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Granular systems on a vibrating wall: the hydrodynamic boundary condition
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Physica A 327 (2003) 88 – 93
www.elsevier.com/locate/physa
Granular systems on a vibrating wall: the hydrodynamic boundary condition Rodrigo Sotoa;∗ , M. Malek Mansourb a Departamento
b Universit e
de F sica, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile Libre de Bruxelles, Bvd. du Triomphe, Campus Plaine, CP 231, B-1050 Brussels, Belgium
Abstract Granular media, )uidized by a vibrating wall, is studied in the limit of high frequency. It is shown that if the product A!−5=4 is kept constant, then di0erent amplitudes A (with the corresponding frequency !) produce the same macroscopic result. Furthermore, it is found that in the hydrodynamic equations the boundary condition associated to the vibrating wall can be replaced by a stationary heat source. Numerical solutions of the full hydrodynamic con3rm these two predictions. c 2003 Elsevier B.V. All rights reserved. Keywords: Granular )uids; Vibrating wall; Energy injection
1. Introduction Granular matter is usually kept )uidized by means of vibrating walls. Commonly, a hydrodynamic approach is used to describe the bulk of the granular )uid (see for example Ref. [1]). However, a detailed consideration of the boundary condition is di?cult because of its explicit time dependence. In the case of high frequency and small amplitude, successive collisions of the grains with the wall are uncorrelated. For this reason, in this limiting case the wall has been usually modeled as a stochastic condition. It has been argued that the wall can be replaced by a thermal wall at a 3xed granular temperature, that scales as Twall ∼ m(A!)2 , where m is the particle mass, and A and ! are the oscillation amplitude and frequency, respectively [2]. Also, kinetic approaches have been used to characterize the vibrating boundary condition [3]. In this ∗
Corresponding author. E-mail address: [email protected] (R. Soto).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(03)00456-4
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
89
article we will show that, instead of a 3xed temperature, the wall imposes a permanent energy in)ux. 2. Global hydrodynamic description Let us consider a granular system )uidized by a wall that oscillates vertically with a frequency ! and an amplitude A. The grains that collide with it, are re)ected elastically, emerging with a normal velocity v = 2Vw − v, where v is the pre-collisional normal velocity and Vw is the instantaneous velocity of the wall. The tangential velocity is preserved in the collision. If the oscillation frequency is signi3cantly smaller than the collision frequency, the wall motion does not produce non-hydrodynamic behavior near it. It is then reasonable to assume that the granular )uid can be described by appropriate hydrodynamic equations, as those used in Ref. [1]. In the reference frame moving with the wall, the structure of hydrodynamic equations do not change except for the vertical component of the momentum equation where an extra non-inertial force term appears: A!2 cos(!t)y, ˆ where is the density and yˆ is the unit vector pointing upwards. For the purpose of obtaining quantitative predictions, we will consider the simplest form of hydrodynamic equations, with a Newtonian stress tensor including the compressible terms [4], the heat )ux given by the Fourier law, and an energy dissipation term as those predicted in Ref. [1]. In the reference frame of the oscillating plate, particles are re)ected elastically, thus no energy is injected into the system through it (note that this is only apparent, because the non-inertial force makes the non-conservative work corresponding to the work of the wall in the inertial frame). Then, the hydrodynamic boundary conditions for the granular )uid at the wall are: no heat )ux and no tangential stress, i.e., Qn = 0; Pnt = 0, where the subscripts n and t represent the normal and tangential component to the wall, respectively. Pnt = 0 is the o0-diagonal component of the stress tensor. 