PHYSICAL REVIEW E 88, 063002 (2013)
Dual role of gravity on the Faraday threshold for immiscible viscous layers W. Batson,1,2,* F. Zoueshtiagh,2 and R. Narayanan1 1
Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA 2 Institut d’Electronique de Microelectronic et de Nanotechnologie (IEMN) UMR CNRS 8520, University of Lille 1, Ave Poincar´e, CS 60069-59652 Villeneuve d’Ascq, France (Received 15 January 2013; revised manuscript received 6 October 2013; published 2 December 2013) This work discusses the role of gravity on the Faraday instability, and the differences one can expect to observe in a low-gravity experiment when compared to an earth-based system. These differences are discussed in the context of the viscous linear theory for laterally infinite systems, and a surprising result of the analysis is the existence of a crossover frequency where an interface in low gravity switches from being less to more stable than an earth-based system. We propose this crossover exists in all Faraday systems, and the frequency at which it occurs is shown to be strongly influenced by layer height. In presenting these results physical explanations are provided for the behavior of the predicted forcing amplitude thresholds and wave number selection. DOI: 10.1103/PhysRevE.88.063002
PACS number(s): 47.20.−k, 68.05.−n, 68.03.−g, 47.54.−r
I. INTRODUCTION
Interfacial instability occurs when a bilayer of two fluids, a light fluid on top of a heavy one, is subject to a time-dependent acceleration in a direction perpendicular to their common interface. This instability, termed the Faraday instability [1] and manifested by a deflecting interface with associated patterns, is a result of a parametric resonance between the imposed frequency of excitation and the natural frequency. The natural frequency depends on the interfacial tension and the product of the density difference with normal gravity. A change in the level of gravity therefore changes the resonant condition. The motivation for the current work arises in the mixing in microfluidic geometries [2] in low gravity via induced Faraday waves. The point that will be made is that reducing gravity does not necessarily produce destabilization. Indeed there is a crossover frequency, where for lower frequencies the stability threshold is higher for an earth-based system, but at frequencies above the crossover the zero-g system has a higher threshold. The general reason is that wave number selection in a gravity-free system is higher than its analogous earthbased system. The gravity-free system therefore experiences greater viscous effects at large frequencies, resulting in higher threshold amplitudes for instability.
where j = 1 indicates the lower layer and j = 2 the upper. The vector V is the velocity field of the fluids with a pressure field P , of density ρ, and viscosity μ. The impermeable rigid upper and lower boundaries lead to the vertical and horizontal components of velocity being zero at those walls. Moreover, continuity of velocity is applied at the interface, which is taken to be a material surface with no mass transfer across it. Thus (V − U) · n = 0 at z = Z(x,y,t), where U · n is the speed of the interface, n being the surface normal pointing into the light fluid, and Z = Z(x,y,t) the position of the interface. The key equation at the interface is the stress balance. Thus [[−P I + 2μ(∇V + (∇V ) )]] · n = γ 2H n at z = Z(x,y,t) obtains where γ is the interfacial tension and 2H is twice the mean surface curvature. The braces represent a jump quantity across the surface evaluated as [[Q]] = Q2 − Q1 . The above system is linearized about the quiescent state with a base pressure gradient that is balanced by the timedependent forced acceleration. The linear system can then be put in terms of the normal component of the stress balance, which, at first order, is written as [[p − 2μ∂z w]] − (ρ2 − ρ1 )(g + Aω2 cos ωτ )ζ + γ (∂xx + ∂yy )ζ = 0,
II. MODEL AND PREDICTIONS
The linear stability of a flat immiscible interface subject to an oscillatory vertical motion of A cos ωt in a horizontally infinite fluid bilayer of depths h1 and h2 with the interface located at z = 0 requires that we express the equations of motion in a reference frame that moves with the imposed motion. This results in the grouping of the forcing acceleration Aω2 cos ωt with the gravitational acceleration, g. Thus we have ∂Vj + V j · ∇V j = −∇Pj + μj ∇ 2 V j ρj ∂t + ρj (g + Aω2 cos ωt)ez and ∇ · V j = 0, *
