Marangoni Shocks in Unobstructed Soap-Film Flows

Abstract

It is widely thought that in steady, gravity-driven, unobstructed soap-film flows, the velocity increases monotonically downstream. Here we show experimentally that the velocity increases, peaks, drops abruptly, then lessens gradually downstream. We argue theoretically and verify experimentally that the abrupt drop in velocity corresponds to a Marangoni shock, a type of shock related to the elasticity of the film. Marangoni shocks induce locally intense turbulent fluctuations and may help elucidate the mechanisms that produce two-dimensional turbulence away from boundaries.

Leading-Order Dynamics of Gravity-Driven Flows in Free-Standing Soap Films

“…1a). The Marangoni shock is represented with a dashed line; its location is determined using the Rayleigh's jump condition, u 0+ u 0− = U 2 M , where the subscripts + and − denote downstream and upstream of the shock, respectively [9]. Using the solution u 0 (x) we compute the other fields by solving equations (27)-(29).…”

“…Nevertheless, in the experimental setup of Fig. 1a there is insufficient time for such a diffusion to occur [9]. Thus, the concentration of soap molecules in the bulk remains constant, and the film is said to be in the "Marangoni elasticity regime," one of two possible limit regimes.…”

“…(In the experimental setup of Fig. 1a, U M and U are of comparable magnitude [9].) The relation between the Marangoni speed U M and the Marangoni elastic modulus E M , U M = √ (E M /2ρh), allows us to rewrite equation (6) in the form:…”

“…1 (a) A common experimental setup used to study the flow in a falling, free-standing soap film [5,6,9]. Wires WL and WR are thin nylon-fishing lines kept taut by weight W. The film hangs from the wires; its width increases from 0 to W over an expansion section of length l, then remains constant and equal to W over a measurement section of length L. Reservoir RT contains a soapy solution which flows through valve V and into the film.…”

“…1a, a free-standing soap film of average thickness H hangs between two long, vertical, mutually parallel wires a few centimeters apart from one another [5,6,9]. The film falls, driven by gravity, and a steady vertical flow of average velocity U soon becomes established within the bulk of the film.…”