SOUND PROPAGATION THROUGH A GAS IN MICROSCALE

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1 Proceedings of the ASME 9 7th International Conference on Nanochannels, Microchannels and Minichannels ICNMM9 Proceedings of 7th International Conferences on Nanochannels, June Microchannels -4, 9, Pohang, and Minichannels South Korea ICNMM9 June -4, 9, Pohang, South Korea ICNMM9-87 ICNMM9-87 SOUND PROPAGATION THROUGH A GAS IN MICROSCALE Felix Sharipov and Denize Kalempa Departamento de F ısica Universidade Federal do Paraná Caixa Postal 944, Curitiba , Brazil s: sharipov@fisica.ufpr.br and kalempa@fisica.ufpr.br ABSTRACT A sound wave propagation through a rarefied gas is investigated on the basis of the linearized kinetic equation by taking into account the influence of the receptor of sound waves on the solution of the problem. In order to do so, a plate oscillating in the normal direction to its own plane is considered as a sound wave source while a stationary one is considered as being the receptor of sound waves. The distance between the plates can be of the order of the molecular mean free path. It is assumed a fully established oscillation so that the solution of the kinetic equation depends on time harmonically. The main parameters of the problem are the oscillation speed parameter, defined as the ratio of intermolecular collision frequency to the sound frequency, and the Knudsen number, defined as the ratio of the molecular mean free path to a characteristic scale of the gas flow. The problem is solved over a wide range of both parameters and the amplitudes and phases of all the macrocharacteristics of the gas flow are calculated. INTRODUCTION The classical problem of sound propagation through a gas, usually, is treated on the basis of the continuum mechanics equations, see e.g. [, ]. However, according to our previous works [3 5], the equations of continuum mechanics are valid under two conditions. First, the characteristic lenght of the gas flow domain must be significantly larger than the molecular mean free path. Second, a characteristic time of the gas flow must be significantly larger than the mean free time. When at least one of these conditions is violated the problem cannot be correctly described via the continuum approach but it must be solved on the basis of the non-stationary kinetic equation. The first condition is characterized by the Knudsen number, defined as the ratio of the molecular mean free path to a characteristic length, and the second condition depends on the ratio of collision frequency to oscillation frequency. The sound propagation was investigated both theoretically, see e.g. [6 ] and experimentally, see e.g. [ 5]. Most of papers consider the propagating medium as being a semi-infinite space and, as a consequence, the results obtained do not take into account the influence of the receptor on the solution of the problem. This is valid when the distance source-receptor is too large in comparison with the intermolecular mean free path so that the influence of the receptor can be neglected. However, according to Ref. [], the receptor of sound waves can significantly change the solution of the problem so that the behavior of the gas flow is totally different from that obtained by considering a semi-infinite space. In some situations even the qualitative behavior of the gas flow properties is changed due to the influence of the reflected waves from the receptor. Furthermore, in many papers it is assumed that the sound wave is harmonic in space and time and this implies constant values for the phase velocity and attenuation. In fact, the classical theory of sound propagation in gases predicts that these two quantities are constant over the space. However, there are situations in which the wave cannot be considered as harmonic in space and the classical theory of sound propagation in gases are not valid. In the present work the sound propagation through a gas in a gap between two infinite and parallel plates is investigated on the basis of the linearized kinetic equation. One of the plates is Copyright c 9 by ASME

