Regularized Stokeslet rings: An efficient method for axisymmetric Stokes flow with application to the growing pollen tube

Abstract

The method of regularized Stokeslets, based on the divergence-free exact solution to the equations of highly viscous flow due to a spatially-smoothed concentrated force, is widely employed in biological fluid mechanics. Many problems of interest are axisymmetric, motivating the study of the azimuthally-integrated form of the Stokeslet which physically corresponds to a ring of smoothed forces. The regularized fundamental solution for the velocity (single layer potential) and stress (double layer potential) due to an axisymmetric ring of smoothed point forces, the `regularized ringlet', is derived in terms of complete elliptic integrals of the first and second kind. The relative errors in the total drag and surrounding fluid velocity for the resistance problem on the translating, rotating unit sphere, as well as the condition number of the underlying resistance matrix, are calculated; the regularized method is also compared to 3D regularized Stokeslets, and the singular method of fundamental solutions. The velocity of Purcell's toroidal swimmer is calculated; regularized ringlets enable accurate evaluation of surface forces and propulsion speeds for non-slender tori. The benefits of regularization are illustrated by a model of the internal cytosolic fluid velocity profile in the rapidly-growing pollen tube. Actomyosin transport of vesicles in the tube is modelled using forces immersed in the fluid, from which it is found that transport along the central actin bundle is essential for experimentally-observed flow speeds to be attained. The effect of tube growth speed on the internal cytosolic velocity is also considered. For axisymmetric problems, the regularized ringlet method exhibits a comparable accuracy to the method of fundamental solutions whilst also allowing for the placement of forces inside of the fluid domain and having more satisfactory convergence properties.