- Email: [email protected]
Acoustic streaming in micro-scale cylindrical channels
Applied Acoustics 65 (2004) 1121–1129 www.elsevier.com/locate/apacoust
Technical note
Acoustic streaming in micro-scale cylindrical channels Kenneth D. Frampton
*,1,
Keith Minor, Shawn Martin
Department of Mechanical Engineering, Vanderbilt University, VU Station 351592 B, Nashville, TN 37235-1592, USA Received 9 September 2003; received in revised form 3 March 2004; accepted 23 March 2004 Available online 19 June 2004
Abstract The focus of this work is to extend the theory of boundary layer induced acoustic streaming to include cylindrical geometries and to highlight the effects of boundary layer induced streaming on flow velocities in micro-scale channels. The work presented here includes the development of a model for streaming in a cylindrical channel by a method of successive approximations. The validity of this model is established by comparison with a well-established model for streaming between parallel plates of infinite extent. This is followed by a discussion on the importance of employing a cylindrical solution including boundary layer induced streaming for the analysis of streaming in micro-scale channels. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Acoustic streaming; Micro-scale channels; Cylindrical geometries
1. Introduction The authors recently published work that highlighted the importance of acoustic boundary layer induced streaming for analysis of micro-fluidic devices [1]. In this publication, it was noted that acoustic streaming offers two compelling advantages for application in micro-fluidic devices. The first advantage is *
Corresponding author. Tel.: +1-615-322-2778. E-mail address: [email protected] (K.D. Frampton). 1 Assistant Professor of Mechanical Engineering, Associate Member of the Acoustical Society of America. 0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.03.005
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that streaming may well be able to overcome the overwhelming viscous forces associated with micro-scale channels. This is because the streaming is generated through acoustic attenuation associated, in part, with the viscous losses and because the streaming force acts wherever high amplitude acoustic waves are present. The second advantage is that streaming forces scale favorably as the channel diameter decreases. This is because of streaming generation in the acoustic boundary layer, which results in very large forces near the channel wall and becomes the dominant streaming mechanism as the channel diameter decreases. All of these conclusions were based on Nyborg’s solution for streaming between parallel plates of infinite extent [2]. The current work is an extension of prior work in that it presents a model for streaming in a cylindrical channel that includes boundary layer effects and, in particular, considers the importance of these effects in micro-fluidic devices. The majority of recent work in acoustic streaming has been done for applications in thermoacoustic devices. Of particular relevance to this manuscript is the previous work by Hamilton et al. [3] and by Bailliet et al. [4]. Both of these investigations dealt with streaming in cylindrical geometries and included boundary layer induced streaming. The work of Bailliet et al. considered streaming in the presence of a steady state temperature and pressure gradient. The work of Hamilton et al. included the effects of non-steady heat transfer within the fluid as well as the temperature dependence of viscosity. Both of these studies considered streaming in a close-ended cylinder. A few investigators have considered the application of acoustic streaming in micro-fluidic devices. Bradely and White [5] considered the use of surface acoustic wave devices for the generation of flow in micro-devices. This is one of the few published results for experimental application of streaming in micro-scale devices achieving flow velocity of 1.15 mm/s. Dohner [6] performed an analytical investigation of an acoustic streaming pump based on traveling plane waves perpendicular to the flow axis. Rife and Bell [14] constructed a small-scale fluid circuit pumped by means of acoustic streaming generated by a planar acoustic source [7]. The transducers were operated at resonance (50 MHz) and produced mean water velocities of 1 mm/s. Flow visualization was accomplished by seeding the fluid with 2 and 10 lm microspheres. By attaching two transducers to the network, Rife et al. were able to control both the nature and direction of the flow. Each of the theoretical investigations into streaming in micro-fluidic devices demonstrated the feasibility such an approach. However, none of them consider the effect that boundary layer induced streaming has on the effectiveness of such a device. The purpose of the work presented here is to consider streaming in a cylindrical channel with open ends. Furthermore, the primary conclusions of this work concern applications of streaming in micro-fluidic devices and the relevance of boundary layer induced streaming to the scaling of streaming induced flow. The solution process begins with the Navier–Stokes equation combined with the continuity equation and dynamic equations resulting in an expression for the net force per unit volume. The method of successive approximations is used to separate the first and second order effects of the field variables. Computational results from this derivation
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are then compared to Nyborg’s solution in order to establish the validity [2]. This is followed by a discussion of the importance of using a cylindrical solution when streaming in micro-scale channels is considered.