[email protected]
1539-3755/2013/88(6)/063002(5)
(1)
where p is the perturbed pressure, w the perturbed vertical component of velocity, and ζ the perturbed free surface. It may be noted that the linearized domain equations and the kinematic conditions relate p and w in terms of ζ thereby yielding a homogeneous problem in ζ . The perturbed system is analyzed by considering horizontal spatial modes with wave number k. A term with two derivatives in time arises from the perturbed pressure field, but the appearance of cos ωt via the base pressure gradient prevents us from expressing the state variables in pure exponential time modes with growth rates, σ . Instead the periodicity of the system must be accounted for by using Floquet theory and the resulting harmonics with an infinite Fourier series. The Fourier series is written in multiples of the basic frequency ω for convenience. Accounting for both the horizontal spatial
063002-1
©2013 American Physical Society
W. BATSON, F. ZOUESHTIAGH, AND R. NARAYANAN
PHYSICAL REVIEW E 88, 063002 (2013)
200
g =9.81 m/s g =0
scaled critical amplitude A ∗
180
2
160 140 120 100 80 60 40
min. threshold decreases
20 0
0
5
scaled wavenumber k ∗
10
15
FIG. 1. Linear stability diagrams for earth-based and zero-gravity systems oscillating at ω = 2 Hz of an h1 = h2 = 3 cm bilayer of FC70 (ρ1 = 1880 kg m−3 , ν1 = 12 cSt) and silicone oil (ρ2 = 846 kg m−3 , ν2 = 1.5 cSt). Interfacial tension was taken to be σ = 7 dyn cm−1 [4]. Identical physical properties are used for all figures. In all figures, solid lines are used to depict g = 9.81 m/s and dashed lines g = 0.
and temporal dependence of the system, for every dependent variable,ψ, we write ψ = eik·r
∞
e[σ +i(α+nω)]τ ψˆn (z).
(2)
n=−∞
The temporal expansion results in an infinite set of equations, for which neighboring Fourier modes are coupled by virtue of the forcing function. The parameter α specifies the response of the system with respect to the parametric frequency ω and is set to 0 for harmonic solutions and ω/2 for subharmonic solutions. Setting the Floquet growth parameter σ to zero and truncating the series at a finite number of equations allows the problem for the neutral stability to be cast in terms of an eigenvalue problem where the critical forcing amplitudes are the eigenvalues. This is the method of Kumar and Tuckerman [3], and all calculations have been carried out using this method. We note that the dissipation of this system
is defined solely by the bulk viscous effects and those arising from enforcement of the no-slip boundary condition, and that interfacial dissipative effects have not been included. The repeated solution of this problem for all wave numbers k forms the familiar tongues of instability, which has been done for 2 Hz in Fig. 1 and 8 Hz in Fig. 2. We choose to scale the resulting curves using the lesser of the fluid heights as a length scale. The frequency is then scaled by the associated earth-based natural frequency, calculated from the inviscid theory dispersion relation for a two-layer system, which, from Kumar and Tuckerman [3], is given by ω02 =
(ρ1 − ρ2 )gk + γ k 3 . ρ1 coth kh1 + ρ2 coth kh2
The above equation states that maintaining a wave of the same natural frequency upon removing gravity entails a larger wave number. This, in turn, suggests a change in the stability limit when gravity is eliminated, particularly for viscous systems. g =9.81 m/s g =0
30
scaled critical amplitude A ∗
(3)
2
25
20
15
10
5
0
min. threshold increases 0
5
10
15
20
scaled wavenumber k ∗
25
30
35
FIG. 2. Linear stability diagram for earth-based (solid line) and zero-gravity (dashed line) h1 = h2 = 3 cm systems oscillating at ω = 8 Hz. 063002-2
DUAL ROLE OF GRAVITY ON THE FARADAY THRESHOLD . . .
PHYSICAL REVIEW E 88, 063002 (2013)
7 60
6
A ∗ω ∗ 2
scaled critical amplitude A ∗
8
5
40 20
4
0
3
0
5
10
ω∗
2 1 0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 3. Minimum threshold predictions for earth-based (solid line) and zero-gravity (dashed line) systems where h1 = h2 = 6 cm.