3. High-frequency limit Solving the complete hydrodynamic equations with these boundary conditions can only be done numerically. However, in the limit of high oscillating frequency, the boundary layer can be studied separately. In fact, in this limit the motion induced by the oscillating wall is rapidly damped (viscous dissipation). The mechanical energy is then transformed into heat, leading to a thermal energy source at some distance from the wall, where no oscillations will be observed. The previous analysis can be put in a quantitative way by means of a multiple scaling analysis. For simplicity, we restrict ourselves to the analysis of a two-dimensional system, but the result is straightforwardly extended to three dimensions. We will take the x direction as tangential to the wall and y direction normal to the wall. We start the analysis by assuming that while ! → ∞, the amplitude A decreases as A = A0 !− . We will then show that a 3nite limiting case exists provided the value of
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R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
the exponent is set to = 5=4. To do so, we also need to assume that in the limiting case there are relevant (fast and short) time and space scales where the oscillations produced by the non-inertial force are described is full detail without divergencies. Trivially, the fast time variable that must be used is s = !t. The problem of viscous damping of shear waves in the boundary layer was studied by Stokes, showing that the size of the boundary layer (where the oscillations are present) scales as !−1=2 [4]. This can be easily checked by noting that in the Navier–Stokes equation there are two space derivatives of v. The case of vertical oscillations is more complicated, but if the non-linear term (v · ∇)v can be neglected, it is easy to check that the size of the boundary layer scales in a way similar to the shear waves. Then, the appropriate (short) length scale variable that describes correctly the oscillations is = !1=2 y. Both s and take values of order one where the oscillations are relevant. Finally, the y component of the hydrodynamic velocity is rescaled as vy = A!uy = A0 !1− uy , while the horizontal component is simply set to vx = ux . The granular temperature T and density , as well as the x coordinate, are not rescaled. The rescaled hydrodynamical equations describe the granular dynamics in the time and length scales of the forcing and the corresponding oscillations, that we will call the short scale. There is also the original scale, the long one, where the experiments are done. Here we adopt a multiple scaling analysis for the study of both type of scales in time and space [5]. For this purpose we de3ne new variables as s0 = s;
0 = ;
s1 = !−1 s;
1 = !−1=2 :
(1)
Note that s0 and 0 describe the fast dynamics whereas s1 and 1 correspond to the slow dynamics, being equal to t and y, respectively. The multiscaling analysis assumes that the hydrodynamic 3elds depend on all these new variables, and that they can be expanded in powers of the small parameter !−1=2 . If X is a hydrodynamic 3eld we take X = X0 (s1 ; x; 1 ) + !−1=2 X1 (s0 ; s1 ; x; 0 ; 1 ) + · · ·, where it is assumed that the main contributions to the macroscopic 3elds do not depend on the fast variables, with the exception of uy where the zeroth-order term also depends on the fast variables. The above multiscaling implies that the space and time derivatives must be computed as 9 9 9 9 9 9 + !−1 ; + !−1=2 : (2) = = 9s 9s0 9s1 9 90 91 The value of the unknown exponent can be found by imposing that the energy equation must describe correctly the transformation of mechanical into thermal energy. Substituting the 3eld expansions into the scaled energy equation it can be directly checked that the only possibility to get both the transformation of mechanical work into heat and its di0usive transport at the dominant order (O(!−1=2 )) is to have =5=4. With this choice the dominant order of the energy equation reads 2 9uy0 9T1 92 T1 2 0 = k( 0 ; T0 ) + A ( ; T ) ; (3) 0 0 0 9s0 90 920 where k is the thermal conductivity and = + , and being the shear and bulk viscosities, respectively. Note that = + 43 in three dimensions. The density 0 and
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91
