Linear response of a viscous liquid sheet to external pressure perturbation. Long wave approximations

Abstract

This paper is concerned with the linear signal response analysis of a thin viscous liquid sheet which is at rest in an appropriate frame of reference and in contact with passive external media, and on which localized external pressures act from the passive media as sources of perturbation. The frame of the analysis is provided by general formulae for the response signals of the sheet in the two excitation modes, sinuous and varicose, which result as the solution of the appropriate fluid dynamic initial-boundary value problems by the Fourier-Laplace transform technique. These formulae display how the signals depend on the nature of the perturbation and on the spectrum of the (linear) eigenmodes of the sheet. The signals can be evaluated either numerically or, as initiated in this paper, analytically, in long wave approximations. The long wave approximation will be seen in the sequel to concentrate on particular eigenmodes of the sheet spectrum, with small values k of the wave vector % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala % aaaa!36F0! $${\vec k}$$ along the sheet. The present paper is devoted mainly to a detailed analysis of the spectrum of eigenmodes of the sheet, and to the formulation of long wave approximations of (linear) response signals of the sheet in the context of this analysis. It turns out that the sheet spectrum of eigenmodes depends on only one characteristic number Γ, which depends on the (positive) fluid parameters: the surface tension γ at (both) the interfaces of the sheet, the density ρ of the fluid, its kinematic viscosity ν, and on the thickness h of the sheet: % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaey % ypa0ZaaOaaaeaadaWcaaqaaiabeo7aNjaadIgaaeaacaaIYaGaeqyW % diNaeqyVd42aaWbaaSqabeaacaaIYaaaaaaaaeqaaOGaaiOlaaaa!40E9! $$\Gamma = \sqrt {\frac{{\gamma h}} {{2\rho \nu ^2 }}} .$$ This number is closely related, by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaey % ypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaaaleqaaaaakiaa % d+eacaWGObWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaiilaaaa!3E58! $$\Gamma = \frac{1} {{\sqrt 2 }}Oh^{ - 1} ,$$ to the Ohnesorge number % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaadI % gacqGH9aqpdaWcaaqaaiabeY7aTbqaamaakaaabaGaeq4SdCMaeqyW % diNaamiAaaWcbeaaaaGccaGGSaaaaa!3FA4! $$Oh = \frac{\mu } {{\sqrt {\gamma \rho h} }},$$ where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey % ypa0JaeqyWdiNaeqyVd4gaaa!3C22! $$\mu = \rho \nu $$ is the dynamic viscosity of the liquid. It will be shown in the sequel that of the infinitely many branches of the sheet spectrum only two (pairs of) ’soft’ branches, one sinuous and one varicose, will be relevant for a long wave approximation. For these branches asymptotic expansions of the dispersion relations ω(k) between the (complex) mode frequencies ω and % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 % da9iaacYhaceWGRbGbaSaacaGG8baaaa!3AE6! $$k = |\vec k|$$ for long waves (i.e. for k → 0), which obey ω(k) → 0 with k → 0, will be derived. The analytic long wave dispersion relations for the (soft) sinuous and varicose spectral branches allow a very favourable insight into the qualitative mode behaviour, including analogies beyond liquid sheets. They show e.g. that to their lowest orders in k the soft sinuous modes are nondispersive, i.e. % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHeSaeq % yYdCNaeyyhIuRaam4AaiaacYcaaaa!3C5A! $$\Im \omega \propto k,$$ and only weakly absorptive, i.e. ℜ ω ∝ k2 for low viscosity and ℜω ∝ k4 for high viscosity (ℜω < 0). Their signals are therefore expected to have some resemblance to signals of a flexible membrane on the one side and those of a (2-dimensional) diffusion process, or a hyperdiffusion process where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaey % 4bIe9aaWbaaSqabeaacaqGYaaaaaaa!3943! $$ - \nabla ^{\text{2}} $$ is replaced by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaW % baaSqabeaacaaI0aaaaOGaaiOlaaaa!391B! $$\nabla ^4 .$$ The behaviour of the soft varicose modes will be seen to depend on the value of Γ: For Γ2 < 4 the asymptotic expansion for k → 0 gives $$ \Im \omega = {\text{0}} $$ and ℜω ∝ k2(ℜω 4 the situation is different: here % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHeSaeq % yYdCNaeyyhIuRaam4AamaaCaaaleqabaGaaGOmaaaaaaa!3C93! $$\Im \omega \propto k^2 $$ and ℜω ∝ k2 (ℜω < 0). This behaviour bears resemblance to the vibration modes of a thin elastic plate with (diffusion-like) damping. The asymptotic insight into the behaviour of the individual branches of spectral modes allows the derivation of rather transparent approximate analytic expressions for the response signals of the sheet to external perturbations.