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Interaction of the spherical sources and drains,  having a common rotation axis



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3.9 Interaction of the spherical sources and drains
having a common rotation axis

Consider the mechanical model of interaction of spherical sources and drains, having a common axis of rotation.

In Fig. 3.9.1 and 3.9.2 are sectional lines of the velocity field of the current cumulative vectors elliptic longitudinal velocity components emitted (absorbed) ether streams rotating ball sources (drains), a source and drain, respectively, on the drawing plane coincident rotation axes obtained by applying the principle of superposition velocity fields.

Note: Given the complete interchangeability of sources and drains, in this case, to simplify, spherical objects in the figures are shown in generalized form.

In Fig. 3.9.1 and 3.9.2 ball sources (drains) or the source and drain, respectively, to face each other, or source stack vectors elliptic longitudinal velocity component streams ether or drains, i.e. the eponymous poles, in the electromagnetic interpretation.

Faraday and Maxwell was proven that the magnetic unit disposed thicker tube, the greater the rate at which ether flows. Therefore ether velocity flows between the sources or drains or source and drain, respectively, indicated in Fig. 3.9.1 and 3.9.2, will be lower than at the ends. Consequently, the pressure in the streams between ether particles will be greater than at the ends. In this case, they will repel each other forces and equal in magnitude but opposite in direction.

 




N S S N

Fig. 3.9.1





S N N S
Fig. 3.9.2

In Fig. 3.9.3 Ball sources (drains) or the source and drain, respectively, to face each other on the one hand sources summary vectors elliptic longitudinal velocity components ether flows, and on the other side of the drain summary vectors elliptic longitudinal velocity components ether streams, i.e. oppositely poles in the electromagnetic interpretation.

Similarly to the above reasoning, we conclude that the rate of ether particles between sources (drains), a source and a drain, respectively, will be greater than at the ends. Consequently, the pressure in the streams between ether particles will be less than at the ends. In this case, they will be attracted to each other the forces and equal in magnitude but opposite in direction.


S N S N

Fig. 3.9.3


"The interaction of the two magnetic poles should occur under the same laws as the interaction of two electricity particles. Therefore, the idea of ​​lines of force can be applied also to magnetism, and his theory is just as electrostatics can be illustrated by means of a moving fluid. Here, however, the study will have solution to the problem of how single cells in the fluid motion can present the polarity of the elementary magnets "Maxwell wrote.

The results of this study make it possible to resolve, previously unknown, the question "how single cells in the fluid motion can present the polarity of the elementary magnets" and justify the mechanical nature of magnetic phenomena predicted by Faraday and Maxwell.



3.10 Interaction spherical sources and drains 
with mutually perpendicular streams ether

Consider a mechanical model of the interaction of the ball source streams with mutually perpendicular ether limited surface of the sphere radius . The source emits ether vortex tube streams perpendicular to the surface at a rate spheres and rotating with an angular velocity . The source is moved at a speed of perpendicularly with respect to the moving speed circulating stream vortex tubes ester, as shown in Fig. 3.10.3. The total whirlwind - , It is acting on the source, characterized by a total angular rotational speed of the circulating vortex tubes together ether stream on the source surface. In this case the source will be exposed to two mutually perpendicular ether streams - the vortex stream moving with velocity and the ram ether stream - , Equal and oppositely directed displacement rate source – , As well as the angular velocity . As shown in the study, 3.7, if we neglect the angular velocity , The axis of rotation of the source - It will coincide with the vortex axis of the source - , But opposite in direction. In this case, the rotation axis is directed toward the incoming stream, i.e. the total velocity vector stream ether and inclined with respect to the vector on the corner , As shown in Fig. 3.10.1.

                                           


S








N

S



N

3.10.1


If we consider the angular velocity , Then the total angular velocity vortices  obliquely with respect to the source of rotational axis by angle clockwise, as shown in Fig. 3.10.2.

         

                    



S









N





S





N

3.10.2
But in this case, according to the evidence presented in the study 3.7, there is a torque that source to be deployed in the angle counterclockwise until it matches the sum of the vortex axis with the direction of the total velocity vector stream ester. In this connection, the rotation axis of the source - also it will turn on the angle counterclockwise, as shown in Fig. 3.10.3.

