PHYSICAL REVIEW LETTERS 121, 094501 (2018) Equilibrium Shapes and Their Stability for Liquid Films in Fast Flows Likhit Ganedi,1 Anand U. Oza,1,* Michael Shelley,1,2 and Leif Ristroph1,† 1Applied Math Lab, Courant Institute, New York University, New York 10012, USA 2Center for Computational Biology, Flatiron Institute, Simons Foundation, New York 10010, USA (Received 19 February 2018; published 31 August 2018) We study how a suspended liquid film is deformed by an external flow en route to forming a bubble through experiments and a model. We identify a family of nonminimal but stable equilibrium shapes for flow speeds up to a critical value beyond which the film inflates unstably, and the model accounts for the observed nonlinear deformations and forces. A saddle-node or fold bifurcation in the solution diagram suggests that bubble formation at high speeds results from the loss of equilibrium and at low speeds from the loss of stability for overly inflated shapes. DOI: 10.1103/PhysRevLett.121.094501 Soap films and bubbles are fascinating from many Here we consider the bubble-blowing problem in which perspectives, not the least being the physics and mathe- a film suspended across a ring is deformed by the uniform matics of their shapes. Dipping a wire frame in soapy water flow of an exterior fluid. Recalling Plateau’s use of yields a thin film that, due to surface tension, forms what immiscible liquid-liquid systems [5], we study oil films can be idealized as a surface of minimal area [1–3]. Its in flowing water, a system that offers advantages over soap shape is dictated by the Young-Laplace equation, which film in air by mitigating the effect of gravity, eliminating relates the pressure difference across an interface to its evaporation, and permitting precise flow control, condi- mean curvature [3,4]. The equality of pressures across an tioning, and visualization. We also numerically construct open film implies an equilibrium shape whose curvature is 2D equilibria of a film distended by an inviscid but everywhere zero, and identifying such minimal surfaces separated flow, with stable shapes qualitatively matching consistent with a boundary has been a classic problem since those seen in experiments and unstable shapes offering first studied by Lagrange and Plateau [5–7]. Similarly, a insight into how a bubble is blown. spherical bubble is a closed surface that minimizes the area Experiments.—We exploit a serendipitous discovery that for a given volume. How a bubble is first formed involves some common oils can form large and long-lived films the necking down and pinching off of a film. This event within water. By passing a wire loop through a layer of may be triggered, e.g., by separating two supports bridged olive oil and into water below, one forms films that last for by a liquid or film, as studied by Maxwell [8]. Recent work minutes before rupturing due to thinning by draining of the has focused on the collapse dynamics of fluid necks in the buoyant oil [25]. We employ olive oil as the interior fluid of 3 approach to pinch off [9–12] and on related topological density ρi 0.91 g=cm immersed in the exterior fluid of transitions in films [13,14]. ¼ Our more common experience of making bubbles by blowing on a film is less studied and involves unique (a) (d) complexities. This event has been visualized in recent velocimeter studies [15,16], and the competition of dynamic and Laplace pressures explains the critical flow speed needed region to form bubbles [17]. Still unresolved is the flow-induced test section reshaping of a film, which even in equilibrium involves laser sheet optics cameraU/U nonlinear deformations, incompressible flows into and (b) (c) * around the film, and flow separation that forms a wake. 1 These effects arise in other natural and industrial contexts, such as the ballooning and breakup of falling raindrops U 0.5 [18,19], water wave generation by wind [20], and the 0 production of foams, emulsions, and sprays [21,22]. The FIG. 1. Experiments. (a) Apparatus for studying the shape of, mutual influence of interface shape and flow is a feature and flow around, an oil film in flowing water. (b) Schematic. shared with many fluid-structure interactions [23,24], (c) Measured profiles of stable shapes for speeds U up to UÃ, at whose study often benefits from clean settings for a which the film ruptures. (d) Flow field near U=Uà 1 for a ring comparison of experiment and theory. of radius R 2 cm. ¼ ¼ 0031-9007=18=121(9)=094501(5) 094501-1 © 2018 American Physical Society PHYSICAL REVIEW LETTERS 121, 094501 (2018) 3 water, ρe 1.00 g=cm ; their coefficient of surface tension