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Transverse sound propagation in superfluid helium inside a carbon nanotube
Journal of Molecular Liquids 127 (2006) 158 – 159 www.elsevier.com/locate/molliq
Transverse sound propagation in superfluid helium inside a carbon nanotube O. Tkachenko ⁎, S. Vilchynsky Taras Shevchenko Kiev National University, Volodymyrska Street, 64, Kiev 01033, Ukraine Available online 30 May 2006
Abstract It is shown in the present work that it is possible of propagation in superfluid helium of the transverse sound caused by geometrical parameters of the helium system. © 2006 Elsevier B.V. All rights reserved. Keywords: Superfluid hydrodynamics; Superfluid helium; Tthermal energy flux; Helium sytem
1. Introduction According to Landau's [1] the two-fluid model of He II the superfluid hydrodynamics, in contrast to the classic hydrodynamics, is characterized by two velocities of the motion — → → ∼ V s and ∼ V n which represent the superfluid and normal components' respectively. The appearance of an additional hydrodynamic variable gives rise to new (in comparison with the classic hydrodynamics) types of sound. Types of the possible waves and their propagation speeds strongly depend on geometrical parameters of the helium system (they also depend on amount of 3He impurity). It is well known that two types of waves are able to propagate in volume of superfluid helium — first sound and second sound. The fourth sound propagates in narrow capillaries or in fine-pored media. The third and fifth sound propagate in thin films of superfluid helium. All these acoustic waves are longitudinal. As is well known, the transverse sounds can also exist in superfluid helium. They are able to propagate in the normal component. These waves are caused by the viscosity of the normal component (wave velocity is directly proportional to the first viscosity coefficient) and they are rapidly damped. 2. Computational method and results The goal of this work is to analyse the possibility of propagation in of the transverse sound superfluid helium which are not caused by effects of viscosity but but geometrical ⁎ Corresponding author. E-mail address: [email protected] (O. Tkachenko). 0167-7322/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2006.03.043
parameters of the helium system. Patterman [2] was the first who pointed at the fundamental possibility of existence of waves of such type. He considered the system with hindered normal component without taking into account boundary conditions and obtained (as a first approximation of equilibrium deviation) the expression for the transverse oscillations amplitude of the normal component velocity qs V′nx ¼ Vs0 ðqn=sqÞ: ð1Þ qn The equations of superfluid hydrodynamics with taking into account the dissipation [3] are known to be → q þ div j ¼ 0 ; ð2Þ Aji APik A Aυni Aυnk 2 Aυni ð3Þ þ ¼ g þ dik Ark 3 At Ark Ark Art Ari
f
g
→ → Þ þ d f divυ → ; þ dik f1 divð j qυ ik 2 n n 2 :→ υ → → Þ þ f divυ → g; ð4Þ y þ j l þ s ¼ jff3 divð j qυ 4 n 2 n S þ div Sυ→ þ →q 1 ð5Þ ¼ R: n T T where → q ¼ &jT and the dissipative function is equal to → Þ2 þ f divð → → Þ 2 þ 2f divυ → divð → → Þ R ¼ f 2 ðdivυ j qυ j qυ n n n n 3 1 2 1 Aυni Asnk 2 Aυnl jT ð6Þ þ dik þ& : þ g Ask Asi As1 2 3 T where ρn i ρs are the density of the normal and superfluid components respectively. ∏ik is the tensor of current impulse density. j is the total mass flux density of He II
ð
→
→
→
j ¼ qs V s þ qn V n
ð7Þ
O. Tkachenko, S. Vilchynsky / Journal of Molecular Liquids 127 (2006) 158–159
Coefficients ζ1, ζ2, ζ3, ζ4 signify the second viscosity coefficients. η is the first viscosity coefficient, it is related to the normal flow. æ is the heat conduction coefficient. As it is known the equations of two fluid hydrodynamics should be supplemented with boundary conditions at the walls of the vessel containing superfluid liquid. Let us consider a fulleren nanotube as a cylinder with radius R. The fluid is not able to escape the vessel walls, we obtain ðqn υnr þ qs υsr Þjr¼R ¼ 0 the 1−st boundary condition:
ð8Þ
Normal fluid as any viscous fluid adheres to a solid surface, therefore the tangential component of the normal velocity should be zero at the surface υnu jr¼R ¼ υnz jr¼R ¼ 0 the 2−nd boundary condition:
ð9Þ
Besides, the normal (radial) component of the thermal energy flux should be continuous at the boundary T qsυnr jr¼R ¼ &
AT j the 3−rd boundary condition: As r¼R
ð10Þ The last boundary condition is dictated by the so called Kapitza temperature jump Tp T jr¼R ¼ Kqsυnr jr¼R the 4−th boundary condition:
ð11Þ
Here Λ is a coefficient assigning the thermal resistance of the boundary. In the complete system of equations of two-fluid hydrodynamics one can analyse the propagation of small oscillations in superfluid helium. Let us analyse acoustic waves propagation in superfluid helium introduced into cylindrical nanotube diameter 2R. All deviations from equilibrium are considered to be small in an acoustic wave, therefore due to linearizing of the system (2)– (5) with respect to the small deviations one can obtain (in cylindrical coordinate system): AqV 1A Aυnz 1A Aυsz þ qn ðrυnr Þ þ ðrυsr Þ þ þ qs ¼0 At r Ar r Ar Az Az → →n qs Aυ ApVY ApVY n qn Aυ er þ ez þ þ Ar Az At At A2 υnr 1 Aυnr A2 υnr υnr Y þ ¼g þ er r Ar Az2 Ar2 r2 2 υnu Y A υnz 1 Aυnz A2 υnz Y þ 2 ee þ þ ez r Ar r Ar2 Az2 1 A 1A Aυnz Y ðrυnr Þ þ þ g er 3 Ar r Ar Az A 1A Aυnz Y ðrυnr Þ þ þ ez ð13Þ Az r Ar Az AsV AqV 1A Aυnz þs þ qs ðrυnr Þ þ q At At r Ar Az A2 T 1 AT A2 T þ þ ð14Þ ¼& Ar2 r Ar Az2
½
½
159
A υ→s 1 ApVY ApVY AT VY AT VY er þ ez s er þ ez ¼ 0 þ q Ar Az Ar Az At
ð15Þ
where the primed variables denote the small deviations from equilibrium state. Assuming that values P and T are independent, let us write down the equations of state in form of ρ = ρ( p,T), s = s( p,T ). From these equations one can obtain for small deviation: Aq Aq qV¼ p Vþ TV ð16Þ Ap T AT p As As sV¼ pVþ TV ð17Þ Ap T AT p Hereinafter let us take into account the existence of thermal Aq f 103 K 1 . It expansion coefficient of superfluid helium q1 AT p
allows us to neglect the second component in relation Eq. (16) and the first component in relation Eq. (17). Firstly we have to consider the case of superfluid helium flow in the cylindrical nanotube in the line of its axis, where we assume that the normal component is braked, and we do not consider effects related to damping i.e. we are neglecting all components containing dissipative coefficients η or æ. As is well known, in the case of linear which describes small deviations from equilibrium state, it is enough to consider the solutions in the form of plain monochromatic wave propagating in the direction of OZ axis i.e. space–time dependencies are given by → → V n ¼ V n VeiðkzxtÞ p ¼ p0 þ p VeiðkzxtÞ ð18Þ → → ð19Þ V s ¼ V ′s0 þ V ′s eiðkzxtÞ T ¼ T0 þ T VeiðkzxtÞ After linearization of the two-fluid hydrodynamic (Eq. (15)) one can obtain that the oscillation of super-fluid component is not
kz pV AT V 1 AV only Vsu . On s0 T V , ixVsr′ ¼ s0 ′ ¼ ′ ¼ 0, but also Vsz x q0→
AT
q0 Ar
′ = 0, in the other hand the condition rot Vs ¼ 0 results in V′sr = Vsφ other words the oscillations of the superfluid component are longitudinal. As it was shown in Ref. [4], in virtue of two-fluid equations without into account dissipative effects the relation rot h → taking → i qn ðV V Þ ¼ 0 takes place in unperturbed state. From this n s sq relation with taking into account the boundary condition in Eq. (9) one can obtain that the amplitude of transverse oscillations of the normal component velocity is non-zero A qn Vs0 Vnr ¼ Alnðq =sqÞ 1 eArð sq ÞðRrÞ : ð20Þ n
Ar
References [1] L.D. Landau, JETP 11 (1941) 592. L.D. Landau, JETP 17 (1947) 91. [2] S.J. Putterman, Superfluid Hydrodynamics, Amer. Elsevier Publish. Comp. Ins., New York, 1974. [3] I.M. Khalatnikov, Theory of Superfluidity, An Introduction to the Theory of Superfluidity, Perseus Books, 2000. [4] P.G. Saffman, Phys. Fluids 11 (1968) 2505.
