Nonlinear generation of vorticity in thin smectic films
PPis’ma v ZhETF
Nonlinear generation of vorticity in thin smectic films
V. M. Parfenyev , S. S. Vergeles and V. V. Lebedev L.D.Landau Institute for Theoretical Physics RAS, 117940 Moscow, Russia
Submitted 11 December 2015
We analyze a solenoidal motion in a vertically vibrated freely suspended thin smectic film. We demon-strate analytically that transverse oscillations of the film generate two-dimensional vortices in the plane of thefilm owing to hydrodynamic nonlinearity. An explicit expression for the vorticity of the in-plane film motionin terms of the film displacement is obtained. The air around the film is proven to play a crucial role, since itchanges the dispersion relation of transverse oscillations and transmits viscous stresses to the film, modifyingits bending motion. We propose possible experimental observations enabling to check our predictions.
1. Introduction.
An interplay between vorticaland wave motions is one of the most exciting and long-standing problem in hydrodynamics. We analyze thisphenomenon for the case of freely suspended thin liquidfilms. On scales larger than the film thickness, such filmcan be considered as a two-dimensional system embed-ded into three-dimensional space. In comparison withtraditional two-dimensional systems, such films possessan additional degree of freedom associated with theirbending distortions. Therefore in addition to the usualin-plane hydrodynamic modes an extra mode can be ex-ited in the freely suspended films associated just withthe bending motion. The non-linear interaction of thebending motion with the in-plane hydrodynamic flowsgives rise to generation of the solenoidal in-plane motionby the bending waves. We examine the phenomenon inour work.One of widely known type of liquid films that canbe freely suspended are soap films. They are consideredas a model system providing an opportunity for testingtwo-dimensional hydrodynamic theory [1]. However thedetailed analysis has shown that the fluid motion in soapfilms is more complex and that a relation to motion de-scribed by the two-dimensional hydrodynamic equationsis not straightforward [2]. The generation of a vorticalmotion in vibrating soap films has a long history. Itwas first observed by Taylor more than a century ago[3]. Quantitative experiments were carried out at theend of the 20th century, see, e.g. [4, 5], and a quali-tative theoretical analysis is presented in the paper [6].To our knowledge there is no quantitative theory of thephenomenon nowadays.In our work, we consider freely suspended thin smec-tic films, see e.g. [7, 8]. Smectic liquid crystals areremarkable layered materials which can form a rich va-riety of structures. The simplest smectic structure isthe smectic A phase, which is solid-like in the direction e-mail: [email protected] perpendicular to the layers and fluid-like within the lay-ers. The smectic C and hexatic liquid crystals possessorientationally ordered layers. Any case, the smecticliquid crystal can be thought as a stack of fluid layers.Due to this layered structure thin films of the smecticliquid crystals consisted of a number of layers can beformed. Such films can be easily pulled from a reservoirof the substance. Due to the absence of the crystallineorder inside the smectic layers such films can be treatedas two-dimensional fluids. We develop a theory for thesimplest case of the smectic A films. However, it can beeasily generalized to the smectic C and hexatic films.In our mind there are no reports about the vor-ticity generation in oscillating smectic films up to thedate, while their transverse (bending) oscillations arewell studied [9, 10]. Fortunately, dynamics of the smec-tic films is simpler than one of the soap films since thereis no analog of Marangoni waves [5] in the case. It al-lows us to construct a quantitative theory of the phe-nomenon. We consider and compare two cases, the filmsurrounded by vacuum and by air. It turns out thatthe air environment plays an important role in the filmdynamics. The vortical fluid motion in the film appearsdue to nonlinear interaction of the bending waves, whichis taken into account in the framework of a perturbationtheory. Thus, our consideration is correct for sufficientlysmall amplitudes of transverse oscillations. We also dis-cuss qualitatively the case of strong non-linearity.
