Discussion of 353(3)
These are all good ideas. Eq. (28) is the equation which FlexPDE can solve. Agreed about constant density, and even in the case del rho not equal to zero Eq. (14) vanishes, giving Eq. (28). So in reducing Eq. (12) to Eq. (28) the baroclinic term is eliminated. The solution of the vorticity equation (12) needs supercomputers and simulation in general, as is well known in atmospherics for example. Eq. (28) seems to be completely new to fluid physics of all kinds. Eq. (30) is a time independent wave equation and Eq. (31) is a time dependent wave equation. For incompressible flows Eq. (30) simplifies with del dot v = 0. Kambe derives wave equations with drastic approximations, but these wave equations do not make such assumptions. FlexPDE may be able to solve them
To: EMyrone@aol.com
Sent: 28/07/2016 21:07:33 GMT Daylight Time
Subj: Re: 353(3): Wave Equation of Fluid Dynamics or Aerodynamics or AtmosphericsIf the density rho is constant, the cross product between grad(rho) and grad(p) vanishes, since according to Eq.(14) this is the curl of a gradient. You come back to this in eq.(29).
The practical problem with eq.(12) is that the equation is of third order in v. one could use both variables w and v for solution wiht definition (7) but the free version of the FlexPDE program only allows four variables, in this case v and p. We discussed this already.Eq.(33) is an inhomogeneous wave equation. I am not sure if the RHS can be written proportional to v with a scalar kappa squared. This would require that the vector of the RHS is parallel to v. But the important result is that a wave equation with turbulence term can be derived. If parts of the RHS can be written in the form kappa^2 v, this would introduce a kind of resonance behaviour in the stationary case:
del2 v + kappa^2 v = f(v)
where f(v) consists of the other RHS terms.
Horst
PS: I am out of Munich at the weekend but will try to follw the emails.
Am 28.07.2016 um 13:55 schrieb EMyrone:
This is equation (33), where an assumed constant speed of sound has been used. This has the format of the ECE wave equation. More generally the speed of sound is variable. So this is a wave equation of spacetime itself in fluid electrodynamics. It looks complicated but it seems that complication is no problem to the contemporary code packages used in UFT349, UFT351 and UFT352. It generalizes the wave equations obtained by Kambe using drastic approximations. The well known vorticity equation (12), which has a vast number of applications, has been simplified in this work to Eq. (28). So Eq. (28) should become a fundamentally useful equation in fluid dynamics, aerodynamics, hydrodynamics, atmospherics, oceanographic physics, and so on. I also found that the homogeneous field equation of Kambe is equivalent to a zero convective derivative of velocity. This means that the viscous force is defined in the Kambe analysis by Eq. (23). In simplifying Eq. (28) to Eq. (12) it has been assumed that if curl A = curl B, then A = B. However it is possible to have A = B + div phi, a gauge like equation, because curl div phi = del x del phi = 0. The solution A = B is certainly valid. In simplifying Eq. (12) to Eq. (28) the baroclinic term is considered as in Eq. (14), whch means that the curl of the baroclinic term is zero. In Kambe’s notation it is – del x del h = 0. So for complete information, use the code to solve both Eqs. (12) and (28), and also Eq. (33). In fluid electrodynamics all this is a property of the source term. There is conservation of total energy by Poynting’s Theorem. In order to emphasize conservation of total energy in fluid electrodynamics, the next note will develop the Poynting Therem of fluid electrodynamics.
Unified Physics Institute of Technology 
No trackbacks yet.