# Fluid dynamic “Vortex Gravitation” Model — derives Kepler’s Laws for Gravitation

Since the Gravitational Vortex has been modeled as a fluid dynamic “Gaseous Aether” it would make more since to derive a relation for Kepler’s Laws for Gravitation, using the mechanics and mathematics of a fluid.

Using the mathematical mechanics of Daniel Bernoulli’s Hydrodynamic Theorems, and the concepts of Steven Rado’s Aethro-Kinematic/Dynamic Ideal “Aether Gas” theory, Kepler’s Laws for Gravitation are derived; and  the results predict the “Aether Gaseous” “Gravitational Vortex” that spins or rotates, and has radial, tangential, and orthogonal components.

Below, complements of Wiki: Propagation of a transverse spherical wave in a two (2) dimensional gradient gravitational field grid.

Starting with the Bernoulli Equation for an Inertial Mass body in uniform motion in a rotating fluid is described below.

Bernoulli Total Pressure Conservation Equation

${P_{Pressure-Static}} \,\, + \,\, \frac{1}{2}{\rho_{Net}}\,{|\vec{v}|^2_{CM}} \,\,\, = \,\,\,\,$  Constant   $\,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}$

Where,

Static Pressure of Fluid${P_{Pressure-Static}}$   $\,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}$

Inertial Mass Density${\rho_{Net}} \,\,\, = \,\,\, \frac{m_{Net}}{V_{olume}} \,\,\, = \,\,\, \frac{{m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}}}{\frac{4\pi}{3}\,\,{r^3}}$   $\,\,\,\,----> \,\, \frac{kg}{m^3}$

Average Rectilinear Velocity of Center of Mass of Fluid particles

${|\vec{v}|_{CM}}$ $\,\,-->\,\, \frac{m}{s}$

${|\vec{v}|^2_{CM}} \,\, = \,\, \frac{(\displaystyle\sum_{i=1}^N {m_{i}}{v_{i}})^2}{m_{Net}^2} = \frac{ ({m_{1}}{v_{1}} + {m_{2}}{v_{2}} + {m_{3}}{v_{3}} + ...... + {m_{N}}{v_{N}})^2}{ ({m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}})^2 }$$\,\,----> \,\, \frac{m^2}{s^2}$

Keeping the Inertial Mass Density (${\rho_{Net}}$) constant for an isolated fluid system and taking the differential of both sides of the above equation.

Letting the Pressure change (${dP_{Pressure-Static}}$) with Velocity changes (${d|\vec{v}|_{CM}}$) yields.

${dP_{Pressure-Static}} \,\, + \,\, {\rho_{Net}}\,{|\vec{v}|_{CM}} \,\, {d|\vec{v}|_{CM}} \,\,\, = 0$    $\,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}$

${dP_{Pressure-Static}} \,\, = \,\, {-}{\rho_{Net}}\,{|\vec{v}|_{CM}} \,\, {d|\vec{v}|_{CM}}$  $\,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}$

Next, introducing the”Pressure Gradient” ($\frac{dP_{Pressure-Static}}{dr}$) of the fluid vortex flow field which is defined as the change in pressure with distance, as described by the equation below:

Pressure Gradient of Vortex Fluid Field

$\frac{dP_{Pressure-Static}}{dr} \,\, = \,\, {-}{\rho_{Net}}\,{|\vec{v}|_{CM}} \,\, \frac{d|\vec{v}|_{CM}}{dr}$  $\,\,\,\,----> \,\, \frac{kg}{m^2\,\,s^2}$

Next, introducing Circulation Force (${F_{Force}}$) which is equal to the centripetal or centrifugal force which is a constant related to the change in distance with pressure which is the “Pressure Gradient” of the fluid flow field; and likewise is a constant related to the velocity changes with distance.

Below, complements of Wiki: Propagation of a transverse spherical wave in a two (2) dimensional gradient gravitational field grid.

