Acoustic Streaming

Synonyms

Steady streaming; Eulerian streaming

Definition

Acoustic streaming is a secondary steady flow generated from the primary oscillatory flow. It includes not only the Eulerian streaming flow caused by the fluid dynamical interaction but also the Stokes drift flow which arises from a purely kinematic viewpoint.

Overview

Acoustic streaming originates from the acoustics. When the compressible fluid experiences a high-frequency oscillation driven by a certain source of sound, the nonlinear interaction causes a steady current in the field. This phenomenon is usually referred to as acoustic streaming. However, it was found that even an incompressible fluid can produce such steady streaming when the fluid oscillates adjacent to a certain obstacle or an interface. So, as pointed out by Riley [1], the term acoustic streaming is misleading. Instead of acoustic streaming he insisted on using steady streaming to encompass the case of incompressible fluid. Acoustic streaming in this article means both of these effects.

Another important issue that must be addressed is, whether or not the Stokes drift flow should be included in the streaming flow category. The Stokes drift flow is purely kinematic and so it is basically different from the Eulerian streaming flow, which is driven by the time average of the Reynolds stress term. However, the mass-transport effect given by the Stokes drift flow is not weaker than the Eulerian streaming. On the contrary, in the case of a progressive wave, the Stokes drift flow plays a dominant role (e. g., application of flexural plate waves to pumping and mixing in microfluidics [2]). Therefore, the Stokes drift flow must be understood as a kind of streaming flow.

Unfortunately, little is known about the historical background to the study of acoustic streaming. Therefore, instead of trying to give the historical background of the general concept, we will briefly survey the modeling and applications of acoustic streaming in microfluidics. A survey of steady streaming flow has been given by Riley [1].

In the early period, study mainly focused on the acoustic streaming of compressible fluids. The simplest and seminal phenomenon associated with acoustic streaming is the quartz wind. When the air experiences a high-frequency oscillation given by an ultrasound source, a progressive wave is established in the air. Due to the attenuation of the wave, a nonzero time averaged Reynolds stress is built in the region close to the sound source, and this stress pushes the air in the direction of wave propagation. The resultant wind is called a quartz wind. Lighthill [3] has provided a physical and simple analytical treatment of this steady streaming flow. Another important phenomenon is the so-called Kundt's dust pattern. When the air within a duct receives a standing wave, a nonzero time average of the Reynolds stress is built inside the duct. Due to the interaction between the air and the duct wall, a steady recirculating flow takes place within the duct. The net effect is that dust or particles accumulate at nodes.

On the other hand, researchers interested in mass transport in water waves have shown that a similar phenomenon occurs for incompressible flow (see [4] for an example). The salient feature is when an obstacle oscillates with high frequency in the quiescent viscous fluid. In this case the steady streaming emerges from both sides of the obstacle along the oscillating direction at small viscosity [5].

In this article, we will describe in some detail the fundamentals of three types of streaming flows associated with the microfluidic applications. First, the quartz wind which corresponds to the one-dimensional compressible flow will be introduced. Second, the Eulerian streaming flow in two-dimensional space will be considered. Kundt's dust phenomenon will also be explained. Finally, we address the flexural plate wave and its net effect, i. e., the Stokes drift flow.

Basic Methodology

One-Dimensional Compressible Flow Model – Quartz Wind

In this and next subsections we closely follow the review by Riley [1] to describe the physical mechanism of acoustic streaming. We first consider the case when the sound is generated from a source and travels along a certain direction, say the $x^{{\ast}}$-direction, in a space. The governing equation for the motion of the compressible fluid is

$$ \begin{aligned}\displaystyle\frac{\partial\rho^{\ast}}{\partial t^{\ast}}+\frac{\partial\rho^{\ast}u^{\ast}}{\partial x^{\ast}}&\displaystyle=0\end{aligned} $$

(1a)

$$ \begin{aligned}\displaystyle\frac{\partial u^{\ast}}{\partial t^{\ast}}+u^{\ast}\frac{\partial u^{\ast}}{\partial x^{\ast}}&\displaystyle=-\frac{1}{\rho^{\ast}}\frac{\partial p^{\ast}}{\partial x^{\ast}}+\frac{4}{3}\nu\frac{\partial^{2}u^{\ast}}{\partial x^{{\ast 2}}}\end{aligned} $$

