Hydrodynamic force on a sphere normal to an obstacle due to a non-uniform flow

For a small sphere suspended in a background fluid flow near an obstacle, we calculate the hydrodynamic force on the sphere in the direction normal to the boundary of the obstacle. Using the Lorentz reciprocal theorem, we obtain analytical expressions for the normal force in the Stokes flow limit, valid for arbitrary separations of the particle from the obstacle, both for solid obstacles and those with free surfaces. The main effect of the boundary is to produce a normal force proportional to extensional flow gradients in the vicinity of the particle. The strength of this force is greatest when the separation between the surfaces of the particle and the obstacle is small relative to the particle size. While the magnitude of the force weakens for large separations between the sphere and the obstacle (decaying quadratically with separation distance), it can significantly modify Faxen's law even at modestly large separation distances. In addition, we find a second force contribution due to the curvature of the background flow normal to the obstacle, which is also important when the sphere is close to the obstacle. The results of the theory are of importance to the dynamics of particles in confined geometries, whether bounded by a solid obstacle, the wall of a channel or a gas bubble.