Capillary waves with surface viscosity

Abstract

Experiments over the last 50 years have suggested a tentative correlation between the surface (shear) viscosity and the stability of a foam or emulsion. We examine this link theoretically using small-amplitude capillary waves in the presence of a surfactant solution of dilute concentrations where the associated Marangoni and surface viscosity effects are modelled via the Boussinesq-Scriven formulation. The resulting integro-differential initial value problem is solved analytically and surface viscosity is found to contribute an overall damping effect on the amplitude of the capillary wave with varying degrees depending on the lengthscale of the system. Numerically, we find the critical damping wavelength to increase for increasing surface concentration but the rate of increase remains different for both the surface viscosity and the Marangoni effect.

Transient structures in rupturing thin films: Marangoni-induced symmetry-breaking pattern formation in viscous fluids

“…Another interesting result of this analytical formulation is that the thin film naturally yields a power relation between film thickness and the instability wavelength  through the requirement that local fluid velocities are finite. In addition, the proximity of the critical instability wavelength  c to the critical damping wavelength of capillary waves,  c w , as observed experimentally with interferometry, together with previous literature on capillary waves in the presence of surfactants (29,30), prompts a relation between the two quantities. We expect the capillary wave theory together with a more sensitive treatment of transient film compositions to be instrumental in understanding film behavior and the nucleation of black spots (36) near the rupture process and that the capillary wave theory provides a useful vehicle through which we can understand the damping behaviors and the roughness of the interface in the neighborhood of the onset of the pattern-forming instability.…”

“…In absence of surfactants, the critical damping wavelength of the system, where the capillary wave transitions from an underdamped to an overdamped regime, is given by , where l vc = μ 2 /(ρσ) is the viscocapillary length scale ( 28 ) and ϵ ⋆ ≃ 1.3115 ( 29 ). For soap films under consideration, the presence of the surfactant solution damps the surface waves of the system and thus increases the value of to within the range of the interferometry techniques used to measure the thickness of thin films and the experimentally observed film thickness in the current study.…”

“…The dispersion relation of the system in the presence of surfactant solutions was recently ( 29 ) derived as where W 0 (ω′, ε) = 1 + (iω′ + 2ε) 2 + 4ε 2 (2 + Φ(ω′, ε)) 1/2 is the dispersion relation of the system in the absence of surfactant solutions, for ε = ν k 2 /ω 0 , β = a Γ 0 k /μω 0 , Φ(ω′, ε) = (1 + iω′/ε) 1/2 − 1, and ω′ = ω/ω 0 , where ω 0 = (σ k 3 /ρ) 1/2 is the frequency of capillary waves in an ideal fluid.…”

Transient structures in rupturing thin-films: Marangoni-induced symmetry-breaking pattern formation in viscous fluids

“…In absence of surfactants, the critical damping wavelength, where the wave transitions from an underdamped to an overdamped regime occurs, is given by λ w c = 2πl vc /ǫ ⋆2 [17], where l vc = µ 2 /(ρσ) is the viscocapillary lengthscale and ǫ ⋆ is the largest positive root [17] of f(ǫ) = 11ǫ 6 − 18ǫ 4 − ǫ 2 − 1 . For soap films under consideration, the presence of the surfactant solution damps the surface waves of the system [18] and thus propels the value of λ w c to within the range of the interferometry techniques employed to measure the thickness of thin films and, indeed, the experimentally observed film thickness in the current study.…”

“…Since the existence of real coefficients in the amplitude equation ( 1) is a necessary condition for the instability dynamics to manifest itself physically, this suggests the pattern-forming region overlaps with the overdamped capillary wave regime. In this overdamped regime, we require iω ∈ R where the general complex frequency ω = ω 1 + iω 2 ∈ iR appears in the dispersion relation of a viscous system with non-trivial surface tension and gravity [17,[19][20][21][22] given by…”