Motion of a confined bubble in a shear-thinning liquid

The motion of a gaseous Taylor bubble in a capillary tube is typical of many biological and engineering systems, such as small-scale reactors and microfluidic devices. Although the dynamics of a bubble in a Newtonian liquid has been the subject of several studies since the seminal works of Taylor (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 161-165) and Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166-188), the case where the fluid exhibits a shear-thinning behaviour is much less understood. To fill this gap, we study the dynamics of a bubble that moves in a shear-thinning fluid whose viscosity is described by the Ellis viscosity model. With this aim, we derive a lubrication model in the film region to identify the scaling laws for the bubble speed, the film thickness and the pressure drop as a function of the Ellis number and the degree of shear thinning. Our model generalizes Bretherton's results to shear-thinning fluids by identification of a universal scaling law for the effective viscosity that accounts for the interplay of the zero-shear-rate and shear-thinning effects. The film thickness follows a scaling law with respect to the capillary number based on the proposed effective viscosity. The ratio between the bubble speed and the average velocity of the fluid ahead of the bubble is a function of the effective capillary number only. We show that some portions of the bubble are dominated by the zero-shear-rate effect discussing the extent to which the use of the power-law viscosity model can be legitimized. Finally, we study the location of the recirculating vortices ahead of the bubble.