The numerical model we have developed above is used to simulate the motionand deformation of a red blood cell in different fluid fields such as shear flow andPoiseuille flow. In shear flow, the tank treading of the RBC membrane is observed atlow enough viscosity contrast. Although the global shape of the RBC is stationary, themembrane circulates along the contour such as a tank tread with the cell orientating toa fixed inclination angle. The angle of the cell depends on the shear ratio andviscosity contrast. The tumbling motion is observed at large enough viscosity contrast,and the frequency of the tumbling motion depends on the shear ratio. Thedeformability of the RBC in Poiseuille flow depends on the rigidity of the membraneand the viscosity contrast. The biconcave shape of the RBC is propitious to createhigh deformation than other shapes. With increased viscosity of the plasma both thevelocity of the blood and the deformability of the cell reduce.