2. Film in vacuum.
Let us describe dynamicalproperties of a freely suspended thin film, pulled fromthe bulk smectic A phase. Such film is an isotropictwo-dimensional system, as we treat it on scales largerthan the film thickness. We assume that in equi-librium the film is parallel to the X − Y plane andthat its bending distortions are characterized by thedisplacement h ( t, x, y ) of the film in the Z -direction,i.e. the film shape is determined by the equation z = h ( t, x, y ). The unit vector perpendicular to the1 a r X i v : . [ phy s i c s . f l u - dyn ] D ec V. M. Parfenyev, S. S. Vergeles and V. V. Lebedev film is l = ( − ∂ x h, − ∂ y h, / √ g , where g = 1 + ( ∇ h ) can be thought as the determinant of the film metrictensor.The film state is described in terms of the film massdensity ρ ( t, x, y ) and of the film momentum density j ( t, x, y ). Both these quantities are two-dimensionaldensities of the film projection to the X − Y plane. Thedynamic equations for the film displacement h and forthe mass density ρ are [10] ρ∂ t h = j z − j α ∂ α h, ∂ t ρ = − ∂ α j α . (1)Here and below Greek indices run over x, y . The firstequation in (1) is the kinematic condition implying thatthe film moves with the velocity v = j /ρ , and the sec-ond equation is the mass conservation law.The equations (1) should be supplemented by theequation for the momentum density j of the film. Firstwe consider the film surrounded by vacuum. In the casethe momentum is conserved, the corresponding equationreads [10] ∂ t j i = − ∂ α (cid:0) v α j i − √ gσδ ⊥ iα − √ gη iαβm ∂ β v m (cid:1) , (2)where Latin subscripts run over x, y, z and δ ⊥ ik ≡ δ ik − l i l k stands for the projector to the film, and σ is its sur-face tension. As above, all quantities are assumed to befunctions of t, x, y . The viscosity tensor can be writtenas [10] η iklm = ( ζ − η ) δ ⊥ ik δ ⊥ lm + η ( δ ⊥ il δ ⊥ km + δ ⊥ im δ ⊥ kl ) . (3)Here η > ζ > |∇ h | (cid:28)
1. All other quantities characterizing the filmare assumed to deviate weakly from their equilibriumvalues as well. Then one can use the perturbationseries in examining the film distortions. In the lin-ear approximation there are two sound-like modes [16].The first one is the bending sound associated with thebending distortions of the film. In the mode, eachfilm element oscillates in the direction transverse to thefilm. The bending sound propagates with the velocity c b = (cid:112) σ /ρ , where ρ and σ are equilibrium values ofthe two-dimensional mass density ρ and of the surfacetension σ , respectively. The second mode is the longi-tudinal sound, which is associated with fluctuations ofthe two-dimensional mass density ρ and does not dis-turb the film shape. The sound propagates with thevelocity c l = (cid:112) − ∂σ/∂ρ . Further we assume that the longitudinal sound is not excited by the pumping forcedirectly.The linear equation for the bending sound reads ∂ t h + 2ˆ α∂ t h − c b ∇ h = 0 , (4)where the operator ˆ α determines the sound attenuation.If the freely suspended film is surrounded by vacuumthen the bending mode has an anomalously weak atten-uation, the property is related to the rotational symme-try of the system [10]. Particularly, there is no contri-bution to the attenuation ˆ α caused by the film viscosity(3). There are some other contributions to the atten-uation. First of all, there is a contribution of higherorder in ∇ , proportional to ∇ . Second, there is a con-tribution related to the thermal fluctuations [10], it iscaused by the non-linear interaction of the fluctuations.However, for films of the thickness of about hundredlayers, we have in mind, the fluctuation contribution isrelatively weak. In addition, there is a contribution tothe bending mode attenuation ˆ α related to processesin the meniscus. The contribution needs an additionalinvestigation and it is beyond the scope of present work.Our goal is to describe the in-plane vortical(solenoidal) motion generated by the bending motionowing to non-linear effects. The in-plane solenoidalmotion is characterized by the vertical componentof the vorticity (cid:36) z = ∂ x v y − ∂ y v x . In the linearapproximation (cid:36) z is zero, that is one should go beyondthe linear approximation to find (cid:36) z . We calculate themain nonlinear contribution to (cid:36) z , which is of thesecond order in the film displacement h . The startingpoint of the calculation is the equation Eq. (2), oneshould take its curl and project it to the Z -direction.Then one obtains an equation for (cid:36) z : ρ ∂ t (cid:36) z − η ∇ (cid:36) z = σ (cid:15) βγ ∂ β h∂ γ ∇ h + η(cid:15) βγ ∂ β ∂ t ( ∂ γ h ∇ h ) , (5)where (cid:15) αβ is the unit two-dimensional antisymmetrictensor.Let us analyze a steady contribution to the vorticity (cid:36) z . Averaging over time the equation (5), one finds ∇ (cid:36) z = ( σ /η ) (cid:15) βγ (cid:104) ∂ γ h∂ β ∇ h (cid:105) , (6)where angle brackets designate time averaging. At thenext step we use the linear equation for the bendingsound (4) to obtain (cid:36) z = 2( ρ /η ) (cid:15) βγ ∇ − (cid:104) ∂ γ h∂ β ∂ t ˆ αh (cid:105) . (7)Let us stress that the expression (7) is proportional tothe attenuation of the bending sound. onlinear generation of vorticity in thin smectic films