(1b)

where $u^{\ast}$ is the flow velocity, $p^{\ast}$ the pressure, $t^{\ast}$ the time, $\rho^{\ast}$ the density of the fluid and $\nu$ the kinematic viscosity of the fluid. The fluid particle is assumed to oscillate back and forth with amplitude $A_{0}$, frequency $\omega$, and wavelength $\lambda$. We then take $1/\omega$, $\lambda/2\pi$, $A_{0}\omega$, and $\rho _{0}^{\ast}A_{0}\omega^{2}(\lambda/2\pi)$ as the reference quantities for the time, length, velocity, and pressure, respectively. Further, we expand the density as

$$ \rho^{\ast}=\rho _{0}^{\ast}\,(1+\varepsilon\rho _{0}+\varepsilon^{2}\rho _{1}+\ldots) $$

(2)

where $\rho _{0}^{\ast}$ corresponds to the undisturbed fluid density. We also expand the dimensionless velocity and pressure, respectively, as

$$ \displaystyle u=u_{0}+\varepsilon u_{1}+\varepsilon^{2}u_{2}+\ldots $$

(3a)

$$ \displaystyle p=p_{0}+\varepsilon p_{1}+\varepsilon^{2}p_{2}+\ldots $$

(3b)

Then the leading-order equations of Eqs. (1a) and (1b) for the small value of the parameter $\varepsilon$ become

$$ \displaystyle\frac{\partial\rho _{0}}{\partial t}+\frac{\partial u_{0}}{\partial x}=0 $$

(4a)

$$ \displaystyle\frac{\partial u_{0}}{\partial t}=-\frac{\partial p_{0}}{\partial x}+\frac{4}{3}\delta\frac{\partial^{2}u_{0}}{\partial x^{2}} $$

(4b)

where the variables without an asterisk are dimensionless. Here, the two dimensionless parameters are

$$ \varepsilon=\frac{A_{0}}{\lambda/2\pi}\,,\quad\delta=\frac{\nu}{\omega(\lambda/2\pi)^{2}} $$

(5)

which can be considered as the dimensionless amplitude of the flow motion and the inverse of the Reynolds number, respectively. As usual, these parameters are assumed to be small. To solve this system of equations we eliminate the perturbed density in Eq. (4a) and the pressure in Eq. (4b) by using the definition of sound

$$ \frac{\partial p^{\ast}}{\partial\rho^{\ast}}=c^{{\ast 2}}=\omega^{2}(\lambda/2\pi)^{2} $$

(6)

We then have

$$ \frac{\partial^{2}u_{0}}{\partial t^{2}}-\frac{\partial^{2}u_{0}}{\partial x^{2}}-\frac{4}{3}\delta\frac{\partial^{3}u_{0}}{\partial t\partial x^{2}}=0 $$

(7)

The solution of this equation is

$$ u_{0}=\exp(-2\delta x/3)\cos(x-t) $$

(8)

Notice that the basic mode is progressive. We need the next higher-order equation of Eq. (1b), which is

$$ \frac{\partial u_{1}}{\partial t}+\frac{\partial p_{1}}{\partial x}-\frac{4}{3}\delta\frac{\partial^{2}u_{1}}{\partial x^{2}}=-u_{0}\frac{\partial u_{0}}{\partial x} $$

(9)

The right-hand side term can be understood as a dimensionless force per unit mass. Substituting Eq. (8) into the right-hand side gives

$$ -u_{0}\frac{\partial u_{0}}{\partial x}=\exp(-4\delta x/3)\\ \left[{\frac{\delta}{3}+\frac{\delta}{3}\cos 2(x-t)+\frac{1}{2}\sin 2(x-t)}\right] $$

Taking the time average over one period (the operation to be denoted as $\langle\rangle$ ) of this provides

$$ -\left\langle{u_{0}\frac{\partial u_{0}}{\partial x}}\right\rangle=\frac{\delta}{3}\exp(-4\delta x/3) $$

(10)

which plays a role as the driving force for the generation of the steady flow $\bar{u}_{1}$. Taking the time average of Eq. (9) results in

$$ \frac{\partial\bar{p}_{1}}{\partial x}-\frac{4}{3}\delta\frac{\partial^{2}\bar{u}_{1}}{\partial x^{2}}=-\left\langle{u_{0}\frac{\partial u_{0}}{\partial x}}\right\rangle $$

(11)

It should be noted that the term (10) can be understood as a kind of body force. Lighthill [3] has presented solutions of this system of equations for the case in which the force (10) acts as a point source in an infinite space. It is seen that when the viscosity is low enough (at high streaming Reynolds numbers) the flow from the source is like a jet as shown in Fig. 1. This flow is sometimes called the quartz wind. The term acoustic streaming refers to this flow in the case of ultrasound. As can be seen from Eq. (10), the driving force for this current vanishes when there is no attenuation, i. e., when $\delta=0$. Therefore we can say that the acoustic streaming in for the case of ultrasound in a compressible fluid is attributed to the attenuation of sound. The photograph in Fig. 2 clearly shows the streaming flow ejected by the transducer that creates the ultrasound.

