Capillary–viscous forcing of surface waves

Abstract

The linear excitation of straight-crested, capillary–gravity waves on the surface (x > 0) of a deep, viscous liquid in response to the sinusoidal, vertical motion of a hydrophilic wall at x = 0 is calculated on the assumptions that: (i) the dynamical variation of the contact angle is proportional to (but not necessarily in phase with) the velocity of the contact line relative to the wall; (ii) the relative tangential velocity (slip) of the fluid below the contact line is proportional to the shear at the wall, (iii) k₀l_v [Lt ] 1 and k₀l_c = O(1), where k₀ is the wavenumber, l_v is the boundary-layer thickness, and l_c is the capillary length. The contact-angle and slip coefficients are complex functions of frequency that are found to be linearly related. Physical considerations suggest that the slip length l_s (≡ slip velocity ÷ shear at wall) should be small compared with l_v, which, in turn, implies that the motion of the contact line must be small in that parametric domain in which linearization provides a viable description of the wave motion near the wall; however, the analysis proceeds from (i) and (ii), qua phenomenological hypotheses, without a priori restrictions on the contact-line and slip coefficients. The present results include those of Wilson & Jones (1973), who assume that the amplitude and phase of the wave slope at the wall are prescribed, and those of Hocking (1987a), who assumes that the variation of the wave slope at the wall is in phase with the contact-line velocity and neglects viscosity. They also include a correction for the dynamical effects of the static meniscus, which is necessarily present for any static contact angle other than ½π but is neglected in the previous analyses, and have counterparts for the closely related problem (cf. Hocking 1987b) of the reflection of a plane wave from a stationary wall.