The frequency scale is obtained upon replacing k by 1/ h1 . Systems of different layer heights will be compared, and in Figs. 1 and 2 we study an h1 = h2 = 3 cm system where the frequency scale evaluates to 1.54 Hz. We note that in the case of h1 = h2 = h, the 1/ coth kh term can be written as tanh kh, a term whose saturation to 1 indicates deep layer conditions where dissipation due to sensing of the bottom surface is minimized. Along with the results for earth’s gravitational acceleration of 9.81 m/s2 , we present the results for zero gravity in Figs. 1 and 2. This is accomplished merely by setting g = 0 in the model. The first tongue in both sets of predictions corresponds to subharmonic solutions of period ω/2, the second tongue to harmonic solutions of period ω, the third to superharmonic solutions of period 3ω/2, and so forth. In a horizontally
infinite system, oscillated at frequency ω, the interface would remain flat until an amplitude is chosen that is equal to the minimum of the first tongue. At this condition the associated wave number, k, would be neutrally stable, as noted in Figs. 1 and 2. An infinitesimal increase in the forcing amplitude would result in that wave number k growing at the interface and oscillating at half of the forcing frequency. Oscillation of the system at an amplitude above the minimum point would result in the excitation of a band of wave numbers and nonlinear competition would then determine which wave pattern is actually observed. Comparing Figs. 1 and 2, one can see that in the 2 Hz results the system with gravity is more stable while at 8 Hz it is less stable, implying the existence of a crossover. The crossover in stability can also be visualized by continuously tracing the motion of the minimum point of the
70
scaled wavenumber, k ∗
60
50
40
30
20
10
0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 4. Excited wave number for earth-based (solid line) and zero-gravity (dashed line) systems of h1 = h2 = 6 cm. Circles indicate frequencies at which tanh kh = 0.99. 063002-3
W. BATSON, F. ZOUESHTIAGH, AND R. NARAYANAN
PHYSICAL REVIEW E 88, 063002 (2013)
50
400
40
A ∗ω ∗ 2
scaled critical amplitude A ∗
500
30
300 200 100 0
20
0
5
10
ω∗
10
0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 5. Minimum threshold predictions for earth-based (solid line) and zero-gravity (dashed line) systems where h1 = h2 = 0.6 cm.
stability diagrams as the imposed frequency is changed. In order to produce an entire family of predictions that could be tested experimentally, one of the model parameters must be varied and an entire set of stability diagrams similar to Fig. 1 can be produced. Then the minimum point in each of these curves forms a set of predictions to be matched. Frequency is an obvious choice for an independent variable because it is an easily controlled experimental parameter. Doing so, it can be seen from Figs. 1 and 2 that an increase of the frequency in the model results in a shift of the resonant tongues toward higher wave numbers. Calculations predict that the general trend of the first excited wave number versus forcing frequency is a monotonic increase. This behavior is also seen in Eq. (3) albeit modified due to the absence of viscosity. Figure 3 presents both the threshold amplitudes and accelerations (inset) for a h1 = h2 = 6 cm system (frequency scale 1.10 Hz) when the minimum threshold is traced for a range of frequencies. Different behavior is seen for both the
threshold amplitudes and accelerations when comparing the results with and without earth’s gravity and is connected to the wave number selection given in Fig. 4. Circled in this figure are the wave numbers at which tanh kh = 0.99, denoting the transition between shallow and deep layer conditions. The presence of a local minimum threshold in the earth-based system can be correlated to its selection of wave numbers in the shallow wave regime. The minimum can then be explained as a competition between the dissipation that arises from the bottom surface in the low-frequency, shallow wave regime and that from the higher wavelengths in the high-frequency, deep layer regime. The local minimum threshold is notably absent in the zero-gravity curve. Removal of gravity is compensated by a higher wave number selection, as can be predicted by either the inviscid dispersion relation or the viscous model calculation in Figs. 1 and 2. Figure 4 indicates for the zero-gravity system that deep layer conditions are met at a much lower frequency than the earth-based system. In this case bulk dissipation dominates
scaled wavenumber, k ∗
15
12
9
6
3
0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 6. Excited wave number for earth-based (solid line) and zero-gravity (dashed line) systems where h1 = h2 = 0.6 cm. 063002-4
DUAL ROLE OF GRAVITY ON THE FARADAY THRESHOLD . . .
