Marangoni elasticity of flowing soap films

We measure the Marangoni elasticity of a flowing soap film to be 22 dyne/cm irrespective of its width, thickness, flow speed, or the bulk soap concentration. We perform this measurement by generating an oblique shock in the soap film and measuring the shock angle, flow speed and thickness. We postulate that the elasticity is constant because the film surface is crowded with soap molecules. Our method allows non-destructive measurement of flowing soap film elasticity, and the value 22 dyne/cm is likely applicable to other similarly constructed flowing soap films. ∗ [email protected] 1 ar X iv :1 61 0. 00 17 8v 1 [ ph ys ic s. fl udy n] 1 O ct 2 01 6 Stationary and flowing soap films are an ideal experimental device to simulate twodimensional (2D) flows. The development of a soap film channel as a scientific instrument[1– 5] expedited the exploration of fundamental physical fluid dynamics problems using 2D hydrodynamics. Applications include investigations of cylinder wake[6–9], the flow past flapping flags[10, 11], 2D decaying and forced turbulence[12–16] and 2D pipe flow[17]. The persistence of a freely suspended soap film and its mechanical stability is because of soap molecules acting as surfactants[18, 19]. The Marangoni effect arising from the surfactant imparts an elasticity E = 2A (dσ/dA) to the soap film; an increase in the area A of a patch of the film, which necessarily accompanies film thinning, causes the surfactant molecules to spread apart and the surface tension σ to increase. This increase provides a restoring force that tends to dynamically recover the original area of the film. In this manner, the same mechanism that stabilizes the soap film also imparts a compressible character to the 2D flow in the film. This compressible character is integral to, and therefore an unavoidable consequence of, the mechanism that stabilizes the film. The degree of compressibility is quantified by comparing the characteristic soap film flow speed u with the Marangoni wave speed vM = √ 2E/ρh, where ρ is the fluid density and h is its thickness [18–20]. If Ma = u/vM 1, then the inertial forces in the film are too weak to overcome the elastic forces and the film is assumed to approach incompressibility[21][22]. The objective of this paper is two-fold; to present a simple method to measure the Marangoni wave speed and to use the measured wave speed to characterize the film elasticity. Despite the widespread use of soap films for simulating 2D fluid system, vM or E is not typically measured or reported. It is desirable to monitor vM given that it may change with the operational parameters of the soap film and that it could be comparable to the typical velocity scale of the simulated 2D flows. Indeed, based on separate measurements of vM (246 to 362 cm/s [23]) and the typical flow speeds (150 to 250 cm/s[10], 100 to 400 cm/s[24] or 270 to 600 cm/s[25]), the soap film flows may not be assumed to be incompressible. However, techniques presented in the literature to measure vM [23, 26] are too cumbersome to be adopted for repeated real-time monitoring. We present a simple technique based on an analogy with compressible gas dynamics [23, 27–31] to measure the Marangoni wave speed. Our technique involves inserting a thin cylinder (a needle) in the soap film, and if required dragging it through, to generate an oblique shock. Then shock angle β is used to determine Ma of the incoming flow. Dragging