Hydrodynamic Doppler effect in flowing soap films

TL;DR

This paper proposes that vorticity in flowing soap films propagates at the Marangoni wave speed, leading to a hydrodynamic Doppler effect, supported by theoretical modifications and experimental validation.

Contribution

It introduces a hypothesis linking vorticity propagation to Marangoni wave speed and demonstrates the resulting Doppler effect through theoretical modeling and experiments.

Findings

Vorticity propagates at Marangoni wave speed in soap films.

Periodic vortex arrays exhibit a redshift similar to Doppler effect.

Experimental results confirm the presence of the hydrodynamic Doppler effect.

Abstract

It is known for many years that the vorticity and the thickness fields in soap films are coupled and that the thickness wave propagates at the Marangoni wave speed. Based on the two observations, we propose a hypothesis that the vorticity field propagates at the Marangoni wave speed in flowing soap film channels. Using the hypothesis, we modify the point vortex model and find that the periodic array of vortices will have an expanded periodicity in the longitudinal direction, similar to the redshift of the Doppler effect. The theoretical prediction is compared with the experiments and we find that the hydrodynamic Doppler effect is present in soap film flows.

Equations

$${\nabla }^2\psi =\omega (z),$$

$$\psi (z,t)=\int G(z,z^{\prime })\omega (z^{\prime },t_r)d{z}^{\prime }.$$

$$\psi (z,t_r)=\frac{1}{2\pi }\frac{\log R}{1+{\vec{v}}^{\prime }\cdot {\hat{n}}^{\prime }/v_M}\int \omega (z_1,t_1)d{z}_1,$$

$$\psi (z,t)=\frac{\kappa }{2\pi }\frac{\log R}{1+M}.$$

$$q=h/\ell =\gamma q_0.$$

Full text

Experiments in physics take place in a model system. A model system is a simplified and approximated version of reality and is designed to capture the key physics under investigation. Ideally, the system is well understood and controlled to avoid unnecessary complication or undue uncertainties that may prevent us from a reproducible and consistent result.

This work concerns a soap film channel as a model system of a two-dimensional (2D) fluid. The soap film channel has been developed as a scientific instrument in late 1980s with seminal contributions by Couder et al. Couder et al. (1989), Beizaie and Gharib Beizaie and Gharib (1997), and Rutgers et al. Rutgers et al. (2001), and since then it has been used to study 2D hydrodynamics. In specific, soap film channels are used to investigate cylinder wakes Vorobieff and Ecke (1999); Roushan and Wu (2005); Kim and Wu (2015); Kim (2019), the flow past elastic structures Jung et al. (2006); Ristroph and Zhang (2008), and 2D pipe flows Tran et al. (2010); Cerbus et al. (2018). Most notably, a great advance has been made in our understanding of 2D decaying and forced turbulence Kellay et al. (1995); Martin et al. (1998), including an important observation of the inverse cascade Rutgers (1998).

Despite numerous success stories, a soap film channel remains as one of capricious systems that requires a great caution to obtain consistent result. It is widely acknowledged by the community that the variation of flow speed alters the outcome of experiments a great deal, and many research groups persistently keep the flow speed constant. It is speculated that the variation of film thickness, accompanied with the variation of flow speed, creates the divergence in results. The apparent correlation between the flow speed and the film thickness is yet to be unpacked Rutgers et al. (1996); Sane et al. (2018), but even if we fully unpack the mystery, the question still remains: why and how does the variation of the film thickness affect the flow structure?

To answer this question, we bring to light two independent experiments. One finding in a flowing soap film is that the vorticity field and the thickness field are strongly correlated. Rivera et al. Rivera et al. (1998) directly measured the vorticity field using the particle image velocimetry and the thickness field using light scattering. They have shown that the two fields are coupled and suggest that the thickness and the vorticity behave as passive scalar, and therefore the correlation persists downstream Vorobieff et al. (1999). Second finding is that the thickness field propagates at a certain speed called the Marangoni wave speed Kim and Mandre (2017). The stability of soap films is because of the elasticity $E$ that arises from the presence of surfactants, and this dynamic process of stabilization naturally produces a 2D-compressible wave Taylor (1959) travelling at the Marangoni wave speed $v_{M}=\sqrt{2E/\rho h}$ Lucassen et al. (1970); Vrij et al. (1970); Couder et al. (1989) where $E\approx 22\,\rm mN/m$ in a typical setup of soap film channels Kim and Mandre (2017).

