D. Halpern et Dp. Gaver, BOUNDARY-ELEMENT ANALYSIS OF THE TIME-DEPENDENT MOTION OF A SEMIINFINITE BUBBLE IN A CHANNEL, Journal of computational physics, 115(2), 1994, pp. 366-375
We present a boundary element method to investigate the time-dependent
translation of a two-dimensional bubble in a channel of width 2a cont
aining a fluid of viscosity mu and surface tension gamma. In our analy
sis, the flow rate, Q(), is specified, and the finger progresses forw
ard 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 t
o surface-tension forces. Steady-state results are given in terms of t
he 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 t
o 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. whos
e 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 t
he pressure drop is directly proportional to a viscous pressure scale.
A regression analysis of the finger width (beta) versus Ca-U yields b
eta approximate to 1 - 0.417(1 - Exp(- 1.69 Ca-U(0.5025))), which give
s 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 ver
y small (Ca-U < 0.02). The result of this regression analysis is consi
stent with Taylor's measurements of residual film thickness in circula
r tubes, which shows a Ca-U(1/2) dependence for values of Ca-U < 0.09.
(C) 1994 Academic Press, Inc.