2. The theory of acoustic streaming The solution process for streaming in a cylindrical channel closely follows that of Nyborg for streaming between parallel plates of infinite extent [2] as well as Eckart’s method of successive approximations [8]. The development begins by assuming an unbounded, homogeneous, isotropic, Newtonian, originally quiescent fluid. The only forces acting in shear are surface stresses due to elasticity and viscosity. Furthermore, bulk and shear viscosity are assumed to be constant. Given these assumptions, the development begins with the compressible Navier–Stokes equation, which is combined with the compressible continuity equation for a Newtonian fluid. Since streaming represents the steady components of the field variables (as opposed to time varying components) these equations are time averaged. Next, following the approach taken by Eckart, the excess pressure, density and velocity terms are expanded as the sum first and second order terms [8]. First order terms describe the oscillating part of the solution, which is related to the acoustic field, and whose time averages will be zero. Second order terms are associated with time independent components of the excess pressure, density, and velocity. These second order components are associated with streaming. In the resulting equations terms of order greater than 2 are assumed negligible and are discarded. This assumption limits the result to the low Reynolds number regime [9]. The result is a pair of equations as follows [2]: hFi ¼ rhp2 i lr2 hu2 i;
ð1Þ
hFi hqðu1 rÞu1 þ u1 ðr qu1 Þi;
ð2Þ
where p2 is the second order pressure, u2 is the second order or streaming velocity, and u1 is the acoustic velocity, and is the viscosity. The braces, hi, indicate the time average of the argument. The acoustic velocity in the region of interest is found by assuming a radially uniform upstream acoustic source then propagating this source down the channel resulting in and acoustic velocity u1 which includes boundary layer effects [1]. Then, given this acoustic velocity one can solve Eq. (2) for the body force in the axial direction, Fz , such that [2]: Fz ¼ Fza þ Fzs ;
ð3Þ
Fza ¼ aqo A2 ;
ð4Þ
Fzs ¼ 12qo A2 k ekz cosðjzÞ þ ejz sinðjzÞ e2jz þ 12qo A2 a e2jz 3ejz cosðjzÞ þ ejz sinðjzÞ ;
ð5Þ
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where Fza is the body force associated with viscous losses and Fzs is the force due to interactions in the acoustic boundary layer, where A is the source p acoustic ffiffiffiffiffiffiffiffiffiffiffiffiffiffivelocity, a is the acoustic absorption coefficient, k is the wavenumber, j ¼ xq=2l, and x is the excitation frequency. Now, Eq. (1) can be solved for the second order or streaming velocity, u2 . Note that Eqs. (1) and (2) (and hence Eqs. (3)–(5)) apply to both streaming between parallel plates and streaming in a cylindrical channel. Now Eqs. (4) and (5) can be used together with (1) to solve for the steady streaming velocity. The development up to this point has followed that of Nyborg and co-worker [2] and Eckart [8]. The cylindrical channel geometry consists of a cylindrical channel of constant cross-section, infinite length, radius a and filled with a viscous fluid. A cylindrical coordinate system is used with the center of the channel as the coordinate origin. An acoustic wave travels in the z-direction (along the axis of the channel) and fills the entire channel. Given such a geometry, and assuming an axisymmetric solution, Eq. (1) reduces to l d dhu2 i Fr ¼ r : r dr dr
ð6Þ