Connecting the two findings, we hypothesize that the vorticity field is transmitted at the speed $v_{M}$ in a flowing soap film. In hydrodynamics, the characteristic velocity of the system is usually much slower than the propagation speed of the field because the high density of water renders both fast acoustic wave and high dynamic viscosity. In soap films, however, the 2D compressibility results in relatively slow propagation, normally at $v_{M}\sim 3\,\rm m/s$, compared to the typical flow speed of $u\sim 1.5\,\rm m/s$.

With the hypothesis in mind, we investigate whether the retardation effect creates a detectable change in the spatial arrangement of vortex street. In our theoretical investigation, we modify the point vortex model for finite propagation. In conventional framework, the flow is expressed as a gradient of a potential function $\phi$ and a stream function $\psi$, which together constitute an analytic function $W=\phi+i\psi$ in a complex plane. When the propagation is assumed to be instantaneous, the field is acquired by the integration using Green’s function. In our modified framework, we assume the weak compressibility to keep the conventional definition of the complex potential Crowdy and Krishnamurthy (2017), but we assume that the field is integrated over the source at the retarded time. It is inferred, as a corollary, that a vortex street is elongated in the longitudinal direction as $\it M\equiv{u}/v_{M}$ increases and produces an effect similar to the redshift of the Doppler effect.

The prediction of the theoretical model has confirmed by experiment. For vortex streets in soap films, we measure the ratio $q\equiv h/\ell$ between the transversal and longitudinal spacing $h$ and $\ell$ between vortices and find that $q$ decreases from 0.42 to 0.35 when $\it M$ is varied from 0.1 to 0.8. The experimental observation strongly indicates that the retardation effect is substantial in flowing soap films and is consistent with the theoretical prediction.

The experiment is carried out in an inclined soap film channel, as discussed in literature Wu et al. (2001); Georgiev and Vorobieff (2002); Kim and Wu (2015). Our channel is approximately 2 m long and inclined by 78∘ from the direction of the gravity. We use the soap solution made of 2% commercial dish-soap (P&G Dawn) and 98% of deionized water. The bulk kinematic viscosity is measured to be 0.0126 ${\rm cm^{2}/s}$.

The thickness and the mean flow speed of the soap film is varied by the flow rate of the soap solution. The continuity of the flow implies that the flow rate $F$ of the soap solution is conserved throughout the channel Tran et al. (2009); Kim and Mandre (2017), i.e. $F=Wu\delta$, where $W$, $u$ and $\delta$ are respectively the width, flow speed and thickness of the soap film. While we fix $W$ at 3.6 cm, we vary $F$ from 0.045 ${\rm{cm}^{3}/s}$ to 0.64 ${\rm{cm}^{3}/s}$, and at the same time $u$ is varied from 30 cm/s to 150 cm/s, measured by the particle tracking velocimetry. The mean thickness of the film is then derived from the relation $\delta=F/(Wu)$ and ranges from 3.5 $\mu\rm{m}$ to 13 $\mu\rm{m}$. For a few data points, we confirm that the estimate of $\delta$ agrees with the direct measurement using the transmittance of the polarized laser Kim and Wu (2010) within 10%.

The vortex streets are generated by inserting a circular cylinder into the soap film. The cylinder is made of titanium and is tapered up to the tip size 50 $\mu\rm{m}$. We vary the diameter $D$ of the cylinder in the soap film from 0.0064 to 0.083 cm by changing the insertion depth which we precisely measure by a long-distance microscope. In this study, the Reynolds number, defined by ${\rm Re}=UD/\nu$, ranges from 50 to 550.

Next, the flow patterns are visualized by a fast video camera (Phantom V5, Vision Research) at 1900 fps. The flow channel is illuminated by the low-pressure sodium lamp (wavelength: 589 mm), and the interferogram is captured by the video recording. As discussed earlier, the thickness variation is correlated with the vorticity variation Rivera et al. (1998), and therefore the interferogram provides us a direct means to measure flow structures without any post-processing of images.

