The Stoke drag force is related to the acoustic streaming field (specifically the streaming velocity; you would first need to determine this). More accurately termed acoustic streaming induced drag force.

The Stokes drag formula for these cases are given by,

F_drag = 6*pi*η*r( - u),

where, r is the particle radius, v_2 is the streaming velocity and u is the particle velocity. 

When dealing, with a standing wave, you would need to determine the dominant forcing mechanism (i.e. acoustic radiation forces or streaming induced drag forces). Strictly speaking the net force of the particle is a combination of both, however, in most standing wave cases, the particle is almost solely driven by the radiation forces. 

On the principle of independence of the boundary conditions of structure direction of turbulence

Lu Panming

Abstract

This is an introduction to the turbulence modeling theory of the present author, which can take the boundary conditions of turbulence structure into consideration ( the main contents have either not been published in normal journals or only published in chinese papers).   It is included that： (I)。Difficulty Problems Encountered in “Second-Moment-Closure” turbulence modelling ，i.e.  “Gao-Ge Anomaly” ;  (II)。The improvement  to the ”Second-Moment-Closure” turbulence modelling in order to allows the boundary conditions of turbulence structure direction could be prescribed;  (III)。An explanation to the principle of  independence  of  the boundary conditions of

turbulence structure direction ;  (IV)。Six suggestions to the possible future work directions;  (V)。 An explanation to the ”Gao-Ge Anomaly”.

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dear professors:

4. The principle of the independence of the boundary conditions is not only to the turbulence modelling, but also suitable to the other subject, because it is frequently meet the equations are not easy to solve, so there is often a need of modelling or predigestion.  For example, both the D‘Alembert anomaly (~1752) and the Stokes anomaly (~1856), are due to breach the  principle of independence  of the boundary conditions, according to today`s point of view .This is because, first, by using the inviscid potential flow`s dynamic equations it is not possible to affiliate the slip-less condition of the real fluid at the wall, and  second, using the Stokes equation it is not possible to affiliate the boundary conditions both at the wall of cylinder and at the infinite to the cylinder.  The first anomaly lead to the discovery of the Prandlt`s boundary layer theory（1904）, and the  second anomaly  was solved by Oseen（1910）with introducing a modification. All these reflect the scientific worthiness and broad latency of the present principle   of the independence of the boundary conditions.

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above is an example paragraph,for more details or need the full text, please go to the end of this letter, where will be an web address （or entrance ）existed.

     http://staff.ustc.edu.cn/~pmlu/

or   http://staff.ustc.edu.cn/~pmlu/%C2%C0%D6%F8%D5%D5%C6%AC%BB%E3%B1%E0/

or   http://staff.ustc.edu.cn/~pmlu/%C2%C0%D6%F8%D5%D5%C6%AC%BB%E3%B1%E0/On%20the%20principle.pdf

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BTW:

my question is that does anyboby can find any other examples of this kind ?(that is any other  anomaly due to breach the  principle of independence  of the boundary conditions.)

Thank for your answers.

Dear Citsabehsan Devendran, 

I have determined that the Stoke's drag force from the acoustic streaming is weaker than the acoustic radiation force. 

I have asked a question what would affect the suspended particle's velocity in a water induced by standing wave field, and someone answered that is drag force. Is that the only thing that affect particle's velocity?

Because from my model results, the magnitude of the particle's velocity is very small. 

Well despite it being weaker, the drag force still ought to play a role. In a standing wave field, the radiation forces are likely (not necessarily though) to be more dominant thus, would be the main factor that contributes to particle migration. The suspended particles would be therefore driven by the radiation forces as well and not solely by drag forces (which from what I gather based on what you said is minimal).

It should be noted that in a standing field, streaming does exist and therefore, drag forces do play a role although insignificant at times. Furthermore, in a pure standing wave, you tend to have velocity fluctuations as well, which are more significant in regions of minimal pressure fluctuations. This at times, tend to cause particle migration within these minimal pressure fluctuation regions as well. 

Hope this helps.