In this work, we apply the boundary element method to describe the fluid velocity profiles in pockets in protein surfaces that are crucial to their function as enzymes. First, we study a simplified model, that of a dimpled sphere, in order to properly interpret the behavior in more complex surfaces such as proteins. In that case, we are able to observe the difference between an unphysical sharp edge for the dimple and a smooth edge. The sharp edge produces extra dissipation in the fluid, accounting for much more friction for all types of body motions. We were able to observe the direct correlation of the stagnation depth with the depth of the dimple in this simple case, allowing us to interpret this feature in a similar fashion for proteins. We have found that the fluid in the protein pockets translates with the body, irrespective of the direction body motion, for a distance comparable to the size of the pocket, and that such stagnation volumes are larger for motions parallel to the pocket axis. Outside of these pockets, the fluid velocity profile decays to that of the surrounding fluid far away from the protein (taken to be zero in Our case, for convenience), as the Oseen tensor requires. We have also found that there is weak local motion of fluid inside of the pockets, with velocities about 1% of those of the body. This Study suggests that there may be a role for the hydrodynamics of solvent inside of pockets for the transport of substrates to protein active sites. If solvent is effectively stagnant inside of a pocket, then transport must occur by diffusion near the pocket surface even if the fluid around the protein is stirred. The weak local motions inside of the pocket may also be relevant in this transport process, but these may be easily overwhelmed by any electrostatic interactions that are likely present at active sites.