 




S






N А
А



S

Fig. 3.10.3

                   

We define the projection of force in the direction of the unit vector - , The observer perpendicular to the plane of Fig. 3.10.3 acting in section A-A of the source.

In Fig. 3.10.4 and 3.10.5 are sectional source in a diametrical plane perpendicular to the vector .

To distinguish four quadrants surface sections I, II, III and IV, as shown in Fig. 3.10.4 and 3.10.5.

In Fig. 3.10.4 specified direction vectors constituting the particle velocities at points on a circle of radius  by sectors I, II, III and IV, respectively.

In Fig. 3.10.5 specified direction vectors constituting the accelerations at the points on a circle of radius by sectors I, II, III and IV, respectively.

  


                                      






II I



III IV









3.10.4










II I



III IV







3.10.5


                                                    

Where:


– the unit vector normal to the outer circumference;

,, and – the angle between the coordinate axes and the normals to the external circumference in sectors I, II, III and IV, respectively;

– circumference radius of the source;

– density ether filling space;

- total velocity vector ether fluxes emitted by the source;

– vector normal velocity component ether streams emitted by the source;

– vector of circular transverse velocity component ether streams emitted by the source;

- vector stream ether angular rotation velocity of the emitted source;

- vector component transverse circular ether stream rate, rotating at an angular velocity according to the source in a projection on the circumference of the cross section;

- total transverse circular velocity vector ether source stream;

– vector of the normal component of acceleration ether fluxes emitted by a source with a rate ;

– vector of circular transverse acceleration component ether streams emitted by the source circle, to the speed ;

- vector centripetal acceleration component ether streams emitted by the source circle, to the speed ;

- vector centripetal acceleration ether streams by rotating the source circle, to the speed ;

- total velocity vector  and и .

Substituting in equation (3.4.3) the received symbols, we define the force projection , Acting in a section A-A in the direction of the source , Perpendicular to its motion, as a sum of terms of the sectors I, II, III and IV given the fact that .

(3.10.1)

First we define the values, and for sectors I, II, III and IV under

provided that

sector I,



sector II



sector III



sector IV



Then, define the product and draw the addition integrand by sectors I, II, III and IV













(3.10.2)
Substituting the value found in equation (3.10.1)

(3.10.3)


Now we define the magnitude of the projection of the force acting generally towards the source - (Fig. 3.10.6).









Fig. 3.10.6
Since the projection acting forces in all sections of the source is proportional to the radius of the circle cross section and the projection of radiation stream velocity ether in the sectional plane, to determine the value of the projection of the force acting on the whole of the source, it is sufficient to carry out the integration of equation (3.10.3) on the surface of the sphere:

  


(3.10.4)

Where: – the angle between the plane of a diametrical sectional areas perpendicular to the rotational axis – and the direction from the center of the sphere to its surface;

- sphere radius of the section perpendicular to the plane , Passing through the intersection of the sphere ray drawn from the center of the sphere at an angle to its surface;



–the projection of the velocity of ether stream to the radiator section plane of a sphere of radius .

As  then  , то

Then , but (3.10.5)

Substituting the value found in (3.10.4), taking into account that , We finally obtain:

(3.10.6)
Now consider the mechanical model of interaction stream with mutually perpendicular to the stream of ether, the limited surface of the sphere radius . Drain absorbs ether vortex tube streams perpendicular to the surface at a rate spheres and rotates with the angular velocity . Drain is moved at a speed of  perpendicularly to the moving speed circulating stream vortex tubes ether as shown in Fig. 3.10.9. The total whirlwind - , acting on the stream characterized by a total angular velocity of rotation of the vortex tubes together ether circulating stream on the surface runoff. In this case, the drain will be exposed to two mutually perpendicular ether streams - the vortex stream moving with velocity and the ram ether stream - , equal and oppositely directed to the velocity moving stream – as well as the angular velocity . As was shown in Section 4.7 that if we neglect the angular velocity , then drain the rotation axis coincides with the axis of the vortex flow, including the direction. In this case, the rotation axis is directed in the same direction as the incoming stream ester, i.e. in the direction of the total velocity vector stream ether and inclined with respect to the vector on the corner , as shown in Fig. 3.10.7.
                                           



S







N

S




N

Fig. 3.10.7

If we consider the angular velocity , then the total angular velocity vortices obliquely with respect to the source of rotational axis by angle clockwise, as shown in Fig. 3.10.8.



