Eulerian Streaming

Fig. 10. Some monthly averaged observed distributions of atmospheric methane from the SAMS satellite, Jones and Pyle, near solstice conditions, compared to the model distribution for July by Solomon and Garcia. Light dashed arrows indicate the residual Eulerian stream function, showing the advection pattern.

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. 

In this article, we will describe in some detail the fundamentals of the 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 will address the flexural plate wave and its net effect, i.e., the Stokes drift flow. 

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 ... 

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. 

Acoustk Streaming, Fig. 3 Sketch of the flow stnictine of the Eulerian streaming around a circular cylinder oscillating at a high Irequency in a viscous fluid... 

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 discemable... 

Acoustic Streaming, Fig. 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... 

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., dx/dt = eu and dy/dt = sv, we substitute Eqs. 32a and 32b. Assuming the variables x and y on the right-hand side are constant, i.e., x = xq and y = yo> we can integrate the equations to obtain... 

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. 

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