Sedimentation motion of sand particles in moving water (I): The resistance on a small sphere moving in non-uniform flow

In hydraulics, when we deal with the problem of sand particles moving relative to the surrounding water, Stokes' formula of resistance has usually been used to render the velocity of sedimentation of the particles. But such an approach has not been proved rigorously, and its accuracy must be carefully considered. In this paper, we discuss the problem of a sphere moving in a non-uniform flow field, on the basis of the fundamental theory of hydrodynamics. We introduce two assumptions: i) the diameter of the sphere is much smaller than the linear dimension of the flow field, and ii) the velocity of the sphere relative to the surrounding water is very small. Using these two assumptions, we solve the linearized Navier-Stokes equations and equations of continuity by the method of Laplace transform, and finally we obtain a formula for the resistance acting on a sphere moving in a non-uniform flow field.