theory formulation [24,30,31], which uses conformal is γ 17¼ 1 dyn/cm, consistent with previous measure- mapping [32] to solve for the potential as well as the path ments¼ [26]Æ. The existence of stable films larger than the of the free or separation streamlines that shed tangentially capillary length, γ= ρe − ρi g 0.4 cm, suggests the from the pinning points and enclose a wake. Following presence of surfactantsð in the oil,Þ ¼ perhaps residuals from previous studies [23,24,33,34], the wake flow is not expli- processing [27]. Top studyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the response to flow, we devise an citly modeled, but rather the wake pressure is assumed apparatus that sits atop a water tunnel and allows for the uniform, consistent with experimental measurements of the repeatable formation of films on wire rings of radius back pressure on bodies at high Re [35]. The film is viewed R 1–3 cm. As shown in Figs. 1(a) and 1(b), the broad- as a 1D curve pinned to two points separated by a (vertical) side¼ of the ring faces the uniform oncoming flow of distance 2R and immersed within the exterior fluid of far-field speed U 1–10 cm=s that can be varied and density ρe ρ. The tension coefficient is γ, and the far-field measured. The external¼ flow is dominated by inertia at flow is horizontal¼ with speed U. The film is assumed static, these high Reynolds numbers: Re ρ UR=μ ∼ 103, and its equilibrium shape arises from the Young-Laplace ¼ e e where μe 1 cP is the viscosity of water. The flow-film law, which prescribes a local balance of differential hydro- interaction¼ involves hydrodynamic and surface tension dynamic pressure and the surface tension pressure asso- pressures, whose ratio is the Weber number We ∝ ciated with curvature. 2 ρeU R=γ ∼ 1, a scaled version of which will serve as We highlight key steps of the model formulation, and the control parameter for this study. supporting details are given in Supplemental Material [28]. When a film is placed within a flow of moderate speed, it Up-down symmetry permits focus on the half-space prob- is deformed from planar and settles to a dome- or para- lem, and the film profile is given by the tangent angle θ σ , ð Þ chutelike shape. The film then seems to be in stable where the parameter σ ∈ 0; π=2 traverses the film. The equilibrium: It persists for minutes and recovers its shape local flow velocity u along½ the film has a direction given by after impulsive perturbations to the support or the flow θ to ensure no penetration, and previous work [24] shows speed. Such shapes exist up to a critical flow speed UÃ, that the tangential speed u satisfies beyond which the film inflates unstably and ruptures (see j j u eτ σ ; where τ H θ 1 Supplemental movie [28]). In Fig. 1(c), we show measured j j¼ ð Þ ¼ ½ ðÞ profiles for a ring of radius R 2.0 cm for several values and H is the Hilbert transform. Crucially, this indicates that of U up to U 6.9 cm=s; 15¼ trials are shown at each of à ¼ the flow speed along the film can be computed from the seven speeds selected from a set of 16 speeds. The film is shape alone. A second flow-shape relationship is furnished nearly planar at low speeds and deforms into a hollow by combining the Young-Laplace and Bernoulli laws. The hemispherelike cavity as U → UÃ. However, the response former relates the local pressure difference across an is nonlinear in U, with the strongest changes occurring in interface to its mean curvature, which for a zero-pressure the immediate lead-up to rupture. Additional analysis of the wake and a film (two interfaces) yields Δp p 2γκ. The profile shapes is presented as Supplemental Material [28]. latter relates the pressure and speed in the¼ exterior¼ flow: Avisualizationstudyrevealsthecharacteroftheexternal 1 2 1 2 p 2 ρ u 2 ρU . Combining to eliminate p yields the flow. The photograph in Fig. 1(d) shows flow path lines dimensionlessþ j j ¼ curvature [29] as revealed by microparticles illuminated by a planar laser sheet shone near the equator of a highly deformed κR Rdθ=ds η 1 − u =U 2 ; η ρU2R=4γ; 2 film. Visual tracing of the incoming flow shows that only ¼ ¼ ½ ðj j Þ ¼ ð Þ fluid very near the midline enters the cavity, where it where s is the arc length and η is akin to the Weber number. abruptly slows down, as indicated by the short path lines. Recasting the above relationship in integral form yields This internal flow slowly reverses course, following the inner contour of the film before abruptly speeding up and π π=2 θ σ − Kη sinh τ σ0 sin 2σ0dσ0; 3 exiting near the support, where it separates from the surface ð Þ¼ 2 þ σ ½ ð Þ ð Þ to leave a large wake downstream.