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

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

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

It follows that

$$n_x \simeq -\frac{\partial\zeta}{\partial x},$$ (11.114)

$$n_z \simeq 1.$$ (11.115)

The Young-Laplace Equation (see Section 3.2) yields

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

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

Hence, Equation (11.116) gives

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

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

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

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

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