Figure 1

Acoustic streaming quartz wind generated from a point source of ultrasound

Figure 2

Acoustic streaming generated from an actual ultrasonic transducer

On the other hand, we can also calculate the Stokes drift velocity from the solution of Eq. (8). We can follow a certain fluid particle's path $x(x_{0},t)$, where $x_{0}$ is a reference point independent of the time. From the definition of the flow velocity we can write

$$ \frac{dx}{dt}=\varepsilon u $$

(12)

We substitute Eq. (8) into the right-hand side of this and let $x=x_{0}+\varepsilon x_{1}(t)$. We then have

$$ x_{1}=-\exp(-2\delta x_{0}/3)\sin(x_{0}-t) $$

(13)

and

$$ \begin{aligned}[b] u_{0}={}&-\exp(-2\delta x_{0}/3)\cos(x_{0}-t)\\ &+\left({\frac{\partial u_{0}}{\partial x}}\right)_{{x=x_{0}}}\varepsilon x_{1}\end{aligned} $$

(14)

Here again the last term contains the steady component and it becomes

$$ \varepsilon u_{\textrm{d}}=\frac{1}{2}\varepsilon\exp(-4\delta x_{0}/3) $$

(15)

which is known as the Stokes drift velocity. The steady streaming flow $\varepsilon\bar{u}_{1}$ given by the solution of Eq. (11) is called the Eulerian streaming flow. Compared with the Eulerian streaming flow, the Stokes drift flow is confined in a region close to the sound source, i. e., within the region $x=O(1/\delta)$. Actually this region corresponds to the one that the force (10) acts upon. That is, within this region the steady flow velocity is composed of the Eulerian streaming velocity and the Stokes drift velocity

$$ u_{\textrm{L}}=\varepsilon\bar{u}_{1}+\varepsilon u_{\textrm{d}} $$

(16)

which is known as the Lagrangian velocity. Beyond this region the Stokes drift flow vanishes and only the Eulerian streaming exists. In many cases the Stokes drift velocity is ignored, but when the location of interest is not far from the sound source it should be considered. In general, the Stokes flow is expected to occur only when the primary wave is progressive, not when it is standing.

Eulerian Streaming

Even the incompressible fluid can also give rise to Eulerian streaming flow when a solid obstacle is in contact with the oscillating fluid. Consider a two-dimensional incompressible flow around a solid body governed by the following dimensionless equations:

$$ \displaystyle\nabla\cdot\mathbf{u}=0 $$

(17a)

$$ \displaystyle\frac{\partial\mathbf{u}}{\partial t}+\varepsilon\mathbf{u}\cdot\nabla\mathbf{u}=-\nabla p+\frac{1}{\textrm{Re}}\nabla^{2}\mathbf{u} $$

(17b)

where $\mathbf{u}=(u,v)$ and $\nabla$ is the two-dimensional gradient operator. We use $1/\omega$, $L$, $A_{0}\omega$ and $\rho^{\ast}A_{0}\omega^{2}L$ as the reference quantities for the time, length, velocity and pressure, respectively. Note that the reference length $L$ used here represents a typical dimension of the obstacle. The Reynolds number Re defined as

$$ \textrm{Re}=\frac{\omega L^{2}}{\nu} $$

is assumed to be large. For the small value of $\varepsilon$, we expand

$$ (\mathbf{u},p)=(\mathbf{u}_{0},p_{0})+\varepsilon(\mathbf{u}_{1},p_{1})+\ldots\,. $$

(18)

The leading-order equation of Eq. (17b) is then

$$ \frac{\partial\mathbf{u}_{0}}{\partial t}=-\nabla p_{0} $$

(19)

Therefore the leading-order solution is of potential flow. Moreover the solutions can take a separable form, e. g.

$$ \mathbf{u}_{0}=\mathbf{f}(x,y)\exp(\textrm{i}t) $$

(20)

where the complex functions f must satisfy the continuity equation (17a) and $\textrm{i}=\sqrt{-1}$ denotes the imaginary unit. Note that the inviscid region governed by these equations comprises most of the flow domain except the thin layer near the solid boundary.

