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Surface Tension
As described in Chapter 3, 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
![$\displaystyle z = \zeta(x,t),$](img4134.png) |
(11.112) |
where
is small. Thus, the unperturbed interface corresponds to the plane
. The unit normal to the interface is
![$\displaystyle {\bf n} = \frac{\nabla(z-\zeta)}{\vert\nabla(z-\zeta)\vert}.$](img4135.png) |
(11.113) |
It follows that
The Young-Laplace Equation (see Section 3.2) yields
![$\displaystyle {\mit\Delta} p = \gamma\,\nabla\cdot{\bf n},$](img1107.png) |
(11.116) |
where
is the jump in pressure seen crossing the interface in the opposite direction to
.
However, from Equations (11.114) and (11.115), we have
![$\displaystyle \nabla\cdot{\bf n} \simeq -\frac{\partial^{\,2}\zeta}{\partial x^{\,2}}.$](img4140.png) |
(11.117) |
Hence, Equation (11.116) gives
![$\displaystyle [p]_{z=0_-}^{z=0_+} = \gamma\,\frac{\partial^{\,2}\zeta}{\partial x^{\,2}}.$](img4141.png) |
(11.118) |
This expression is the generalization of Equation (11.99) 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 (11.118) yields
![$\displaystyle \rho\,g\,\zeta - \left.p_1\right\vert _{z=0} = \gamma\,\frac{\partial^{\,2}\zeta}{\partial x^{\,2}},$](img4143.png) |
(11.119) |
where
is the surface tension at an air/water interface. However,
,
where
is the perturbed velocity potential of the water. Moreover, from Equation (11.9),
.
Hence, the previous expression gives
![$\displaystyle g\left.\frac{\partial\phi}{\partial z}\right\vert _{z=0} + \left....
...left.\frac{\partial^{\,3}\phi}{\partial z\,\partial^{\,2} x}\right\vert _{z=0}.$](img4146.png) |
(11.120) |
This relation, which is a generalization of Equation (11.15), is the condition satisfied at a free surface in the presence of
non-negligible surface tension. Applying this boundary condition to the general solution, (11.19) (which already satisfies the boundary
condition at the bottom), we obtain the dispersion relation
![$\displaystyle \omega^2 =\left(g\,k+\frac{\gamma\,k^{\,3}}{\rho}\right) \tanh(k\,d),$](img4147.png) |
(11.121) |
which is a generalization of Equation (11.21) that takes surface tension into account.
Next: Capillary Waves
Up: Waves in Incompressible Fluids
Previous: Steady Flow over a
Richard Fitzpatrick
2016-03-31