temperature T0 that appear in the energy Eq. (3) must be evaluated at 1 = 0. This is so because in the scaling (1) whenever 0 is 3nite, the corresponding value of 1 vanishes. Inserting the 3eld expansions into the scaled momentum equations we get Eq. (3) and 0
92 uy0 9uy0 = ( 0 ; T0 ) + 0 cos s0 : 9s0 920
(4)
Eq. (4) can be solved giving, after the transient time, uy0 = A0 e−0 =‘ sin(0 =‘ − s0 ) + sin(s0 ) + Uy0 (s1 ; x; 1 ) ; (5) where ‘= 2( 0 ; T0 )= 0 . The term Uy0 (s1 ; x; 1 ) is the contribution to the zeroth order of uy which depends only on the slow variables. The velocity pro3le uy0 contributes with a positive energy source in Eq. (3) due to the viscous heating. Integrating Eq. (3) in all space, we get that a non-divergent temperature 3eld can be achieved in the fast scale if 2 ∞ 9uy0 9T1 A20 ( 0 ; T0 ) (6) = d0 ; 90 90 y=0 k( 0 ; T0 ) 0
where it was used that 9T1 =90 must vanish at in3nity. This results seems to be in contradiction with the BC of vanishing heat )ux, but in fact the heat )ux is computed as Qy = −k 9T1 =90 − k 9T0 =91 = 0. Then, Eq. (6) implies that the heat )ux at the macroscopic scale (beyond the boundary layer) Qymacro ≡ −k 9T0 =91 is computed as 9T1 Qymacro = k : (7) 90 Replacing the explicit solution (5) into (6) we get the macroscopic boundary condition for the heat )ux ( 0 ; T0 ) 0 macro 2 5=2 =A ! : (8) Qy 8 y=0
Note that the viscosity coe?cient appears in (8) because the mechanism responsible for the generation of the heat source is viscous dissipation of the density waves. At the next non-trivial order (O(!−1 )) we recover the original hydrodynamic equations without the non-inertial force term. These equations are for the macroscopic 3elds 0 , ux0 , Uy0 , and T0 . The procedure to obtain these equations is lengthy but straightforward. In words, the equations are written to order !−1 and the terms that depend only on the slow variables are separated from the rest. The sum of these terms proves to be a constant which must be set to zero since otherwise the equation for the fast variables is divergent in s0 (secular divergence). This condition in turn gives rise to the hydrodynamic equations (see Ref. [5] for details). In summary, at the macroscopic spatio-temporal scale the oscillations are not seen, and the oscillating wall is replaced by a stationary wall that injects heat as given by Eq. (8).
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To check the validity of our results, we have solved numerically the hydrodynamic equations for two values of the amplitude and frequency, keeping constant the value of A!5=4 . These values are chosen such that the granular )uid gives rise to buoyancy convection (see Ref. [6] for a description of this instability). The time evolution of the order parameter characterizing the instability in the two cases show quantitative agreement, con3rming the predicted form of the scaling. We have also performed a third simulation, without vibration, where an incoming heat )ux given by (8) was imposed. Here again the evolution of the order parameter shows quantitative agreement with the previous simulations. More details will be published elsewhere. 4. Comments on the scaling The validity of our analysis can only be guaranteed in the limit of high frequencies. In this limit, the length of the boundary layer 2( 0 ; T0 )= 0 ! remains much smaller than the system length in the corresponding direction, whereas the maximum velocity of the wall VW = A! = A0 !−1=4 becomes vanishingly small. As a consequence, the fast oscillations of the wall do not induce supersonic waves. However, as mentioned in the introduction, the analysis remains valid as long as the frequencies are not larger than the grain mean collision frequency, where the very vaildity of hydrodynamics becomes questionable. Also, the vanishing limit of the wall velocity implies that in the fast scale, the nonlinear term (v · ∇)v vanishes. This validates a posteriori the scaling of the boundary layer size. Finally, it is interesting to note that in the fast scale, the granular dissipative mechanism does not play any role, it appears only in the slow (hydrodynamic) scale. In the fast scale, the only macroscopic mechanism that is involved is viscosity. 