THE MOTION OF LONG BUBBLES IN POLYGONAL CAPILLARIES .2. DRAG, FLUID PRESSURE AND FLUID-FLOW

This work determines the pressure-velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke and Morris 1995), we find that the drag scales as Ca-2/3 in the limit Ca-->0 (Ca = mu U/sigma, where mu is the liquid viscosity and sigma the surface tension). Thus, the pressure drop also scales as Ca-2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary. The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure-velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by sigma a(2)/mu, where alpha is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Q(c), whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Q(c) much less than 10(-6). For Q(c) much less than 1, the plug flow dominates, and the gradient in liquid pressure varies with Q(2/3). For Q much less than Q(c), the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.