The body force components, Fxa and Fxs , of Eq. (3) are equally applicable in this axisymmetric cylindrical geometry. However, the coordinate systems for each expression must be transformed, moving the origin to the center of the channel, and converting them to cylindrical coordinates. Solving for the streaming due to the viscous affects first, Eq. (4) is substituted into Eq. (6) to yield l d dhu2a i 2 r aqA ¼ ; ð7Þ dr r dr where u2a is the streaming velocity associated with viscous effects. Eq. (7) is integrated twice with respect to r and the boundary conditions for continuity at the center of the channel, du2 ð0Þ=dr ¼ 0, and no slip at the wall, u2 ðaÞ ¼ 0, are applied. This results in u2a ðrÞ ¼
aqA2 2 r a2 ; 4l
ð8Þ
which is the streaming velocity profile resulting from viscous affects. Boundary layer streaming is induced by the body force given in Eq. (5). The assumptions used in Nyborg’s parallel plate solution are also reasonable here [2]. These are that al 1 so that e2az 1 and a k. The result of the second assumption is that the second term in Eq. (5) is negligible and therefore Eq. (5) can be expressed in cylindrical coordinates as [10,11] Fzs ¼ 12qo A2 k ejðarÞ cosðjða rÞÞ þ ejðarÞ sinðjða rÞÞ e2jðarÞ :
ð9Þ
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Eq. (9) is substituted into Eq. (6) resulting in d dhu2s ðrÞi q A2 kr jðarÞ ¼ o cosðjða rÞÞ þ ejðarÞ sinðjða rÞÞ e2jðarÞ : r e dr dr 2l ð10Þ Eq. (10) is integrated with respect to r and the no slip boundary condition is again applied. The result is that dhu2s ðrÞi q A2 k r 1 ¼ o 2 ejðarÞ cosðjða rÞÞ dr 2lr j 2j q o A2 k 1 qo A2 k 1 r jðarÞ 2jðarÞ e e sinðjða rÞÞ 2lr 2j2 2lr 4j2 2j qA2 k ja 1 ð11Þ e sinðjaÞ cosðjaÞ þ eja : þ 4lj2 r 2 The final step toward the solution for boundary layer induced streaming velocity requires the numerical integration of Eq. (11) and the application of the no slip boundary condition, uðaÞ ¼ 0. The total solution for streaming in a cylindrical channel is simply the sum of the streaming velocities due to viscous effects, given in -3
x 10 8
Streaming Velocity, m/s
7
6
5
4
3
2
Cylindrical solution
1
Parallel plate solution 0 0
0.2
0.4
0.6
0.8
1
Normalized Radius Fig. 1. Velocity profile for 3-D and 2-D streaming solutions, as a function of non-dimensional channel radius, for a radius of 0.01 m.
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Eq. (8), and boundary layer streaming obtained by integrating Eq. (11). The total solution can then be written as u2 ¼ u2a þ u2s :
ð12Þ
3. A Comparison of 2-dimensional and 3-dimensional solutions A comparison of the velocity profiles obtained from Nyborg’s solution and the cylindrical solution are shown in Figs. 1 and 2 for channels diameters of 1 102 and 1 105 m, respectively. These plots were constructed assuming that the fluid is water excited by a 15 kW intensity incident acoustic wave at 1 MHz. The water was assumed to have a speed of sound of 1480 m/s, a viscosity of 0.000893 Ns/m, an absorption coefficient of 0.025 Ns/m and a density of 1000 kg/m3 . The first clear difference noted between Nyborg’s solution and cylindrical solutions is that Nyborg’s solution is twice the magnitude of the cylindrical solution for the 1 102 m case. This result is not unexpected and can be explained from fundamental fluid mechanics. First of all, it has been noted that streaming flow in large-scale channels is identical to pressure driven flow [10,12]. In comparing the velocity profile equa-5
x 10 1.4
Streaming Velocity, m/s
1.2
1.0
0.8
Cylindrical solution 0.6
Parallel plate solution 0.4
0.2
0.0 0
0.2
0.4
0.6
0.8
1
Normalized Radius Fig. 2. Velocity profile for 3-D and 2-D streaming solutions, as a function of non-dimensional channel radius, for a radius of 10 lm.
