Surface Tension

As described in Chapter 4, 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

$$z = \zeta(x,t),$$ (1218)

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

$${\bf n} = \frac{\nabla(z-\zeta)}{\vert\nabla(z-\zeta)\vert}.$$ (1219)

It follows that

$$n_x \simeq -\frac{\partial\zeta}{\partial x},$$ (1220)

$$n_z \simeq 1.$$ (1221)

Now, the Young-Laplace Equation yields

$$\Delta p = \gamma\,\nabla\cdot{\bf n},$$ (1222)

where $\Delta p$ is the jump in pressure seen crossing the interface in the opposite direction to ${\bf n}$. See Section 4.2. However, from (1220) and (1221), we have

$$\nabla\cdot{\bf n} \simeq -\frac{\partial^2\zeta}{\partial x^2}.$$ (1223)

Hence, Equation (1222) gives

$$[p]_{z=0_-}^{z=0_+} = \gamma\,\frac{\partial^2\zeta}{\partial x^2}.$$ (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 $\rho$ 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 (1224) yields

$$\rho\,g\,\zeta - \left.p_1\right\vert _{z=0} = \gamma\,\frac{\partial^2\zeta}{\partial x^2},$$ (1225)

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 (1115), $p_1=-\rho\,(\partial \phi/\partial t)$. Hence, the above expression gives

$$g\left.\frac{\partial\phi}{\partial z}\right\vert _{z=0} + \... ...c{\partial^3\phi}{\partial z\,\partial^2 x}\right\vert _{z=0}.$$ (1226)

This relation, which is a generalization of Equation (1121), is the condition satisfied at a free surface in the presence of non-negligible 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

$$\omega^2 =\left(g\,k+\frac{\gamma\,k^3}{\rho}\right) \tanh(k\,d),$$ (1227)

which is a generalization of (1127) that takes surface tension into account.