5. Conclusion In this article, we have performed an asymptotic analysis of granular media, )uidized by a vibrating wall, in the limit of high frequencies. We have shown that two experiments, done with di0erent oscillation frequencies and amplitudes, will produce the same macroscopic )ows provided the value of A!5=4 is preserved. Also, in the limiting case, we have shown that the time dependent boundary condition can be replaced by a simple stationary boundary that injects heat. The value of the injected heat depends on the local density and temperature at the wall, and is proportional to A2 !5=2 . Acknowledgements The authors would like to thank R. RamJKrez, M. Argentina, and M. Mareschal for fruitful discussions. They acknowledge the hospitality of CECAM (Lyon) where an important part of this work was done. R.S. thanks the support of Programa de inserci on
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
93
de cient 8cos chilenos of FundaciJon Andes, the support of FONDAP grant 11980002, and the FONDECYT project 1010416. The simulations were done in the CIMAT’s parallel cluster. References [1] J.T. Jenkins, S.B. Savage, A theory for the rapid )ow of identical, smooth, nearly elastic, spherical particles, J. Fluid Mech. 130 (1983) 187–202; J.T. Jenkins, M.W. Richman, Arch. Rat. Mech. 87 (1985) 355; J.J. Brey, M.J. Ruiz-Montero, D. Cubero, Homogeneous cooling state of a low-density granular )ow, Phys. Rev. E 54 (1996) 3664–3671; C. Bizon, M.D. Shattuck, J.B. Swift, H.L. Swinney, Transport coe?cients for granular media from molecular dynamics simulations, Phys. Rev. E 60 (1999) 4340. [2] S. Warr, J.M. Huntley, Energy input and scaling laws for a single particle vibrating in one dimension, Phys. Rev. E 52 (5) (1995) 5596–5601. [3] S. McNamara, J.-L. Barrat, Energy )ux into a )uidized granular medium at a vibrating wall, Phys. Rev. E 55 (6) (1997) 7767–7770; S. McNamara, S. Luding, Energy )ows in vibrated granular media, Phys. Rev. E 58 (1) (1998) 813–822. [4] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Elmsford, New York, 1975; G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1991. [5] C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, Berlin, 1999. [6] R. RamJKrez, D. Risso, P. Cordero, Thermal convection in )uidized granular systems, Phys. Rev. Lett. 85 (2000) 1230–1233.
Physica A 327 (2003) 88 – 93
www.elsevier.com/locate/physa
Granular systems on a vibrating wall: the hydrodynamic boundary condition Rodrigo Sotoa;∗ , M. Malek Mansourb a Departamento
b Universit e
de F sica, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile Libre de Bruxelles, Bvd. du Triomphe, Campus Plaine, CP 231, B-1050 Brussels, Belgium
Abstract Granular media, )uidized by a vibrating wall, is studied in the limit of high frequency. It is shown that if the product A!−5=4 is kept constant, then di0erent amplitudes A (with the corresponding frequency !) produce the same macroscopic result. Furthermore, it is found that in the hydrodynamic equations the boundary condition associated to the vibrating wall can be replaced by a stationary heat source. Numerical solutions of the full hydrodynamic con3rm these two predictions. c 2003 Elsevier B.V. All rights reserved. Keywords: Granular )uids; Vibrating wall; Energy injection
1. Introduction Granular matter is usually kept )uidized by means of vibrating walls. Commonly, a hydrodynamic approach is used to describe the bulk of the granular )uid (see for example Ref. [1]). However, a detailed consideration of the boundary condition is di?cult because of its explicit time dependence. In the case of high frequency and small amplitude, successive collisions of the grains with the wall are uncorrelated. For this reason, in this limiting case the wall has been usually modeled as a stochastic condition. It has been argued that the wall can be replaced by a thermal wall at a 3xed granular temperature, that scales as Twall ∼ m(A!)2 , where m is the particle mass, and A and ! are the oscillation amplitude and frequency, respectively [2]. Also, kinetic approaches have been used to characterize the vibrating boundary condition [3]. In this ∗
Corresponding author. E-mail address: [email protected] (R. Soto).
