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The effect of surface tension on nonpropagating hydrodynamic solitons
2 September
1996
PHYSICS ELSEVIER
Physics Letters A 220
LETTERS
A
(I 996) 87-W
The effect of surface tension on nonpropagating hydrodynamic solitons Guoqing Miao, Rongjue Wei Modern Acoustic State Key Laboratory. Institute of Acousrics, Nanjing Received
University,
Nanjing
11 January 1996; revised manuscript received 7 May 1996; accepted for publication Communicated by A.R. Bishop
210093.
Chinu
IO June 1996
Abstract The distribution of breather and kink solitons in the kd versus CT plane is measured experimentally, where kd is the dimensionless wavenumber and u the dimensionless surface tension coefficient of the liquid. The surface tension shows little effect on the distribution. An analysis is made on the effect of surface tension on the distribution, as well as on the stability region for the existence of breather solitons. PACS:
43.25. + y; 47.35. + i
The discovery of nonpropagating or forced standing hydrodynamic breather solitons from the Faraday oscillating water channel experiment by Wu, Keolian and Rudnick [ll, recently described as to have “opened the wide field of the existence and dynamics of resonantly excited solitary water wave” [2], has currently brought about quite a few interesting and even unique nonlinear phenomena, such as the observation of a kink soliton on an oscillating water channel by Denardo, Keolian and Putterman [3], the transition of a soliton to chaos [4,5], complicated multisoliton collisions [6], etc. The basic theory was developed by Miles [7], Larraza and Putterman [8]. However, the physics of some of these nonlinear phenomena is not yet well understood. Among these unsolved problems, the present paper will emphasise the very basic one, i.e. the effect of surface tension on these two kinds of solitons, although it has been taken into account by some authors [7,9,10]. 0375-9601/96/$12.00 Copyright PII SO375-9601(96)00488-4
For a long rectangular channel of length 1 and breadth b filled with a liquid to depth d that is subjected to the vertical oscillation z0 = u0 cos 2 wr, both breather and kink solitons are a modulation of the (0, 1) mode of the channel. On the weakly nonlinear level, the surface height of the liquid is [7] T ) cos wt + q( X, T) sin 6~11
rl= {&(XT +a*k
tanh kd [ A,( X, T) cos 2wr
+B,( X, T) sin 2wt + C,( X, T)]} cos ky (k=
n/b),
(1) where a is the length scale. The governing equation for the complex amplitude, r = p + iq, of the dominant cross mode is a nonlinear Schriidinger (NLS) equation i( r7 + 6r)
0 1996 Elsevier Science B.V. All rights reserved.
+ Br,,
+(p+Ajrj2)r+yr’
=O. (2)
88
G. Miuo, R. Wei /Physics
where r * = p - iq is the complex conjugate of r, S is the damping parameter, B = T + kcrll - T2>, p = (&I2 - o:1/2+, y = w2a,/eg, where o, = (gkTY’2 is the natural frequency of the (0, 1) mode and T= tanh kd, E is a small scaling parameter, and g the acceleration due to gravity. The nonlinear coefficient is A=$(6T4-5T2+16-9T-2).
(3)
When the correction by Miles for the surface tension is made, Eq. (3) is replaced by A,
=;[I
+(l
-T2)2+f(l
+4+‘(1
Letters A 220 (1996) 87-90
into breather solitons described by hyperbolic secants, l/2 r = e”@lsech
+$vT2,
(5)
1
For A, < 0, i.e. the parameters are in region II, the free oscillations of the cross mode harden, and the mode is consequently stable. However, there can exist a 180” kink in the phase of the mode. The stable kink solution to Eq. (2) is
+T2)* r=Me’+Z
-$(T’-~v)-‘(~-T~)~]
X .
tanh[
tl4( !-$-!)"*X],
(4)
while B and p are replaced by B, = B + 2aT and /3* = p - tr/2e, respectively, where v = crk’/pg and a is the surface tension coefficient. Miles’ theory successfully explained Wu’s and Denardo’s experiments. Let us draw the phase diagram of A * in the kd-cr plane (Fig. 11, where A * > 0 in regions I and III, and A * < 0 in region II. Along the border line (dashed) between regions I and II A * = 0, while along the border line (solid) between regions II and III T* - 4a = 0, corresponding to internal resonance between the first and second modes. For A, > 0, i.e. the parameters are in regions I and III, the free oscillations of the cross mode soften (i.e. the frequency decreases as the amplitude is increased), in this case the mode is subject to a Benjamin-Feir instability, and initial disturbances evolve
l
in
A:>0 l
Fig. 1. Phase diagram of A, and the experimentally measured distribution of breather and kink solitons in the kd- v plane. (a) Breather, Cm) kink, (0) Wu’s breather, ( q ) Denardo’s kink.
