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Discussion on the Similarity Between Maxwell Equations and the Mechanical Equations of Continuous Mass

Author: Xintie Yang
Airfoil Center, Aerodynamic Teaching and Research Group NPU 710072
Xian, China:
Shuangren Zhao,

Abstract:

A new system of equations can be deduced from the continuous viscosity medium flow. The system is similar to Maxwell's equations, therefore a connection is considered probable. This conclusion is reached by observing that an electromagnetic field exhibits the same characteristics as those in the mechanics of a continuous medium. The vortex in continuous medium flow plays a role equal to that of the magnet field and the role of force in mechanics plays a role equal to that of the electric field. This approach may be useful for further advances in gravitational field theory.

Keywords: Maxwell's equations, Gravity Field, Relativity Theory, NS equations, Non-Newtonian Fluids

1. The Equivalence of Maxwell’s and NS Equations

A generalization of the equations of electrodynamics and continuous medium mechanics was attempted by Maxwell. This was followed by investigation by P. W. Brigiman. The basic question was: Could equations for inertial mass be found that were similar to those of the electromagnetic field? Brigiman ventured an hypothesis just before his death.

In 1969, J. Carstoiu <3> presented a gravitational hypothesis. Carstoiu considered the variable E in the electromagnetic field to be very similar to the force F in the gravitational field, and therefore introduced a second field W , as the so-called vortex field of gravitational force, comparable to e and m in the electromagnetic system. Two Gravitational constants e g and m e defined the gravitational field.

e g = - 1/G (G = the universal gravitational constant)

m e = - G/C**2

This allowed the equations to be written similar to those of Maxwell. But this is only a formal extension. <4>

div (F) = -G r e

rot W = 1/c**2 F/ t -G/c**2 Je

rot F = - W / t

div (W ) = 0

He called W the vortex of gravity. If it does indeed exist, the question arises as to what physical meaning we may attribute to it and how it might be measured. These questions were not answered by Carstoiu.

Is it possible that such a simple system exists in continuous mechanics? The following is an analysis and deduction from the fundamental equations.

2. Analysis of the Similarity Between the Fundamental Equations of Electromagnetism and a Continuous Medium

Let us write the fundamental electromagnetic equations as follows:

div (D) = r 1

rot H = d + D/ t 2

rot E = - B/ t 3

div (m H) = 0 4

Here, D = e E, B =m H, d = g E, E is strength of the electric field ，H is the strength of the magnetic field，m is the magnetic conductivity rate, e is the constant of the electric medium，g is the electric conductivity rate. Now we must find similar equations in viscous fluid mechanics. The following are similar to eq. 1.and 4.

divW = 0 5

div F= a M（ Mass of Earth ） 5*

Here, a is related to the universal gravitation force.

In the following we deduce the equation of variation of vortex product rotation of force for a viscous fluid. We consider the incompressible momentum expression of Chromic-Lame：

V/ t + div (V.V/2) + W XV = F -1/r div P + 1/r div {2m [e ]} 6

Note: In a continuous medium, field e expresses the tension of the rate of strain《5》

| e 11 e 12 e 13 |

[e ] = | e 21 e 22 e 23 |

| e 31 e 32 e 33 |

Take the rot for two sides of equation 6:

W / t + rot (W XV) = rot F – rot (1/r  div P) + rot { 1/r  div {2m [e ]}}

i.e.

W / t = rot F + rot {-(1/r div P) – rot (W XV) + { 1/ div {2m [e ]}}}

The first bracket on the right of the equation is equal in meaning to force and has the same scale, so we may simplify and denote it as F2. We then have:

W / t = rot F +rot {F2} 7

The physical meaning of this equation is that the time variation of the vortex equals the rotation of force and another equal part of force in the medium. To correspond with the electromagnetic equation, we require another equation that provides a force to produce the vortex. We first consider the expression of the incompressible NS equation:

V / t + （div V）V = F -1/r div P + g rot W 8

that is ：

F =+ g rot W + V / t -1/r div P -（div V）V 9

The three terms to the right of equation V / t -1/r div P - (div V) is also a force, We denote it as F3:

F =+ g rot W + F3 10

It is unfortunate that the left of this equation is a derivative of speed du/dt = F, since we want a time derivative of force dF/dt. The discrepancy is a dimension of time. To find the time factor, we determine the difference between the two side of the equation which results in:

F/ t = {g rot W }/ t + F3/ t 11

To arrive at an equation completely consistent with electromagnetic theory, we must modify the first term at the right of equation 11. We must give up the constraint of the relation between stress and rate of strain from Newton’s hypothesis since it is the linear relation between stress and rate of strain. i.e. In the direction of main stress, the relation can be expressed as:

 

s = - P+a e 11+ｂe 22+c e 33 12

Here, a, b, c are constants, εij is the element of tension in the main direction of the rate of strain. In these hypotheses is the absence of the relation that we needed; that the stress is not only line related to rate of strain, but also line related to the strain itself. Obviously this absence is due to the hypothesis of Newton’s fluids. If we consider non-Newtonian flow, this conflict can be resolved. In non-Newtonian fluid, its stress is also linear related to strain, so the expression in the main strain direction is:

s = - P +a h 11+b h 22+c h 33

+a e 11+ b e 22+ｃe 33 13

Here, a*, b*, c* are constants,ηｉｊ is the element of tension of the main direction of strain. Through the same deductive process, we can get a similar equation as in the viscous equation 10 as follows:

F = x rot q + F3 14

Here,ζis the viscous coefficient based on the strain,q is angle of strain. By taking the difference between the two sides of equation 14 to time, we get:

F/ t = { z rot q } / t + F3/ t 15

And simplify with {z rot q }/ t = x rot W which becomes:

F/ t + F3/ t = z rot W 16

3. Comparing Maxwell equations and the mechanical equations of continuous mass, we write as follows:

Maxwell Equations                Mechanical Equations of Continuous Medium

div (e E) = r div F = a M（Mass of Earth）

(εE)/ t = rot H +g E / t = z rot W + F3/ t

(m H)/ t = -rot E W / t = rot F + rot F2

div (m H) = 0 div W = 0

 

We see that there are obvious similarities between the continuous medium equations and those for electromagnetism. The variation in force produces a vortex in the medium, and the pulse of Vortex can produce the vortex of force for the continuous medium, carried by a "Cross wave". This is similar to a magnetic wave. The analysis of this wave was done many years ago.

In consideration of the above, we find that electric and magnetic effects have a correspondence in vortex and force in a continuous medium. However, the magnitude of this effect in nature is very small, so we regrettably neglect its existence and by doing so, overlook its application to gravitational theory and neglect a beautiful corresponding relationship. then asked we, can we find same deep physical connection in this correspondence? from the given equation , Obviously the same characteristics exist. magnetic field strength is similarly to a vortex in incompressible flow, and electric field strength is similar to the force in incompressible flow.

Electromagnet field theory is still expressed as a linear one order equation system, but continuous equations have had major development in the last century, up to a non-liner equation system including compressible flow and boundary layer flow. If electromagnetism can be incorporated in this century, then the vigor of wholly compressible viscous continual mechanics may be infused into the theory of electromagnetism. Clearly in this new theory opens the possibility of speeds that exceed light. This is beyond the goal of this paper and will be discussed in the next.

4. Conclusion:

In Continuous medium mechanics there exists a system of equations similar to electromagnetic theory. This equation system may been called the Extended Maxwell Equation System. It is nonlinear and its use may supply impetus for development in Maxwell’ equations.《11》

5. My thanks to professor Mingzhu Xu; Jiaxiang Yan; Lirong Du; Zhide Qiao and Zhongyin Zhang for the discussions and suggestions in the preparation of this paper.

Reference:

1 Heaviside. O., Electromagnetic Theory, New York, 1893

2 Brigiman.P.W, A Sophisticates Primer of Relativity, Middeltown,

Connection, 1962.

3 Carstoiu J. Compute. Rend., 268,201 (1969).

4 Brillouin L.,Lucas R., Journ. Physics. Radium, 27, 229, 1966.

5 Mechanics of Continuous Medium, Landau, 1962

6 Non Newtonian Fluids W.l.Wilkinson, 1960

7 Vortex Flow in Nature and Technology, Hans J. Lugt, 1982

8 Non Newtonian Fluid Mechanics G. Boehme, 1986

9 Experimental Basis of Special Relativity, Zhang Yuanzhong, China,1979

10 Conflict in Relativity Theory，Zhang Quan JSBN7-80045-038-2