We propose two possible mechanisms for self-propulsion via acoustic streaming


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  • The quartz wind effect. Here the attenuation takes place in the bulk of the fluid. Streaming is normal to the source. (When piezoelectrically excited, the faces of a quartz crystal vibrate, creating an ultrasonic beam. AS generates a turbulent jet with velocities reaching 10’s of cm/s.)

  • Boundary induced streaming. Here the attenuation takes place near a solid surface. The induced streaming is tangential to the surface.





Let U be an irrotational oscillatory vector field in a fluid representing an acoustic wave.

  • Let U be an irrotational oscillatory vector field in a fluid representing an acoustic wave.

  • Owing to the no slip boundary condition, U must vanish at a solid boundary. There is a thin layer (the Stokes boundary layer) where U is rotational. The thickness of the Stokes boundary layer is 5√(where is the kinematic viscosity and is the frequency of U.

  • Within the Stokes boundary layer shear stresses cause strong attenuation of U leading to streaming.



In the late 19th century Lord Rayleigh showed that the streaming velocity at the edge of the boundary layer due to an oscillatory vector field U=U(x) is

  • In the late 19th century Lord Rayleigh showed that the streaming velocity at the edge of the boundary layer due to an oscillatory vector field U=U(x) is

  • -3/(4U(x)U’(x)

  • and that the streaming is in the direction of the nodes:





  • We propose two possible mechanisms for self-propulsion via acoustic streaming:

  • The Quartz Wind (QW) model.

  • The Surface Acoustic Wave (SAW) model based on boundary induced streaming



In this model the spicules in a small region vibrate at a high frequency buckling the crystalline shell in a manner similar to an electric door buzzer. Our inspiration for this mechanism came from the cuica: aBrazilian samba instrument

  • In this model the spicules in a small region vibrate at a high frequency buckling the crystalline shell in a manner similar to an electric door buzzer. Our inspiration for this mechanism came from the cuica: aBrazilian samba instrument

  • A flow, normal to the cell, is generated by attenuation in the bulk of the fluid.

  • Problem: Low efficiency. (Quartz wind swimmers are Hummers!)



Lighthill defines the efficiency to be the ratio of the power required to push

  • Lighthill defines the efficiency to be the ratio of the power required to push

  • the cell through the water to the power required by the mechanism (P):

  • η = viscosity, for water η = 0.01 g / cm sec

  • a = radius, for Synechococcus a = 10^(-4) cm

  • V = velocity, for Synechococcus V = 2.5×10^(-3) cm / sec

  • The force (F) exerted on the fluid by the QW effect and acoustic power (P)

  • are related by P=Fc where c is the speed of sound.

  • The force required to drive the cell with velocity V is

  • F = 6πμaV

  • making the power output for Synechococcus P=7×10^(-10) watts.



The efficiency of the QW mechanism for Synechococus is then

  • The efficiency of the QW mechanism for Synechococus is then

  • η=1.7×10^(-6) %

  • The squirming and boundary induced streaming mechanisms have

  • efficiencies between 0.1-1%.

  • We have not ruled out the QW mechanism completely. There are

  • possible power enhancement mechanisms.

  • Example: Bubble induced streaming. Here submicro-bubbles adhere

  • to the CS. Being of characteristic size, the bubbles resonate

  • enhancing the local streaming.



In this model, the cell propagates a high frequency traveling wave

  • In this model, the cell propagates a high frequency traveling wave

  • along the CS. Attenuation of the wave within the Stokes boundary layer

  • generates a mean flow just outside this layer creating an effective ‘slip’

  • velocity.

  • Longuet-Higgins (1953) derived a generalization of Rayleigh’s law for

  • streaming due to a traveling wave. The limiting streaming velocity at

  • the edge of the Stokes boundary layer is the real part of

  • (* = complex conjugate)

  • where is the tangential velocity at the CS and is the solution to the

  • linearized NS equations outside the Stokes boundary layer (=0 for us).



  • We model the cell as a sphere of radius a with spherical coordinates

  • where is the azimuthal coordinate. The traveling wave is

  • where ϕm is a material point on the CS. The slip velocity due to

  • streaming leads to a swimming velocity of

  • which is 2.5 times that predicted by the squirming mechanism.



  • The efficiency compares well with other known strategies.

  • But … are the required frequencies biologically feasible?



Question: Is singing biologically feasible?

  • Question: Is singing biologically feasible?

  • Bacterial flagellar motors are large membrane embedded structures

  • and have been observed to rotate at 300Hz when unloaded.

  • From E-Coli in Motion

  • HC Berg (2004, Springer)

  • The required frequency for acoustic streaming is biologically feasible.





























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