BOUNDARY-ELEMENT ANALYSIS OF THE TIME-DEPENDENT MOTION OF A SEMIINFINITE BUBBLE IN A CHANNEL

We present a boundary element method to investigate the time-dependent translation of a two-dimensional bubble in a channel of width 2a containing a fluid of viscosity mu and surface tension gamma. In our analysis, the flow rate, Q(*), is specified, and the finger progresses forward at a nonconstant velocity until it reaches a steady-state velocity U-*. The primary dimensionless parameter in the unsteady formulation is Ca-Q = mu Q(*)/2a gamma, representing the ratio of viscous forces to surface-tension forces. Steady-state results are given in terms of the conventional form of the capillary number, Ca-U = mu U-*/gamma. The steady-state shape of the finger, the pressure drop across the tip of the finger, and its radius of curvature are presented for a range of Ca-U much larger than has previously been published (0.05 less than or equal to Ca-U less than or equal to 10(4)). Good agreement is shown to exist with the finited-difference results of Reinelt and Saffman in the range of their studies (0.05 less than or equal to Ca-U less than or equal to 3), and with the experimental data of Tabeling et al. whose studies extend to Ca-U = 0.2. Beyond Ca-U = 20, we predict that the steady-state meniscus interface shape is insensitive to Ca, and that the pressure drop is directly proportional to a viscous pressure scale. A regression analysis of the finger width (beta) versus Ca-U yields beta approximate to 1 - 0.417(1 - Exp(- 1.69 Ca-U(0.5025))), which gives the correct behavior for both small and large Ca-U. This regression result may be considered an extension of the low-capillary asymptotic predictions of Bretherton, who found a Ca-U(2/3) dependence for Ca very small (Ca-U < 0.02). The result of this regression analysis is consistent with Taylor's measurements of residual film thickness in circular tubes, which shows a Ca-U(1/2) dependence for values of Ca-U < 0.09. (C) 1994 Academic Press, Inc.