The motion of long bubbles in polygonal capillaries. Part 1. Thin films

Abstract

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 (Ca ≡ μU/σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 [Lt ] x [Lt ] Ca−1, the film profile is frozen with a thickness of order Ca2/3 at the centre and order Ca at the sides. For x ∼ Ca−1, surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca4/3. For x [Gt ] Ca−1, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca−5/3, the height of the parabola is order Ca2/3. Finally, for x [Gt ] Ca−5/3, the height decreases as Ca1/4x−1/4.