Minimum drag shape of a semi-ellipsoid exposed to shear flow and its possible relation to the shape of endothelial cell

Abstract

A semi-ellipsoid attached on a wall is considered as a model problem for the study of blood flow effect on the shape of an endothelial cell. Under the condition that the volume is fixed and one axis of the semi-ellipsoid is aligned to the flow direction, the shape for the minimum drag under the applied shear flow is determined. Both the analytical and numerical approaches are adopted for computation of the minimum drag shape. Since analytical solution is not available for the original model problem, analytical solution to a closely related problem is used to compute the approximate value of the drag force. Adopted is the classical result on the motion of an ellipsoidal particle in a viscous fluid [Jeffery, Proc. Roy. Soc. A (1922)]. To corroborate the analytically obtained results, the model problem has also been studied numerically by using the finite element method (FEM). The minimum drag shape predicted analytically by using Jeffery’s solution is ( a, b, c) = (1.71, 0.67, 0.88), where a is the dimensionless semi-axis in the flow direction, b the height, and c the half width of the semi-ellipsoid. The numerical approach predicts the minimal drag shape as ( a, b, c) = (1.96,0.64,0.80). This result obtained by the FEM method shows good agreement with the result obtained by the boundary integral method [Hazel and Pedley, Biophys. J. (2000)].