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tions for viscous, pressure driven flow between parallel plates to the solution for pressure driven flow in a cylindrical channel (or Poiseuille flow), it is observed that the viscous force acting on a cross section within the cylindrical field is twice that acting on a cross section between parallel plates [13]. This leads to flow velocities that are twice as large for flow between parallel plates as for Poiseuille flow. This same observation is noted for acoustic streaming in relatively large-scale channels such as that shown in Fig. 1. Thus, it was expected that the factor of two would appear when comparing Nyborg’s solution to the cylindrical solutions for sufficiently large diameters. A similar comparison yields different results in small-scale channels as shown in Fig. 2. Note that the two solutions differ only slightly in this case. However, the streaming induced in the boundary layer is very different. This difference can be attributed to the different geometries. A particle in the boundary layer of the cylindrical geometer will ‘‘see’’ more channel wall nearby than a similar particle near a flat surface. This is because the curved wall of the cylindrical solution ‘‘wraps’’ around the particle. This effect becomes larger as the channel diameter decreases. The influence of boundary layer induced flow, and the implications of employing a cylindrical solution, can be seen more clearly when the volume velocity is considered. This solution is obtained by revolving the velocity profiles and integrating over the channel area. The volume velocity as a function of channel radius is shown 0
80 Volume Velocity, m3/s
-5
10
Parallel plate solution Cylindrical Solution
60
Percent Boundary -10
10
Layer Contribution 40
Percent Boundary Layer Contribution
100
10
-15
10
20
-6
10
-5
10
-4
10
-3
10
-2
10
-1
0
10
Diameter, m
Fig. 3. Volume velocity and relative contribution of boundary layer induced streaming to overall volume velocity as a function of channel diameter.
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in Fig. 3 for both solutions. Also shown in Fig. 3 is the percent contribution to the total volume velocity by boundary layer induced flow (based on the cylindrical solution). As shown in Fig. 3, Nyborg’s solution predicts twice the flow rate when the channel diameter is greater than about 0.01 m. Below about 0.001 m the two solutions merge. In the region between 0.01 m and 100 lm the ratio transitions from a ratio of two to a ratio of one. This transition region corresponds to the transition from bulk streaming dominated flow to boundary layer dominated flow as shown in Fig. 5. When the boundary layer induced streaming is dominant (between about 3 106 and 1 104 m for this case) the two solutions predict identical volume velocities. This is interesting considering that the velocity flow profiles predicted by the two solutions (as shown in Fig. 2) are quite different. In previous work, Frampton and co-workers [1] noted that the volume velocity does not continue to decrease as the channel diameter decreases due to boundary layer streaming. This conclusion, which was based on Nyborg’s solution, is supported by results presented in Fig. 3 for the cylindrical solution. However, while this general conclusion is supported by the cylindrical solution, the use of Nyborg’s parallel plate theory will be in error for larger scale channels. Furthermore, the specific behavior in the acoustic boundary layer will be significantly different from that predicted by parallel plate theory, as demonstrated in Fig. 2.
4. Conclusions A model for acoustic streaming in a cylindrical channel has been presented. The accuracy of this model was established by comparison with existing models for streaming. The inaccuracy of predicting streaming velocities by revolving a solution for streaming between parallel plates was also noted. It was noted that, when channel diameters become very small the boundary layer induced streaming dominates the flow. Therefore, accurate representation of boundary layer streaming, and in particular the use of a solution based on cylindrical coordinates, is critical for small-scale channels.
Acknowledgements This work was supported by support of the National Science Foundation Grant No. CMS-0070108 and the Tennessee Space Grant Consortium.
References [1] Frampton Kenneth D, Martin S, Minor Keith. The scaling of acoustic streaming for application in micro-fluidic devices. Appl Acoust 2003;64(7):681–92. [2] Nyborg Wesley L. Acoustic streaming. In: Hamilton Mark, Blackstock David, editors. Nonlinear acoustics. San Diego, California: Academic Press; 1998. p. 207–31.
