  Most classical work on the hydrodynamics of low-Reynolds-number swimming
addresses deterministic locomotion in quiescent environments. Thermal
fluctuations in fluids are known to lead to a Brownian loss of the swimming
direction. As most cells or synthetic swimmers are immersed in external flows,
we consider theoretically in this paper the stochastic dynamics of a model
active particle (a self-propelled sphere) in a steady general linear flow. The
stochasticity arises both from translational diffusion in physical space, and
from a combination of rotary diffusion and run-and-tumble dynamics in
orientation space. We begin by deriving a general formulation for all
components of the long-time mean square displacement tensor for a swimmer with
a time-dependent swimming velocity and whose orientation decorrelates due to
rotary diffusion alone. This general framework is applied to obtain the
convectively enhanced mean-squared displacements of a steadily-swimming
particle in three canonical linear flows (extension, simple shear, and
solid-body rotation). We then show how to extend our results to the case where
the swimmer orientation also decorrelates on account of run-and-tumble
dynamics. Self-propulsion in general leads to the same long-time temporal
scalings as for passive particles in linear flows but with increased
coefficients. In the particular case of solid-body rotation, the effective
long-time diffusion is the same as that in a quiescent fluid, and we clarify
the lack of flow-dependence by briefly examining the dynamics in elliptic
linear flows. By comparing the new active terms with those obtained for passive
particles we see that swimming can lead to an enhancement of the mean-square
displacements by orders of magnitude, and could be relevant for biological
organisms or synthetic swimming devices in fluctuating environmental or
biological flows.