that of the bottom surface and the minimum in Fig. 3 for the zero-gravity threshold amplitudes appears at an unnoticeably small frequency. The crossover in stability presents itself and stems from the higher wave number selection of the zero-gravity system. With aim to explore the role of layer height on the predictions, Figs. 5 and 6 present results analogous to Figs. 3 and 4 for the same fluids, now with heights of h1 = h2 = 0.6 cm. While the system height is much lower, suggesting shallow layer conditions, it is important to recognize that the frequency scale has increased to 3.50 Hz and that the unscaled frequency where the earth-based system enters deep layer conditions is delayed considerably relative to the 6 cm case. What is seen in both Figs. 3 and 5 is that the crossover occurs in the frequency range where the earth-based system excites shallow layer modes while the zero-gravity excites deep layer modes. For the 0.6 cm system presented in Fig. 5, this occurs at a scaled frequency of 1.98, slightly higher than that of Fig. 3 where the crossing occurs at 1.78. We attribute this change to the increased frequency range in which the zero gravity system excites deep layer modes, up to a scaled frequency of 1.4, while the larger height system almost exclusively excites deep layer modes when gravity is absent. Nonetheless we believe the proximity of the two crossovers in the scaled domain validates the chosen length and frequency scales. Comparison of Figs. 3 and 5 also reveals that at high frequencies, the gap in instability due to gravity has diminished. This behavior is due to the decreased difference in wave number selection relative to the larger height case. This narrowing can be qualitatively inferred by the inviscid dispersion relation and is verified by the viscous calculation in Figs. 4 and 6. In even shallower layers we continue to expect crossovers, with the gaps in the thresholds and wave numbers diminishing further. However, this behavior may become more complicated, as excitation of harmonic or even larger period responses occurs in especially shallow layers, first shown by Kumar [5].
PHYSICAL REVIEW E 88, 063002 (2013)
magnitude of effective gravity switches sign, and the system bears resemblance to the classic Rayleigh-Taylor (RT) instability where two fluids are placed in a heavy-over-light configuration. Kumar [6] conducted a study which compared the selected Faraday wavelength to the RT wavelength assuming the time-averaged level of gravity at instability. He concluded both wavelengths were comparable in deep layer situations for viscous fluids however discrepancy arose when the depths were decreased. Subthreshold parametric excitation has also been considered as a means to stabilize RT systems experimentally by Wolf et al. [7] and theoretically by Lapuerta et al. [8]. The resemblance the Faraday instability shares with the RT instability is important to this work as RT intuition suggests that instability is encouraged by increasing the level of gravity. Therefore it is insightful for space experiments to observe that the effect of gravity on the natural frequency of a mode causes that intuition to fail at sufficiently high forcing frequencies. The explanation arises from the resonant nature of the instability and the contribution of gravity toward the natural frequency of a given pattern. Removing gravity therefore promotes the selection of a higher wave number mode, and, in turn greater viscous effects are felt, which suppress the instability. Finally we note this theory has been verified by experimental reproduction of the predicted wavelengths and threshold amplitudes by Bechhoefer et al. [9] and for laterally small systems by Batson et al. [10] via incorporation of a sidewall stress-free condition into the model. The theory has not been verified in either laterally large or small systems in reduced gravity. Complications in performing a traditional Faraday experiment in reduced gravity were observed by Falcon et al. [11], where the more wetting fluid was observed to coat the walls during parabolic flights. ACKNOWLEDGMENTS
A physical consideration to make in studying the Faraday instability is when Aω2 > g. In this case the time-averaged
W.B. and R.N. acknowledge support from NASA NNX11AC16G, Space Research Initiative and NSF 0968313. F.Z. acknowledges support from CNRS-CNES and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Program.
[1] M. Faraday, Philos. Trans. R. Soc. London 121, 299 (1831). [2] J. Friend and L. Y. Yeo, Rev. Mod. Phys. 83, 647 (2011). [3] K. Kumar and L. Tuckerman, J. Fluid Mech. 279, 49 (1994). [4] S. Someya and T. Munakata, J. Cryst. Growth 275, e343 (2005). [5] K. Kumar, Proc. R. Soc. London A 452, 1113 (1996). [6] S. Kumar, Phys. Rev. E 62, 1416 (2000). [7] G. H. Wolf, Phys. Rev. Lett. 24, 444 (1970).
[8] V. Lapuerta, F. J. Mancebo, and J. M. Vega, Phys. Rev. E 64, 016318 (2001). [9] J. Bechhoefer, V. Ego, S. Manneville, and B. Johnson, J. Fluid Mech. 288, 325 (1995). [10] W. Batson, F. Zoueshtiagh, and R. Narayanan, J. Fluid Mech. 729, 496 (2013). [11] C. Falcon, E. Falcon, U. Bortolozzo, and S. Fauve, Europhys. Lett. 86, 14002 (2009).
