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/(4U(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 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) % 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 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 (* = 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 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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