Swimming of a sphere in a viscous incompressible fluid with inertia

Abstract

The swimming of a sphere immersed in a viscous incompressible fluid with inertia is studied for surface modulations of small amplitude on the basis of the Navier–Stokes equations. The mean swimming velocity and the mean rate of dissipation are expressed as quadratic forms in term of the surface displacements. With a choice of a basis set of modes the quadratic forms correspond to two Hermitian matrices. Optimization of the mean swimming velocity for given rate of dissipation requires the solution of a generalized eigenvalue problem involving the two matrices. It is found for surface modulations of low multipole order that the optimal swimming efficiency depends in intricate fashion on a dimensionless scale number involving the radius of the sphere, the period of the cycle, and the kinematic viscosity of the fluid.