III. CONCLUSION
063002-5
Dual role of gravity on the Faraday threshold for immiscible viscous layers W. Batson,1,2,* F. Zoueshtiagh,2 and R. Narayanan1 1
Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA 2 Institut d’Electronique de Microelectronic et de Nanotechnologie (IEMN) UMR CNRS 8520, University of Lille 1, Ave Poincar´e, CS 60069-59652 Villeneuve d’Ascq, France (Received 15 January 2013; revised manuscript received 6 October 2013; published 2 December 2013) This work discusses the role of gravity on the Faraday instability, and the differences one can expect to observe in a low-gravity experiment when compared to an earth-based system. These differences are discussed in the context of the viscous linear theory for laterally infinite systems, and a surprising result of the analysis is the existence of a crossover frequency where an interface in low gravity switches from being less to more stable than an earth-based system. We propose this crossover exists in all Faraday systems, and the frequency at which it occurs is shown to be strongly influenced by layer height. In presenting these results physical explanations are provided for the behavior of the predicted forcing amplitude thresholds and wave number selection. DOI: 10.1103/PhysRevE.88.063002
PACS number(s): 47.20.−k, 68.05.−n, 68.03.−g, 47.54.−r
I. INTRODUCTION
Interfacial instability occurs when a bilayer of two fluids, a light fluid on top of a heavy one, is subject to a time-dependent acceleration in a direction perpendicular to their common interface. This instability, termed the Faraday instability [1] and manifested by a deflecting interface with associated patterns, is a result of a parametric resonance between the imposed frequency of excitation and the natural frequency. The natural frequency depends on the interfacial tension and the product of the density difference with normal gravity. A change in the level of gravity therefore changes the resonant condition. The motivation for the current work arises in the mixing in microfluidic geometries [2] in low gravity via induced Faraday waves. The point that will be made is that reducing gravity does not necessarily produce destabilization. Indeed there is a crossover frequency, where for lower frequencies the stability threshold is higher for an earth-based system, but at frequencies above the crossover the zero-g system has a higher threshold. The general reason is that wave number selection in a gravity-free system is higher than its analogous earthbased system. The gravity-free system therefore experiences greater viscous effects at large frequencies, resulting in higher threshold amplitudes for instability.
where j = 1 indicates the lower layer and j = 2 the upper. The vector V is the velocity field of the fluids with a pressure field P , of density ρ, and viscosity μ. The impermeable rigid upper and lower boundaries lead to the vertical and horizontal components of velocity being zero at those walls. Moreover, continuity of velocity is applied at the interface, which is taken to be a material surface with no mass transfer across it. Thus (V − U) · n = 0 at z = Z(x,y,t), where U · n is the speed of the interface, n being the surface normal pointing into the light fluid, and Z = Z(x,y,t) the position of the interface. The key equation at the interface is the stress balance. Thus [[−P I + 2μ(∇V + (∇V ) )]] · n = γ 2H n at z = Z(x,y,t) obtains where γ is the interfacial tension and 2H is twice the mean surface curvature. The braces represent a jump quantity across the surface evaluated as [[Q]] = Q2 − Q1 . The above system is linearized about the quiescent state with a base pressure gradient that is balanced by the timedependent forced acceleration. The linear system can then be put in terms of the normal component of the stress balance, which, at first order, is written as [[p − 2μ∂z w]] − (ρ2 − ρ1 )(g + Aω2 cos ωτ )ζ + γ (∂xx + ∂yy )ζ = 0,
II. MODEL AND PREDICTIONS
The linear stability of a flat immiscible interface subject to an oscillatory vertical motion of A cos ωt in a horizontally infinite fluid bilayer of depths h1 and h2 with the interface located at z = 0 requires that we express the equations of motion in a reference frame that moves with the imposed motion. This results in the grouping of the forcing acceleration Aω2 cos ωt with the gravitational acceleration, g. Thus we have ∂Vj + V j · ∇V j = −∇Pj + μj ∇ 2 V j ρj ∂t + ρj (g + Aω2 cos ωt)ez and ∇ · V j = 0, *
