Boundary conditions for plane flows of smooth, nearly elastic, circular disks

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Abstract
We consider plane flows of identical, smooth, nearly elastic, circular disks interacting with a boundary formed by attaching halves of similar disks at equal intervals along a flat wall. The roughness of the boundary is given in terms of the diameters of the two types of disks and the spacing of the wall disks. We suppose that the velocity distribution of the flow disks is Maxwellian and calculate the rates at which momentum and energy are supplied to the flow disks in collisions over a unit length of the boundary. At the boundary we balance these supplies with the stress and the total flux of energy in the flow and obtain boundary conditions on the shear stress, pressure, and flux of fluctuation energy. We find that the boundary can either supply fluctuation energy to the flow or absorb it, depending on the relative magnitudes of the rate of working of the boundary tractions through the slip velocity and the rate at which energy is dissipated in collisions. As an example we solve the boundary-value problem for the steady shearing flow maintained by the relative motion of parallel plates a fixed distance apart. When the dimensions and properties of the flow disks and the boundary are given, the specification of the distance between the plates and their relative velocity determines the slip velocity, the shear stress and pressure necessary to maintain the flow, and the distributions of mean velocity, fluctuation energy, and density.

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