Circulation Force of Vortex Fluid Field

${F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, {V_{olume}}\,\,\frac{dP_{Pressure-Static}}{dr}$  $\,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}$

${F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, {-}({V_{olume}} \,\, {\rho_{Net}})\,{|\vec{v}|_{CM}} \,\, \frac{d|\vec{v}|_{CM}}{dr}$  $\,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}$

The Inertial Mass and Density equal, (${m_{Net}} \,\, = \,\, {V_{olume}} \,\, {\rho_{Net}}$)

Circulation Force of Vortex Fluid Field

${F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, {-}{m_{Net}} \,{|\vec{v}|_{CM}} \,\, \frac{d|\vec{v}|_{CM}}{dr}$  $\,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}$

Dividing both sides of the above equation by the Net Inertial Mass (${m_{Net}}$), and rearranging the above yields,

$\frac{d\vec{r}}{\vec{r}} \,\, + \,\, \frac{d|\vec{v}|_{CM}}{|\vec{v}|_{CM}} \,\,\, = \,\,\, 0$

Integrating both sides of the above equation yields,

$\int{\frac{d\vec{r}}{\vec{r}}} \,\, + \,\, \int{\frac{d|\vec{v}|_{CM}}{|\vec{v}|_{CM}}} \,\,\, = \,\,\, \int{0}$

Using the logarithmic law and the Integral of a constant law

$\ln{\vec{r}} \,\, + \,\, \ln{|\vec{v}|_{CM}} \,\,\, = \,\, {Constant}$

$\ln{(\vec{r}} \,\, {|\vec{v}|_{CM})} \,\,\, = \,\, {Constant}$

The above produces a new constant,

$\vec{r} \,\, {|\vec{v}|_{CM}} \,\,\, = {||\vec{r}||} \,\, {||\vec{v}||_{CM}} \,\,\, = \,\, 2({Constant'})$

It follows that, the constant becomes equal to Kepler’s Second Law equation which states that planets orbits in gravitational fields in “Equal Areas in Equal Times.”

Where the Constant equals – (${Constant''} \,\, = \,\, {Constant'}\sqrt{1\,\,\, - \,\,\, e^2}$)

Kepler’s Second Law – Gravitational Angular Momentum & Aerial Velocity – Circular

${(\frac{dA_{rea}}{dt})_{Gravity}} \,\,\, = \,\,\, \frac{L_{Angular-Momentum}}{2{m_{Mass}}} \,\, = \,\,\, {Constant'} \,\,\, = \,\,\frac{1}{2}\, {||\vec{r}||} \,\, {||\vec{v}||_{CM}}$ —> $\frac{\, m^2}{s}$

${(\frac{dA_{rea}}{dt})_{Gravity}} \,\,\, = \,\,\, \frac{1}{2}\sqrt{{K_{Gravity}}\,\,{r}} \,\,\, = \,\,\, \frac{1}{2}\sqrt{({m_{Net}\,G})\,\,{r}}$  —-> $\frac{m^2}{s}$

Kepler’s Second Law – Gravitational Angular Momentum & Aerial Velocity – Elliptical

${(\frac{dA_{rea}}{dt})_{Ellipse-Gravity}} \,\,\, = \,\,\,\frac{L_{Angular-Elliptical}}{2{m_{Mass}}} \,\,\, = \,\,\, {Constant''} \,\,\, = \,\,\, \frac{1}{2}{||\vec{r}||} \,\, {||\vec{v}||_{CM}} \,\, \sqrt{1\,\,\, - \,\,\, e^2}$ —–> $\frac{\, m^2}{s}$

${(\frac{dA_{rea}}{dt})_{Ellipse-Gravity} \, = \, \frac{1}{2}\sqrt{{K_{Gravity}}\,\,{(r)}\, (1\,\,\, - \,\,\, e^2)}} = \, \frac{1}{2}\sqrt{({m_{Net}\,G})\,\,{(r)}\,\, (1\,\,\, - \,\,\, e^2)}$ —–> $\frac{m^2}{s}$

Therefore the Force now becomes

Circulation Force of Vortex Fluid Field

${F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, \frac{m_{Net}\,\, {(Constant')}^2} {\vec{r}^3}$   $\,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}$

${F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, \frac{m_{Net}\,\, {(\frac{dA_{rea}}{dt}_{Gravity})}^2} {\vec{r}^3}$  $\,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}$

${F_{Force}} \,\,\, = \,\,\, \frac{{m_{Net}}\,\,K_{Gravity}}{\vec{r^2}} \,\,\, = \,\,\, \frac{m^2_{Net}\,\,G}{\vec{r^2}}$  $\,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}$

Kepler’s First Law – (Space)

We also know from Kepler’s First law that the planets or bodies in gravitational fields orbit in elliptical and not circular orbits.

