MMarangoni elasticity of flowing soap films
Ildoo Kim ∗ and Shreyas Mandre School of Engineering, Brown University, Providence, Rhode Island 02912 (Dated: September 26, 2018)
Abstract
We measure the Marangoni elasticity of a flowing soap film to be 22 dyne/cm irrespective ofits width, thickness, flow speed, or the bulk soap concentration. We perform this measurementby generating an oblique shock in the soap film and measuring the shock angle, flow speed andthickness. We postulate that the elasticity is constant because the film surface is crowded withsoap 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] a r X i v : . [ phy s i c s . f l u - dyn ] O c t tationary and flowing soap films are an ideal experimental device to simulate two-dimensional (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 2Dhydrodynamics. Applications include investigations of cylinder wake[6–9], the flow pastflapping 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 becauseof soap molecules acting as surfactants[18, 19]. The Marangoni effect arising from thesurfactant imparts an elasticity E = 2 A ( 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 surfactantmolecules to spread apart and the surface tension σ to increase. This increase provides arestoring 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 the2D flow in the film. This compressible character is integral to, and therefore an unavoidableconsequence of, the mechanism that stabilizes the film. The degree of compressibility isquantified by comparing the characteristic soap film flow speed u with the Marangoni wavespeed v M = (cid:113) E/ρh , where ρ is the fluid density and h is its thickness [18–20]. If Ma = u/v M (cid:28) , then the inertial forces in the film are too weak to overcome the elastic forcesand the film is assumed to approach incompressibility[21][22].The objective of this paper is two-fold; to present a simple method to measure theMarangoni 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, v M or E is nottypically measured or reported. It is desirable to monitor v M given that it may change withthe operational parameters of the soap film and that it could be comparable to the typicalvelocity scale of the simulated 2D flows. Indeed, based on separate measurements of v M (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 v M [23, 26] are too cumbersometo 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 thincylinder (a needle) in the soap film, and if required dragging it through, to generate anoblique shock. Then shock angle β is used to determine Ma of the incoming flow. Dragging2he needle through the film increases the relative speed and allows the shock to form evenwhen the soap film flow is subcritical, which is analogous to subsonic flow. We have foundthat, with some practice, the needle can be dragged with bare hands, therefore no additionalexperimental setup is required.The relationship between β and Ma is derived from a simple geometric construction. Theshock is formed by the envelope of circular wavefronts, that originate at the obstacle, areadvected by the free stream u and expand at speed v M (see Fig 1(a)); this simple constructionleads to the relation sin β = 1Ma = v M u . (1)Thus, Ma may be estimated by measuring the oblique shock angle β , and an independentmeasurement of u yields v M . Eq. (1) is the special case of a more general α − β − Ma relation for oblique shocks formed around wedges of angle α [32]. According to the gasdynamic analogy, the surface tension acts analogous to gas pressure, the film thickness isanalogous to gas density, and the ratio Ma plays a role identical to that played by the Machnumber in compressible gas dynamics; then the soap film flow is shown to be analogous togas flow with the heat capacity ratio γ = 1 . The corresponding oblique shock relation dueto a wedge is presented in a previous study[29].We also present the Marangoni wave speed measured for soap films created using thecommonly used solutions of commercial detergent. We find that the Marangoni wave speedis between 330 and 200 cm/s as the film thickness varies from 4 to 11 µ m.Our measurement of v M allows us to conveniently probe the elasticity of soap films;we find that in our setup the soap film elasticity remains constant at E = 22 dyne / cm . Wepropose that the constant value of the elasticity is due to the overcrowding of soap moleculeson the film surface.Our soap film channel setup is similar to those previously used by various groups[4, 13, 24,33]. The channel is vertical and is approximately 1.8 m long and 3 to 6 cm wide dependingon experimental conditions. We use 2% solution of commercial dish soap (Dawn, P&G) indistilled water to form soap films. The soap solution flux F is controlled by adjusting thevalve opening at the top of the channel. The valve opening is calibrated to F by directlycollecting the solution per unit time and weighing them, and it gives the measurement of F up to 5% of uncertainty. We work in a section of the soap film where the thickness doesnot vary with downstream distance, far from any “hydraulic jump” that can form near the3 u' (a) (b) ut v m t β Figure 1. (Color online) (a) If a source of the wave is moved by ut for a time interval t and the waveis expanded by v M t for the same interval, a simple trigonometry shows the relation sin β = v M /u .