S









N





S





N

Fig. 3.10.8


But in this case, according to the evidence presented in Section 3.7, there is a torque that is to be deployed in the angle counterclockwise until it matches the sum of the vortex axis with the direction of the total velocity vector stream ether . In this connection, the rotation axis Photo also will turn on the angle counterclockwise, as shown in Fig. 3.10.9.

 

  




S








N А
А



S





N
Fig. 3.10.9

                                    

                   

We define the projection of force in the direction of the unit vector - , The observer perpendicular to the plane of Fig. 3.10.9 operating in section A-A in drain.

In Fig. 3.10.10 and 3.10.11 are sectional stream in the diametrical plane perpendicular to the vector .

To distinguish four quadrants surface sections I, II, III and IV, as shown in Fig. 3.10.10 and 3.10.11.

In Fig. 3.10.10 direction vectors are velocity components ether streams at points on a circle of radius by sectors I, II, III and IV, respectively.

In Fig. 3.10.11 specified direction vectors constituting accelerations ether streams at points on a circle of radius by sectors I, II, III and IV, respectively.










II I



III IV









Fig. 3.10.10









II I



III IV







 Fig. 3.10.11

       

                                                                                    

                                                                                                                                             

                                            

                                                        

Where:


– the unit vector normal to the outer circumference;

,, and – the angle between the coordinate axes and the normals to the external circumference in sectors I, II, III and IV, respectively;

– circle radius of the flow;

– density ether filling space;

- total velocity vector ether streams absorbed drain;

– vector of the normal component of the stream velocity ether absorbed drain;

– vector component transverse circular ether stream rate absorbed by the drain;

– vector ether angular velocity absorbed drain;

- vector component transverse circular ether stream rate, rotating at an angular velocity stream circumferentially in a projection on the cross section;

- total vector ether circular transverse velocity stream streams;

– vector of the normal component of acceleration ether fluxes absorbed by the drain at a rate ;

- vector centripetal acceleration component streams ether absorbed by the drain and the rotating circumferential section with a velocity ;

– vector of circular transverse acceleration component streams ether absorbed drain and rotating circumferentially in a section at a rate ;

- vector stream centripetal acceleration ether rotating circumferentially in a section at a rate ;

- total velocity vector and .

Substituting in equation (3.4.3) the received symbols, we define the force projection , acting in a section A-A in stream direction , perpendicular to its motion, as a sum of terms of the sectors I, II, III and IV given the fact that .

(3.10.7)

we define the values, and for sectors I, II, III and IV

provided that we get:

sector I,




sector II



sector III



sector IV



Then, define the product and draw the addition integrand by sectors I, II, III and IV












(3.10.8)
Substituting the value found in equation (3.10.7)

(3.10.9)

Now we define the magnitude of the projection of the force acting as a whole on drain in the direction - (See Fig. 3.10.12).









    

Fig. 3.10.12

Since the projections acting force value in all sections Photo proportional to the radius of the circle and the projection of the absorption stream rate ether section on the plane, to determine the values ​​of the projection of the force acting as a whole on drain, it is sufficient to carry out the integration of equation (3.10.9) over a sphere:

(3.10.10)

Where: – the angle between the plane of a diametrical sectional areas perpendicular to the rotational axis – and the direction from the center of the sphere to its surface;

- sphere radius of the section perpendicular to the plane , passing through the intersection of the sphere ray drawn from the center of the sphere at an angle to its surface;

– the projection of ether stream velocity of radiation source on the plane sections of the sphere radius .

As  then , то

Then , but (3.10.11)

Substituting the value found in (3.10.10), taking into account that , we finally obtain:

(3.10.12)

This study is set, not previously known dependence of the forces acting on the sources and drains in the interaction with mutually perpendicular streams ester

the velocities and accelerations, both the sources and drains, and the impinging ether streams.



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