Semi-Major Radius – (${r}$)
Semi-Minor Radius – (${r_{Minor}}$)
elliptical eccentricity – (${e}$)

${r_{Minor}} = {r}\,\, \sqrt{1\,\,\, - \,\,\, e^2}$  $\,\,\,\,----> \,\, {m}$

The Area of the Circular Orbit is given by

${A_{rea-Circle}} = \pi \,{r^2}\,\,$ $\,\,\,\,----> \,\, {m^2}$

The Area of the Elliptical Orbit is given by

${A_{rea-Ellipse}} = \pi \,{r}{r_{Minor}}\,\, = \pi \,{r^2}\,\, \sqrt{1\,\,\, - \,\,\, e^2}$ $\,\,\,\,----> \,\, {m^2}$

Kepler’s First Law & Second Law – (Time)

The Orbital Period of the fluid particles in a “Gravitational Vortex” spin or rotate around the center of the vortex equal areas in equal times.

Orbital “Time” Period – (${T_{Period}}$) is defined

${T_{Period}} \,\,\, = \,\,\, \frac{A_{rea-Ellipse}}{(\frac{dA_{rea}}{dt})_{Ellipse-Gravity}}\,\,=\,\,\, \frac{A_{rea-Circle}}{(\frac{dA_{rea}}{dt})_{Gravity}} \,\,$  —–> ${s}$

${T_{Period}} \,\,\,=2\pi \,\,\sqrt{\frac{r^3}{m_{Net}\,G}}\,\,\,=\,\,\, \frac{2\pi \,\,{r}}{{||\vec{v}||_{CM}}} \,\,\,$  —–> ${s}$

Kepler’s Third Law – Evolution Attraction Rate

${K_{Gravity}}= \, 4\pi^2 (\frac{r^3_{Schwarzschild}}{T^2_{Schwarzschild}}) \, = \, 4\pi^2 (\frac{r^3_{1}}{T^2_{1}}) = 4\pi^2 (\frac{r^3_{2}}{T^2_{2}})= ...=4\pi^2 (\frac{r^{3}}{T^{2}_{Period}}) \,$

${K_{Gravity}}\,\,\, = \,\,\, \frac{(2{(\frac{dA_{rea}}{dt})_{Gravity}})^2}{r} \,\,\, = \,\,\, \frac{(2{(\frac{dA_{rea}}{dt})_{Ellipse-Gravity}})^2}{{r}\, (1\,\,\, - \,\,\, e^2)} \,\,\,$

${K_{Gravity}} \,\, = {m_{Net}}\,G \, \, = ( {m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}})G \,\,$   $---> \frac{m^3}{s^2}$

Concluding, Kepler’s Laws of the “Gravitational Vortex” using Fluid Mechanics have been derived.

************************************************************

### Citation

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Fluid dynamic “Vortex Gravitation” Model — derives Kepler’s Laws for Gravitation– Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

************************************************************

Author: Robert Louis Kemp

http://www.SuperPrincipia.com

This entry was posted in The General Theory of Relativity and tagged , , , , , , , . Bookmark the permalink.

### 10 Responses to Fluid dynamic “Vortex Gravitation” Model — derives Kepler’s Laws for Gravitation

1. I reckon something really interesting about your site so I bookmarked .

2. I see something truly interesting about your web blog so I saved to my bookmarks .

3. Utterly written written content , regards for entropy.

4. You are my intake , I own few web logs and often run out from to post : (.

5. I have read several excellent stuff here. Certainly worth bookmarking for revisiting. I wonder how a lot effort you set to make this sort of excellent informative web site.

6. Tabetha Daus says:

My husband and i ended up being quite contented Albert could do his investigation from the precious recommendations he discovered from your own web pages. It is now and again perplexing to simply choose to be freely giving tactics that many others might have been selling. So we already know we need the blog owner to appreciate for this. All the illustrations you’ve made, the easy website menu, the friendships you will help foster – it’s all incredible, and it’s assisting our son and the family feel that the concept is awesome, and that’s highly essential. Many thanks for everything!