Since the potential flow solution (Eq. (20)) does not satisfy the no-slip condition on the solid surface, we must consider a thin layer (called the Stokes layer) adjacent to the surface $n=0$, where $n$ refers to the local coordinate normal to the wall. In this thin layer, we use the strained coordinate $Y$ defined as

$$ n=\sqrt{2/\textrm{Re}}\, Y $$

We also use the velocity components $U$ and $V$ along the local coordinates $s$ and $n$, respectively (the coordinate $s$ is along the surface). Then the boundary layer equation becomes

$$ \frac{\partial U}{\partial t}-\frac{\partial u_{\textrm{e}}}{\partial t}=\frac{1}{2}\frac{\partial^{2}U}{\partial Y^{2}}-\varepsilon\left({U\frac{\partial U}{\partial s}+V\frac{\partial U}{\partial n}}\right) $$

(21)

where $u_{\textrm{e}}(s)$ denotes the tangential component of the potential flow velocity evaluated at the solid surface. We expand the velocity components as before.

$$ (U,V)=\left(U_{0},\sqrt{2/\textrm{Re}}V_{0}\right)+\varepsilon\left(U_{1},\sqrt{2/\textrm{Re}}V_{1}\right)+\ldots $$

(22)

The leading-order equation is

$$ \frac{\partial U_{0}}{\partial t}-\frac{\partial u_{\textrm{0e}}}{\partial t}=\frac{1}{2}\frac{\partial^{2}U_{0}}{\partial Y^{2}} $$

whose solution takes the following form:

$$ U_{0}=u_{\textrm{0e}}(s)\left[{1-\exp(-(1+\textrm{i})Y)}\right]\exp(\textrm{i}t) $$

(23)

The normal component $V_{0}$ can also be obtained from the continuity equation. Note that this flow is still time periodic. The $O(\varepsilon)$ equation (21) then becomes

$$ \frac{\partial U_{1}}{\partial t}-\frac{\partial u_{\textrm{1e}}}{\partial t}=\frac{1}{2}\frac{\partial^{2}U_{1}}{\partial Y^{2}}-\left({U_{0}\frac{\partial U_{0}}{\partial s}+V_{0}\frac{\partial U_{0}}{\partial Y}}\right) $$

(24)

Here our interest is the time independent flow. Taking the time average of the above equation over one period of oscillation gives

$$ \frac{1}{2}\frac{\partial^{2}\bar{U}_{1}}{\partial Y^{2}}=\left\langle{U_{0}\frac{\partial U_{0}}{\partial s}+V_{0}\frac{\partial U_{0}}{\partial Y}}\right\rangle $$

(25)

The solution of this equation yields the streaming velocity at the edge of the boundary layer,

$$ \bar{U}_{{1\infty}}=-\frac{3}{8}\left[{(1-\textrm{i})u_{\textrm{0e}}^{\#}\frac{\textrm{d}u_{\textrm{0e}}}{\textrm{d}s}+(1+\textrm{i})u_{\textrm{0e}}\frac{\textrm{d}u_{\textrm{0e}}^{\#}}{\textrm{d}s}}\right] $$

(26)

where the superscript # denotes the complex conjugate. This velocity then acts as a boundary condition for the exterior bulk region. The governing equation of the steady streaming flow takes the following form [1]:

$$ (\bar{\mathbf{u}}+\bar{\mathbf{u}}_{\textrm{d}})\cdot\nabla\,\bar{\mathbf{{u}}}=-\nabla\,\bar{p}+\frac{1}{\textrm{Re}_{\textrm{s}}}\nabla^{2}\bar{\mathbf{u}} $$

(27)

where $\textrm{Re}_{\textrm{s}}$ is the streaming Reynolds number based on the streaming velocity at the edge of the Stokes layer; $\textrm{Re}_{\textrm{s}}=\varepsilon^{2}\textrm{Re}$. This equation looks very similar to the Navier–Stokes equation, but here the convective velocity is replaced by the Lagrangian velocity $\bar{\mathbf{u}}_{\textrm{L}}=\bar{\mathbf{u}}+\bar{\mathbf{u}}_{d}$.

The above formulation is effective and suitable when the streaming Reynolds number is large so that the Reynolds stress action is confined within the thin Stokes layer. In the microfluidic application, however, $\textrm{Re}_{\textrm{s}}$ is usually small. So, the Reynolds stress term may be added to the streaming-flow equation (27) so that

$$ \begin{aligned}\displaystyle(\bar{\mathbf{u}}+\bar{\mathbf{u}}_{\textrm{d}})\cdot\nabla\,\bar{\mathbf{u}}=-\nabla\,\bar{p}+\frac{1}{\textrm{Re}_{\textrm{s}}}\nabla^{2}\bar{\mathbf{u}}-\mathbf{F}\end{aligned} $$

(28a)

$$ \begin{aligned}\displaystyle\mathbf{F}=\langle\,{(\mathbf{U}_{0}\cdot\nabla)\mathbf{U}_{0}+\mathbf{U}_{0}(\nabla\cdot\mathbf{U}_{0})}\,\rangle\end{aligned} $$

(28b)

where the second term within $\langle\,\rangle$ on the right-hand side is nonzero for the compressible fluid case; for the compressible fluid case Eq. (28a) itself must be modified. This means that for the case with a low streaming Reynolds number, Eq. (28a) must be solved over the whole domain including the Stokes layer. Another important point for the incompressible flow is that the Reynolds stress vanishes when the primary oscillating flow is of the progressive-wave type. On the other hand, it should be noted that in the microfluidic area no literature has taken into account the Stokes drift flow in the convecting velocity (i. e., the velocity $(\bar{\mathbf{u}}+\bar{\mathbf{u}}_{\textrm{d}})$ in Eq. (28a)) in the numerical simulation of the streaming flow.