2 oscillating in a normal direction to its own plane with arbitrary oscillation frequency and is the source of sound waves. The other plate is stationary and is the receptor of sound waves. Diffuse scattering of gaseous particles on both plates is assumed. In our previous paper [5] a similar work was investigated but, instead of a bounded gas flow domain, a semi-infinite space was considered and, as a consequence, the results are valid to describe the sound propagation only in situations where the influence of the receptor can be neglected. Nowadays the solution of this kind of problem is very important for the development and performance improvement of microdevices with some movable part which can oscillate in the normal direction to its own plane and a gas is used as a damping medium [6]. For instance, it happens in micro-accelerometers [7]. In these kinds of devices a sound wave propagates through the gas and affects its macrocharacteristics such as bulk velocity, temperature, etc. Since the dimension of some part of these devices has the same order of the mean free path of gas molecules and the oscillation frequency can be very close to the collision frequency of gas molecules, the equations of continuum mechanics fail in describing the gas flow properly. Consequently, in order to improve the performance of these devices it is necessary to understand and predict the behavior of the gas flow in the gap between surfaces. In fact, the understanding of damping mechanisms in such devices is the most important since the performance can be significantly increased in rarefied conditions, i.e. at low pressure. Therefore, since the results obtained in the present work are valid at arbitrary oscillation frequency and dimension of the gas domain, which is characterized by the Knudsen number, the results can be applied to correctly describe the behavior of the gas flow in such kinds of devices. STATEMENT OF THE PROBLEM Consider a monoatomic gas confined between two flat, infinite and parallel plates located at x = and x = L. The plate at x = oscillates harmonically in the normal direction to its own plane, i.e. in the x direction, with frequency ω so that its velocity depends on time t as U w t) = R[U m exp iωt)], ) where R denotes the real part of a complex quantity and U m is the velocity amplitude of the plate which is assumed to be very small when compared with the most probable molecular speed v m, i.e., U m v m, v m = ) kb T /, ) m where m is the molecular mass of the gas, T is its equilibrium temperature and k B is the Boltzmann constant. The sound wave generated by the oscillating plate propagates through the gas and causes a gas flow characterized by its bulk velocity U x, density n and temperature T deviations from their equilibrium values n and T, respectively. The stationary plate located at x = L represents the receptor of the sound wave. Depending on the gas rarefaction and sound frequency the gas flow can be significantly affected by the sound wave reflected by the stationary plate. In practice, the only quantity measured is the pressure in the wave propagation direction which is denoted by P xx. All the macrocharacteristics of the gas flow depend on time harmonically and can be written as U