An analysis is given for the slow motion of a sphere in a viscous fluid with a solid particle suspension. It is assumed that the relaxation time of the suspended particles is small enough for a perturbation analysis to be performed on the steady Stokes solution for the flow past the sphere. The analysis is similar to that given by Michael for the motion of a sphere in a dusty gas, at large Reynolds number, but the physical conditions under which this flow may be realized are very different. If it is assumed that "dusty gas" conditions apply, in which the density of the solid suspension is much greater than that of the gas, the effect of gravity on the motion of particles will be small compared with the Stokes resistance only when [Formula: see text], where a is the sphere radius, v the kinematic viscosity of the gas, and g the gravitational acceleration. For air this condition limits the sphere size to 10−2 cm. In this context the conditions therefore apply to small length scale effects in which a particle moves through a suspension of much smaller particles.Perhaps a more interesting situation arises when the two phases have comparable densities, and in particular when the particles are neutrally buoyant in a very viscous liquid, when gravity effects do not enter the problem, and the sphere size need not be so restricted. Our analysis assumes, following Saffman, that the bulk concentration of particles is small enough to be neglected, and it would follow in this case that the mass concentration is small enough to leave the liquid flow unaffected. However, the formal analysis given here is not restricted to this case.The final section of this paper is concerned with a related problem of finding the critical value of the Stokes number σ for which a particle will collide with the sphere, for Stokes flow. This number was previously given as σ = 1.214 by Langmuir, using a step-by-step integration. Our method gives σ = 1.21194, correct to five decimal places. This calculation, however, does not take account of the buildup of resistance as a particle approaches the sphere, and the number must be regarded as the value of σ at which a particle will approach to within a distance of the order of its diameter to the sphere, in a finite time.
