Interdisciplinary Applied Mathematics

tion regions, i.e., between the caps and the middle parallel section of the bubble. Balancing the curvatures in the transition region, we obtain that

x ос VRh.

On the other hand, using the momentum equation and balancing viscous stresses with pressure drop, we obtain the classical Bretherton scaling. Specifically,

pU    dp y

h?    dx    Rx

where the last equation defines the capillary number. Combining the above two equations, we obtain

h oc RCa^2/3 and x oc RCa^1/3.

Typical bubble speeds are in the range of 1 pm/s to 1 mm/s, and thus for aqueous solutions we have that the corresponding capillary number is 10~⁷ to 10~⁴. This, in turn, implies that using the above Bretherton scalings, the film thickness is about 10,000 times smaller then the radius of the micropipe R, while the transition layer is about 100 times less than R.

Bretherton has used asymptotic analysis to obtain accurate expressions for the above scalings including the coefficients. The starting point is to use the quasi-one-dimensional mass conservation equation

dh dQ dt dx

where Q = (Yh³h_xxx)/(3p) is the flowrate across the film obtained from the parabolic velocity profile. In the Lagrangian reference frame moving with the bubble    velocity    we    have    that    h_t    =    -Uh_x,    and    integrating    the    above

equation from the middle point (where h ^ h_TO) to the transition region, we obtain

Ca^-1h³h_xx = 3(h — h_x).

Defining now h = h/h^ and £ = (x/h_x)(3Ca)¹/³, we obtain the Bretherton equation

h³h_m = h — 1.    (8.22)

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