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Surface Tension
As described in Chapter 4, there is a positive excess energy per unit area, , associated with an interface between two immiscible fluids. The quantity can also be interpreted as a surface tension. Let us now incorporate surface tension into
our analysis. Suppose that the interface lies at

(1218) 
where is small. Thus, the unperturbed interface corresponds to the plane . The unit normal to the interface is

(1219) 
It follows that
Now, the YoungLaplace Equation yields

(1222) 
where is the jump in pressure seen crossing the interface in the opposite direction to . See Section 4.2.
However, from (1220) and (1221), we have

(1223) 
Hence, Equation (1222) gives

(1224) 
This expression is the generalization of (1205) that takes surface tension into account.
Suppose that the interface in question is that between a body of water, of density and depth , and the atmosphere. Let the unperturbed water lie
between and , and let the unperturbed atmosphere occupy the region . In the limit in which the density
of the atmosphere is neglected, the pressure in the atmosphere takes the fixed value , whereas the pressure
just below the surface of the water is
. Here, is the pressure perturbation
due to the wave. The relation (1224) yields

(1225) 
where is the surface tension at an air/water interface. However,
,
where is the perturbed velocity potential of the water. Moreover, from (1115),
.
Hence, the above expression gives

(1226) 
This relation, which is a generalization of Equation (1121), is the condition satisfied at a free surface in the presence of
nonnegligible surface tension. Applying this boundary condition to the general solution, (1125) (which already satisfies the boundary
condition at the bottom), we obtain the dispersion relation

(1227) 
which is a generalization of (1127) that takes surface tension into account.
Next: Capillary Waves
Up: Waves in Incompressible Fluids
Previous: Steady Flow over a
Richard Fitzpatrick
20120427