[email protected]
1539-3755/2013/88(6)/063002(5)
(1)
where p is the perturbed pressure, w the perturbed vertical component of velocity, and ζ the perturbed free surface. It may be noted that the linearized domain equations and the kinematic conditions relate p and w in terms of ζ thereby yielding a homogeneous problem in ζ . The perturbed system is analyzed by considering horizontal spatial modes with wave number k. A term with two derivatives in time arises from the perturbed pressure field, but the appearance of cos ωt via the base pressure gradient prevents us from expressing the state variables in pure exponential time modes with growth rates, σ . Instead the periodicity of the system must be accounted for by using Floquet theory and the resulting harmonics with an infinite Fourier series. The Fourier series is written in multiples of the basic frequency ω for convenience. Accounting for both the horizontal spatial
063002-1
©2013 American Physical Society
W. BATSON, F. ZOUESHTIAGH, AND R. NARAYANAN
PHYSICAL REVIEW E 88, 063002 (2013)
200
g =9.81 m/s g =0
scaled critical amplitude A ∗
180
2
160 140 120 100 80 60 40
min. threshold decreases
20 0
0
5
scaled wavenumber k ∗
10
15
FIG. 1. Linear stability diagrams for earth-based and zero-gravity systems oscillating at ω = 2 Hz of an h1 = h2 = 3 cm bilayer of FC70 (ρ1 = 1880 kg m−3 , ν1 = 12 cSt) and silicone oil (ρ2 = 846 kg m−3 , ν2 = 1.5 cSt). Interfacial tension was taken to be σ = 7 dyn cm−1 [4]. Identical physical properties are used for all figures. In all figures, solid lines are used to depict g = 9.81 m/s and dashed lines g = 0.
and temporal dependence of the system, for every dependent variable,ψ, we write ψ = eik·r
∞
e[σ +i(α+nω)]τ ψˆn (z).
(2)
n=−∞
The temporal expansion results in an infinite set of equations, for which neighboring Fourier modes are coupled by virtue of the forcing function. The parameter α specifies the response of the system with respect to the parametric frequency ω and is set to 0 for harmonic solutions and ω/2 for subharmonic solutions. Setting the Floquet growth parameter σ to zero and truncating the series at a finite number of equations allows the problem for the neutral stability to be cast in terms of an eigenvalue problem where the critical forcing amplitudes are the eigenvalues. This is the method of Kumar and Tuckerman [3], and all calculations have been carried out using this method. We note that the dissipation of this system
is defined solely by the bulk viscous effects and those arising from enforcement of the no-slip boundary condition, and that interfacial dissipative effects have not been included. The repeated solution of this problem for all wave numbers k forms the familiar tongues of instability, which has been done for 2 Hz in Fig. 1 and 8 Hz in Fig. 2. We choose to scale the resulting curves using the lesser of the fluid heights as a length scale. The frequency is then scaled by the associated earth-based natural frequency, calculated from the inviscid theory dispersion relation for a two-layer system, which, from Kumar and Tuckerman [3], is given by ω02 =
(ρ1 − ρ2 )gk + γ k 3 . ρ1 coth kh1 + ρ2 coth kh2
The above equation states that maintaining a wave of the same natural frequency upon removing gravity entails a larger wave number. This, in turn, suggests a change in the stability limit when gravity is eliminated, particularly for viscous systems. g =9.81 m/s g =0
30
scaled critical amplitude A ∗
(3)
2
25
20
15
10
5
0
min. threshold increases 0
5
10
15
20
scaled wavenumber k ∗
25
30
35
FIG. 2. Linear stability diagram for earth-based (solid line) and zero-gravity (dashed line) h1 = h2 = 3 cm systems oscillating at ω = 8 Hz. 063002-2
DUAL ROLE OF GRAVITY ON THE FARADAY THRESHOLD . . .
PHYSICAL REVIEW E 88, 063002 (2013)
7 60
6
A ∗ω ∗ 2
scaled critical amplitude A ∗
8
5
40 20
4
0
3
0
5
10
ω∗
2 1 0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 3. Minimum threshold predictions for earth-based (solid line) and zero-gravity (dashed line) systems where h1 = h2 = 6 cm.
