Rectilinear plastic flow of a Bingham solid

Abstract

It is apparent from the two previous papers of the same main title (1, 2) that velocity distributions in steady rectilinear plastic flow of a Bingham solid between moving boundaries are not easy to determine, even when the boundaries are of simple shape. In the familiar case of a Newtonian liquid (which can be regarded as a special case of a Bingham solid with zero yield value) the velocity ω is a harmonic function and can be obtained by a conformal transformation of the region of flow of the typeIn order to generalize and include materials of finite yield value, in which ω is not harmonic, one must first regard the Newtonian case from a slightly different point of view. The transformationmust be regarded as defining a change of coordinates, from Cartesian coordinates x, y to the natural curvilinear coordinates for a particular problem ω, W, one of which is the velocity. For a Bingham solid in general, it is shown in the present paper that natural coordinates ω, W exist, but are not obtainable by a conformal transformation from Cartesian coordinates, except in the limit when the yield value tends to zero. In practice it may be difficult to determine the natural coordinate system in a given problem, but when it is found the velocity distribution is automatically known.