We can experience a typical example of the Eulerian streaming flow around a circular cylinder [5]. Here, the fluid surrounding the cylinder oscillates with high frequency; or the cylinder may oscillate in the otherwise quiescent fluid without fundamental difference in the results. The steady flow within the Stokes layer at high streaming Reynolds numbers shows four-cell structure around the circular cylinder as shown in Fig. 3. There are two streams coming out of the cylinder from both sides in the direction of oscillation.

Figure 3

Sketch of the flow structure of the Eulerian streaming around a circular cylinder oscillating at a high frequency in a viscous fluid

Kundt's dust pattern manifests another simple example of the Eulerian streaming flow given by the two-dimensional standing wave in a duct. When an acoustic standing wave is established in the duct with a compressible fluid, the steady streaming reveals four-cell structure over a half wavelength (or over the space between two neighboring nodes) as shown in Fig. 4. Near the duct wall, the steady streaming is toward the nodes, and near the duct center it is coming out of the nodes. Therefore dust within the duct should cluster near the nodal points of the standing wave. The detailed solution for this case has been given by Riley [1].

Figure 4

Eulerian streaming flow occurring between two parallel plates when a standing wave takes place inside the duct. Distance between the nodes corresponds to a half wavelength of the standing wave

Stokes Drift Flow

Usually, by acoustic streaming we mean Eulerian streaming. However, the recent application of the flexural plate wave in microfluidics implies that the Stokes drift flow should also be considered as a type of steady streaming flow, because the net effect in the fluid transport is not discernable between the two. Here we derive the equations as well as the solutions for the Stokes drift flow given by the flexural plate wave. The case of one-dimensional compressible flow has already been treated in this article.

Consider a viscous fluid confined between a flat wall at $y^{\ast}=0$ and a flexural plate at $y^{\ast}=h^{\ast}$. The flexural plate oscillates showing a traveling wave with a high frequency $\omega$ by a well established piezoelectric vibration system. Fig. 5 illustrates the geometry of the flow model investigated in this article. The wavelength is $\lambda$ and the wave propagation speed is $c^{\ast}=\omega\lambda/2\pi$. A particle attached on the flexural plate describes an ellipse with major $A_{0}$ and minor $B_{0}=rA_{0}$ rotating clockwise when the wave travels toward the right-hand side as shown in Fig. 5 [2]. The ratio $r$ is given from $r=\pi d/\lambda$, where $d$ is the membrane thickness. A typical value of $r$ is $\mathrm{0.1}$. The spatial coordinate $x^{\ast}$ is along the wave propagation direction and $y^{\ast}$ normal to it.

Figure 5

Microfluidic device utilizing the flexural plate wave. (a) Fabricated device; (b) shape of the flexural plate and the coordinates for analysis

As before we use $1/\omega$ as the time scale, $\lambda/2\pi$ as the length scale, $A_{0}\omega$ as the velocity scale and $\rho A_{0}\omega^{2}\lambda/2\pi$ as the pressure scale, then the dimensionless continuity and momentum equations take the form given in Eqs. (17a) and (17b). The equations for the particle motion on the flexural plate are

$$ \begin{aligned}\displaystyle\delta x&\displaystyle=-r\varepsilon\sin(x_{0}-t)\end{aligned} $$

(29a)

$$ \begin{aligned}\displaystyle\delta y&\displaystyle=\varepsilon\cos(x_{0}-t)\end{aligned} $$

(29b)

As a typical example, for $\lambda={\mathrm{100}}{\mathrm{\mu m}}$, $d={\mathrm{3}}{\mathrm{\mu m}}$, $A_{0}={\mathrm{6}}{\mathrm{nm}}$, and the frequency $\omega/2\pi={\mathrm{3}}{\mathrm{MHz}}$, we have $A_{0}\omega={\mathrm{113}}{\mathrm{mm/s}}$, $\varepsilon={\mathrm{3.8\cdot 10^{{-4}}}}$, and $\textrm{Re}={\mathrm{4760}}$. The boundary conditions are

$$ \begin{aligned}\displaystyle u&\displaystyle=0&\displaystyle\textrm{at }y=0\end{aligned} $$

