Abstract

We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex-possibly nonsmooth-geometries in two dimensions. The key ingredient is a set of panel quadrature rules capable of evaluating weakly-singular, nearly-singular and hyper-singular integrals to high accuracy. Near-singular integral evaluation, in particular, is done using an extension of the scheme developed in J. Helsing and R. Ojala, J. Comput. Phys. 227 (2008) 2899-2921. The boundary of the given geometry is "panelized" automatically to achieve user-prescribed precision. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9-digit solution accuracy.

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Many biological microswimmers locomote by periodically beating the densely packed cilia on their cell surface in a wave-like fashion. While the swimming mechanisms of ciliated microswimmers have been extensively studied both from the analytical and the numerical point of view, optimisation of the ciliary motion of microswimmers has received limited attention, especially for non-spherical shapes. In this paper, using an envelope model for the microswimmer, we numerically optimise the ciliary motion of a ciliate with an arbitrary axisymmetric shape. Forward solutions are found using a fast boundary-integral method, and the efficiency sensitivities are derived using an adjoint-based method. Our results show that a prolate microswimmer with a $2\,{:}\,1$ aspect ratio shares similar optimal ciliary motion as the spherical microswimmer, yet the swimming efficiency can increase two-fold. More interestingly, the optimal ciliary motion of a concave microswimmer can be qualitatively different from that of the spherical microswimmer, and adding a constraint to the cilia length is found to improve, on average, the efficiency for such swimmers.