x t,x ) = R [ U x x )exp iωt) ], 3) nt,x ) = R [ ñx )exp iωt) ], 4) Tt,x ) = R [ Tx )exp iωt) ], 5) P xx t,x ) = R [ P xx x )exp iωt) ], 6) where U x x ), ñx ), Tx ), and P xx x ) are complex quantities. The solution of the problem is determined by two main parameters. First, the oscillation speed parameter defined as θ = ν ω, ν = P µ where ν is the intermolecular collision frequency, P is the equilibrium pressure and µ is the shear viscosity of the gas. Second, the rarefaction parameter which is defined as 7) δ = P L µv m Kn, 8) where L is the distance between the plates and Kn is the Knudsen number. It is worthwhile to note that, since the quantity l = µv m /P is the equivalent free path, the rarefaction parameter δ characterizes a relation between the mean free path l and the characteristic scale L of the gas flow, but it does not contain any information concerning the oscillation speed. On the other hand, Copyright c 9 by ASME

3 the oscillation speed parameter defined by 7) does not contain any information about the characteristic lenght of the gas flow. For further derivations it is more convenient to introduce the dimensionless coordinate as x = ω v m x. 9) The quantities given in 3)-6) are also written in a dimensionless form as ux) = Ũxx), ρx) = ñx) v m, ) U m U m τx) = Tx) T v m U m, n Πx) = P xx P v m U m. ) Since these quantities are complex, they can be written as ux) = A u x)exp[iϕ u x)], ) where ft,x,v) is the distribution function which depends on the time t, x coordinate and molecular velocity v, Q f f ) denotes the collision integral [8, 9]. For our purpose the Shakhov model [3] of the collision integral is most suitable, because it describes correctly both mass and heat transfer. In the non-linear form the model reads Q f f ) = P µ { [ f M + mq V) mv 5nk B T) k B T 5 )] } f, 7) where V = v U is the peculiar velocity and f M is the local Maxwellian given by ) m 3/ ] f M mv U) = n exp[, 8) πk B T k B T where n, U, T and Q are the local number density, bulk velocity, temperature and heat flux vector, respectively. Note, for the onedimensional problem under question both U and Q have only the x component. All these quantities are calculated via the distribution function as ρx) = A n x)exp[iϕ n x)], 3) nt,x ) = ft,x,v)dv, 9) τx) = A T x)exp[iϕ T x)], 4) U x t,x ) = n v x ft,x,v)dv, ) Πx) = A P x)exp[iϕ P x)], 5) where A i x) i = u,n,t,p) are the amplitudes, while ϕ i x) i = u, n, T, P) are the corresponding phases. In the present work we are going to calculate the amplitudes and phases as a function of both oscillation speed θ and rarefaction δ parameters and to analyze the influence of the receptor on these quantities. KINETIC EQUATION In order to consider arbitrary values of oscillation θ and rarefaction δ parameters the problem must be solved on the basis of the one-dimensional Boltzmann equation that reads f t + v f x x = Q f f ), 6) Tt,x ) = m v U) ft,x,v)dv, ) 3nk B Q x t,x ) = m V v x U x ) ft,x,v)dv. ) The pressure P xx is also calculated via the distribution function as P xx t,x ) = m v x U x ) ft,x,v)dv. 3) Since it is assumed the velocity amplitude of the plate U m to be small when compared with the most probable molecular 3 Copyright c 9 by ASME

