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

Understanding the movement of drops and bubbles in microchannels isincreasingly important in the design and operation of microfluidicdevices that involve two-phase flows. Thus, Bretherton's analysis ofthe motion of long bubbles in tubes and the associated profile ofthe wetting film around it are relevant. In this work, steady motionof a long bubble through a cylindrical tube is revisited in the casewhere the wetting film between the bubble interface and thecapillary wall is non-Newtonian and described by the power-lawconstitutive relation. Using the standard lubrication analysis, theequation for the thickness of the wetting film as a function ofaxial distance is derived and integrated to find the film thickness.The film thickness and pressure drop across the bubble are found toscale with the capillary number as Ca^2/3, with a proportionality factor that depends on the power-law index.