c 2003 Elsevier B.V. All rights reserved. 0378-4371/03/$ - see front matter doi:10.1016/S0378-4371(03)00456-4
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
89
article we will show that, instead of a 3xed temperature, the wall imposes a permanent energy in)ux. 2. Global hydrodynamic description Let us consider a granular system )uidized by a wall that oscillates vertically with a frequency ! and an amplitude A. The grains that collide with it, are re)ected elastically, emerging with a normal velocity v = 2Vw − v, where v is the pre-collisional normal velocity and Vw is the instantaneous velocity of the wall. The tangential velocity is preserved in the collision. If the oscillation frequency is signi3cantly smaller than the collision frequency, the wall motion does not produce non-hydrodynamic behavior near it. It is then reasonable to assume that the granular )uid can be described by appropriate hydrodynamic equations, as those used in Ref. [1]. In the reference frame moving with the wall, the structure of hydrodynamic equations do not change except for the vertical component of the momentum equation where an extra non-inertial force term appears: A!2 cos(!t)y, ˆ where is the density and yˆ is the unit vector pointing upwards. For the purpose of obtaining quantitative predictions, we will consider the simplest form of hydrodynamic equations, with a Newtonian stress tensor including the compressible terms [4], the heat )ux given by the Fourier law, and an energy dissipation term as those predicted in Ref. [1]. In the reference frame of the oscillating plate, particles are re)ected elastically, thus no energy is injected into the system through it (note that this is only apparent, because the non-inertial force makes the non-conservative work corresponding to the work of the wall in the inertial frame). Then, the hydrodynamic boundary conditions for the granular )uid at the wall are: no heat )ux and no tangential stress, i.e., Qn = 0; Pnt = 0, where the subscripts n and t represent the normal and tangential component to the wall, respectively. Pnt = 0 is the o0-diagonal component of the stress tensor. 3. High-frequency limit Solving the complete hydrodynamic equations with these boundary conditions can only be done numerically. However, in the limit of high oscillating frequency, the boundary layer can be studied separately. In fact, in this limit the motion induced by the oscillating wall is rapidly damped (viscous dissipation). The mechanical energy is then transformed into heat, leading to a thermal energy source at some distance from the wall, where no oscillations will be observed. The previous analysis can be put in a quantitative way by means of a multiple scaling analysis. For simplicity, we restrict ourselves to the analysis of a two-dimensional system, but the result is straightforwardly extended to three dimensions. We will take the x direction as tangential to the wall and y direction normal to the wall. We start the analysis by assuming that while ! → ∞, the amplitude A decreases as A = A0 !− . We will then show that a 3nite limiting case exists provided the value of
90
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
the exponent is set to = 5=4. To do so, we also need to assume that in the limiting case there are relevant (fast and short) time and space scales where the oscillations produced by the non-inertial force are described is full detail without divergencies. Trivially, the fast time variable that must be used is s = !t. The problem of viscous damping of shear waves in the boundary layer was studied by Stokes, showing that the size of the boundary layer (where the oscillations are present) scales as !−1=2 [4]. This can be easily checked by noting that in the Navier–Stokes equation there are two space derivatives of v. The case of vertical oscillations is more complicated, but if the non-linear term (v · ∇)v can be neglected, it is easy to check that the size of the boundary layer scales in a way similar to the shear waves. Then, the appropriate (short) length scale variable that describes correctly the oscillations is = !1=2 y. Both s and take values of order one where the oscillations are relevant. Finally, the y component of the hydrodynamic velocity is rescaled as vy = A!uy = A0 !1− uy , while the horizontal component is simply set to vx = ux . The granular temperature T and density , as well as the x coordinate, are not rescaled. The rescaled hydrodynamical equations describe the granular dynamics in the time and length scales of the forcing and the corresponding oscillations, that we will call the short scale. There is also the original scale, the long one, where the experiments are done. Here we adopt a multiple scaling analysis for the study of both type of scales in time and space [5]. For this purpose we de3ne new variables as s0 = s;