4 speed v m of the gas, the kinetic equation 6) can be linearized by representing the distribution function as { ft,x n,v) = πv m ) 3 exp c ) +R [ hx,c)e iωt] } U m, v m 4) where hx,c) is the complex perturbation function and c = v/v m is the dimensionless molecular velocity. Substituting 4) into Eq.6) with Eq.7) the linearized kinetic equation is obtained [ h θ i)h+c x x = θ ρ+c x u+τ c 3 ) qc x c 5 )], 5) exp c x )hl,c x)c x dc x =. 3) After substituting 7) and 8) in 9) and 3) the constants ν and ν L are determined as ν = π π ν L = π c x > c x < c x h,c)exp c )dc, 3) c x hl,c)exp c )dc. 3) In order to eliminate the velocity variables c y and c z the following new functions are introduced Φx,c x ) = π hx,c)exp c y c z)dc y dc z, 33) where ρx) ux) τx) qx) = Πx) π 3/ c x 3 c c x c 5 ) hx,c)e c dc. 6) c x The last moment does not appear in the kinetic equation 5) but its calculation is necessary in order to determine the pressure tensor given in Eq.). It is assumed diffuse scattering of gaseous particles on both plates. Therefore, the boundary condition on the oscillating and stationary plates are given by h,c x ) = ν + c x ), c x >, 7) Ψx,c x ) = c y π + c z )hx,c)exp c y c z )dc ydc z. 34) Multiplying Eq.5) by exp c y c z)/π and by c y + c z )exp c y c z )/π and then integrating it with respect to c y and c z the following equations are obtained [ Φ θ i)φ+c x x = θ ρ+c x u+τ c x ) qc x c x 3 )], 35) hl,c x ) = ν L, c x <, L = δ θ. 8) The quantities ν and ν L are calculated from the impenetrability condition on the plates which means that the bulk velocity of the gas at x = and x = L is equal to the velocity of the plate located at these positions. Therefore, according to [3], in order to satisfy the impenetrability condition in both plates the following conditions must be satisfied π exp c x)h,c x )c x dc x =, 9) [ Ψ θ i)ψ+c x x = θ τ+ 4 ] 5 qc x. 36) The moments defined by Eq.6) are calculated via the new perturbation functions as ρx) ux) τx) qx) = π Πx) c x 3 c x 3 c x c x 3 ) c x Φx,c x)+ 3 c x Ψx,c x) 4 Copyright c 9 by ASME

5 exp c x )dc x. 37) Eqs. 35) and 36) satisfy the following boundary conditions which are obtained from 7) and 8) where the functions Φ and Ψ satisfy the equations θ i)φ + c x Φ x =, 46) Φ,c x ) = c x + ν, c x >, 38) ΦL,c x ) = ν L, c x <, 39) Ψ,c x ) =, c x >, 4) with the following boundary conditions θ i)ψ + c x Ψ x = 47) Φ = c x + ν, at x = and c x >, 48) ΨL,c x ) =, c x <, 4) From 3) and 3) the constants ν and ν L are written in terms of the new perturbation functions Φ and Ψ as follows ν = π c x Φ,c x )exp c x)dc x, 4) c x < ν L = c x ΦL,c x )exp c x)dc x. 43) c x > Eqs. 35) and 36) with the boundary conditions 38) and 4) were solved by the discrete velocity method in which a nonregular distribution of points c xi i N c ) in the velocity space was chosen so that a higher density of points for small values of c xi and a lower density for its large values was used. A regular distribution of points in the x-coordinate was introduced as x k = x k + x, k N x, x =, x = L N x, 44) where N x is an integer. Eqs. 35) and 36) are approximated by the same central finite difference scheme presented in our previous work [5]. The calculations were carried out with the computational error less than.% estimated by varying the parameters N x and N c. Like our previous work [4], in order to reduce the number of points in the velocity space the solution of the kinetic equation is split. Therefore, the perturbation functions Φ and Ψ are presented as Φ = Φ + Φ, Ψ = Ψ + Ψ, 45) Φ = ν, at x = and c x <, 49) Ψ =, at x = and c x >, 5) Ψ =, at x = L and c x <. 5) The constants ν and ν are obtained from 4) and 43) as ν = π c x Φ,c x )exp c x)dc x, 5) c x < ν = c x Φ L,c x )exp c x)dc x. 53) c x > The integration of Eqs. 46) and 47) with the boundary conditions 48)-5) yields the expressions ] Φ x,c x ) = c x + ν )exp [ θ i) xcx, c x >, 54) [ ] x L) Φ x,c x ) = ν exp θ i), c x <, 55) c x Ψx,c x ) =, 56) 5 Copyright c 9 by ASME

