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The flow field created by swimming microorganisms not only enables theirlocomotion but also leads to advective transport of nutrients. In this paper weaddress analytically and computationally the link between unsteady feeding andunsteady swimming on a model microorganism, the spherical squirmer, actuatingthe fluid in a time-periodic manner. We start by performing asymptoticcalculations at low P\'eclet number (Pe) on the advection-diffusion problem forthe nutrients. We show that the mean rate of feeding as well as itsfluctuations in time depend only on the swimming modes of the squirmer up toorder Pe^(3/2), even when no swimming occurs on average, while the influence ofnon-swimming modes comes in only at order Pe^2. We also show that genericallywe expect a phase delay between feeding and swimming of 1/8th of a period.Numerical computations for illustrative strokes at finite Pe confirmquantitatively our analytical results linking swimming and feeding. We finallyderive, and use, an adjoint-based optimization algorithm to determine theoptimal unsteady strokes maximizing feeding rate for a fixed energy budget. Theoverall optimal feeder is always the optimal steady swimmer. Within the set oftime-periodic strokes, the optimal feeding strokes are found to be equivalentto those optimizing periodic swimming for all values of the P\'eclet number,and correspond to a regularization of the overall steady optimal.

Unsteady feeding and optimal strokes of model ciliates