Mass transport of interfacial waves in a two-layer fluid system

Abstract

Effects of viscous damping on mass transport velocity in a two-layer fluid system are studied. A temporally decaying small-amplitude interfacial wave is assumed to propagate in the fluids. The establishment and the decay of mean motions are considered as an initial-boundary-value problem. This transient problem is solved by using a Laplace transform with a numerical inversion. It is found that thin ‘second boundary layers’ are formed adjacent to the interfacial Stokes boundary layers. The thickness of these second boundary layers is of O(ε1/2) in the non-dimensional form, where ε is the dimensionless Stokes boundary layer thickness defined as $\epsilon = \hat{k}\hat{\delta}=\hat{k}(2\hat{v}/\hat{\sigma})^{1/2}$ for an interfacial wave with wave amplitude â, wavenumber $\hat{k}$ and frequency $\hat{\sigma}$ in a fluid with viscosity $\hat{v}$. Inside the second boundary layers there exists a strong steady streaming of O(α2ε−1/2), where $\alpha = \hat{k}\hat{a}$ is the surface wave slope. The mass transport velocity near the interface is much larger than that in a single-layer system, which is O(α2) (e.g. Longuet-Higgins 1953; Craik 1982). In the core regions outside the thin second boundary layers, the mass transport velocity is enhanced by the diffusion of the mean interfacial velocity and vorticity. Because of vertical diffusion and viscous damping of the mean interfacial vorticity, the ‘interfacial second boundary layers’ diminish as time increases. The mean motions eventually die out owing to viscous attenuation. The mass transport velocity profiles are very different from those obtained by Dore (1970, 1973) which ignored viscous attenuation. When a temporally decaying small-amplitude surface progressive wave is propagating in the system, the mean motions are found to be much less significant, O(α2).