Rectilinear plastic flow of a Bingham solid

The velocity distribution in steady rectilinear plastic flow of a Bingham solid between coaxial circular cylinders in relative motion can be obtained exactly by simple arguments. On the basis of the general equations of plastic flow, an approximate velocity distribution is derived below for the case of eccentric circular cylinders as rigid boundaries and zero pressure gradient by making use of a conformal transformation. This approximation is valid only if a dimensionless parameter S (which is proportional to the yield value) is sufficiently small, and there is flow in the whole region between the cylinders. The differences in velocity distribution in the cases of Bingham solid (S finite) and Newtonian liquid (S zero) are illustrated by a particular example. It is shown that the effect of a finite yield value is to cause the contours of equal velocity in a normal section to become more nearly concentric with the inner boundary as the yield value increases from zero.