As fish, micro-organisms, or other bodies move through a fluid, they stir their surroundings. This can be beneficial to some fish, since the plankton they eat depends on a well-stirred medium to feed on nutrients. Bacterial colonies also stir their environment, and this is even more crucial for them since at small scales there is no turbulence to help mixing. It has even been suggested that the total biomass in the ocean makes a significant contribution to large-scale vertical transport, but this is still a contentious issue. We propose a simple model of the stirring action of moving bodies through both inviscid and viscous fluids. In the dilute limit, this model can be solved using Einstein and Taylor's formula for diffusion (Brownian motion). We compare to direct numerical simulations of objects moving through a fluid. This is joint work with Steve Childress and George Lin.