(30a)

$$ \begin{aligned}\displaystyle v&\displaystyle=0&\displaystyle\textrm{at }y=0\end{aligned} $$

(30b)

$$ \begin{aligned}\displaystyle u&\displaystyle=r\cos(x-t)&\displaystyle\textrm{at }y=h\end{aligned} $$

(30c)

$$ \begin{aligned}\displaystyle v&\displaystyle=\sin(x-t)&\displaystyle\textrm{at }y=h\end{aligned} $$

(30d)

where $h$ is the dimensionless depth of the fluid layer. The velocity conditions, Eqs. (30c) and (30d), are derived from Eqs. (29a) and (29b), respectively. In this derivation we simply replaced $x_{0}$ by $x$, which is valid for small $r$; later we will consider the higher-order effect.

As before, we neglect the nonlinear term in Eq. (17b). We also assume that Re is very large. Equation. (17b) then becomes

$$ \frac{\partial\mathbf{u}}{\partial t}=-\nabla p $$

(31)

This indicates that the velocity field can be described by a potential function $\phi(x,y,t)$ in such a way that $\mathbf{u}=\nabla\phi$. As usual, this inviscid solution must satisfy the boundary condition for the normal component of the velocity at each of the walls but not the tangential component. This is the most important part of the analysis, because otherwise the leading-order solution is not of the progressive-wave type. Applying the conditions given in Eqs. (30b) and (30d) to the Laplace equation $\nabla^{2}\phi=0$ and solving for $\phi$, we obtain $\phi=(\cosh y/\sinh h)\sin(x-t)$ and

$$ \begin{aligned}\displaystyle u=\frac{\cosh y}{\sinh h}\cos(x-t)\end{aligned} $$

(32a)

$$ \begin{aligned}\displaystyle v=\frac{\sinh y}{\sinh h}\sin(x-t)\end{aligned} $$

(32b)

The bulk flow solution given by Eqs. (32a) and (32b) does not satisfy the condition for the tangential component of the velocity on the walls. So, we must expect boundary layers near each of the walls. First, near the top wall, $y=h$, we use the strained variable $Z=(h-y)\sqrt{\textrm{Re}/2}$. Then we obtain the appropriate $x$-momentum equation and solve for $u$ to obtain

$$ \begin{aligned}[b] u={}&\coth h\,\cos(x-t)\\ &+(r-\coth h)\exp(-Z)\cos(x+Z-t)\end{aligned} $$

(33)

A similar process leads to the following solution for $u$ in the bottom boundary layer:

$$ u=\frac{1}{\sinh h}\left[{\cos(x-t)-\exp(-Y)\cos(x+Y-t)}\right] $$

(34)

where $Y=y\sqrt{\textrm{Re}/2}$.

The analytical solutions obtained so far imply that the flow field driven by the oscillating flexural plate is progressive. This means that the Reynolds stress should vanish and so there should be no Eulerian streaming. On the other hand, it produces the nonzero Stokes drift flow. First we consider the bulk flow. Into the definition of the velocity components $u$ and $v$, i. e., $\textrm{d}x/\textrm{d}t=\varepsilon u$ and $\textrm{d}y/\textrm{d}t=\varepsilon v$, we substitute Eqs. (32a) and (32b). Assuming the variables $x$ and $y$ on the right-hand side are constant, i. e. $x=x_{0}$ and $y=y_{0}$, we can integrate the equations to obtain

$$ \begin{aligned}\displaystyle x&\displaystyle=x_{0}-\varepsilon\frac{\cosh y_{0}}{\sinh h}\sin(x_{0}-t)\end{aligned} $$

(35a)

$$ \begin{aligned}\displaystyle y&\displaystyle=y_{0}+\varepsilon\frac{\sinh y_{0}}{\sinh h}\cos(x_{0}-t)\end{aligned} $$

(35b)

Now we substitute Eqs. (35a) and (35b) into Taylor expansions for $u$ and $v$, i. e., Eqs. (32a) and (32b) about $(x,y)=(x_{0},y_{0})$ and take the time average over one period to obtain the steady velocity components $\bar{v}=0$ and

$$ \bar{u}=\frac{\varepsilon}{2\sinh^{2}h}(\cosh^{2}y+\sinh^{2}y) $$

(36)