3. Air Environment.
Here we consider the casewhere the freely suspended smectic film is surroundedby air and analyze its influence to the film dynamics. Aspreviously, we assume that some pumping force excitesthe bending oscillations. One can think about soundas the pumping source, the corresponding experimentaltechnique is described in Ref. [11]. It is well knownthat the air around the soap film considerably changesits dynamics. Inspired by papers [5, 6], we consider aninfluence of air to the motion of the thin smectic filmand, particular, to the mechanism of the vorticity gen-eration.We assume that the air sound velocity c a is muchlarger than c b . It is reasonably to expect since c b = σ /ρ is inversely proportional to the number of thesmectic layers in the film, and we have in mind the filmconsisting of about hundred layers. At the condition c a (cid:29) c b the air motion can be described in terms of thethree-dimensional Navier-Stokes equation [12], supple-mented by the incompressibility condition div v = 0,where v is the air velocity.The film separates two regions of space filled withair. In the presence of air, Eq. (2) has to be modified,since the air influences the liquid motion in the film.The stress tensor in the air is σ ik = − pδ ik + ρ a ν a ( ∂ k v i + ∂ i v k ) , (8)where ρ a and ν a are the air mass density and its kine-matic viscosity coefficient, respectively, and p is pres-sure. Then the correct dynamic equation for the film is[12] ∂ t j i = − ∂ α (cid:0) v α j i − √ gσδ ⊥ iα − √ gη iαβm ∂ β v m (cid:1) + √ g ( σ IIik − σ Iik ) l k , (9)where the unit vector l , normal to the film, is pointedout from the region I ( z < h ) to the region II ( z > h ).Note that the velocity of air v is continuous in the wholespace and it coincides with the film velocity at the filmsurface. The equation (9) can be treated as the bound-ary condition for the three-dimensional air motion.The air motion around the film can be easily an-alyzed in the linear approximation. Exploiting theNavier-Stokes equation and the boundary condition (1),one obtains [16] v α = ∓ ν a ˆ κ (ˆ κ + ˆ k )ˆ k (cid:16) e ∓ ˆ kz − e ∓ ˆ κz (cid:17) ∂ α h, (10) v z = ν a (ˆ κ + ˆ k ) (cid:16) ˆ κe ∓ ˆ kz − ˆ ke ∓ ˆ κz (cid:17) h, (11)where the upper/lower signs correspond to the regionsII/I and we have introduced the following (non-local) operators ˆ k = ( − ∂ x − ∂ y ) / , ˆ κ = ( ∂ t /ν a + ˆ k ) / . Thefirst terms in Eqs. (10, 11) correspond to the potentialpart of the velocity, while the second terms belong to itssolenoidal part. Further we assume that the air viscousterm is weak in comparison with the frequency ω of theexternal force, that is γ ≡ (cid:112) ν a k /ω (cid:28) , (12)where k is the wave vector of the bending mode excitedin the film. Then the vortical (solenoidal) velocity islocated in a much thinner layer of depth 1 /κ ∼ γ/ | k | than the potential one, penetrating to the length | k | − .Hereinafter we assume that the bending mode decayis supplied mainly by the air viscosity, that is correct if | k | − is much larger than the film thickness. Then thebending dispersion law is [16] ω = ω (cid:18) − iγ √ (cid:19) , ω = σ | k | ρ + 2 ρ a / | k | , (13)where Θ = (1 + ρ | k | / ρ a ) − . The term 2 ρ a / | k | corre-sponds to the associated mass of air involved into thebending motion of the film. The factor 1 / | k | in 2 ρ a / | k | stands for the penetration depth of the air potential ve-locity. The expression (13) is obtained at the condition(12).Now we turn to the vorticity (cid:36) = curl v . Thebulk equation can be obtained by taking the curl of theNavier-Stokes equation [12], it is ∂ t (cid:36) = ( (cid:36) ∇ ) v − ( v ∇ ) (cid:36) + ν a ∇ (cid:36) . (14)In the linear approximation the vertical component ofthe vorticity, (cid:36) z , is zero, it is generated due to nonlin-earity. As before, we are interested in a contribution to (cid:36) z which is independent of