Faraday instability in a linear viscoelastic fluid - Theoretische Physik I

170 EUROPHYSICS LETTERSl**in**ear **viscoelastic** model” is broadly accepted as an appropriate description for **in**compressible**viscoelastic** **fluid**s [12]. Nowadays quantitative measurements of the l**in**ear **viscoelastic** flowproperties are available by standard rheological techniques. Accord**in**gly, those experiments,for which the l**in**ear **viscoelastic** model applies, are expected to compare quantitatively withthe theory.The present paper deals with the onset of surface waves on a **viscoelastic** **fluid** layer subjectedto a vertical vibration, the so-called **Faraday** ** instability**. For Newtonian liquids

h. w. müller et al.: faraday **in**stability**in** a **viscoelastic** **fluid** 171driv**in**g if β = 0 and subharmonic (S) if β = 1. Then the marg**in**al stability criterion, σ g =0,leads to a tridiagonal system for the Fourier coefficients[ω20 − ω 2 + ω 2 X(k, ω) ] ˆη(k, ω) +a |k| [ˆη(k, ω +2Ω)+ˆη(k, ω − 2Ω)] = 0,2( ν ⋆ k 2 ) ( ν ⋆ k 2 ) 2 (√with X(k, ω) =4i +41+iω/(ν⋆ kωω2 ) − 1). (3)The solvability condition determ**in**es the neutral vibration amplitude a(k). A subsequentm**in**imization with respect to the lateral wave vector k yields the threshold a c . In eq. (3) wehave ω =2nΩ for the harmonic response and ω =(2n+ 1)Ω for the subharmonic one, wheren runs form −∞ to +∞. Furthermore, ω0 2 = g 0 | k | +(α/ρ) | k | 3 and the complex k**in**ematicviscosity isν ⋆ = ν ′ (ω) − iν ′′ (ω) =(1/ρ)∫ ∞0G(t)e −iωt dt. (4)The real and imag**in**ary parts of ν ⋆ account for the dissipative and the elastic character ofthe **fluid**. For a = 0, eq. (3) co**in**cides with the complex dispersion relation for free (=unforced)**viscoelastic** surface waves studied **in** [13]. The last term on the right-hand side of eq. (3) reflectsthe parametric drive and couples different temporal Fourier modes.Truncated solution. – The thresholds for the harmonic and the subharmonic branch arefairly good approximated by truncations of the system of equations (3). For the subharmonicbranch a two-mode truncation (n = −1, 0) leads to a driv**in**g force a (S) (k), which is m**in**imalat the wave number k S :a (S)c = a (S) (k S ) ≃ 2Ω2| k S | I[X(k S, Ω)] , where k S solves ω0(k 2 S )/Ω 2 + R[X(k S , Ω)] ≃ 1 . (5)R[X] andI[X] are the real and imag**in**ary part of X(k, Ω), respectively. For the harmonicresponse a three-mode truncation (n = −1, 0, 1) yields√a (H)c = a (H) 40 Ω ω(k H ) ≃ √ 0 (k H ) √I[X(kH , 2Ω)] , (6)6 | k H |with the critical wave number k H solv**in**g(√ )ω02 24Ω 2 + R[X(k H , 2Ω)] −3 I[X(k H, 2Ω)] ≃ 1. (7)L**in**ear **viscoelastic** materials. – The complex k**in**ematic viscosity ν ⋆ **in** many polymericsolutions exhibits an extended power-law region with ν ′ ,ν ′′ ∝ Ω −γ and γ around unity [12].The dissipative real part ν ′ starts at the zero-shear rate viscosity ν 0 = ν ′ (Ω → 0) and decaysby orders of magnitude to a saturation value at large frequencies. For low frequencies theimag**in**ary part is proportional to Ω but vanishes proportionally to Ω −1 at large frequencies.Hence the strongest **in**fluence of the fad**in**g memory is expected at **in**termediate frequencies,around the maximum of ν ′′ (Ω). This is the frequency range we are mostly **in**terested **in**. Afamiliar model which mimics the viscosity spectra of ν ′ and ν ′′ is the Maxwell model, whichadopts an exponentially decay**in**g memory, G(t) =(ν 0 /λ)e −t/λ . With eq. (4) one getsν ⋆ (Ω) =ν 01+iΩλ = ν 01+(Ωλ) 2 −iΩλ ν 01+(Ωλ) 2 , (8)