where
The experimental results of Wu’s (breather) and Denardo’s (kink) appear in regions I and II of Fig. 1 according to their experimental parameters, respectively, which is in agreement with the above theory. This investigation was stimulated by Wu et al., Denardo et al. and Miles’ work on solitons. We have extended our study to the entire /cd-v plane in Fig. 1, and measured the distribution of breather and kink solitons experimentally in the plane. An analysis was made for the effect of surface tension on the distribution, as well as on the stability region for the existence of breather solitons. The channel was made of Plexiglas, and has a length of f = 35 cm and a variable breadth b by placing a movable plate lengthwise into a wider channel of breadth 8 cm. Thus we can change kd and D continuously over a range of 0 to 4 and 0 to 0.4, respectively, by changing 6 and d. To oscillate the channel vertically, we fixed the channel to an aluminum table that was attached to a Briiel & Kjzr type 4812 General Purpose exciter. A Babel & Kjrer type 1050 vibration control was used to control the vibration of the table with a Briiel & KjEr type 4393 accelerometer as vibrating sensor. The working liquid was either alcohol or water. For water a few drops of saponin were added to minimize the friction
89
G. Mkw. R. Wei / Physics Letrers A 220 (1996) 87-90
f
potential energy of the free surface due to surface tension is (to O(Vq12))
. 3.00
-
.
.
.
SA>O
=:zoQ-. i
.
.
.
.
l
l
l
.
0.00 0.00
.
l
.
. l
1.M) ‘2 0:
.
and the time-averaged ~tentia1 energy of the free surface due to displacement is
kd=lAt?2
A<0 0.10
0.20 (I
030
1 0.40
Fig. 2. Phase diagram of A and the same measured dis~b~tion as in Fig. 1 of the breather and kink solitons in the ~&-CT plane. The symbols are also the same as in Fig. 1.
where To = Z?T/W. For the breather, we take I = 35 cm, b = 3 cm, a, = 0.1 cm, w = 28.9 rad/s, g = 980 cm/s2, and the surface height 77of a single breather soliton,
between the water and the wall, so as to facilitate the soliton generation and to keep it stable. For a series of given sets of parameters (kd, cr 1, the drive amplitude and frequency were changed carefully, such that the breathers (one or more) or kinks can be generated in the channel. The results are also marked in Fig. 1. The data reported here are for alcohol. They are similar for water. We see from Fig. I that in the part kd> 1 in the entire kd-cr plane, only breather solitons can be generated experimentally; but for kd < 1, only in a small part of region II, kink solitons can be generated, while in most parts neither breather nor kink solitons can be generated. Thus the experimental result shows a disagreement with the prediction of Miles’ theory. At the same time, we have also found that the breather soliton becomes less stable as kd approaches 1 from above (i.e. kd > 1). We have drawn the phase diagram of A in the k&-rr plane according to Eq. (3) in Fig. 2 (the straight line of kd = 1.022 represents A = 0, above which, A > 0, a breather soliton is admitted, while below it, A < 0, a kink soliton is admitted), and also mark Wu’s, Denardo’s and our experimental results in this diagram. As a result, we see a good agreement between the experimental dist~bution and the theoretical one according to Eq. (3). This means that the surface tension has little effect on the distribution of breather and kink solitons in the kd-a plane. To explain this experimental finding, we calculated the ratio of the potential energy of the free surface due to surface tension to that due to the displacement by using typical experimental data. The time-averaged
q(x.
y, t) =D cosysech
CX sin wt.
(10)
where we take D = 1 cm, C = I cm-‘. For alcohol we take p = 0.813 g/cm”, (Y= 22 dyne/cm, calculate it numerically with Mathematics, and obtain v,//= 0.03; and for water plus some drops of saponion, p= 1 g/cm3, Q!= 40 dyne/cm, the result is q/F= 0.05. At the same time, for both alcohol and water we have t.o*a,/g = 0.08. These are all of the same order of magnitude. For the kink, we take I= 35 cm, b = 5.3 cm, a0 = 0.07 cm, o = 21 rad/s, and q of a single kink, q=F
btanh coswy
Ex sin wt,
(11)
where we take F = 0.4 cm, E = 0.1 cm- ‘, the -- calculation results are q/5= 0.01 for alcohol, V,/ V(!= 0.01 for water, and w2a,/g = 0.03. These are also all of the same order of magni~de. Therefore we take the potential energy per unit area of the free surface, V,, due to a surface tension to be O( E) in comparison with that due to displacement, i.e. v,= &z(~.; + 7$) = fEc+f
+ II’)
(CY,= (Y/e> (12)
Averaging over y and t, to O(r), we obtain (V,)=~[~a,k~a~(p’+q’) --&uIk4a4(p2+qz)z].