Figure 1 shows the flow structure captured at various $u$ and $D$. From (a) to (e), $u=32$, 38, 52, 64 and 77 cm/s, and $D=0.056$, 0.047, 0.035, 0.027 and 0.024 cm, respectively. Even though $u$ and $D$ are varied, ${\rm Re}$ is controlled to be almost invariant at 138($\pm 4$). The red rectangles highlight the longitudinal and transversal spacing between vortices, $\ell$ and $h$, respectively.

In Fig. 2, we show the measurements of the $\ell$ and $h$ with respect to $D$. We find that both $\ell$ and $h$ are linearly proportional to $D$ as demonstrated in Refs. Roushan and Wu (2005); Kim and Wu (2015) after the data are grouped by their respective film thickness as presented in Fig. 2(a,c). The scaling relation follows the linear form $\ell=\ell_{0}(\delta)+\alpha D$ and $h=h_{0}(\delta)+\beta D$, where $\alpha$ and $\beta$ are the proportionality constants and $\ell_{0}$ and $h_{0}$ are the intercepts. The intercepts, both $\ell_{0}$ and $h_{0}$, are decreasing functions of $\delta$, as depicted in Fig. 2(b,d). For a soap film of $\delta\simeq 3\,\mu\rm{m}$ and $u\simeq 34\,\rm cm/s$, the intercepts can be as high as $\ell_{0}=0.27\,\rm cm$ and $h_{0}=0.13\,\rm cm$. We also calculate the values of the slopes: $\alpha=5.00\pm 0.15$ and $\beta=1.66\pm 0.12$. Using the above information, we infer that the ratio $q\equiv h/\ell$ can be as high as 0.48($\approx{h_{0}}/{\ell_{0}}$) for small $\delta$ and $D$, and that as small as 0.33($\approx\beta/\alpha$) in the other extreme. We remark that this range of $q$ is greater than the prediction of the original point vortex model von Kármán (1911); von Karman and Henze (2013); Saffman (1992), which implies that only vortex streets with $q\simeq 0.28$ are stable.

Finally, we obtain $q$ from the individual measurement of $\ell$ and $h$. The calculated $q$ is plotted in Fig. 3 with respect to a dimensionless number $\it M=u/v_{M}\propto\sqrt{\delta}u$, that is analogous to Mach number in aerodynamics. The data clearly indicate that the spatial arrangement of vortex streets, quantified by $q$, is affected by either $u$ or $\delta$, and that the change is characterized by the expansion of the periodic structure in the longitudinal direction.

To reveal the physical origin of the change, we develop a modified point vortex model which explores a hypothesis that the vorticity field propagates at the Marangoni wave speed $v_{M}$. This hypothesis is based on the experimentally observed facts: i) in soap films, it is well understood that the vorticity field is strongly correlated with the thickness field Rivera et al. (1998), and ii) it is known that the thickness field propagates at the Marangoni wave speed, which depends on the mean thickness $\delta$ and the elasticity $E$, that is, $v_{M}=\sqrt{2E/\rho\delta}$ Kim and Mandre (2017).

Throughout the current theoretical analysis we assume that the soap film is compressible, yet the compressibility is weak enough so that we can still use the point vortex model with a minimum modification. For such a flow, the stream function $\psi(z)$ of the flow satisfies the inhomogeneous Laplace equation

[TABLE]

where $\omega(z)$ is the vortex density function. In the conventional point vortex model, Eq. (1) is solved by using Green’s function, which in 2D is $G(z,z^{\prime})=(2\pi)^{-1}\log{|z-z^{\prime}|}$, and it follows that $\psi(z)=\int G(z,z^{\prime})\omega(z^{\prime})dz^{\prime}$. In the modified point vortex model, the integration is evaluated at the retarded time $t_{r}=t-|z-z^{\prime}|/v_{M}$ as

[TABLE]

For a vortex street drifting at $u$ along the real axis, $\omega(z,t)=\kappa\sum_{n}\delta(z-ut-nl)-\sum\kappa\delta(z-ut-nl-l-ih)$, where $\kappa$ is the circulation of each vortex.