172 EUROPHYSICS LETTERSacceleration a/g 01005000 5 10Ω/π=20Hzλ=0(a) (b) 100(c)Ω/π=20HzΩλ=0.250Ω/π=100HzΩλ=10 5010 0 10 20wave number k [1/cm]Fig. 1. – The neutral curves a(k) of **Faraday** waves are given for the subharmonic (solid) and theharmonic (dashed) branches at different vibration frequencies, f =Ω/π. In part (a) a Newtonian**fluid** is considered for the same viscosity ν 0 =6×10 −4 m 2 /s and surface tension α =0.02 N/m as forthe Maxwell model with the relaxation time λ =(0.01/π) s **in** parts (b), (c).giv**in**g an easy two-parameter approach for the complex viscosity ν ⋆ . For real polymericsolutions the Maxwell model applies only qualitatively. However, it covers the pr**in**cipalfeatures of **viscoelastic** liquids, which are important for the effect considered here. Moreover,our results turn out to be **in**sensitive to the actual power laws. For a special class of aqueoussurfactant solutions [15] the Maxwell model applies even quantitatively.Numerical results. – The static viscosities ν 0 = ν ′ (Ω = 0) of polymer solutions often exceedthose of their Newtonian solvents by several orders of magnitude. In our numerical example weuse ν 0 =6×10 −4 m 2 /s (600 times that of pure water). Systematic **in**vestigations [16,11] revealthat the relaxation time λ of surfactant or polymeric solutions can be varied considerably bythe salt concentration, the viscosity of the solvent, or the temperature. For our calculation wechoose the value λ =(0.01/π) s, which can be conveniently probed **in** a **Faraday** experiment.The values α = 0.02 N/m for the surface tension and ρ = 1 g/cm 3 for the density aretypical for most **fluid**s. In our frequency–wave number range the follow**in**g **in**equalities hold:αk 3 /ρ ≪ ν0 2k4 and αk 3 /ρ ≪ 2ν 0 k 2 /λ. This parameter regime of negligible surface tensionis also addressed by Ple**in**er et al. [13]. They **in**vestigate surface waves without driv**in**g **in** thetransition range between elastic Rayleigh waves at Ωλ ≫ 1 and overdamped surface modes atΩλ ≪ 1. Here however, the supercritical parametric drive guarantees stand**in**g surface wavesover the whole parameter range, even **in** the overdamped region.In Newtonian **fluid**s (λ = 0) as **in** fig. 1a the neutral stability chart exhibits an alternat**in**gsuccession of neutral subharmonic and harmonic resonance tongues [5]. Viscoelasticity witha f**in**ite relaxation time λ =(0.01/π)s reduces the vibration amplitudes along the neutralcurves (cf. fig. 1b). Simultaneously, a new harmonic tongue develops. On **in**creas**in**g the drivefrequency to Ω ∝ 1/λ (while keep**in**g the rest of the parameters fixed) the new harmonic branchfalls down below the neighbor**in**g subharmonic one and determ**in**es the primary threshold (seefigs. 1c and 2a). Further elevation of the frequency beyond Ω ∼ 1/λ makes the subharmonictongue re-overtake the primary stability onset.The harmonic branch as the first ** instability** is

h. w. müller et al.: faraday **in**stability**in** a **viscoelastic** **fluid** 173critical acceleration a c /g 010080604020(a)a c(S) (subharmonic)a c (H) (harmonic)00 100 200 300vibration frequency f= Ω/π [Hz]critical acceleration a c /g 015010050(b)a c(S) (subharmonic)a c (H) (harmonic)00 100 200vibration frequency f= Ω/π [Hz]Fig. 2. – In part (a) the critical vibration amplitudes a (S)c and a (H)c for the subharmonic (thick solidl**in**e) and harmonic (dashed) **Faraday** **in**stability**in** a Maxwell **fluid** are shown as a function of thevibration frequency (parameters as **in** fig. 1). Th**in** l**in**es are the respective truncated solutionsaccord**in**g to eqs. (5)-(7). Part (b) shows the onset amplitude for the harmonic and subharmonic**Faraday** ** instability** calculated for the viscosity spectra of 1.5% polyacrylamide