(13)
As a result, we have a NLS equation in which the only difference from Eq. (21 is the replacement of p
90
G. Miao. R. Wei / Physics Letters A 220 (1996) 87-90
by /I* in Eq. (2), while A remains of the form of Eq. (3). This means, to O(E), that the surface tension has no effect on the distribution of breather and kink solitons in the k&a plane, as shown in Fig. 2. As for no soliton being admitted for most of the region kd < 1.022, we consider this as due to the fact that the dispersion of the system becomes weaker and weaker as the depth of the liquid becomes smaller and smaller. The stability analysis by Laedke et al. [1 1] points out that for S2
interval of X, of the NLS equation (2) analytically and numerically, from which probably a better explanation can be obtained for the effect of surface tension. We are grateful to Professor Wei-zhong Chen for the helpful discussion. This work was supported by the Chinese National Science Foundation.
References
111J. Wu, R. Keolian and I. Rudnick, Phys. Rev. Lett. 52 (1984) 1421.
121H. Friedel, E.W. Laedke and K.H. Spatschek, J. Fluid Mech. 284 (1995) 341. [31 B. Denardo, W. Wright and S. Putterman, Phys. Rev. Lett.
64 (1990) 1518. [41R. Wei, B. Wang, Y. Mao, X. Zheng and G. Miao, J. Acoust.
Sot. Am. 88 (1990) 469. [51 X. Chen and R. Wei, J. Fluid Mech. 259 (1994) 291.
b1 X. Wang and R. Wei. Phys. Lett. A 192 (1994) I. [71 J.W. Miles, J. Fluid Mech. 148 (1984) 451; J. FluidMech. 154 (1985) 535. RI A. Larraza and S. Putterman, J. Fluid Mech. 148 (1984) 443. [91 Huang Guo-xiang, Yan Jia-ren and Dai Xian-xi, Acta Phys. Sin. 39 (1990) 1234. II01 Zhou Xian-chu, Cui Hong-nong, Scientia Sinica A 12 (1992) 1269. [Ill E.W. Laedke and K.H. Spatschek, J. Fluid Mech. 223 (1991)589.
1996
PHYSICS ELSEVIER
Physics Letters A 220
LETTERS
A
(I 996) 87-W
The effect of surface tension on nonpropagating hydrodynamic solitons Guoqing Miao, Rongjue Wei Modern Acoustic State Key Laboratory. Institute of Acousrics, Nanjing Received
University,
Nanjing
11 January 1996; revised manuscript received 7 May 1996; accepted for publication Communicated by A.R. Bishop
210093.
Chinu
IO June 1996
Abstract The distribution of breather and kink solitons in the kd versus CT plane is measured experimentally, where kd is the dimensionless wavenumber and u the dimensionless surface tension coefficient of the liquid. The surface tension shows little effect on the distribution. An analysis is made on the effect of surface tension on the distribution, as well as on the stability region for the existence of breather solitons. PACS:
43.25. + y; 47.35. + i
The discovery of nonpropagating or forced standing hydrodynamic breather solitons from the Faraday oscillating water channel experiment by Wu, Keolian and Rudnick [ll, recently described as to have “opened the wide field of the existence and dynamics of resonantly excited solitary water wave” [2], has currently brought about quite a few interesting and even unique nonlinear phenomena, such as the observation of a kink soliton on an oscillating water channel by Denardo, Keolian and Putterman [3], the transition of a soliton to chaos [4,5], complicated multisoliton collisions [6], etc. The basic theory was developed by Miles [7], Larraza and Putterman [8]. However, the physics of some of these nonlinear phenomena is not yet well understood. Among these unsolved problems, the present paper will emphasise the very basic one, i.e. the effect of surface tension on these two kinds of solitons, although it has been taken into account by some authors [7,9,10]. 0375-9601/96/$12.00 Copyright PII SO375-9601(96)00488-4
For a long rectangular channel of length 1 and breadth b filled with a liquid to depth d that is subjected to the vertical oscillation z0 = u0 cos 2 wr, both breather and kink solitons are a modulation of the (0, 1) mode of the channel. On the weakly nonlinear level, the surface height of the liquid is [7] T ) cos wt + q( X, T) sin 6~11
rl= {&(XT +a*k
tanh kd [ A,( X, T) cos 2wr
+B,( X, T) sin 2wt + C,( X, T)]} cos ky (k=
n/b),
(1) where a is the length scale. The governing equation for the complex amplitude, r = p + iq, of the dominant cross mode is a nonlinear Schriidinger (NLS) equation i( r7 + 6r)
0 1996 Elsevier Science B.V. All rights reserved.