The frequency scale is obtained upon replacing k by 1/ h1 . Systems of different layer heights will be compared, and in Figs. 1 and 2 we study an h1 = h2 = 3 cm system where the frequency scale evaluates to 1.54 Hz. We note that in the case of h1 = h2 = h, the 1/ coth kh term can be written as tanh kh, a term whose saturation to 1 indicates deep layer conditions where dissipation due to sensing of the bottom surface is minimized. Along with the results for earth’s gravitational acceleration of 9.81 m/s2 , we present the results for zero gravity in Figs. 1 and 2. This is accomplished merely by setting g = 0 in the model. The first tongue in both sets of predictions corresponds to subharmonic solutions of period ω/2, the second tongue to harmonic solutions of period ω, the third to superharmonic solutions of period 3ω/2, and so forth. In a horizontally
infinite system, oscillated at frequency ω, the interface would remain flat until an amplitude is chosen that is equal to the minimum of the first tongue. At this condition the associated wave number, k, would be neutrally stable, as noted in Figs. 1 and 2. An infinitesimal increase in the forcing amplitude would result in that wave number k growing at the interface and oscillating at half of the forcing frequency. Oscillation of the system at an amplitude above the minimum point would result in the excitation of a band of wave numbers and nonlinear competition would then determine which wave pattern is actually observed. Comparing Figs. 1 and 2, one can see that in the 2 Hz results the system with gravity is more stable while at 8 Hz it is less stable, implying the existence of a crossover. The crossover in stability can also be visualized by continuously tracing the motion of the minimum point of the
70
scaled wavenumber, k ∗
60
50
40
30
20
10
0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 4. Excited wave number for earth-based (solid line) and zero-gravity (dashed line) systems of h1 = h2 = 6 cm. Circles indicate frequencies at which tanh kh = 0.99. 063002-3
W. BATSON, F. ZOUESHTIAGH, AND R. NARAYANAN
PHYSICAL REVIEW E 88, 063002 (2013)
50
400
40
A ∗ω ∗ 2
scaled critical amplitude A ∗
500
30
300 200 100 0
20
0
5
10
ω∗
10
0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 5. Minimum threshold predictions for earth-based (solid line) and zero-gravity (dashed line) systems where h1 = h2 = 0.6 cm.
stability diagrams as the imposed frequency is changed. In order to produce an entire family of predictions that could be tested experimentally, one of the model parameters must be varied and an entire set of stability diagrams similar to Fig. 1 can be produced. Then the minimum point in each of these curves forms a set of predictions to be matched. Frequency is an obvious choice for an independent variable because it is an easily controlled experimental parameter. Doing so, it can be seen from Figs. 1 and 2 that an increase of the frequency in the model results in a shift of the resonant tongues toward higher wave numbers. Calculations predict that the general trend of the first excited wave number versus forcing frequency is a monotonic increase. This behavior is also seen in Eq. (3) albeit modified due to the absence of viscosity. Figure 3 presents both the threshold amplitudes and accelerations (inset) for a h1 = h2 = 6 cm system (frequency scale 1.10 Hz) when the minimum threshold is traced for a range of frequencies. Different behavior is seen for both the
threshold amplitudes and accelerations when comparing the results with and without earth’s gravity and is connected to the wave number selection given in Fig. 4. Circled in this figure are the wave numbers at which tanh kh = 0.99, denoting the transition between shallow and deep layer conditions. The presence of a local minimum threshold in the earth-based system can be correlated to its selection of wave numbers in the shallow wave regime. The minimum can then be explained as a competition between the dissipation that arises from the bottom surface in the low-frequency, shallow wave regime and that from the higher wavelengths in the high-frequency, deep layer regime. The local minimum threshold is notably absent in the zero-gravity curve. Removal of gravity is compensated by a higher wave number selection, as can be predicted by either the inviscid dispersion relation or the viscous model calculation in Figs. 1 and 2. Figure 4 indicates for the zero-gravity system that deep layer conditions are met at a much lower frequency than the earth-based system. In this case bulk dissipation dominates
scaled wavenumber, k ∗
15
12
9
6
3
0
0
5
10
15
scaled frequency, ω ∗
20
25
FIG. 6. Excited wave number for earth-based (solid line) and zero-gravity (dashed line) systems where h1 = h2 = 0.6 cm. 063002-4
DUAL ROLE OF GRAVITY ON THE FARADAY THRESHOLD . . .
