Fundamentals of steady state, non-Newtonian rimming flow

Rimming flow on the inner surface of a horizontal rotating cylinder is investigated. Using a scale analysis, a theoretical description is obtained for steady-state non-Newtonian flow. Simple lubrication theory is applied since the Reynolds number is small and the liquid film is thin. Since the Deborah number is very small the flow is viscometric. The shear-thinning number, which characterizes the shear-thinning effect, may be small or large. A general constitutive law for this kind of flow requires only a single function relating shear stress and shear rate that corresponds to a generalized Newtonian liquid. For this case the run-off condition for rimming flow is derived. Provided the run-off condition is satisfied, the existence of a continuous steady-state solution is proved. The rheological models, which show Newtonian behavior at low shear rates with transition to power-law shear thinning at moderate shear rates, are considered. Numerical results are carried out for the Carreau and Ellis models, which exhibit Newtonian behavior near the free surface and power-law behavior near the wall of the rotating cylinder.