Transverse sound propagation in superfluid helium inside a carbon nanotube O. Tkachenko ⁎, S. Vilchynsky Taras Shevchenko Kiev National University, Volodymyrska Street, 64, Kiev 01033, Ukraine Available online 30 May 2006
Abstract It is shown in the present work that it is possible of propagation in superfluid helium of the transverse sound caused by geometrical parameters of the helium system. © 2006 Elsevier B.V. All rights reserved. Keywords: Superfluid hydrodynamics; Superfluid helium; Tthermal energy flux; Helium sytem
1. Introduction According to Landau's [1] the two-fluid model of He II the superfluid hydrodynamics, in contrast to the classic hydrodynamics, is characterized by two velocities of the motion — → → ∼ V s and ∼ V n which represent the superfluid and normal components' respectively. The appearance of an additional hydrodynamic variable gives rise to new (in comparison with the classic hydrodynamics) types of sound. Types of the possible waves and their propagation speeds strongly depend on geometrical parameters of the helium system (they also depend on amount of 3He impurity). It is well known that two types of waves are able to propagate in volume of superfluid helium — first sound and second sound. The fourth sound propagates in narrow capillaries or in fine-pored media. The third and fifth sound propagate in thin films of superfluid helium. All these acoustic waves are longitudinal. As is well known, the transverse sounds can also exist in superfluid helium. They are able to propagate in the normal component. These waves are caused by the viscosity of the normal component (wave velocity is directly proportional to the first viscosity coefficient) and they are rapidly damped. 2. Computational method and results The goal of this work is to analyse the possibility of propagation in of the transverse sound superfluid helium which are not caused by effects of viscosity but but geometrical ⁎ Corresponding author. E-mail address: [email protected] (O. Tkachenko). 0167-7322/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2006.03.043
parameters of the helium system. Patterman [2] was the first who pointed at the fundamental possibility of existence of waves of such type. He considered the system with hindered normal component without taking into account boundary conditions and obtained (as a first approximation of equilibrium deviation) the expression for the transverse oscillations amplitude of the normal component velocity qs V′nx ¼ Vs0 ðqn=sqÞ: ð1Þ qn The equations of superfluid hydrodynamics with taking into account the dissipation [3] are known to be → q þ div j ¼ 0 ; ð2Þ Aji APik A Aυni Aυnk 2 Aυni ð3Þ þ ¼ g þ dik Ark 3 At Ark Ark Art Ari
f
g
→ → Þ þ d f divυ → ; þ dik f1 divð j qυ ik 2 n n 2 :→ υ → → Þ þ f divυ → g; ð4Þ y þ j l þ s ¼ jff3 divð j qυ 4 n 2 n S þ div Sυ→ þ →q 1 ð5Þ ¼ R: n T T where → q ¼ &jT and the dissipative function is equal to → Þ2 þ f divð → → Þ 2 þ 2f divυ → divð → → Þ R ¼ f 2 ðdivυ j qυ j qυ n n n n 3 1 2 1 Aυni Asnk 2 Aυnl jT ð6Þ þ dik þ& : þ g Ask Asi As1 2 3 T where ρn i ρs are the density of the normal and superfluid components respectively. ∏ik is the tensor of current impulse density. j is the total mass flux density of He II
ð
→
→
→
j ¼ qs V s þ qn V n
ð7Þ
O. Tkachenko, S. Vilchynsky / Journal of Molecular Liquids 127 (2006) 158–159
Coefficients ζ1, ζ2, ζ3, ζ4 signify the second viscosity coefficients. η is the first viscosity coefficient, it is related to the normal flow. æ is the heat conduction coefficient. As it is known the equations of two fluid hydrodynamics should be supplemented with boundary conditions at the walls of the vessel containing superfluid liquid. Let us consider a fulleren nanotube as a cylinder with radius R. The fluid is not able to escape the vessel walls, we obtain ðqn υnr þ qs υsr Þjr¼R ¼ 0 the 1−st boundary condition:
ð8Þ
