Boundary Element Analysis of the Time-Dependent Motion of a Semi-infinite 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 μ and surface tension γ. 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 Cau = μQ*/2aγ , 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, Cau = μU*/γ. 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 Cau much larger than has previously been published (0.05 ≤ Cau ≤ 104). Good agreement is shown to exist with the finite-difference results of Reinelt and Saffman in the range of their studies (0.05 ≤ Cau ≤ 3), and with the experimental data of Tabeling et al. whose studies extend to Cau = 0.2. Beyond Cau = 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 (β) versus Cau yields β ≈ 1 - 0.417(1 - Exp(- 1.69 Ca0.5025u )), which gives the correct behavior for both small and large Cau. This regression result may be considered an extension of the low-capillary asymptotic predictions of Bretherton, who found a Ca2/3u dependence for Ca very small (Cau < 0.02). The result of this regression analysis is consistent with Taylor's measurements of residual film thickness in circular tubes, which shows a Ca1/2u dependence for values of Cau < 0.09.