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[3] Hamilton Mark F, Ilinskii Yurii A, Zabolotskaya Evgenia A. Thermal effects on acoustic streaming in standing waves. J Acoust Soc America 2003;114(6):3092–101. [4] Helene Bailliet, Vitalyi Gusev, Richard Raspet, Robert A Hiller. Acoustic streaming in closed thermoacoustic devices. J Acoust Soc Am 2001;110(4):1808–21. [5] Bradley CE, White RM. Acoustically driven flow in flexural plate wave devices: theory and experiment. In: Proceedings of the 1994 Ultrasonic Symposium. p. 593–7. [6] Dohner JL. The analysis of a micro-scale pump which uses controlled acoustic streaming for fluid locomotion, Sandia Report SAND98-0207. Sandia National Laboratories; 1998. [7] Rife JC, Bell MI, Horwitz JS, Kabler MN, Auyeung RCY, Kim WJ. Miniature valveless ultrasonic pumps and mixers. Sensor Actuator 2000;86:135–40. [8] Eckart Carl. Vortices and streams caused by sound waves. Phys Rev 1948;73(1):68–76. [9] Lighthill J. Acoustic streaming. J Sound Vib 1978;61(3):391–418. [10] Shawn Martin. Parametric analysis of acoustic streaming pumps utilizing planar ultrasonic acoustic beams, MS Thesis. Vanderbilt University; 2001. [11] Keith Minor. Acoustic streaming micro-pumps, MS thesis. Vanderbilt University; 2002. [12] Shawn Martin, Frampton KD. Micro-fluidic control based on acoustic streaming. In: Proceedings of IMECE 2000, Orlando, FL, November. Noise Control and Acoustics Division; 2000. [13] Currie IG. In: Corrigan J, Moriss J, editors. Fundamental mechanics of fluids. 2nd ed.; 1993. p. 217– 22. [14] Rife JC, Bell MI. Acousto- and electroosmotic microfluidic controllers. In: Proceedings of the SPIE Conference on Microfluidic Devices and Systems, Santa Clara, CA, September. vol. 3515. SPIE; 1998. p. 125–35.
Technical note
Acoustic streaming in micro-scale cylindrical channels Kenneth D. Frampton
*,1,
Keith Minor, Shawn Martin
Department of Mechanical Engineering, Vanderbilt University, VU Station 351592 B, Nashville, TN 37235-1592, USA Received 9 September 2003; received in revised form 3 March 2004; accepted 23 March 2004 Available online 19 June 2004
Abstract The focus of this work is to extend the theory of boundary layer induced acoustic streaming to include cylindrical geometries and to highlight the effects of boundary layer induced streaming on flow velocities in micro-scale channels. The work presented here includes the development of a model for streaming in a cylindrical channel by a method of successive approximations. The validity of this model is established by comparison with a well-established model for streaming between parallel plates of infinite extent. This is followed by a discussion on the importance of employing a cylindrical solution including boundary layer induced streaming for the analysis of streaming in micro-scale channels. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Acoustic streaming; Micro-scale channels; Cylindrical geometries
1. Introduction The authors recently published work that highlighted the importance of acoustic boundary layer induced streaming for analysis of micro-fluidic devices [1]. In this publication, it was noted that acoustic streaming offers two compelling advantages for application in micro-fluidic devices. The first advantage is *
Corresponding author. Tel.: +1-615-322-2778. E-mail address: [email protected] (K.D. Frampton). 1 Assistant Professor of Mechanical Engineering, Associate Member of the Acoustical Society of America. 0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.03.005
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that streaming may well be able to overcome the overwhelming viscous forces associated with micro-scale channels. This is because the streaming is generated through acoustic attenuation associated, in part, with the viscous losses and because the streaming force acts wherever high amplitude acoustic waves are present. The second advantage is that streaming forces scale favorably as the channel diameter decreases. This is because of streaming generation in the acoustic boundary layer, which results in very large forces near the channel wall and becomes the dominant streaming mechanism as the channel diameter decreases. All of these conclusions were based on Nyborg’s solution for streaming between parallel plates of infinite extent [2]. The current work is an extension of prior work in that it presents a model for streaming in a cylindrical channel that includes boundary layer effects and, in particular, considers the importance of these effects in micro-fluidic devices. The majority of recent work in acoustic streaming has been done for applications in thermoacoustic devices. Of particular relevance to this manuscript is the previous work by Hamilton et al. [3] and by Bailliet et al. [4]. Both of these investigations dealt with streaming in cylindrical geometries and included boundary layer induced