Looks good! These equations will give some very interesting orbits. The other equations of fluid dynamics can also be used, conservation of matter (continuity equation), conservation of angular momentum (vorticity equation) and conservation of energy. I will proceed to these next. In fact, standard fluid dynamics already contains a gravitational term in the Navier Stokes equation, plus a viscous force and another force. For this work, only the gravitational force has been used.
Sent: 04/04/2017 11:21:37 GMT Daylight Time
Subj: Re: Checking Note374(5) by Computer
Thanks for these important hints. I did not include the transition of the fluid velocity to zero when checking the transition to Newtonian result. I introduced a “switching parameter” s in the calculation:
v_r = s*x*r dot,
v_phi = s*r*phi dot.
The limit check now executes
x –> 1
s –> 0.
Am 04.04.2017 um 09:15 schrieb EMyrone:
Many thanks, these checks are important for accuracy and self consistency. The equations can be checked step by step.
1) Eq. (16) reduces to the classical result when x = 1, x dot = 0 and when the convective terms of Eq. (7) all vanish. So the convective terms must all vanish in these limits and that could be checked by computer.
2) Eq. (17) reduces to the classical result when x = 1 and when the convective terms all vanish.
3) In the limit x goes to 1 and x dot goes to zero, it should be checked that Eq. (18) is true.
4) In the limit x goes to 1, partial r dot / partial r = r double dot / r dot must vanish in order to regain the Newtonian result.
Eq. (21) for the velocity field reduces correctly to the Newtonian result as x goes to one. The answer is that v sub r cannot be a function of r in the Newtonian limit, so partial r dot / partial r must vanish in this limit. So in order to regain the Newtonian limit the additional conditions (18) must be imposed as x goes to 1. The reason for this is that the velocity field v is a function of r(t), phi(t) and t, but the classical dynamic velocity v is a function only of v. So I think that this is the correct way of approaching the Newtonian limit. In any other situation R is a function in general of r(t), phi(t) and t and v is also a function of r(t), phi(t) and t. In general teh velocity field is defined by the convective derivative of R:
v = DR/Dt
and the gravitational Navier Stokes equation is:
Dv / Dt = – (MG / r squared) e sub r
in plane polar coordinates. In the Newtonian limit, v = dr / dt, and a = dv / dt = – MG / r squared e sub r. Gnuplot for example could eb used to produce some very interesting orbits. The basic idea is that the above is ECE2 unified field theory, and that spacetime or the aether is a fluid controlled by the equations of fluid dynamics. The methods of solving these are well known and can be all adapted for orbital theory. This leads to a vast number of possibilities.
Sent: 03/04/2017 17:34:20 GMT Daylight Time
Subj: Re: 374(5): The General Planar Orbit of Fluid Gravitation
I checked by computer. The time derivative dx/dt is taken as a total derivative by Maxima:
dx/dt = dx/dr * dr/dt + dx/dphi * dphi/dt.
Even if avoiding this, the result for x–>1 gives the right equation for r, but a factor of 3 instead of 2 in the phi equation. Is anywhere a wrong definiton?
Am 03.04.2017 um 15:47 schrieb EMyrone:
This is found from the gravitational Navier Stokes equation (4), which is a particular case of the general Navier Stokes equation (8). The velocity field is given by Eq. (21) as derived in UFT363. The orbit is worked out entirely in terms of the radial component R sub r of the position element of fluid spacetime, its r derivative and second derivative, and its time derivative. Additional equations are available from fluid dynamics: notably the continuity equation and conservation of angular momentum. These can be developed in future notes, in the meantime model functions can be used. It is already known from Horst’s numerical analysis of yesterday and this morning that a fluid spacetime or aether gives a precessing orbit, a major discovery in my opinion. In this model a planet or object of mass m around an object of mass M moves in a fluid spacetime or aether. The structure of the theory is that of Cartan geometry.