Gas displacing liquids from non-circular tubes: high capillary number flow of a shear-thinning liquid

Transient, three-dimensional finite element analysis has been used to investigate the displacement of a shear thinning liquid from prismatic channels of square, rectangular and trapezoidal cross sections. Inertia, gravity and surface tension effects are neglected and the results therefore apply in the limits of low Reynolds and high capillary numbers. The analysis is carried out in a fixed frame of reference and gas penetration is modelled as the bubble moves down the tube, which is long relative to its transverse dimension. Results are provided for the thickness of the layers left on the channel walls under developed conditions, and the fraction of the cross section occupied by liquid, as a function of the channel cross-sectional geometry and the degree of shear thinning, modelled using the power law. Interface contours on the channel cross sections are displayed. It is found for the Newtonian liquid that the fingering instability arises in the rectangular channel when the aspect ratio reaches about five. Shear thinning delays the onset of the instability to higher aspect ratios. The results are systematized, and insights gained into the influence of channel geometry and shear thinning, by noting a qualitative, inverse relationship between the deposited layer thickness and the shear rate at the wall in the flow ahead of the bubble.