Stokeslet arrays in a pipe and their application to ciliary transport

Abstract

The problem of fluid transport by cilia in a circular cylinder is investigated. The discrete-cilia approach is used in building the model, using the Green function due to an infinite periodic Stokeslet array in a pipe. Two different expressions are obtained for the Green function, one via a residue method and the other using the Poisson summation formula each amenable for computation in a different region. Interaction of the Stokeslets is investigated to see how, as distance decreases, interaction changes from initially separated closed vortices to a continuous flow. The singular integral equations for the forces in this model are now replaced by non-singular equations, thus overcoming the numerical difficulties in earlier works. It is found that in the pipe core the flow is time-independent and varies between a plug flow and a negative parabolic profile, in the pumping range. These results are seen to be local results due to the near field. Streamlines in the sublayer show eddies near the cilia bases blending into a uniform flow near the cilia tips.