Interdisciplinary Applied Mathematics

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mm; thus all microbubbles are spherical. Similarly, for a silicone oil droplet we have that    ls    «    1.5    mm, and    thus    micron-size    oil    droplets    are spherical


as well. The Bond number defined as


(8.4)



FIGURE 8.3. Surface tension balance at equilibrium.


L2


where L is a characteristic length (here the bubble diameter), determines the relative importance of gravity. For Bo < 10~3 the effects of gravity can be neglected.


When gravity is important, the condition for equilibrium at points belonging to the interface is


pgz y



1 1


b[ + R~2



C,



(8.5)


which expresses hydrostatic equilibrium with C a constant. The radii of curvature are taken positive if the centers of curvature are located on the gas side of the interface. This equation is useful in studying the rise of a liquid in a micropipe partially immersed in a liquid. In particular, depending on


(a)    (b)


FIGURE 8.4. A sketch illustrating the definition of the contact angle. Panel (a) corresponds to a wetting surface (в < 90°), and panel (b) corresponds to a nonwetting surface (в > 90°).


the liquid-pipe    interaction,    the free    surface    of the    liquid    may    rise    or    fall


into the pipe. This phenomenon is called capillarity. We can compute the height of liquid rise H by a simple hydrostatic balance following the sketch of Figure 8.2. The hydrostatic pressure is Ap = pgH, and for a micropipe of radius r we assume that the radius of curvature is R = r/ cos в, where в is the contact angle; see below. Therefore, equilibrium implies that


pgH

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