This paper presents a statistical mechanical description of the random motion of particles in suspension and some results derived there-from for the viscosity of concentrated suspensions of solid spherical particles. In statistical mechanics the information about the probability of dynamical states of the particles of a fluid （i.e., their positions, translational velocities and rotational velocities） is embedded in a particle distribution function, and all of the macroscopic properties of the fluid are specified as integral noments of the distribution function. In particular, the local stress in a suspension is defined as the integral over all particle states of the distribution weighted with the stress at a point for a given instantaneous total particle state. Thus, the calculation of the rheological properties of the suspension comes down to a determination of the distribution function and this local stress function.
If the overall shear rate in the suspension is steady and not too large and the particles are neutrally buoyant solid spheres, the net forces and torques on the particles are zero and the local fluid motion is governed by the Stokes equations. It is shown that under these conditions 1） the particle distribution function is independent of the linear and angular velocities of the particles and so is uniquely determined by the particle positions, 2) the viscosity of the suspension must be independent of the shear rate for all concentrations, and 3) the viscosity can be written in a power series in the concentration of particles in which the term of the nth power is the incremental contribution to the viscosity from interactions involving particles. In order to calculate the coefficients of these terms the fluid is velocity near the surface of a sphere moving with its neighbors in a shear field required. Since an exact determination of the flow field is impractical to obtain for groups of more than two particles, an approximate solution of the Stokes equations obtained by the method of weighted residuals is used in the calculations of the suspension viscosity reported here. The particle configuration is assumed to be simple cubic, and the distribution of interparticle distances is obtained from an approximate statistical model. The viscosity-concentration relationship calculated under these assumptions is found to agree closely with the compilations of experimental data reported by Rutgers and Thomas.