+ Br,,
+(p+Ajrj2)r+yr’
=O. (2)
88
G. Miuo, R. Wei /Physics
where r * = p - iq is the complex conjugate of r, S is the damping parameter, B = T + kcrll - T2>, p = (&I2 - o:1/2+, y = w2a,/eg, where o, = (gkTY’2 is the natural frequency of the (0, 1) mode and T= tanh kd, E is a small scaling parameter, and g the acceleration due to gravity. The nonlinear coefficient is A=$(6T4-5T2+16-9T-2).
(3)
When the correction by Miles for the surface tension is made, Eq. (3) is replaced by A,
=;[I
+(l
-T2)2+f(l
+4+‘(1
Letters A 220 (1996) 87-90
into breather solitons described by hyperbolic secants, l/2 r = e”@lsech
+$vT2,
(5)
1
For A, < 0, i.e. the parameters are in region II, the free oscillations of the cross mode harden, and the mode is consequently stable. However, there can exist a 180” kink in the phase of the mode. The stable kink solution to Eq. (2) is
+T2)* r=Me’+Z
-$(T’-~v)-‘(~-T~)~]
X .
tanh[
tl4( !-$-!)"*X],
(4)
while B and p are replaced by B, = B + 2aT and /3* = p - tr/2e, respectively, where v = crk’/pg and a is the surface tension coefficient. Miles’ theory successfully explained Wu’s and Denardo’s experiments. Let us draw the phase diagram of A * in the kd-cr plane (Fig. 11, where A * > 0 in regions I and III, and A * < 0 in region II. Along the border line (dashed) between regions I and II A * = 0, while along the border line (solid) between regions II and III T* - 4a = 0, corresponding to internal resonance between the first and second modes. For A, > 0, i.e. the parameters are in regions I and III, the free oscillations of the cross mode soften (i.e. the frequency decreases as the amplitude is increased), in this case the mode is subject to a Benjamin-Feir instability, and initial disturbances evolve
l
in
A:>0 l
Fig. 1. Phase diagram of A, and the experimentally measured distribution of breather and kink solitons in the kd- v plane. (a) Breather, Cm) kink, (0) Wu’s breather, ( q ) Denardo’s kink.
where
The experimental results of Wu’s (breather) and Denardo’s (kink) appear in regions I and II of Fig. 1 according to their experimental parameters, respectively, which is in agreement with the above theory. This investigation was stimulated by Wu et al., Denardo et al. and Miles’ work on solitons. We have extended our study to the entire /cd-v plane in Fig. 1, and measured the distribution of breather and kink solitons experimentally in the plane. An analysis was made for the effect of surface tension on the distribution, as well as on the stability region for the existence of breather solitons. The channel was made of Plexiglas, and has a length of f = 35 cm and a variable breadth b by placing a movable plate lengthwise into a wider channel of breadth 8 cm. Thus we can change kd and D continuously over a range of 0 to 4 and 0 to 0.4, respectively, by changing 6 and d. To oscillate the channel vertically, we fixed the channel to an aluminum table that was attached to a Briiel & Kjzr type 4812 General Purpose exciter. A Babel & Kjrer type 1050 vibration control was used to control the vibration of the table with a Briiel & KjEr type 4393 accelerometer as vibrating sensor. The working liquid was either alcohol or water. For water a few drops of saponin were added to minimize the friction
89
G. Mkw. R. Wei / Physics Letrers A 220 (1996) 87-90
f
potential energy of the free surface due to surface tension is (to O(Vq12))
. 3.00
-
.
.
.
SA>O
=:zoQ-. i
.
.
.
.
l
l
l
.
0.00 0.00
.
l
.
. l
1.M) ‘2 0:
.
and the time-averaged ~tentia1 energy of the free surface due to displacement is
kd=lAt?2
A<0 0.10
0.20 (I
030
1 0.40
Fig. 2. Phase diagram of A and the same measured dis~b~tion as in Fig. 1 of the breather and kink solitons in the ~&-CT plane. The symbols are also the same as in Fig. 1.