(b) Typical oblique shock in a soap film flow. To enhance experimental range, the thin plate ismoved at u (cid:48) in a flow of speed u . Shock is formed at a sharply defined angle (the dashed line servesas a guide to the eye). end of the channel[24]. The flow speed u of the channel is determined by particle trackingusing high speed imaging (Photron SA4). Our measurements of u are compared to separateparticle image velocimetry (PIV), and the two measurements agree within 1%. Once F , thewidth W and u are determined, we can calculate the thickness of the film using h = q/u ,where q ≡ F/W is the flux per unit width. We take all of our measurements at the centerof the channel, although our method in principle can be used at any part of the soap filmchannel, provided that h is considered as local thickness where we measure u . Our separatemeasurements of h ( x ) using low pressure sodium lamp interferogram and u ( x ) using PIV,where x is the span-wise coordinate of the channel, reveal that q = h ( x ) u ( x ) is independentof x . This is also indicated by other studies[24, 34].In the usual experimental conditions, q is varied from 0.1 cm / s to 0.4 cm / s . Undersuch conditions, u varies from 250 to 330 cm/s, and h varies from 4 to 11 µ m . A simpledimensional analysis to balance the gravitational force and air friction implies u ∼ q / and h ∼ q / [34], and this is roughly consistent with our observations.To simulate a wedge of α = 0 , we place a thin plate in the middle of the soap film. Thethin plate is 0.4 cm long in longitudinal direction and 25 µ m wide in thickness. If the flowspeed is greater than the Marangoni wave speed, namely u > v M , an oblique shock is formedon both sides of the α = 0 wedge. Otherwise, when u < v M , no shock is observed; we then4 (cid:1) h (cid:13)(cid:8)(cid:4)(cid:9)(cid:1)(cid:17)(cid:1)(cid:6)(cid:4)(cid:7)(cid:1) m (cid:15)(cid:1) h (cid:13)(cid:10)(cid:4)(cid:6)(cid:1)(cid:17)(cid:1)(cid:6)(cid:4)(cid:7)(cid:1) m (cid:15)(cid:1) h (cid:13)(cid:11)(cid:4)(cid:6)(cid:1)(cid:17)(cid:1)(cid:6)(cid:4)(cid:7)(cid:1) m (cid:15)(cid:1) h (cid:13)(cid:12)(cid:4)(cid:11)(cid:1)(cid:17)(cid:1)(cid:6)(cid:4)(cid:7)(cid:1) m (cid:15) b v (cid:1)(cid:2)(cid:14)(cid:15)(cid:5)(cid:16)(cid:3) Figure 2. (Color online) ( / sin β ) vs. v at different film thicknesses. Eq. (1) suggests thatthese two quantities are linearly proportional to each other when the Marangoni wave speed v M is constant. Such linearity is observed when experimental data points are grouped by the filmthickness h . For a fixed h , ( / sin β ) is directly proportional to v with zero intercept, and the slopedecreases as h decreases. This relationship implies that the Marangoni wave speed increases as thefilm gets thinner. move the thin plate against the soap film flow along a translational stage at the speed u (cid:48) inthe lab frame. A simple Galilean transformation gives the relative speed v between the flowand the wedge as v = u + u (cid:48) . This technique grants us two important features: to observeoblique shocks when the flow is naturally subcritical, and to achieve a greater range of Ma .We note that our results are reproducible when the α = 0 wedge is replaced by a thinneedle. Unlike a thin plate shock, a thin needle shock is insensitive to the angle of attack.Therefore our scheme can be adopted at no cost, even without a translational stage.Figure 1(b) shows a typical oblique shock formed at an angle β relative the α = 0 wedge inserted in a soap film channel. Analyses of images like the one in Fig. 1(b) givethe measurements of β as a function of flow conditions. To reduce the uncertainty, themeasurements are repeated six times per flow condition.In the prescribed setup, the gasdynamic analogy in Eq. (1) implies a linear relationbetween (1 / sin β ) and the relative speed v , and we find that the linearity is observed onlywhen we group data by their corresponding film thickness. For example, data with h =5 . ± . µ m are grouped together and displayed in Fig. 2 as circles. As the corresponding5olid line indicates, (1 / sin β ) and v are linearly proportional to each other. By groupingsimilar cases for other film thicknesses, we find that v M is faster in thin films than in thickfilms. Fig. 2 also shows experimental data for h =7.0, 8.0 and 9.8 µ m . Here we find thatfor all cases the intercept is zero as expected, but the slope varies by the thickness. Theslope is the most gradual for the thinnest film (see circles in Fig. 2) and the steepest for thethickest film (upside-down triangles). In our model, the slope is reciprocal to v M , thereforeour observation implies that v M is faster in thinner films than in thicker films.Figure 3 shows our measurement of the Marangoni wave speed using v M = v sin β asa function of h . In our experiments, F , W , and v are independently varied, however themeasured v M depends only on h . All data points, each collected using different soap solutionflux ranging . ≤ F ≤ .