Next, we consider the top boundary layer. After some algebra we obtain $\bar{v}=0$ and

$$ \begin{aligned}[b]\bar{u}&=\frac{\varepsilon}{2}\left\{\left[{\frac{\cosh y}{\sinh h}(1-A)+rA}\right]^{2}\right.\\ &\quad+\frac{\sinh y}{\sinh h}(1-A)\left[{\frac{\sinh y}{\sinh h}(1-A)+A+r(y-h)A}\right]\\ &\quad+B^{2}\left({r-\frac{\cosh y}{\sinh h}}\right)^{2}\\ &\quad+\left.B^{2}\frac{\sinh y}{\sinh h}\left[{\frac{\sinh y}{\sinh h}-r(y-h)}\right]\vphantom{\left[{\frac{\cosh y}{\sinh h}(1-A)+rA}\right]^{2}}\right\}\end{aligned} $$

(37)

where functions $A$ and $B$ are

$$ (A,B)=\exp(-Z)(\cos Z,\sin Z) $$

Similar treatment can be done for the bottom boundary layer and we obtain $\bar{v}=0$ and

$$ \bar{u}=\frac{\varepsilon}{2\sinh^{2}h}\left[{1-2\exp(-Y)\cos Y+\exp(-2Y)}\right] $$

(38)

Now, the complete formula for the streaming velocity applicable to all the regions including the two boundary layers is $\bar{v}=0$ and

$$ \begin{aligned}[b]\bar{u}&=\frac{\varepsilon}{2}\left\{\left[{\frac{\cosh y}{\sinh h}(1-A)+rA}\right]^{2}\right.\\ &\quad+\frac{\sinh y}{\sinh h}(1-A)\left[{\frac{\sinh y}{\sinh h}(1-A)+A+r(y-h)A}\right]\\ &\quad+B^{2}\left({r-\frac{\cosh y}{\sinh h}}\right)^{2}\\ &\quad+B^{2}\frac{\sinh y}{\sinh h}\left[{\frac{\sinh y}{\sinh h}-r(y-h)}\right]\\ &\quad+\left.{\frac{1}{\sinh^{2}h}\left[{\exp(-2Y)-2\exp(-Y)\cos Y}\right]-r^{2}\frac{y}{h}}\vphantom{\left[{\frac{\cosh y}{\sinh h}(1-A)+rA}\right]^{2}}\right\}\end{aligned} $$

(39)

As a typical example, for the case with $\lambda={\mathrm{100}}{\mathrm{\mu m}}$ and $h^{\ast}={\mathrm{50}}{\mathrm{\mu m}}$, we have $h=\pi$, and for $\varepsilon={\mathrm{3.8\cdot 10^{{-4}}}}$ and $\textrm{Re}=4760$ we evaluate $\bar{u}={\mathrm{3.8\cdot 10^{{-4}}}}$ as the approximately maximum streaming velocity obtained from the bulk-flow solution (Eq. (36)) at $y=h$. When the velocity scale is $A_{0}\omega={\mathrm{113}}{\mathrm{\mu m}}$, this becomes $\bar{u}^{\ast}={\mathrm{43}}{\mathrm{\mu m/s}}$. This is not much different from the measured velocity (less than $\mathrm{100}$ $\mathrm{\mu m/s}$) reported by Nguyen et al. [2], considering that the numerical result $\mathrm{1000}$ $\mathrm{\mu m/s}$ obtained by them is more than 10 times the measured value for the same parameter set.

Figure 6 shows distributions of the streaming velocity for three depths of the fluid layer with the parameter set presented previously. The magnitude of streaming velocity increases on the whole as the depth decreases. Further decrease of $h$ results in the overlap of the top and bottom boundary layers, and so the analytical solutions presented so far may not hold in that case.

Figure 6

Distributions of the Stokes drift velocity across the fluid layer for three channel depths given by the flexural plate wave

The flow rate through the channel is given by

$$ q=\int _{0}^{h}{\bar{u}\textrm{d}y} $$

Substituting the streaming velocity distribution (Eq. (39)) into this yields the following formula.

$$ q=\frac{\varepsilon}{2\sinh^{2}h}\\ \left[{\sinh h\cosh h-\sqrt{2/\textrm{Re}}\left({1+\frac{7}{8}\sinh^{2}h}\right)}\right] $$

(40)

At high Reynolds numbers, the second term within [ ] on the right-hand side of the above equation can be neglected and so the flow rate increases as the channel depth decreases, in line with the numerical result of Nguyen et al. [2]. On the other hand, when the second term is considered the flow rate shows a maximum value at a critical depth. Figure 7 shows the dependence of the flow rate on the depth at three Reynolds numbers and at the other parameter values the same as those presented previously. It indeed shows that at each of the critical depths, the flow rate has a maximum value. The critical depth increases as the Reynolds number decreases. It can be shown that the critical depth is given by the formula

$$ h_{\textrm{c}}=\sinh^{{-1}}\left({\sqrt{\frac{8}{\textrm{Re}-8}}}\right) $$

(41)

For instance, at $\textrm{Re}=4760$ we obtain $h_{\textrm{c}}=0.041$ which corresponds to $h_{\textrm{c}}^{\ast}={\mathrm{0.65}}{\mathrm{\mu m}}$ at $\lambda={\mathrm{100}}{\mathrm{\mu m}}$. This is very small compared with the wavelength, but it is still far larger than the amplitude of the oscillation of the plate, $A_{0}={\mathrm{6}}{\mathrm{nm}}$.