where To = Z?T/W. For the breather, we take I = 35 cm, b = 3 cm, a, = 0.1 cm, w = 28.9 rad/s, g = 980 cm/s2, and the surface height 77of a single breather soliton,
between the water and the wall, so as to facilitate the soliton generation and to keep it stable. For a series of given sets of parameters (kd, cr 1, the drive amplitude and frequency were changed carefully, such that the breathers (one or more) or kinks can be generated in the channel. The results are also marked in Fig. 1. The data reported here are for alcohol. They are similar for water. We see from Fig. I that in the part kd> 1 in the entire kd-cr plane, only breather solitons can be generated experimentally; but for kd < 1, only in a small part of region II, kink solitons can be generated, while in most parts neither breather nor kink solitons can be generated. Thus the experimental result shows a disagreement with the prediction of Miles’ theory. At the same time, we have also found that the breather soliton becomes less stable as kd approaches 1 from above (i.e. kd > 1). We have drawn the phase diagram of A in the k&-rr plane according to Eq. (3) in Fig. 2 (the straight line of kd = 1.022 represents A = 0, above which, A > 0, a breather soliton is admitted, while below it, A < 0, a kink soliton is admitted), and also mark Wu’s, Denardo’s and our experimental results in this diagram. As a result, we see a good agreement between the experimental dist~bution and the theoretical one according to Eq. (3). This means that the surface tension has little effect on the distribution of breather and kink solitons in the kd-a plane. To explain this experimental finding, we calculated the ratio of the potential energy of the free surface due to surface tension to that due to the displacement by using typical experimental data. The time-averaged
q(x.
y, t) =D cosysech
CX sin wt.
(10)
where we take D = 1 cm, C = I cm-‘. For alcohol we take p = 0.813 g/cm”, (Y= 22 dyne/cm, calculate it numerically with Mathematics, and obtain v,//= 0.03; and for water plus some drops of saponion, p= 1 g/cm3, Q!= 40 dyne/cm, the result is q/F= 0.05. At the same time, for both alcohol and water we have t.o*a,/g = 0.08. These are all of the same order of magnitude. For the kink, we take I= 35 cm, b = 5.3 cm, a0 = 0.07 cm, o = 21 rad/s, and q of a single kink, q=F
btanh coswy
Ex sin wt,
(11)
where we take F = 0.4 cm, E = 0.1 cm- ‘, the -- calculation results are q/5= 0.01 for alcohol, V,/ V(!= 0.01 for water, and w2a,/g = 0.03. These are also all of the same order of magni~de. Therefore we take the potential energy per unit area of the free surface, V,, due to a surface tension to be O( E) in comparison with that due to displacement, i.e. v,= &z(~.; + 7$) = fEc+f
+ II’)
(CY,= (Y/e> (12)
Averaging over y and t, to O(r), we obtain (V,)=~[~a,k~a~(p’+q’) --&uIk4a4(p2+qz)z].
(13)
As a result, we have a NLS equation in which the only difference from Eq. (21 is the replacement of p
90
G. Miao. R. Wei / Physics Letters A 220 (1996) 87-90
by /I* in Eq. (2), while A remains of the form of Eq. (3). This means, to O(E), that the surface tension has no effect on the distribution of breather and kink solitons in the k&a plane, as shown in Fig. 2. As for no soliton being admitted for most of the region kd < 1.022, we consider this as due to the fact that the dispersion of the system becomes weaker and weaker as the depth of the liquid becomes smaller and smaller. The stability analysis by Laedke et al. [1 1] points out that for S2
interval of X, of the NLS equation (2) analytically and numerically, from which probably a better explanation can be obtained for the effect of surface tension. We are grateful to Professor Wei-zhong Chen for the helpful discussion. This work was supported by the Chinese National Science Foundation.
References
111J. Wu, R. Keolian and I. Rudnick, Phys. Rev. Lett. 52 (1984) 1421.
121H. Friedel, E.W. Laedke and K.H. Spatschek, J. Fluid Mech. 284 (1995) 341. [31 B. Denardo, W. Wright and S. Putterman, Phys. Rev. Lett.
64 (1990) 1518. [41R. Wei, B. Wang, Y. Mao, X. Zheng and G. Miao, J. Acoust.
Sot. Am. 88 (1990) 469. [51 X. Chen and R. Wei, J. Fluid Mech. 259 (1994) 291.
b1 X. Wang and R. Wei. Phys. Lett. A 192 (1994) I. [71 J.W. Miles, J. Fluid Mech. 148 (1984) 451; J. FluidMech. 154 (1985) 535. RI A. Larraza and S. Putterman, J. Fluid Mech. 148 (1984) 443. [91 Huang Guo-xiang, Yan Jia-ren and Dai Xian-xi, Acta Phys. Sin. 39 (1990) 1234. II01 Zhou Xian-chu, Cui Hong-nong, Scientia Sinica A 12 (1992) 1269. [Ill E.W. Laedke and K.H. Spatschek, J. Fluid Mech. 223 (1991)589.