6 where ν = 8I [θ i)l]i [θ i)l]+4 πi [θ i)l] 4I [θ i)l], 57) ν = 4I [θ i)l]+ πi [θ i)l] 4I [θ i)l]. 58) The special functions I n z) are defined as I n z) = c n exp c z ) dc. 59) c Substituting 45) into Eqs. 35) and 36) the following equations are obtained for the functions Φ and Ψ [ Φ θ i) Φ+ c x x = θ ρ+c x u+τ c x ) The moments given by Eq. 37) are also decomposed into two parts as ρx) ρx) ρ x) ux) τx) qx) = ũx) τx) qx) + u x) τ x) q x), 66) Πx) Πx) Π x) where the terms ρ, ũ, τ, q, Π are calculated by Eq.37) using Φ and Ψ instead of Φ and Ψ, respectively. The second terms in the right hand side of Eq.66) are given by the following expressions ρ x) = ν π I [ θ i)x L)]+ π I [θ i)x] + + ν )I [θ i)x], 67) π qc x c x 3 )], 6) u x) = ν π I [ θ i)x L)]+ π I [θ i)x] θ i) Ψ+ c x Ψ x [ = θ τ+ 4 ] 5 qc x. 6) The boundary conditions to Eqs. 6) and 6) are given by Φ,c x ) = c x Φ,c x )exp c x)dc x, c x >, 6) c x < ΦL,c x ) = c x ΦL,c x )exp c x )dc x, c x <, 63) c x > Ψ,c x ) =, c x >, 64) + + ν )I [θ i)x], 68) π τ x) = ν { 3 I [ θ i)x L)] } π I [ θ i)x L)] π I 3[θ i)x] 3 π I [θ i)x]+ + ν ) 3 π { I [θ i)x] } 3 I [θ i)x], 69) ΨL,c x ) =, c x <. 65) q x) = ν {I 3 [ θ i)x L)] 3 } π I [ θ i)x L)] 6 Copyright c 9 by ASME

7 + I 4 [θ i)x] 3 I [θ i)x]+ + ν ) π π π { I 3 [θ i)x] 3 } I [θ i)x], 7) Eqs. 6) and 6) are approximated in the same way as in [5]. The splitting of the functions Φ and Ψ allows us to decrease their oscillatory behavior on the velocity c x and, as a consequence, to reduce significantly the number of points N c necessary to achieve an accuracy of.% in the calculations. RESULTS AND DISCUSSION The numerical calculations were carried out for a wide range of both parameters δ and θ with accuracy of.%. The Figures and show the profile of the amplitude and phase of the bulk velocity ux) for rarefaction parameter δ =., and, respectively. In each figure, for a specific value of rarefaction parameter δ, three values of oscillation parameter are considered: θ =., and. From these figures one can see that for a fixed value of rarefaction parameter δ the profiles of the amplitude A u and phase ϕ u of the bulk velocity change both qualitatively and quantitatively by increasing the oscillation parameter θ. When δ < θ the amplitude A u has a linear behavior on the distance. In other situations the behavior of the amplitude A u on the distance is not linear and by increasing the distance sound-receptor the profile of the amplitude tends to the one obtained in our previous work [5] without considering the influence of the receptor on the solution of the problem. The same occurs to the phase ϕ u, i.e. when δ θ it tends to the one given in [5]. It is worthwhile noting that even for δ θ, near the stationary plate the phase presents a behavior different from that obtained in [5] due to the fact that near the receptor there is always the influence of the reflected waves from it. ACKNOWLEDGMENT The authors acknowledge the Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq, Brazil) and also the Coordenação de Aperfeiçoamento de Pessoal de Nivel Superior CAPES, Brazil) for the support of their research. REFERENCES [] Lighthill, J., 978. Waves in Fluids. Cambridge University Press, New York. [] Landau, L. D., and Lifshitz, E. M., 989. Fluid Mechanics. Pergamon, New York. [3] Sharipov, F., and Kalempa, D., 7. Gas flow near a plate oscillating longitudinally with an arbitrary frequency. Phys. Fluids, 9), p. 7. [4] Sharipov, F., and Kalempa, D., 8. Oscillatory Couette flow at arbitrary oscillation frequency over the whole range of the knudsen number. Microfluidics and Nanofluidics, 45), pp [5] Sharipov, F., and Kalempa, D., 8. Numerical modelling of the sound propagation through a rarefied gas in a semi-infinite space on the basis of linearized kinetic equation. J. Acoust. Soc. Am., 