Linear response of a viscous liquid sheet to oscillatory external pressure perturbation in the long wave approximation. Varicose excitation

Summary.

In this paper, a plane viscous liquid sheet is considered in contact with two passive media, from which two external perturbations of small intensity, given by local pressure distributions, act on the sheet. Of the two possible ways of action of the sources of perturbation only the symmetric one is pursued here, for which the sheet reacts in the so called varicose mode. The action of the sources is taken either as instantaneous, or as persisting infinitely long. During the persistent action the sources are considered to move with constant velocity along the sheet and the perturbation is changing harmonically with circular frequency ω _e . In the paper, the linear response of the viscous sheet to the external perturbation is computed and analyzed. The starting point of the computation is a general formula for the exact linear response of the sheet as an inverse Fourier-Laplace transform, the derivation of which is reproduced in Appendix A of this paper. The formula shows the response to depend on several characteristic numbers, of which one is intrinsic to the unperturbed sheet, whereas the others also imply the properties of the source. The intrinsic number can be taken as in which the sheet thickness h, the density ρ of the liquid, its kinematic viscosity ν, and it surface tension γ with respect to the two ambient media enter. It is related by to the Ohnesorge number Oh. First the concept of the long wave approximation used is discussed in the context: it implies the neglect of all the hard branches ω _n (k),n=1,..., of spectral modes of the sheet, for which On the remaining two soft branches ω_±(k) of the spectrum, for which lim _k →0ω_±(k)=0, the asymptotic expansion of ω(k) fork→0 is considered and the whole branches are approximated by the (extrapolation of the) lowest order of the expansion. Finally, the whole Fourier integrand of the signal is substituted by its lowest order asymptotic expression aroundk=0. Within this approximation the response of the sheet to an instantaneous point perturbation is computed in its rest frame of reference. It is a long time asymptotic form of the exact response to the instantaneous point perturbation, the kernel of the general signal evolution of the sheet. The computed asymptotic form of this kernel bears a strong formal resemblance to the evolution kernel of the heat equation. The analogy is, however, not physical: whereas for Γ ²<4 the long wave modes of the two soft branches of the sheet spectrum are purely absorptive, like the modes of the heat equation, for Γ²>4 their modes are dispersive and dissipative waves, with their peculiar propagation properties. By adequate application of the superposition principle the responses to instantaneous line perturbation and to harmonically persistent perturbations by moving point and line sources are computed from the asymptotic evolution kernel. Because of the specific form of this kernel, the long time surface deflection responses to the harmonic excitations do not depend separately on the additional characteristic numbers ω _e andW=||/Γ (where ω _e and || are considered in units of 4ν/h² and 2ν/h, respectively), but only on their ratio ω _e /W². The properties of these signals in dependence on Γ and ω _e /W², as well as their implications for technical processes involving liquid sheets, like courtain coating, are discussed in detail.