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Next: Capillary Waves Up: Waves in Incompressible Fluids Previous: Steady Flow over a


Surface Tension

As described in Chapter 3, there is a positive excess energy per unit area, $ \gamma$ , associated with an interface between two immiscible fluids. The quantity $ \gamma$ can also be interpreted as a surface tension. Let us now incorporate surface tension into our analysis. Suppose that the interface lies at

$\displaystyle z = \zeta(x,t),$ (11.112)

where $ \vert\zeta\vert$ is small. Thus, the unperturbed interface corresponds to the plane $ z=0$ . The unit normal to the interface is

$\displaystyle {\bf n} = \frac{\nabla(z-\zeta)}{\vert\nabla(z-\zeta)\vert}.$ (11.113)

It follows that

$\displaystyle n_x$ $\displaystyle \simeq -\frac{\partial\zeta}{\partial x},$ (11.114)
$\displaystyle n_z$ $\displaystyle \simeq 1.$ (11.115)

The Young-Laplace Equation (see Section 3.2) yields

$\displaystyle {\mit\Delta} p = \gamma\,\nabla\cdot{\bf n},$ (11.116)

where $ {\mit\Delta} p$ is the jump in pressure seen crossing the interface in the opposite direction to $ {\bf n}$ . However, from Equations (11.114) and (11.115), we have

$\displaystyle \nabla\cdot{\bf n} \simeq -\frac{\partial^{\,2}\zeta}{\partial x^{\,2}}.$ (11.117)

Hence, Equation (11.116) gives

$\displaystyle [p]_{z=0_-}^{z=0_+} = \gamma\,\frac{\partial^{\,2}\zeta}{\partial x^{\,2}}.$ (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 $ \rho$ and depth $ d$ , and the atmosphere. Let the unperturbed water lie between $ z=-d$ and $ z=0$ , and let the unperturbed atmosphere occupy the region $ z>0$ . In the limit in which the density of the atmosphere is neglected, the pressure in the atmosphere takes the fixed value $ p_0$ , whereas the pressure just below the surface of the water is $ p_0-\rho\,g\,\zeta+\left.p_1\right\vert _{z=0}$ . Here, $ p_1$ is the pressure perturbation due to the wave. The relation (11.118) yields

$\displaystyle \rho\,g\,\zeta - \left.p_1\right\vert _{z=0} = \gamma\,\frac{\partial^{\,2}\zeta}{\partial x^{\,2}},$ (11.119)

where $ \gamma$ is the surface tension at an air/water interface. However, $ \partial\zeta/\partial t = -(\partial\phi/\partial z)_{z=0}$ , where $ \phi$ is the perturbed velocity potential of the water. Moreover, from Equation (11.9), $ p_1=\rho\,(\partial \phi/\partial t)$ . Hence, the previous expression gives

$\displaystyle g\left.\frac{\partial\phi}{\partial z}\right\vert _{z=0} + \left....
...left.\frac{\partial^{\,3}\phi}{\partial z\,\partial^{\,2} x}\right\vert _{z=0}.$ (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

$\displaystyle \omega^2 =\left(g\,k+\frac{\gamma\,k^{\,3}}{\rho}\right) \tanh(k\,d),$ (11.121)

which is a generalization of Equation (11.21) that takes surface tension into account.


next up previous
Next: Capillary Waves Up: Waves in Incompressible Fluids Previous: Steady Flow over a
Richard Fitzpatrick 2016-01-22