44), pp [6] Maidanik, G., Fox, H. L., and Heckl, M., 965. Propagation and reflection of sound in rarefied gases. I. Theoretical. Phys. Fluids, 8), pp [7] Sirovich, L., and Thurber, J. K., 965. Propagation of forced sound waves in rarefied gasdynamics. J. Acoust. Soc. Am., 37), pp [8] Kahn, D., and Mintzer, D., 965. Kinetic theory of sound propagation in rarefied gases. Phys. Fluids, 86), pp. 9. [9] Kahn, D., 966. Sound propagation in rarefied gases. Phys. Fluids, 99), pp [] Buckner, J. K., and Ferziger, J. H., 966. Linearized boundary value problem for a gas and sound propagation. Phys. Fluids, 9), pp [] Toba, K., 968. Kinetic theory of sound propagation in a rarefied gas. Phys. Fluids, ), pp [] Toba, K., 968. Effect of gas-surface interaction on sound propagation. Phys. Fluids, 3), pp [3] Hanson, F. B., and Morse, T. F., 967. Kinetic models for a gas with internal structure. Phys. Fluids,, pp [4] Wang Chang, C. S., and Uhlenbeck, G. E., 97. On the propagation of sound in monatomic gases. Studies in Statistical Mechanics, 5, pp [5] Thomas, J. R., and Siewert, C. E., 979. Sound-wave propagation in a rarefied-gas. Trans. Theory Stat. Phys,, 84), pp [6] Aoki, K., and Cercignani, C., 984. A technique for timedependent boundary value problems in the kinetic theory of gases part II. Application to sound propagation. J. Appl. Math. Phys. ZAMP), 35, pp [7] Marques Jr, W., 999. Dispersion and absorption of sound in monatomic gases: An extended kinetic description. J. Acoust. Soc. Am., 66), pp [8] Hadjiconstantinou, N. G.,. Sound wave propagation in a transition regime micro and nanochannels. Phys. Fluids, 4), pp [9] Hadjiconstantinou, N. G., and Simek, O., 3. Sound propagation at small scales under continuum and noncontinuum transport. J. Fluid Mech., 488, pp [] Sharipov, F., Marques Jr, W., and Kremer, G. M.,. Free molecular sound propagation. J. Acoust. Soc. Am., 7 Copyright c 9 by ASME

8 ), pp [] Garcia, R. D. M., and Siewert, C. E., 6. The linearized Boltzmann equation: Sound-wave propagation in a rarefied gas. Z. angew. Math. Phys. ZAMP), 57, pp. 94. [] Greenspan, M., 956. Propagation of sound in five monatomic gases. J. Acoust. Soc. Am., 8, pp [3] Meyer, E., and Sessler, G., 957. Schallausbreitung in Gasen bei hohen Frequenzen und sehr niedrigen Drucken. Z. Phys., 49, pp [in German]. [4] Maidanik, G., and Heckl, M., 965. Propagation and reflection of sound in rarefied gases. II. Experimental. Phys. Fluids, 8), pp [5] Schotter, R., 974. Rarefied gas acoustics in the noble gases. Phys. Fluids, 76), pp [6] Karniadakis, G. E., Beskok, A., and Narayan, A., 5. Microflows and Nanoflows - Fundamentals and Simulation. Springer-Verlag, New York. [7] Suzuki, Y., and Tai, Y.-C., 6. Micromachined highaspect-ratio parylene spring and its application to lowfrequency accelerometers. J. Microelectromechanical Systems, 55), pp [8] Cercignani, C., 975. Theory and Application of the Boltzmann Equation. Scottish Academic Press, Edinburgh. [9] Ferziger, J. H., and Kaper, H. G., 97. Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company, Amsterdam. [3] Shakhov, E. M., 974. Method of Investigation of Rarefied Gas Flows. Nauka, Moscow. [in Russian]. [3] Sharipov, F. M., and Subbotin, E. A., 993. On optimization of the discrete velocity method used in rarefied gas dynamics. Z. Angew. Math. Phys. ZAMP), 44, pp Copyright c 9 by ASME

9 ,,8,5 A u,6 ϕ u,4, θ=. θ= θ= -,5 -, θ=. θ= θ=,,4,6,8 a) δ =. -,5,,4,6,8 a) δ =. A u,8,6,4 θ=. θ= θ= 6 5 ϕ u 4 3 θ=. θ= θ=,,,4,6,8 b) δ =,,4,6,8 b) δ = 4 A u,8,6 θ=, θ= θ= 3 ϕ u θ=. θ= θ=,4,,,4,6,8 c) δ =,,4,6,8 c) δ = Figure. AMPLITUDE OF BULK VELOCITY ux). Figure. PHASE OF BULK VELOCITY ux). 9 Copyright c 9 by ASME