Deformation of two-dimensional red blood cell in linear shear flow is simulated using the immersed boundary method,in which the cell is modeled as a force source instead of a real body.The effect of three constitutive laws,i.e.Hookean,Neo-Hookean and Skalak elasticity,on the deformation is studied by simulating the cell movement in two linear shear flows.The results show that the effect of the constitutive laws gets more obvious as the shear rate increases.Both the aspect ratio and the inclination of the steady shapes get bigger, and the differences between the periods of the cell tank-treading motion become larger.For the same shear flow, the period with Hookean elasticity is less than the period with Neo-Hookean elasticity and bigger than the period with Skalak elasticity.
Deformation of the spherical capsule in 3D simple shear fow is simulated using the immersed boundary method. The capsule membrane is regarded as an elastic medium satisfying the Neo-Hookean or Skalak elasticity. The motions of the capsule under various capillary numbers are studied. The results show that the deformation of the capsule becomes larger as the capillary number increases;in the same shear fow,the deformation under Skalak law is smaller than that under Neo-Hookean;for small capillary number the Taylor parameter agrees well with the analytical solution,whereas for large capillary number it is less than the analytical solution. Those results are validated by previous works obtained by the boundary integral method and the immersed boundary method.
The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics.This paper analyzes the accuracy of the immersed boundary method.The procedure contains two parts,i.e.,the code verification and the accuracy analysis.The code verification provides the confidence that the code used is free of mistakes,and the accuracy analysis gives the order of accuracy of the immersed boundary method.The method of manufactured solutions is taken as a means for both parts.In the first part,the numerical code employs a second-order discretization scheme,i.e.,it has second-order accuracy in theory.It matches the calculated order of accuracy obtained in the numerical calculation for all variables.This means that the code contains no mistake,which is a premise of the subsequent work.The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations.By analyzing the discretization errors,the accuracy of the immersed boundary method is proven to be first order even though the discretization scheme is second order.It has been found that the coarser mesh may not be sensitive enough to capture the influence of the immersed boundary,and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.
In this paper,we analyze the stability of the Immersed Boundary Methodapplied to a membrane-fluid system with a plasma membrane immersed in an incompressibleviscous fluid.We show that for small deformations,the planar rest state isstable for a membrane with bending rigidity.The smoothed version,using a standardregularization technique for the singular force,is also shown to be stable.Furthermore,we show that the coupled fluid-membrane system is stiff and smoothing helpsto reduce the stiffness.Compared to the system of elastic fibers immersed in an incompressiblefluid,membrane with bending rigidity consist of a wider range of decayrates.Therefore numerical instability could occur more easily for an explicit methodwhen the time step size is not sufficiently small,even though the continuous problemis stable.