Equation (2) is not easily integrated because $t_{r}$ is not fixed. We invoke the procedure to solve the Liénard-Wiechert potential in relativistic electromagnetic theory Jackson (1998); Reitz et al. (1992). First we choose a time $t_{1}$, which is the retarded time for some point in the interior of the vortex distribution. Next, we change the integration over $z^{\prime}$ to an integration over $z_{1}$. Then, Eq. (2) becomes

[TABLE]

where $R\hat{n}^{\prime}=z^{\prime}-z$ and $\vec{v}^{\prime}$ the velocity of the vortex distribution at $t_{1}$. The integration of Eq. (3) for a pair of vortex arrays is intractable, so, instead, we solve for a simpler case where a single vortex is moving at $u$ along the real axis, i.e. $\omega(z,t)=\kappa\delta(z-ut)$. Then we get

[TABLE]

To geometrically interpret Eq. (4), we use a diagram analogous to the light cone of relativity. Figure 4 depicts two vortices separated by $\ell$ traveling at the speed $u$ (a pair of thick solid lines). When the wave speed is $v_{M}$, the flow field at the origin (O) is determined by the location $x_{1}$ and $x_{2}$ of the vortices at the retarded time $t_{1}$ and $t_{2}$. The geometric analysis suggests that $t_{1}-t_{2}=(x_{2}-x_{1})/{v_{M}}=(\ell-x_{2}+x_{1})/{U}$, implying $(x_{2}-x_{1})=\ell\gamma$, where $\gamma=(1+M)^{-1}$. At the origin, the spatial arrangement of vortices is determined to satisfy $q_{0}=h/(x_{2}-x_{1})$, where $q_{0}$ is the value of $q$ when $M=0$. Therefore, $q$ is, in the lab frame,

[TABLE]

Physical interpretation regards this phenomenon similar to the redshift of the Doppler effect because it occurs only when the vortices recede from the origin.

We have compared the theoretical prediction of the modified point vortex model with experimental measurements. In Fig. 3, we evaluate Eq. (5) using $E=22\text{ mN/m}$ Kim and Mandre (2017) and $q_{0}=0.48$, and the calculation is plotted with the solid line. We find that the theoretical curve matches with data within the margin of error, and this overall trend shows that the retardation effect may be substantial in a soap film flow.

Before closing the discussion, we also review the possibility that the variation of $q$ is solely due to the change of the viscosity of the film which in effect is due to the change in its thickness. In literature, the viscosity of a soap film is described by the Trapeznikov relation $\eta=2\eta_{s}+\eta_{b}\delta$, where $\eta_{b}$ is the dynamic viscosity of the bulk liquid and $\eta_{s}$ is the contribution of the surface viscosity Prasad and Weeks (2009). The relation indicates that the dynamic viscosity is higher for thicker films. Contrary to literature Hooker (1936) which states that the increasing viscosity acts to increase $q$, our thicker film yields to lower $q$. Also, when the Trapeznikov relation is divided by the 2D density $\rho_{2}=\rho_{3}\delta$, it becomes the kinematic viscosity relation $\nu=2\nu_{s}+\nu_{b}$, which does not depend on thickness. This concludes that the effect of the viscosity is not present, or even if it does, not significant.

To summarize, we have proposed a hypothesis that the vorticity field propagates at the Marangoni wave speed in flowing soap films. This hypothesis is based on two experiments: one is that the vorticity field is coupled with the thickness field, and the other is that the thickness variation propagates at the Marangoni wave speed. Based on the hypothesis, we propose the retarded point vortex model to describe the morphological change of the vortex arrangement in flowing soap films. The model predicts that the longitudinal periodicity of the vortex shedding increases by a factor of $(1+u/v_{M})^{-1}$, as the flow speed $u$ of the soap film flow approaches the wave speed $v_{M}$. The model is compared with the experimental data using the cylinder wake, and we find that they agree within the margin of error.

The study suggests that the retardation effect of the field may present in soap film flows. If we regard the caveats of the study as true statements, the discussed hypothesis is indeed true. This new finding is potentially useful for improving measurement techniques in fluid experiments. If the current conclusion is determined to be false, then either postulate is false. I speculate that in such case the general belief that the vorticity field is captured by the interferogram is in peril and needs reconsideration.

In any case, the soap film channel should be considered as two-dimensional Navier-Stokes system only under a specific context.