Bjerknes force

In Fields of Force Bjerknes lay out geometrical and dynamical analogies between the Maxwell's theory of electromagnetism and hydrodynamics. In the light of these analogies the Bjerknes forces are being predicted. Principle of kinematic buoyancy Bjerknes writes:"Any body which participates in the translatory motion of a fluid mass is subject to a kinematic buoyancy equal to the product of the acceleration of the translatory motion multiplied by the mass of the water displaced by the body" Where \bold{u} is the fluid velocity and \rho is the fluid density. Using conservation of momentum for incompressible non-viscous fluid one can find that to first order: \rho \frac{\partial \bold{u}}{\partial t}=-\nabla P , Concluding that F = - V \nabla P . Charge and oscillating particles Bjerknes realized that the velocity field generated by an expanding particle in an incompressible fluid has the same geometrical structure as the electric field generated by a positively charged particle, and that the same applies for contracting particle and a negatively charged particle. In the case of an oscillating motion, Bjerknes argued that two particles that oscillate in phase generate a velocity field that is geometrically equivalent to the electric field generated by two particles with the same charge, whereas two particles that oscillate in an opposite phase will generate a velocity field that is geometrically equivalent to the electric field generated by particles with an opposite sign. Bjerknes then writes:"Between Bodies pulsating in the same phase there is an apparent attraction; between bodies pulsating in the opposite phase there is an apparent repulsion, the force being proportional to the product of the two intensities of pulsating, and proportional to the inverse square of the distance." This result is counter to our intuition, as it demonstrates that bodies oscillating in phase exert an attractive force on each other, despite creating a field akin to that of identically charged particles. This result was described by Bjerknes as "Astonishing". == Primary Bjerknes force ==

The force on a small particle in a sound wave is given by: : F = - V \nabla P where V is the volume of the particle, and \nablaP is the acoustic pressure gradient on the bubble. Assuming a sinusoidal standing wave, \omega_0^2 = \frac{1}{\rho R_0^2} \Bigl( 3\kappa \bigl(P_0 - P_V \bigr) + 2(3\kappa - 1) \frac{\sigma}{R_0} \Bigr) Where \rho is the fluid density, R_0 is the rest radius of the bubble, \kappa is the polytropic index, P_0 is the ambient pressure, P_V is the vapor pressure and \sigma is the surface tension constant. Bubbles with resonance frequency above the acoustic driving frequency travel up the pressure gradient, while those with a lower resonance frequency travel down the pressure gradient. ==References==