Confined motion of a long bubble through a power-law fluid

Understanding the movement of drops and bubbles in microchannels is increasingly important in the design and operation of microfluidic devices that involve two-phase flows. Thus, Bretherton’s analysis of the motion of long bubbles in tubes and the associated profile of the wetting film around it are relevant. In this work, steady motion of a long bubble through a cylindrical tube is revisited in the case where the wetting film between the bubble interface and the capillary wall is non-Newtonian and described by the power-law constitutive relation. Using the standard lubrication analysis, the equation for the thickness of the wetting film as a function of axial distance is derived and integrated to find the film thickness. The film thickness and pressure drop across the bubble are found to scale with the capillary number as Ca 2/3 , with a proportionality fac