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1 6 1 6 Section 7.8
Curved Interfaces
5
8
7
h

2 5 2
3 4 3 4
Figure 7.23
(a) (b) Capillary rise.

in the range 0° u 90° (Fig. 7.22a). When the cohesive forces exceed the adhesive
forces, then 90° u 180°.
Suppose that 0° u 90°. Figure 7.23a shows the situation immediately after a
capillary tube has been inserted into a wide dish of liquid b. Points 1 and 6 are at the
same height in phase a (which is commonly either air or vapor of liquid b), so P P .
1 6
Points 2 and 5 are located an equal distance below points 1 and 6 in phase a, so P P .
2 5
Points 2 and 3 are just above and just below the planar interface outside the capillary
tube, so P P . Hence, P P . Because the interface in the capillary tube is curved,
2 3 5 3
we know from (7.34) that P P P . Since P P , phase b is not in equilibrium,
4 5 3 4 3
and fluid will flow from the high-pressure region around point 3 into the low-pressure
region around point 4, causing fluid b to rise into the capillary tube.
The equilibrium condition is shown in Fig. 7.23b. Here, P P , and since points
1 6
8 and 5 are an equal distance below points 1 and 6, respectively, P P . Also, P P ,
8 5 3 4
since phase b is now in equilibrium. Subtraction gives P P P P . The pres-
8 3 5 4
sures P and P are equal, so
2 3
P P P P 1P P 2 1P P 2 (7.35)
7
7
5
5
4
4
8
2
where P was added and subtracted. Equation (1.9) gives P P r gh and P
7 2 8 a 4
P r gh, where r and r are the densities of phases a and b and h is the capillary
7 b a b
rise. Provided the capillary tube is narrow, the interface can be considered to be a seg-
ment of a sphere, and (7.34) gives P P 2g/R, where R is the sphere’s radius.
5 7
Substitution in (7.35) gives r gh 2g/R r gh and
a b
g 1 2 1r r 2ghR (7.36)
b
a
When phases b and a are a liquid and a gas, the contact angle on clean glass is
usually 0 (liquid Hg is an exception). For u 0, the liquid is said to wet the glass com-
pletely. With a zero contact angle and with a spherically shaped interface, the interface
is a hemisphere, and the radius R becomes equal to the radius r of the capillary tube R
(Fig. 7.24b). Here,
r
r
1
g 1r r 2ghr for u 0 (7.37)
r
a
2
b
(For a slightly more accurate equation, see Prob. 7.57.) For u 0, we see from
Fig. 7.24a that r R cos u, so g 1 2 (r r )ghr/cos u. Since contact angles are
b
a
hard to measure accurately, the capillary-rise method is only accurate when u 0.
For liquid mercury on glass, the liquid–vapor interface looks like Fig. 7.22b with (a) (b)
u 140°. Here, we get a capillary depression instead of a capillary rise.
Capillary action is familiar from such things as the spreading of a liquid dropped Figure 7.24
onto cloth. The spaces between the fibers of the cloth act as capillary tubes into which Contact angles: (a) u 0;
the liquid is drawn. When fabrics are made water-repellent, a chemical (for example, (b) u 0.