that of the bottom surface and the minimum in Fig. 3 for the zero-gravity threshold amplitudes appears at an unnoticeably small frequency. The crossover in stability presents itself and stems from the higher wave number selection of the zero-gravity system. With aim to explore the role of layer height on the predictions, Figs. 5 and 6 present results analogous to Figs. 3 and 4 for the same fluids, now with heights of h1 = h2 = 0.6 cm. While the system height is much lower, suggesting shallow layer conditions, it is important to recognize that the frequency scale has increased to 3.50 Hz and that the unscaled frequency where the earth-based system enters deep layer conditions is delayed considerably relative to the 6 cm case. What is seen in both Figs. 3 and 5 is that the crossover occurs in the frequency range where the earth-based system excites shallow layer modes while the zero-gravity excites deep layer modes. For the 0.6 cm system presented in Fig. 5, this occurs at a scaled frequency of 1.98, slightly higher than that of Fig. 3 where the crossing occurs at 1.78. We attribute this change to the increased frequency range in which the zero gravity system excites deep layer modes, up to a scaled frequency of 1.4, while the larger height system almost exclusively excites deep layer modes when gravity is absent. Nonetheless we believe the proximity of the two crossovers in the scaled domain validates the chosen length and frequency scales. Comparison of Figs. 3 and 5 also reveals that at high frequencies, the gap in instability due to gravity has diminished. This behavior is due to the decreased difference in wave number selection relative to the larger height case. This narrowing can be qualitatively inferred by the inviscid dispersion relation and is verified by the viscous calculation in Figs. 4 and 6. In even shallower layers we continue to expect crossovers, with the gaps in the thresholds and wave numbers diminishing further. However, this behavior may become more complicated, as excitation of harmonic or even larger period responses occurs in especially shallow layers, first shown by Kumar [5].
PHYSICAL REVIEW E 88, 063002 (2013)
magnitude of effective gravity switches sign, and the system bears resemblance to the classic Rayleigh-Taylor (RT) instability where two fluids are placed in a heavy-over-light configuration. Kumar [6] conducted a study which compared the selected Faraday wavelength to the RT wavelength assuming the time-averaged level of gravity at instability. He concluded both wavelengths were comparable in deep layer situations for viscous fluids however discrepancy arose when the depths were decreased. Subthreshold parametric excitation has also been considered as a means to stabilize RT systems experimentally by Wolf et al. [7] and theoretically by Lapuerta et al. [8]. The resemblance the Faraday instability shares with the RT instability is important to this work as RT intuition suggests that instability is encouraged by increasing the level of gravity. Therefore it is insightful for space experiments to observe that the effect of gravity on the natural frequency of a mode causes that intuition to fail at sufficiently high forcing frequencies. The explanation arises from the resonant nature of the instability and the contribution of gravity toward the natural frequency of a given pattern. Removing gravity therefore promotes the selection of a higher wave number mode, and, in turn greater viscous effects are felt, which suppress the instability. Finally we note this theory has been verified by experimental reproduction of the predicted wavelengths and threshold amplitudes by Bechhoefer et al. [9] and for laterally small systems by Batson et al. [10] via incorporation of a sidewall stress-free condition into the model. The theory has not been verified in either laterally large or small systems in reduced gravity. Complications in performing a traditional Faraday experiment in reduced gravity were observed by Falcon et al. [11], where the more wetting fluid was observed to coat the walls during parabolic flights. ACKNOWLEDGMENTS
A physical consideration to make in studying the Faraday instability is when Aω2 > g. In this case the time-averaged
W.B. and R.N. acknowledge support from NASA NNX11AC16G, Space Research Initiative and NSF 0968313. F.Z. acknowledges support from CNRS-CNES and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Program.
[1] M. Faraday, Philos. Trans. R. Soc. London 121, 299 (1831). [2] J. Friend and L. Y. Yeo, Rev. Mod. Phys. 83, 647 (2011). [3] K. Kumar and L. Tuckerman, J. Fluid Mech. 279, 49 (1994). [4] S. Someya and T. Munakata, J. Cryst. Growth 275, e343 (2005). [5] K. Kumar, Proc. R. Soc. London A 452, 1113 (1996). [6] S. Kumar, Phys. Rev. E 62, 1416 (2000). [7] G. H. Wolf, Phys. Rev. Lett. 24, 444 (1970).
[8] V. Lapuerta, F. J. Mancebo, and J. M. Vega, Phys. Rev. E 64, 016318 (2001). [9] J. Bechhoefer, V. Ego, S. Manneville, and B. Johnson, J. Fluid Mech. 288, 325 (1995). [10] W. Batson, F. Zoueshtiagh, and R. Narayanan, J. Fluid Mech. 729, 496 (2013). [11] C. Falcon, E. Falcon, U. Bortolozzo, and S. Fauve, Europhys. Lett. 86, 14002 (2009).
III. CONCLUSION
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