The viscosity of a dilute suspension

We consider the flow of a dilute suspension of equisized solid spheres in a viscous fluid. The viscosity of such a suspension is dependent on the volume fraction, c, of solid particles. If the particles are perfectly smooth, then solid spheres will not come into contact, because lubrication forces resist their approach. In this paper, however, we consider particles with microscopic surface asperities such that they are able to make contact. For straining motions we calculate the O(c2) coecient of the resultant viscosity, due to pairwise interactions. For shearing motions (for which the viscosity is undetermined because of closed orbits on which the probability distribution is unknown) we calculate the c2 contribution to the normal stresses N1 and N2. The viscosity in strain is shown to be slightly lower than that for perfectly smooth spheres, though the increase in the O(c) term caused by the increased eective radius due to surface asperities will counteract this decrease. The viscosity decreases with increasing contact friction coecient. The normal stresses N1 and N2 are zero if the surface roughness height is less than a critical value of 2:11 10−4 times the particle radius, and then become negative as the roughness height is increased above this value. N1 is larger in magnitude than N2. 1