Non-Newtonian and viscoplastic models of a vertically aligned thick liquid film draining due to gravity

Abstract

We consider theoretically the two-dimensional flow in a vertically aligned thick liquid film supported at the top and bottom by wire frames. The film gradually thins as the liquid drains due to gravity. We focus on investigating the influence of non-Newtonian and viscoplastic effects, such as shear thinning and yield stress, on the draining and thinning of the liquid film, important in metallic and polymeric melt films. Lubrication theory is employed to derive coupled equations for a generalized Newtonian liquid describing the evolution of the film's thickness and the extensional flow speed. We use the non-Newtonian (power-law and Carreau) and viscoplastic (Bingham and Herschel–Bulkley) constitutive laws to describe the flow rheology. Numerical solutions combined with asymptotic solutions predict the late-time power-law thinning rate of the middle section of the film. For a Newtonian liquid, a new power law thinning rate of t−2.25 is identified. This is in comparison with a thinning rate of t−2 predicted for a thin Newtonian liquid film neglecting gravity, suggesting a weak dependence on gravity for the drainage of thicker films. For a non-Newtonian and viscoplastic liquid, varying the power law index and the yield stress influences the timescale of the thinning, but has weak dependence on the late-time thinning rate relative to the Newtonian thinning rate. The shortcomings of the power-law model are exposed when the shear rate is low and these are resolved using the Carreau model.