time. Using Eq. (14) andaveraging over time one finds( ∂ z − ˆ k ) (cid:36) z = − ν − a (cid:104) (cid:36) α ∂ α v z (cid:105) . (15)The term in the right hand side of the equation is asource for (cid:36) z . The equation (15) has to be supple-mented by the boundary condition (9). The internalfilm viscosity η can be estimated as η ∼ η s d , where d isthe thickness of the film and η s is the dynamic viscositycoefficient of the bulk smectic. Thus the term with thefilm viscosity can be neglected for sufficiently thin films, ρ a ν a /η (cid:29) | k | . Then, using Eq. (9), we obtain (cid:10) ( ∂ z (cid:36) z ) II − ( ∂ z (cid:36) z ) I (cid:11) = 0 . (16)The condition (16) should be posed at z = 0. A solutionof Eqs. (15, 16) is (cid:36) z = (cid:15) αβ (cid:28)(cid:18) ˆ κ ˆ k e ∓ ˆ κz ∂ α h (cid:19) e ∓ ˆ kz ∂ β ∂ t h (cid:29) +( ν a ˆ k ) − e ∓ ˆ kz (cid:15) αβ (cid:68) (ˆ k − ∂ β ∂ t h ) ∂ α ∂ t h (cid:69) . (17) V. M. Parfenyev, S. S. Vergeles and V. V. Lebedev
Similar to the case of the film surrounded by vacuum,the last term in the expression (17) is non-zero only ifone takes into account the attenuation of the bendingmode. For this reason, in calculations we have kept twofirst terms of the expansion in parameter γ . As a resultwe have obtained the first term in the expression (17).Details of the calculations can be found in [16].
4. Discussion.
We considered the case, wherebending oscillations of the freely suspended smectic filmare excited by an external pumping. We had in mind thesituation where the pumping has frequencies in a narrowfrequency range. Then some steady vortical (solenoidal)motions are excited in the film due to non-linear effects.We developed a quantitative theory enabling one to ob-tain an explicit expressions (7) or (17) for the vorticityof the film surrounded by vacuum and by air, respec-tively. Note that the expression (17) is correct whenthe air surrounding the film essentially influences thefilm dynamics. Particularly, the bending dispersion lawis strongly modified in comparison with the sound one ω = (cid:112) σ /ρ k , see Eq. (13).Now we analyze the expression (7) and (17) to calcu-late the vertical steady vorticity (cid:36) z of the film for somespecific case, where the film is bounded by a rectangu-lar frame with dimensions L x and L y and a monochro-matic pumping is applied to the film. Then the modesof the system are standing waves, which have zero dis-placement at the edges of the frame. Let us considera superposition of two standing bending waves excitedby the external pumping. Thus, we assume the filmdisplacement to be h = H sin( k x x ) sin( k y y ) cos( ωt )++ H sin( q x x ) sin( q y y ) cos( ωt + φ ) , (18)where k x + k y = q x + q y = k , and the absolute value ofthe wave vector k is determined by the frequency ω ofthe external force via the resonance condition ω = c b k or (13).Taking into account the attenuation of the mode,and using Eq. (7), we obtain the steady vorticity at thefilm embedded in vacuum (cid:36) z = 2 αρ η ωH H ˆ k − sin φ × (cid:104) k y q x sin( k x x ) sin( q y y ) cos( q x x ) cos( k y y ) −− k x q y cos( k x x ) cos( q y y ) sin( q x x ) sin( k y y ) (cid:105) . (19)Above we have discussed different mechanisms con-tributed to the attenuation constant α . Because ofuncertainty in the relative effectiveness of the mecha-nisms, we do not know a dependence of the constant The spatial distribution of the steady vorticity (cid:36) z fortwo standing waves (18). For all cases the second di-mension of the frame L y = π . The other parametersare shown in the figure. α on the wave vector k . The obtained vorticity (cid:36) z isproportional to the attenuation constant, therefore weconclude that the experimental study of the generatedvorticity would provide a useful information concerningthe attenuation of the transverse oscillations of the filmsurrounded by vacuum.In the case of the film embedded in air, we can obtainthe attenuation constant from the dispersion law (13), α = ωγ Θ / √
