Interdisciplinary Applied Mathematics

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This nonlinear equation can be solved either analytically or numerically, and its solutions need to be matched with the static solutions of the Laplace-Young equation (8.2) at the ends of the transition region, i.e., at the two caps. The solution is different at the two limits, i.e., as £ ^ ±oo, and this asymmetry at the back and front caps produces the pressure difference that drives the bubbles into the wetting annular liquid film.


The    above    solution    gives    the    film    thickness    at    the    midsection    of    the


bubble h= 0.64R(3Ca)2/3, and also the pressure difference between the back and front caps


Ap = 10 —Ca2/3.


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This equation has been modified for large values of capillary number, 10-1 > Ca > 10-2, in (Ratulowski and Chang, 1989), as follows:


Ap = — [10Ca2/3 — 12.6Ca°’95],


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but this correction is negligible for microfluidic applications, where the capillary number is very small. The pressure estimate helps us in quantifying the length scales in applications in which a train of spacer bubbles is used to transport slugs    or    drops    of    liquid    drugs.    Clearly,    the pressure    drop    Ap


along the length L of the liquid slug is equal to the difference between the pressure at the front cap and the pressure at the back of the leading bubble, which is given by the above expression. We also have that


— oc => L ос ДСа~1/3,


L    R2


so the above scaling shows that the size of the liquid slug is about 100 times the size of the micropipe radius. The aforementioned scalings are realizable in experiments. However, surfactant contaminants, which are particularly effective at low Ca, may affect the results. In (Ratulowski and Chang, 1990), a comparison between theory and experiments is presented, and corrections due to Marangoni traction at the liquid-air interface are proposed. Also, for noncircular cross-section, the Bretherton-Chang theoretical solutions are not valid, and numerical solutions should be obtained using the formulation outlined in (Ratulowski and Chang, 1989).

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