Non-Newtonian and non-isothermal effects in the gravity-driven draining of a vertically-aligned thin liquid film

Abstract

The drainage and thinning of liquid films are important in a variety of applications,such as in liquid and solid foam networks, relevant in the manufacture of metallic and ceramic foams, the food industry, processing in the petro-chemical industry, and the biological and life sciences. In a liquid foam network, there are gas bubbles separated by thin liquid lamellae. If one is interested in predicting the lifetime of a foam or its overall stability then, as a starting point, understanding the drainage within the lamella is important.

Motivated by the above, in this thesis, we consider a two-dimensional model system to investigate the draining and thinning of the lamella relevant to metallic and polymeric melts. Lubrication theory is employed to derive two master Partial Differential Equa-tions (PDEs) for a generalised Newtonian liquid describing the evolution of the film thickness and the extensional flow speed. The PDEs include the effects due to gravity, extensional viscous and surface tension forces. We use the non-Newtonian (Power-law and Carreau) and viscoplastic (Bingham and Herschel-Bulkley) constitutive laws to describe the flow rheology.

We first describe the evolution of a Newtonian liquid film in the limit of large Capillary number, Ca=ρ*g*L*2/(eγ*)>>1. We derive early and late-time similarity solutions for the draining and thinning of the lamella. A new power law thinning rate of t−2.25 in the lamella is identified at late times. This is in comparison to a thinning rate of t−2 predicted for a Newtonian film without gravity, suggesting a weak-dependence on gravity.

Next, we perform numerical simulations to investigate the influence of non-Newtonian and viscoplastic effects by varying the power-law index and the yield stress. We observe that the power law index and the yield stress affects the time scale of the thinning, but has weak dependence on the late-time thinning rate relative to the Newtonian thinning rate. We identify the limitations of the power-law model when the shear rate is low and how these can be resolved using the Carreau model.

We extend the Newtonian model to include non-isothermal effects, such as temperature-dependent viscosity and surface tension. We perform numerical simulations to describe the evolution for a variety of parameter values, such as the reduced P ́eclet number and those related to the exponential viscosity-temperature model and the linear surface tension-temperature model. Our results indicate that the resulting temperature drop in the film due to cooling from the free surface, particularly in the lamella, and the corresponding viscosity and surface tension contrast, significantly influence the draining and thinning of the film. Preliminary results show that the viscosity variation has greater influence compared to surface tension variations; however additional work is required to confirm this.

The new knowledge will enhance the current understanding to a wider class of thin liquid film draining flows associated with metallic and polymeric melts.