0 = ;
s1 = !−1 s;
1 = !−1=2 :
(1)
Note that s0 and 0 describe the fast dynamics whereas s1 and 1 correspond to the slow dynamics, being equal to t and y, respectively. The multiscaling analysis assumes that the hydrodynamic 3elds depend on all these new variables, and that they can be expanded in powers of the small parameter !−1=2 . If X is a hydrodynamic 3eld we take X = X0 (s1 ; x; 1 ) + !−1=2 X1 (s0 ; s1 ; x; 0 ; 1 ) + · · ·, where it is assumed that the main contributions to the macroscopic 3elds do not depend on the fast variables, with the exception of uy where the zeroth-order term also depends on the fast variables. The above multiscaling implies that the space and time derivatives must be computed as 9 9 9 9 9 9 + !−1 ; + !−1=2 : (2) = = 9s 9s0 9s1 9 90 91 The value of the unknown exponent can be found by imposing that the energy equation must describe correctly the transformation of mechanical into thermal energy. Substituting the 3eld expansions into the scaled energy equation it can be directly checked that the only possibility to get both the transformation of mechanical work into heat and its di0usive transport at the dominant order (O(!−1=2 )) is to have =5=4. With this choice the dominant order of the energy equation reads 2 9uy0 9T1 92 T1 2 0 = k( 0 ; T0 ) + A ( ; T ) ; (3) 0 0 0 9s0 90 920 where k is the thermal conductivity and = + , and being the shear and bulk viscosities, respectively. Note that = + 43 in three dimensions. The density 0 and
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
91
temperature T0 that appear in the energy Eq. (3) must be evaluated at 1 = 0. This is so because in the scaling (1) whenever 0 is 3nite, the corresponding value of 1 vanishes. Inserting the 3eld expansions into the scaled momentum equations we get Eq. (3) and 0
92 uy0 9uy0 = ( 0 ; T0 ) + 0 cos s0 : 9s0 920
(4)
Eq. (4) can be solved giving, after the transient time, uy0 = A0 e−0 =‘ sin(0 =‘ − s0 ) + sin(s0 ) + Uy0 (s1 ; x; 1 ) ; (5) where ‘= 2( 0 ; T0 )= 0 . The term Uy0 (s1 ; x; 1 ) is the contribution to the zeroth order of uy which depends only on the slow variables. The velocity pro3le uy0 contributes with a positive energy source in Eq. (3) due to the viscous heating. Integrating Eq. (3) in all space, we get that a non-divergent temperature 3eld can be achieved in the fast scale if 2 ∞ 9uy0 9T1 A20 ( 0 ; T0 ) (6) = d0 ; 90 90 y=0 k( 0 ; T0 ) 0
where it was used that 9T1 =90 must vanish at in3nity. This results seems to be in contradiction with the BC of vanishing heat )ux, but in fact the heat )ux is computed as Qy = −k 9T1 =90 − k 9T0 =91 = 0. Then, Eq. (6) implies that the heat )ux at the macroscopic scale (beyond the boundary layer) Qymacro ≡ −k 9T0 =91 is computed as 9T1 Qymacro = k : (7) 90 Replacing the explicit solution (5) into (6) we get the macroscopic boundary condition for the heat )ux ( 0 ; T0 ) 0 macro 2 5=2 =A ! : (8) Qy 8 y=0
Note that the viscosity coe?cient appears in (8) because the mechanism responsible for the generation of the heat source is viscous dissipation of the density waves. At the next non-trivial order (O(!−1 )) we recover the original hydrodynamic equations without the non-inertial force term. These equations are for the macroscopic 3elds 0 , ux0 , Uy0 , and T0 . The procedure to obtain these equations is lengthy but straightforward. In words, the equations are written to order !−1 and the terms that depend only on the slow variables are separated from the rest. The sum of these terms proves to be a constant which must be set to zero since otherwise the equation for the fast variables is divergent in s0 (secular divergence). This condition in turn gives rise to the hydrodynamic equations (see Ref. [5] for details). In summary, at the macroscopic spatio-temporal scale the oscillations are not seen, and the oscillating wall is replaced by a stationary wall that injects heat as given by Eq. (8).