Normal fluid as any viscous fluid adheres to a solid surface, therefore the tangential component of the normal velocity should be zero at the surface υnu jr¼R ¼ υnz jr¼R ¼ 0 the 2−nd boundary condition:
ð9Þ
Besides, the normal (radial) component of the thermal energy flux should be continuous at the boundary T qsυnr jr¼R ¼ &
AT j the 3−rd boundary condition: As r¼R
ð10Þ The last boundary condition is dictated by the so called Kapitza temperature jump Tp T jr¼R ¼ Kqsυnr jr¼R the 4−th boundary condition:
ð11Þ
Here Λ is a coefficient assigning the thermal resistance of the boundary. In the complete system of equations of two-fluid hydrodynamics one can analyse the propagation of small oscillations in superfluid helium. Let us analyse acoustic waves propagation in superfluid helium introduced into cylindrical nanotube diameter 2R. All deviations from equilibrium are considered to be small in an acoustic wave, therefore due to linearizing of the system (2)– (5) with respect to the small deviations one can obtain (in cylindrical coordinate system): AqV 1A Aυnz 1A Aυsz þ qn ðrυnr Þ þ ðrυsr Þ þ þ qs ¼0 At r Ar r Ar Az Az → →n qs Aυ ApVY ApVY n qn Aυ er þ ez þ þ Ar Az At At A2 υnr 1 Aυnr A2 υnr υnr Y þ ¼g þ er r Ar Az2 Ar2 r2 2 υnu Y A υnz 1 Aυnz A2 υnz Y þ 2 ee þ þ ez r Ar r Ar2 Az2 1 A 1A Aυnz Y ðrυnr Þ þ þ g er 3 Ar r Ar Az A 1A Aυnz Y ðrυnr Þ þ þ ez ð13Þ Az r Ar Az AsV AqV 1A Aυnz þs þ qs ðrυnr Þ þ q At At r Ar Az A2 T 1 AT A2 T þ þ ð14Þ ¼& Ar2 r Ar Az2
½
½
159
A υ→s 1 ApVY ApVY AT VY AT VY er þ ez s er þ ez ¼ 0 þ q Ar Az Ar Az At
ð15Þ
where the primed variables denote the small deviations from equilibrium state. Assuming that values P and T are independent, let us write down the equations of state in form of ρ = ρ( p,T), s = s( p,T ). From these equations one can obtain for small deviation: Aq Aq qV¼ p Vþ TV ð16Þ Ap T AT p As As sV¼ pVþ TV ð17Þ Ap T AT p Hereinafter let us take into account the existence of thermal Aq f 103 K 1 . It expansion coefficient of superfluid helium q1 AT p
allows us to neglect the second component in relation Eq. (16) and the first component in relation Eq. (17). Firstly we have to consider the case of superfluid helium flow in the cylindrical nanotube in the line of its axis, where we assume that the normal component is braked, and we do not consider effects related to damping i.e. we are neglecting all components containing dissipative coefficients η or æ. As is well known, in the case of linear which describes small deviations from equilibrium state, it is enough to consider the solutions in the form of plain monochromatic wave propagating in the direction of OZ axis i.e. space–time dependencies are given by → → V n ¼ V n VeiðkzxtÞ p ¼ p0 þ p VeiðkzxtÞ ð18Þ → → ð19Þ V s ¼ V ′s0 þ V ′s eiðkzxtÞ T ¼ T0 þ T VeiðkzxtÞ After linearization of the two-fluid hydrodynamic (Eq. (15)) one can obtain that the oscillation of super-fluid component is not
kz pV AT V 1 AV only Vsu . On s0 T V , ixVsr′ ¼ s0 ′ ¼ ′ ¼ 0, but also Vsz x q0→
AT
q0 Ar
′ = 0, in the other hand the condition rot Vs ¼ 0 results in V′sr = Vsφ other words the oscillations of the superfluid component are longitudinal. As it was shown in Ref. [4], in virtue of two-fluid equations without into account dissipative effects the relation rot h → taking → i qn ðV V Þ ¼ 0 takes place in unperturbed state. From this n s sq relation with taking into account the boundary condition in Eq. (9) one can obtain that the amplitude of transverse oscillations of the normal component velocity is non-zero A qn Vs0 Vnr ¼ Alnðq =sqÞ 1 eArð sq ÞðRrÞ : ð20Þ n
Ar
References [1] L.D. Landau, JETP 11 (1941) 592. L.D. Landau, JETP 17 (1947) 91. [2] S.J. Putterman, Superfluid Hydrodynamics, Amer. Elsevier Publish. Comp. Ins., New York, 1974. [3] I.M. Khalatnikov, Theory of Superfluidity, An Introduction to the Theory of Superfluidity, Perseus Books, 2000. [4] P.G. Saffman, Phys. Fluids 11 (1968) 2505.