  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.