174 EUROPHYSICS LETTERSdifferent values of α. S**in**ce harmonic waves are preferred below each curve **in** fig. 3b, largestatic viscosities and small relaxation times are favorable for a harmonic response.Conclusion. – The **Faraday** **in**stability**in** **viscoelastic** surface waves occurs **in** harmonicresonance with the external drive if the vibration frequency comes close to the **in**verse elasticrelaxation time. We have shown this **in** detail for Maxwell **fluid**s, but it is not a specific propertyof this model. Calculations with empiric viscosity data confirm this observation. A large staticviscosity, often met **in** polymeric solutions, and a small relaxation time are favorable for theobservation of the harmonic response. It should be emphasized that the predicted harmonicresponse is a **viscoelastic** bulk effect. This is **in** contrast to a similar observation **in** a th**in** layerof a Newtonian **fluid** due to **in**creas**in**g boundary layer dissipation [7].***Discussions with V. Ste**in**berg and H. Rehage are gratefully acknowledged.REFERENCES[1] Wesfreid J. E., Brand H. R., Manneville P., Alb**in**et G. and Boccara N., Propagation**in** Systems Far from Equilibrium (Spr**in**ger, Berl**in**) 1988; Cross M. C. and Hohenberg P. C.,Rev. Mod. Phys., 65 (1993) 851.[2] Busse F. H., **in**Hydrodynamic Instabilities and the Transition to Turbulence, edited by H. L.Sw**in**ney and J. P. Gollub (Spr**in**ger, New York) 1981.[3] DiPrima R. C. and Sw**in**ney H. L., **in**Hydrodynamic Instabilities and the Transition to Turbulence,edited by H. L. Sw**in**ney and J. P. Gollub (Spr**in**ger, New York) 1981.[4] Kudrolli A. and Gollub J. P., Physica D, 97 (1996) 133; B**in**ks D. and van de Water W.,Phys. Rev. Lett., 78 (1997) 4043; Chen P. and V**in**als J., Phys. Rev. Lett., 79 (1997) 2670;Zhang W. and V**in**als J., J. Fluid Mech., 336 (1997) 301; 341 (1997) 225.[5] Kumar K. and Tuckerman L. S., J. Fluid Mech., 279 (1994) 49; Kumar K., Proc. R. Soc.London, Ser. A, 452 (1996) 1113.[6] Cerda E. and Tirapegui E., Phys. Rev. Lett., 78 (1997) 859.[7] Müller H. W., Wittmer H., Wagner C., Albers J. and Knorr K., Phys. Rev. Lett., 78(1997) 2357.[8] Vest C. M. and Arpaci V. S., J. Fluid Mech., 36 (1969) 613; Sokolov M. and TannerR. I., Phys. Fluids, 15 (1972) 534.[9] Brand H. R. and Ziel**in**ska B. J. A., Phys. Rev. Lett., 57 (1986) 3167; Ziel**in**ska B. J. A.,Mukamel D. and Ste**in**berg V., Phys. Rev. E, 33 (1986) 1454.[10] Larson R. G., Shaqfeh E. S. and Muller S. J., J. Fluid Mech., 218 (1990) 537; LarsonR. G., Rheol. Acta, 31 (1992) 213.[11] Groisman A. and Ste**in**berg V., Phys. Rev. Lett., 77 (1996) 1480; 78 (1997) 1460; Elastic vs.**in**ertial **in**stability**in** Couette-Taylor flow of a polymer solution: Review, prepr**in**t.[12] Bird R. B., Armstrong R. C. and Hassager O., Dynamics of Polymeric Liquids, Vol. I(Wiley, New York) 1987.[13] Ple**in**er H., Harden J. L. and P**in**cus P., Europhys. Lett., 7 (1988) 383.[14] Huppler J. D., Ashare E. and Holmes L. A., Trans. Soc. Rheol., 11 (1967) 159. Theviscometric data are also presented on p. 118 of ref. [12].[15] Rehage H. and Hoffmann H., J. Phys. Chem., 92 (1988) 4712; Mol. Phys., 74 (1991) 933.[16] Fischer P. and Rehage H., Langmuir, 13 (1997) 7012.[17] Wagner C., Müller H. W. and Knorr K., **in** preparation.