2, and using then Eq. (17) we find the filmvorticity (cid:36) z (0) = ωH H γ √ (cid:18) | k | ˆ k − (cid:19) sin φ × (cid:104) k y q x sin( k x x ) sin( q y y ) cos( q x x ) cos( k y y ) −− k x q y cos( k x x ) cos( q y y ) sin( q x x ) sin( k y y ) (cid:105) . (20)Let us stress, that both answers, (19) and (20), are pro-portional to sin φ , where φ is a phase shift between thestanding waves.Next we consider the film stretched in a nearlysquare frame. It this case k x = q y , k y = q x and we dealwith two degenerate modes. If the frame is perfectlysquare, then sin φ = 0 and there is no steady vorticity.However, slightly changing the aspect ratio of the frameone can produce the phase shift between the modes andthe vertical vorticity becomes non-zero. When pass-ing through the ‘point of compensation’ the vorticitychanges its sign. Some possible spatial vorticity distri-butions are presented in Fig. 1. They were obtainednumerically from the Eq. (20) in the limit Θ →
1. The onlinear generation of vorticity in thin smectic films v (2) , have to be taken into ac-count. It follows from Eq. (17) that v (2) ∼ ωkh /γ .Therefore the nonlinear terms with v (2) are small if( v (2) ∇ ) (cid:36) z (cid:28) ν ∆ (cid:36) z . Thus, we arrive at the condition kh (cid:28) γ / , that is stronger than the small tilt condition kh (cid:28)
5. Conclusion.
To summarize, we examined thenonlinear mechanism of the vorticity generation in thinfreely suspended smectic A films. We considered twocases, where the film is surrounded by vacuum and byair. In the first case the generated vorticity appears tobe sensitive to the attenuation of the bending sound.We obtained the explicit formula (7) for the steady vor-ticity and speculated that an experimental study of thevorticity would provide a useful information about theanomalously weak attenuation of the bending sound.In the second case we found that the air considerablychanges dynamics of the bending mode and affects themechanism of the vorticity generation. We obtained themodified dispersion law (13) and the explicit formula(17) for the vertical vorticity. We analyzed the formulaand formulated some predictions which can be checkedexperimentally.Though our final answers (7) and (17) are obtainedfor the case of monochromatic pumping, our theoreticalscheme can be used for an arbitrary time dependenceof pumping. Particularly, one can think about pumpingcontaining two close frequencies. Then some beatingeffects in the generated vorticity are anticipated. Theeffects can be useful for experimental observations andto an experimental verification of our predictions.Note that the applicability condition of our theoryis |∇ h | (cid:28) γ / , where |∇ h | is the film tilt and γ is thesmall parameter of the theory (12). If the tilt |∇ h | be-comes larger than γ / then the applicability conditionof our theory that is weakness of nonlinear effects in thefilm is violated. Then one expects a strong non-linearityof the solenoidal motions exited in the film. That couldlead to formation of an analog of the inverse cascadeand to appearing of large coherent vortices in the film,like in Refs. [13, 14, 15]. Let us stress that the effectcan be observed even for a weak tilt |∇ h | .We have developed our theory for the case of thesmectic A films that are isotropic. However, practicallyall the obtained results can be carried to the case of the orientationally ordered films of the smectics C andof the hexatic smectics due to their weak anisotropy.Note that for such cases the vortical in-plane motionwould generate some non-trivial orientational patternsthat could help to investigate the motion.We are grateful to S. Yablonskii and E. Kats forvaluable discussions. This work was funded by RussianScience Foundation, Grant No. 14-22-00259.
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