streaming. The work of Bailliet et al. considered streaming in the presence of a steady state temperature and pressure gradient. The work of Hamilton et al. included the effects of non-steady heat transfer within the fluid as well as the temperature dependence of viscosity. Both of these studies considered streaming in a close-ended cylinder. A few investigators have considered the application of acoustic streaming in micro-fluidic devices. Bradely and White [5] considered the use of surface acoustic wave devices for the generation of flow in micro-devices. This is one of the few published results for experimental application of streaming in micro-scale devices achieving flow velocity of 1.15 mm/s. Dohner [6] performed an analytical investigation of an acoustic streaming pump based on traveling plane waves perpendicular to the flow axis. Rife and Bell [14] constructed a small-scale fluid circuit pumped by means of acoustic streaming generated by a planar acoustic source [7]. The transducers were operated at resonance (50 MHz) and produced mean water velocities of 1 mm/s. Flow visualization was accomplished by seeding the fluid with 2 and 10 lm microspheres. By attaching two transducers to the network, Rife et al. were able to control both the nature and direction of the flow. Each of the theoretical investigations into streaming in micro-fluidic devices demonstrated the feasibility such an approach. However, none of them consider the effect that boundary layer induced streaming has on the effectiveness of such a device. The purpose of the work presented here is to consider streaming in a cylindrical channel with open ends. Furthermore, the primary conclusions of this work concern applications of streaming in micro-fluidic devices and the relevance of boundary layer induced streaming to the scaling of streaming induced flow. The solution process begins with the Navier–Stokes equation combined with the continuity equation and dynamic equations resulting in an expression for the net force per unit volume. The method of successive approximations is used to separate the first and second order effects of the field variables. Computational results from this derivation
K.D. Frampton et al. / Applied Acoustics 65 (2004) 1121–1129
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are then compared to Nyborg’s solution in order to establish the validity [2]. This is followed by a discussion of the importance of using a cylindrical solution when streaming in micro-scale channels is considered.
2. The theory of acoustic streaming The solution process for streaming in a cylindrical channel closely follows that of Nyborg for streaming between parallel plates of infinite extent [2] as well as Eckart’s method of successive approximations [8]. The development begins by assuming an unbounded, homogeneous, isotropic, Newtonian, originally quiescent fluid. The only forces acting in shear are surface stresses due to elasticity and viscosity. Furthermore, bulk and shear viscosity are assumed to be constant. Given these assumptions, the development begins with the compressible Navier–Stokes equation, which is combined with the compressible continuity equation for a Newtonian fluid. Since streaming represents the steady components of the field variables (as opposed to time varying components) these equations are time averaged. Next, following the approach taken by Eckart, the excess pressure, density and velocity terms are expanded as the sum first and second order terms [8]. First order terms describe the oscillating part of the solution, which is related to the acoustic field, and whose time averages will be zero. Second order terms are associated with time independent components of the excess pressure, density, and velocity. These second order components are associated with streaming. In the resulting equations terms of order greater than 2 are assumed negligible and are discarded. This assumption limits the result to the low Reynolds number regime [9]. The result is a pair of equations as follows [2]: hFi ¼ rhp2 i lr2 hu2 i;
ð1Þ
hFi hqðu1 rÞu1 þ u1 ðr qu1 Þi;
ð2Þ
where p2 is the second order pressure, u2 is the second order or streaming velocity, and u1 is the acoustic velocity, and is the viscosity. The braces, hi, indicate the time average of the argument. The acoustic velocity in the region of interest is found by assuming a radially uniform upstream acoustic source then propagating this source down the channel resulting in and acoustic velocity u1 which includes boundary layer effects [1]. Then, given this acoustic velocity one can solve Eq. (2) for the body force in the axial direction, Fz , such that [2]: Fz ¼ Fza þ Fzs ;
ð3Þ
Fza ¼ aqo A2 ;
ð4Þ
Fzs ¼ 12qo A2 k ekz cosðjzÞ þ ejz sinðjzÞ e2jz þ 12qo A2 a e2jz 3ejz cosðjzÞ þ ejz sinðjzÞ ;
ð5Þ