92
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
To check the validity of our results, we have solved numerically the hydrodynamic equations for two values of the amplitude and frequency, keeping constant the value of A!5=4 . These values are chosen such that the granular )uid gives rise to buoyancy convection (see Ref. [6] for a description of this instability). The time evolution of the order parameter characterizing the instability in the two cases show quantitative agreement, con3rming the predicted form of the scaling. We have also performed a third simulation, without vibration, where an incoming heat )ux given by (8) was imposed. Here again the evolution of the order parameter shows quantitative agreement with the previous simulations. More details will be published elsewhere. 4. Comments on the scaling The validity of our analysis can only be guaranteed in the limit of high frequencies. In this limit, the length of the boundary layer 2( 0 ; T0 )= 0 ! remains much smaller than the system length in the corresponding direction, whereas the maximum velocity of the wall VW = A! = A0 !−1=4 becomes vanishingly small. As a consequence, the fast oscillations of the wall do not induce supersonic waves. However, as mentioned in the introduction, the analysis remains valid as long as the frequencies are not larger than the grain mean collision frequency, where the very vaildity of hydrodynamics becomes questionable. Also, the vanishing limit of the wall velocity implies that in the fast scale, the nonlinear term (v · ∇)v vanishes. This validates a posteriori the scaling of the boundary layer size. Finally, it is interesting to note that in the fast scale, the granular dissipative mechanism does not play any role, it appears only in the slow (hydrodynamic) scale. In the fast scale, the only macroscopic mechanism that is involved is viscosity. 5. Conclusion In this article, we have performed an asymptotic analysis of granular media, )uidized by a vibrating wall, in the limit of high frequencies. We have shown that two experiments, done with di0erent oscillation frequencies and amplitudes, will produce the same macroscopic )ows provided the value of A!5=4 is preserved. Also, in the limiting case, we have shown that the time dependent boundary condition can be replaced by a simple stationary boundary that injects heat. The value of the injected heat depends on the local density and temperature at the wall, and is proportional to A2 !5=2 . Acknowledgements The authors would like to thank R. RamJKrez, M. Argentina, and M. Mareschal for fruitful discussions. They acknowledge the hospitality of CECAM (Lyon) where an important part of this work was done. R.S. thanks the support of Programa de inserci on
R. Soto, M.M. Mansour / Physica A 327 (2003) 88 – 93
93
de cient 8cos chilenos of FundaciJon Andes, the support of FONDAP grant 11980002, and the FONDECYT project 1010416. The simulations were done in the CIMAT’s parallel cluster. References [1] J.T. Jenkins, S.B. Savage, A theory for the rapid )ow of identical, smooth, nearly elastic, spherical particles, J. Fluid Mech. 130 (1983) 187–202; J.T. Jenkins, M.W. Richman, Arch. Rat. Mech. 87 (1985) 355; J.J. Brey, M.J. Ruiz-Montero, D. Cubero, Homogeneous cooling state of a low-density granular )ow, Phys. Rev. E 54 (1996) 3664–3671; C. Bizon, M.D. Shattuck, J.B. Swift, H.L. Swinney, Transport coe?cients for granular media from molecular dynamics simulations, Phys. Rev. E 60 (1999) 4340. [2] S. Warr, J.M. Huntley, Energy input and scaling laws for a single particle vibrating in one dimension, Phys. Rev. E 52 (5) (1995) 5596–5601. [3] S. McNamara, J.-L. Barrat, Energy )ux into a )uidized granular medium at a vibrating wall, Phys. Rev. E 55 (6) (1997) 7767–7770; S. McNamara, S. Luding, Energy )ows in vibrated granular media, Phys. Rev. E 58 (1) (1998) 813–822. [4] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Elmsford, New York, 1975; G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1991. [5] C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, Berlin, 1999. [6] R. RamJKrez, D. Risso, P. Cordero, Thermal convection in )uidized granular systems, Phys. Rev. Lett. 85 (2000) 1230–1233.