Similarly, a fluid body with a closed surface on which no net external
forces act will always adopt a spherical shape. This was already concluded in
Section 10.1 on thermodynamic grounds; it can yet also be shown by
invoking the Laplace pressure. Figure 10.21 shows a spherical drop that is
deformed to give a prolate ellipsoid. Near the ends of the long axis (near a)
we havepL¼ 2 g/Ra, whereas near b,pL¼g/(1/R1,bþ1/R2,b). The latter
value is the smaller one, since bothR1,b andR2,b are larger thanRa.
Consequently liquid will flow from the pointed ends to the middle of the
drop, until a spherical shape is attained. Only for a sphere is the Laplace
pressure everywhere the same.
A general conclusion then is that, in a fluid body with a closed surface,
the fluid always wants to move from regions with a high curvature and
hence highpLto those of a lowpL. This also holds for a fluid body that is
partly confined, as for instance an amount of water in an irregularly-shaped
solid body. Only by applying external forces can gradients in pL be
established. The larger the equilibrium value ofpL, the higher the external
stress needed to obtain nonequilibrium shapes.
10.5.2 Capillarity
Curved fluid surfaces can give rise to a number of capillary phenomena. It is
well known that water rises in a narrow or ‘‘capillary’’ glass tube (if the glass
FIGURE10.21 Illustration of the increase in Laplace pressure when a spherical
drop (or bubble) is deformed into a prolate ellipsoid. Cross sections are shown in
thick lines; the axis of revolution of the ellipsoid is in the horizontal direction. Two
tangent circles to the ellipse are also drawn.