17 cm / s and and channel width ranging from ≤ W ≤ ,collapse into a single scaling relation v M ∝ h − / in the range for h spanning little less than adecade. This clear trend that v M is a function of h but not of F and W allows us to calculatethe soap film’s elasticity using v M = (cid:113) E/ρh [18–20]. The proportionality constant impliesthat the elasticity of our soap film is E = 22 ± / cm , being independent of h , F , and W , within our measurement error.The Marangoni elastic wave is not the only wave that can propagate through a soap film;the bending wave derived by Taylor[20] describes motion in which two interfaces of a filmmove together and moves with a speed v b = (cid:113) σ/ρh [18–20]. As the surface tension has thesame dimensions as elasticity, the bending wave speed has the same functional dependenceon the film thickness as the Marangoni wave speed. Using previous measurements[33] of σ (cid:39) . / cm , we plotted the resulting v b in Fig. 3. The distinction of the bendingwave speed from the Marangoni wave speed determined from the shock wave implies thatwe excite the Marangoni shock waves.Our experimental results furnish us with new insight into physics of flowing soap films.The soap film possesses Gibbs elasticity if sufficient time is available for perturbations in soapfilm concentrations to equilibrate, and possesses Marangoni elasticity otherwise. The Gibbselasticity E G = 2 RT c/ (1 + c b h ) [18–20, 35, 36] depends on h and the bulk concentration c b of surfactant in equilibrium with the surface concentration c . The Marangoni elasticity, E M = 2 RT c , depends on c but not on h or c b . We find that in the range of parameters weare able to establish the soap film, its elasticity does not depend on the soap film width orits thickness. The independence of the measured elasticity on film thickness implies that6
10 15 20150200250300350400450 v M ( c m / s ) h ( µ m) ! = E=
22 dyne/cm F , W F , W F , W F , W F , W F , W F , W F , W F , W F , W F , W F , W -1/2 Figure 3. (Color Online) The Marangoni wave speed v M vs. the film thickness h (closed symbols).The solid (blue) line shows v M ∼ h − . corresponding to E = 22 dyne / cm . The symbol andcolor stands for different flux and width settings: Flux F = 0 .
38 cm / s , F = 0 .
56 cm / s , F =0 .
65 cm / s , F = 0 .
75 cm / s , F = 0 .
85 cm / s , F = 0 .
95 cm / s and F = 1 .
17 cm / s . Width W = 3 cm , W = 3 . , W = 4 cm , W = 5 cm and W = 6 cm . Open circles show measurementof the bending wave speed[33], and the dashed line corresponds to σ = 32 . / cm . our flowing soap film possesses Marangoni elasticityFurthermore, we experiment with bulk soap concentrations of 1% and 4% in the overheadreservoir and by changing the size of the nozzle that feeds the soap solution to the film inan attempt to influence the soap film elasticity. We find no noticeable difference in ourobservation; such modifications vary the elasticity less than 4%, falling within the marginof error.The constant value for the Marangoni elasticity we measure and its independence onthe operational parameters of the flowing soap film implies one of two possibilities: theinterface is crowded with soap molecules or σ ∼ σ − E (ln c ) / in the parameter regimewe examine. The surface tension, σ ( c ) , is a function of surface soap concentration, and theaccompanying Marangoni elasticity is derived using dc/c = − dA/A to be E = − cdσ/dc .The first possibility is that in the parameter regime we explored, the soap molecules crowdthe interface leading to a limiting value c = c ∞ . The soap solution concentration in theoverhead reservoir, and as the solution flows out and forms the film, is above the criticalmicellar concentration, and the surface concentration of soap molecules rapidly approachesthe limiting value c ∞ . Consequently, the elasticity approaches the value of − cdσ/dc at7 Ma R e f e r e n c e s
Figure 4. Elastic Mach numbers are calculated for recent studies[7, 9, 10, 17, 24, 37–43]. Using thefilm thickness and E = 22 dyne / cm , the Marangoni wave speed is calculated and used to normalizedthe flow speed as they are specified in the articles. The bar graph shows the lower and upper limitof the Mach number in each studies. Horizontal lines indicate Ma = 0 . and 1.0 to guide readers. c = c ∞ . The alternative is that the form of σ ( c ) is such that the elasticity E is a constant,implying σ ∼ σ − E (ln c ) / . While we cannot strictly rule out the latter possibility, theformer is more likely because it is the simplest explanation consistent with the observations.We postulate that our observation of the constant elasticity can be generalized, giventhat most soap film channel setups reported in the literature used the same soap, the sameconcentration, similar flow rates, and comparable dimensions for the soap film. For suchpublished articles which also report the film thickness, we estimate the Marangoni wavespeed by assuming that the elasticity of any soap film channel is
22 dyne / cm . The range ofMach number is then calculated using the range of flow speed cited in each article[7, 9, 10,17, 24, 37–43]. Fig. 4 shows the estimated range of Ma for these studies, which indicates thatfor the vast majority of the cases the flow is clearly of a compressible nature. One study[24]recognized the compressible nature of the flow and used the Marangoni shock to estimatethe compressibility, while the others do not attempt to measure the compressible characterof the flow. Our method presents a non-intrusive and low cost method for estimating theMarangoni Mach number in situ for a complete characterization of flowing soap films infuture investigations. Furthermore, the value E = 22 dyne/cm may be used to determinethe Marangoni Mach number without any experimentation.To summarize, we provide an experimental method for in situ measurement of theMarangoni wave speed. In our method, we artificially generate oblique shocks in soap filmflows by introducing an obstruction, and determine the Marangoni Mach number by mea-8uring the shock angle. Our measurements show that the Marangoni wave speed dependson the film thickness, and the elasticity is constant for films in our range of experimentsindependent of film thickness, width, flow rate, or the bulk concentration of surfactants. Wesuspect that the elasticity is constant in our soap films because soap concentration is higherthan the critical micelle concentration. Considering that it is hard to establish a soap filmusing a dilute soap solution, we suspect that the reported value
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