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where Fza is the body force associated with viscous losses and Fzs is the force due to interactions in the acoustic boundary layer, where A is the source p acoustic ffiffiffiffiffiffiffiffiffiffiffiffiffiffivelocity, a is the acoustic absorption coefficient, k is the wavenumber, j ¼ xq=2l, and x is the excitation frequency. Now, Eq. (1) can be solved for the second order or streaming velocity, u2 . Note that Eqs. (1) and (2) (and hence Eqs. (3)–(5)) apply to both streaming between parallel plates and streaming in a cylindrical channel. Now Eqs. (4) and (5) can be used together with (1) to solve for the steady streaming velocity. The development up to this point has followed that of Nyborg and co-worker [2] and Eckart [8]. The cylindrical channel geometry consists of a cylindrical channel of constant cross-section, infinite length, radius a and filled with a viscous fluid. A cylindrical coordinate system is used with the center of the channel as the coordinate origin. An acoustic wave travels in the z-direction (along the axis of the channel) and fills the entire channel. Given such a geometry, and assuming an axisymmetric solution, Eq. (1) reduces to l d dhu2 i Fr ¼ r : r dr dr
ð6Þ
The body force components, Fxa and Fxs , of Eq. (3) are equally applicable in this axisymmetric cylindrical geometry. However, the coordinate systems for each expression must be transformed, moving the origin to the center of the channel, and converting them to cylindrical coordinates. Solving for the streaming due to the viscous affects first, Eq. (4) is substituted into Eq. (6) to yield l d dhu2a i 2 r aqA ¼ ; ð7Þ dr r dr where u2a is the streaming velocity associated with viscous effects. Eq. (7) is integrated twice with respect to r and the boundary conditions for continuity at the center of the channel, du2 ð0Þ=dr ¼ 0, and no slip at the wall, u2 ðaÞ ¼ 0, are applied. This results in u2a ðrÞ ¼
aqA2 2 r a2 ; 4l
ð8Þ
which is the streaming velocity profile resulting from viscous affects. Boundary layer streaming is induced by the body force given in Eq. (5). The assumptions used in Nyborg’s parallel plate solution are also reasonable here [2]. These are that al 1 so that e2az 1 and a k. The result of the second assumption is that the second term in Eq. (5) is negligible and therefore Eq. (5) can be expressed in cylindrical coordinates as [10,11] Fzs ¼ 12qo A2 k ejðarÞ cosðjða rÞÞ þ ejðarÞ sinðjða rÞÞ e2jðarÞ :
ð9Þ
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Eq. (9) is substituted into Eq. (6) resulting in d dhu2s ðrÞi q A2 kr jðarÞ ¼ o cosðjða rÞÞ þ ejðarÞ sinðjða rÞÞ e2jðarÞ : r e dr dr 2l ð10Þ Eq. (10) is integrated with respect to r and the no slip boundary condition is again applied. The result is that dhu2s ðrÞi q A2 k r 1 ¼ o 2 ejðarÞ cosðjða rÞÞ dr 2lr j 2j q o A2 k 1 qo A2 k 1 r jðarÞ 2jðarÞ e e sinðjða rÞÞ 2lr 2j2 2lr 4j2 2j qA2 k ja 1 ð11Þ e sinðjaÞ cosðjaÞ þ eja : þ 4lj2 r 2 The final step toward the solution for boundary layer induced streaming velocity requires the numerical integration of Eq. (11) and the application of the no slip boundary condition, uðaÞ ¼ 0. The total solution for streaming in a cylindrical channel is simply the sum of the streaming velocities due to viscous effects, given in -3
x 10 8
Streaming Velocity, m/s
7
6
5
4
3
2
Cylindrical solution
1
Parallel plate solution 0 0
0.2
0.4
0.6
0.8
1
Normalized Radius Fig. 1. Velocity profile for 3-D and 2-D streaming solutions, as a function of non-dimensional channel radius, for a radius of 0.01 m.
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Eq. (8), and boundary layer streaming obtained by integrating Eq. (11). The total solution can then be written as u2 ¼ u2a þ u2s :
ð12Þ
3. A Comparison of 2-dimensional and 3-dimensional solutions A comparison of the velocity profiles obtained from Nyborg’s solution and the cylindrical solution are shown in Figs. 1 and 2 for channels diameters of 1 102 and 1 105 m, respectively. These plots were constructed assuming that the fluid is water excited by a 15 kW intensity incident acoustic wave at 1 MHz. The water was assumed to have a speed of sound of 1480 m/s, a viscosity of 0.000893 Ns/m, an absorption coefficient of 0.025 Ns/m and a density of 1000 kg/m3 . The first clear difference noted between Nyborg’s solution and cylindrical solutions is that Nyborg’s solution is twice the magnitude of the cylindrical solution for the 1 102 m case. This result is not unexpected and can be explained from fundamental fluid mechanics. First of all, it has been noted that streaming flow in large-scale channels is identical to pressure driven flow [10,12]. In comparing the velocity profile equa-5
x 10 1.4
Streaming Velocity, m/s
1.2
1.0
0.8
Cylindrical solution 0.6
Parallel plate solution 0.4
0.2
0.0 0
0.2
0.4
0.6
0.8
1
Normalized Radius Fig. 2. Velocity profile for 3-D and 2-D streaming solutions, as a function of non-dimensional channel radius, for a radius of 10 lm.