Figure 7

Effect of the channel depth on the flow rate at three Reynolds numbers for the flexural-plate-wave pumping. Dashed line corresponds to $\textrm{Re}=\infty$

Key Research Findings

Acoustic streaming has been applied to various microfluidic areas. White's group [2] have developed micropumps that utilize the flexural plate wave as the actuator. They have conducted a numerical simulation but the deviation was significant; their simulation results had to be corrected by a factor of 7 in order to fit the measurement data. To date, no report has been given of a satisfactory agreement between the experimental measurements and numerical or theoretical predictions. The numerical computation for acoustic streaming is not as simple as it looks. There are two key factors which may degrade the numerical results: one arises from the thin Stokes layer and the other from the extremely low levels of the streaming velocity compared with the primary flow. These can be overcome when we put very fine grids near the Stokes layer and simultaneously use high-precision numerical values.

In order to apply acoustic streaming to fluid mixing, Yaralioglu et al. [7] designed patterns of piezoelectric transducers on the bottom wall of a microchannel. They verified experimentally that the transducers, by generating acoustic streaming inside the channel normal to the main flow, enhanced the mixing. The mixing effect was, of course, affected by the configuration of the electrode patterning. They also asserted that the fundamental problem of the formation of bubbles and unintended heating of the fluid was not expected to occur in their system.

Acoustic streaming has also been applied to electrochemistry. Compton's group [8] has tested the use of ultrasound to study the effect of the various configurations of the acoustic streaming on the limiting current from an electrode. It was shown that there was a critical diffusion layer thickness below which the layer no longer became thinner. Recently they performed a numerical simulation on the acoustic streaming over an electrode within an axi-symmetric space. The streaming flow was generated by the Reynolds stress terms formulated by other investigators. They successfully verified the previous experimental results that acoustic streaming was the principle mechanism of enhanced mass transport in sono-electrochemical cells.

Ultrasonics and its universal effect (acoustic streaming) have been used to manipulate particles in microfluidic devices, e. g., trapping, collection, separation, and deposition. For instance, Petersson et al. [9] employed a microchannel system with one center and two side channels at the inlet and outlet, respectively, of the channel. Through the side inlet channels they put a medium, say fluid A, containing particles, and through the center inlet channel the fluid B containing no particles. They then applied ultrasound normal to the channel in such a way that a standing wave was established in the main channel where the two media contacted each other. Due to the Reynolds stress and recirculating streaming flows, particles were collected near the channel center and this pattern persisted all the way through the main channel. At the outlet, the particles flow with the fluid $B$. They implied that such a medium-exchange effect could be applied to blood washing.

Marmottant and Hilgenfeldt [10] have considered bubble-driven microfluidic transport for bioengineering applications. In their experiment, bubbles were adsorbed on the bottom wall of the microchannel. When the standing ultrasound wave was introduced within the channel, steady streaming flow was generated. They stressed that the flow pattern in bubble streaming was quite different from that induced by an oscillating solid object. More interestingly, close contact of a foreign solid particle to the bubble showed a significant effect in the overall steady-flow field due to superposition of the bubble and particle streaming fields.

Future Directions for Research

To date no numerical simulation of the acoustic streaming flow associated with application to the microfluidics has been successfully performed. The simulation can be performed only with the streaming flow by using the governing equation given as Eq. (27), but we also need to solve the full unsteady Navier–Stokes equations including not only the primary oscillatory flow component but also the secondary steady flow component. In this case locally fine grids must be adopted to deal with the thin Stokes layer adjacent to the solid surface.

We also need to analyze the case for the flexural-plate-wave flow in more detail and to simultaneously perform numerical simulations of the resultant flows. The results of the full numerical simulation may be used to prove that the steady flow components are generated by the mechanism known as the Stokes drift flow.

We also need to study the dissipation effect caused by the oscillatory flow. At present, in Lab-on-a-Chip design, we are usually interested in the feasibility of the functions provided by the transducers, but sooner or later the energy consumption may be one of the most important factors.

Cross References

Acoustics Based Biosensors

Piezoelectric Microdispenser

Transport of Droplets by Acoustics

Ultrasonic Pumps