On the gravity-driven draining of a rivulet of a viscoplastic material down a slowly varying substrate

We use the lubrication approximation to investigate the steady locally unidirectional gravity-driven draining of a thin rivulet of viscoplastic material, modeled as a biviscosity fluid (or, as a special case, as a Bingham material), down a slowly varying substrate. In contrast to the earlier work on viscoplastic rivulets we consider small-scale flows, such as those found in many industrial coating and printing processes, in which surface-tension effects play a significant role. We interpret our results as describing a slowly varying rivulet draining in the azimuthal direction from the top to the bottom of a large horizontal circular cylinder. Provided that the yield stress is nonzero we find that the flow is always unyielded near the top of the cylinder (where the rivulet becomes infinitely wide in the transverse direction), and, except in the special case when the viscosity ratio is zero, near the bottom of the cylinder (where it becomes infinitely deep). For sufficiently small values of the prescribed volume flux the flow is unyielded everywhere, but for larger values of the flux the flow near the substrate in the center of the rivulet is yielded. We obtain numerically calculated values of the semiwidth of the rivulet and of the yielded region as well as of the maximum height of the rivulet and of the yielded region for a range of parameter values, and describe the asymptotic behavior of the solution in the limits of large and small yield stress, large and small flux, and small viscosity ratio. In the special case of a Bingham material the flow near the top of the cylinder consists of an infinitely wide rigid and stationary plug, while elsewhere it consists of two rigid and stationary levees at the edges of the rivulet and a central region in which the flow near the free surface is a pseudoplug whose velocity does not vary normally to the substrate, separated from the fully plastic flow near the substrate by a pseudoyield surface. (C) 2002 American Institute of Physics.