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tions for viscous, pressure driven flow between parallel plates to the solution for pressure driven flow in a cylindrical channel (or Poiseuille flow), it is observed that the viscous force acting on a cross section within the cylindrical field is twice that acting on a cross section between parallel plates [13]. This leads to flow velocities that are twice as large for flow between parallel plates as for Poiseuille flow. This same observation is noted for acoustic streaming in relatively large-scale channels such as that shown in Fig. 1. Thus, it was expected that the factor of two would appear when comparing Nyborg’s solution to the cylindrical solutions for sufficiently large diameters. A similar comparison yields different results in small-scale channels as shown in Fig. 2. Note that the two solutions differ only slightly in this case. However, the streaming induced in the boundary layer is very different. This difference can be attributed to the different geometries. A particle in the boundary layer of the cylindrical geometer will ‘‘see’’ more channel wall nearby than a similar particle near a flat surface. This is because the curved wall of the cylindrical solution ‘‘wraps’’ around the particle. This effect becomes larger as the channel diameter decreases. The influence of boundary layer induced flow, and the implications of employing a cylindrical solution, can be seen more clearly when the volume velocity is considered. This solution is obtained by revolving the velocity profiles and integrating over the channel area. The volume velocity as a function of channel radius is shown 0
80 Volume Velocity, m3/s
-5
10
Parallel plate solution Cylindrical Solution
60
Percent Boundary -10
10
Layer Contribution 40
Percent Boundary Layer Contribution
100
10
-15
10
20
-6
10
-5
10
-4
10
-3
10
-2
10
-1
0
10
Diameter, m
Fig. 3. Volume velocity and relative contribution of boundary layer induced streaming to overall volume velocity as a function of channel diameter.
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in Fig. 3 for both solutions. Also shown in Fig. 3 is the percent contribution to the total volume velocity by boundary layer induced flow (based on the cylindrical solution). As shown in Fig. 3, Nyborg’s solution predicts twice the flow rate when the channel diameter is greater than about 0.01 m. Below about 0.001 m the two solutions merge. In the region between 0.01 m and 100 lm the ratio transitions from a ratio of two to a ratio of one. This transition region corresponds to the transition from bulk streaming dominated flow to boundary layer dominated flow as shown in Fig. 5. When the boundary layer induced streaming is dominant (between about 3 106 and 1 104 m for this case) the two solutions predict identical volume velocities. This is interesting considering that the velocity flow profiles predicted by the two solutions (as shown in Fig. 2) are quite different. In previous work, Frampton and co-workers [1] noted that the volume velocity does not continue to decrease as the channel diameter decreases due to boundary layer streaming. This conclusion, which was based on Nyborg’s solution, is supported by results presented in Fig. 3 for the cylindrical solution. However, while this general conclusion is supported by the cylindrical solution, the use of Nyborg’s parallel plate theory will be in error for larger scale channels. Furthermore, the specific behavior in the acoustic boundary layer will be significantly different from that predicted by parallel plate theory, as demonstrated in Fig. 2.
4. Conclusions A model for acoustic streaming in a cylindrical channel has been presented. The accuracy of this model was established by comparison with existing models for streaming. The inaccuracy of predicting streaming velocities by revolving a solution for streaming between parallel plates was also noted. It was noted that, when channel diameters become very small the boundary layer induced streaming dominates the flow. Therefore, accurate representation of boundary layer streaming, and in particular the use of a solution based on cylindrical coordinates, is critical for small-scale channels.
Acknowledgements This work was supported by support of the National Science Foundation Grant No. CMS-0070108 and the Tennessee Space Grant Consortium.
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