Thin-film flow of a viscoplastic material round a large horizontal stationary or rotating cylinder

We consider the steady two-dimensional thin-film flow of a viscoplastic material, modelled as a biviscosity fluid with a yield stress, round the outside of a large horizontal stationary or rotating cylinder. In both cases we determine the leading- order solution both when the ratio of the viscosities in the 'yielded' and 'unyielded' regions is of order unity and when this ratio approaches zero in the appropriate distinguished limit. When the viscosity ratio is of order unity the flow consists, in general, of a region of yielded fluid adjacent to the cylinder and a region of unyielded fluid adjacent to the free surface, separated by the yield surface. In the distinguished limit the flow consists, in general, of a region of yielded fluid adjacent to the cylinder whose stress is significantly above the yield stress and a pseudo-plug region adjacent to the free surface, in which the leading-order azimuthal component of velocity varies azimuthally but not radially, separated by the pseudo-yield surface; the pseudo-plug is itself, in general, divided by the yield surface into a region of yielded fluid whose stress is only just above the yield stress and a region of unyielded fluid adjacent to the free surface whose stress is significantly below the yield stress. The solution for a stationary cylinder represents a curtain of fluid with prescribed volume flux falling onto the top of and off at the bottom of the cylinder. If the flux is sufficiently small then the flow is unyielded everywhere, but when it exceeds a critical value there is a yielded region. In the distinguished limit the yielded region always extends all the way round the cylinder, but the unyielded region does so only when the flux is sufficiently small. For a rotating cylinder a film with finite thickness everywhere is possible only when the flux is sufficiently small. Depending on the value of the flux and the speed of rotation the flow may be unyielded everywhere, have a yielded region on the right of the cylinder only, or have yielded regions on both the right and left of the cylinder. At the critical maximum flux the maximum supportable weight of fluid on the cylinder is attained and the pseudo-yield, yield and free surfaces all have a corner. In the distinguished limit there are rigid plugs (absent in the stationary case) near the top and bottom of the cylinder