fdi in danisha hall thruster using gravitational collapse pattern in saiph star(kappa orion)

8/3/2019 FDI in DANISHA hall thruster using gravitational collapse pattern in Saiph star(Kappa orion)

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FUZZY DIFFERENTIAL INCLUSION (FDI) SIMULATION FOR

GRAVITIATIONAL COLLAPSE IN DIFFUSION ASSOCIATED

NEOCLASSICAL INDEGINOUS SYSTEM OF HALL ASSEMBLY

(DANISHA)

A.B.R.HAZARIKA

DEPT. OF MATHEMATICS,

DIPHU GOVT. COLLEGE, DIPHU,

ASSAM, INDIA-782462

E-mail:

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

ABSTRACT

Fuzzy differential inclusion(FDI) simulation is studied for gravitational collapse processin Diffusion associated neoclassical indigenous system of Hall assembly (DANISHA) in

toroidal geometry coordinates for derived magneto hydrodynamic formulation to get the

thrust effect by using such magnetic device used for first time .The DANISHA hallthruster works for 86800(eighty-six thousand eight hundred hours) instead of 8000 hrs. in

case of SPT-100.The gravitational collapse pattern theory is used to study the fusion

process of Saiph star (Orion Kappa) in this case .

Keywords: Fuzzy logic, computer modeling and simulations, Z-pinch implosions,

gravitational instability, gravitational collapse, finite larmor radius, sheared axial flow,

MHD formulation, plasma propulsion, Hall thruster, plasma bridge, Hall effect devices,Hall effect hybrid, Magnetic device for propulsion, mathematical procedures and

computer techniques.

PACS: 7.05Mh, 7.05Tp, 53.35Py, 52.58Lq, 52.65Kj, 52.75Di, 85.30Fg, 85.75Nn,85.70Rp, 95.75Pq
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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INTRODUCTION

Bhatia and Hazarika (1995,1996) did some work on gravitational instability using the

viscosity term showing influence on the gravitational collapse ,in the later paper the twosuperposed fluid(plasma) flow past each other in horizontal magnetic field with rotation

and viscosity playing vital role for the collapse rotating neutrons stars Schwarzschild

radius is derived. Jeans instability leading to gravitational collapse of stars is studiedearlier by many authors Jeans (1902); El-Sayed and Mohamed (2011) tried to prove the

gravitational collapse with finite conductivity plain crucial role.

PRESENTWORK

The present work is based on low beta plasma having low frequency fluctuation whichis being stabilized by sheared velocity, finite conductivity and with other parameters. The

induced gravitational instability by above mentioned parameters and as a whole the

classical transport phenomenon is taken into consideration and is studied as pattern forfusion process in DANISHA instead of old version SPT-100. The heat conduction term is

calculated ,Banana regime (Bootstrap) , ware effect is calculated where an important

result Hazarika regime for Saiph star (Kappa orionis) which is R(1+4 -sin3 sin -2sin -2sin ) times of solar radius is found to be correct as 10.8503(approx. 11 )which is same as for DANISHA,with right accession (RA) of 05h 47m 45.4s(72.79594 degrees)

as the radial angle and declination of -09deg 40m 11sec(-9.669723 deg)as the azimuthal

angle. The above facts compels one to study the phenomena of gravitational collapsepattern based fusion process in Saiph star (Orion Kappa) and thereby the suppression by

different parameters.

Hazarika constant for Saiph star (Kappa Orionis) and for DANISHA is

sin2sin2sin3sin41 +++=hC =10.8503


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Cross-sectional view of DANISHA hall thruster

DANISHA

DANISHA

DANISHAHALL

THRUSTER


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Lateral view of DANISHA Hall thruster

BASICEQUATION

h

h

h

h

E RC

FRCp

RC

FpRCR

eNE

2

)(

2

1 22 +==

.. (1)

[ ]

=

22

222

2

h

th

t

i

CR

BCR

B

.. (2)

ee

ee nqd t

d Tn+ =

2

3.. (3)

XenonGasc

hamber


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( )

2

s i n22

)s i n22(

122

22

t

e

BR

Rq

+

+= X {

+

+22

)sin22(

1)sin22(

R

R } ..(4)

( )

+=

iie

i pen

EB

Bv .

12

.. (5)

GENERALMHDMODEL:

[ ] =

+

2

0

2

0

0 ,B

c

tB

cmn i

( ) zGnmzmGnB

cnm

R

zrpXj

c

Bii

.1..cos2

00

0

4

0

0

0 ++

+

+

(6)

jjtc

A 2

+=

.. (7)

[ ] ppB

c

t

p 2

0

, =+

(8)

Where Ac

j 2

4=

, [ ] BAXzBA = ., is Poissons bracket

Eigen mode equation is obtained by linearizing the eqn. (6)-(8) in toroidal coordinates

(r, , ) for RTI as

( ) ( )( ) ( ) +++=

= 0

1

0 s i n2e x p2),,( qi nrri n qlrqi nlr m nl

..

(9)

The Eigen value equation reduces to a differential equation

( ) 0)(

1

222

2

2

2

=+

++

sC

s

d

dh ..

(10)

After using Poissons equation in equation (10) and FDI we get

( )2222

2

)(

1

sCs

G

h +

+

= ... (11)

On further solving eqn. (11) we get the growth as follows


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.. (12)

Where

( )

[ ]

( )

( )

[ ]( ) ( )

( )

sin2sin2sin3sin41,

ln,

ln

,,4

,,ln

,,

124

3

,

4

,ln,,

8

,

,sin2sin2sin3sin41

,,,

2

0101

2

04

2

2

02

0

05

1

5

45

4

5

35

4

0

2

0

2

02

2

00

2

22

2222244

+=

=

=

====

==+

+

=

=====

+====

R

cg

dr

ndLn

dr

pdLp

Ln

rnn

r

c

rB

c

gLngLnrmn

Br

AAt

t

ls

a

B

rmn

gLngLnrmnB

p

pqn

R

rqnqnqn

s

mnmn

mn

pa

mn

pa

mni

pa

mnpa

mni

pa

mni

pa

pa

DISCUSSION

The growth rate can be found by using the first order derivative to analyse that thefunction increases or decreases :

( ){ }

( ){ }

0

12

2

1

2222

12

22

1222

22

=

+

+

+

+

=

sCs

s

ssC

s

Gd

d

h

h

. (13)

As the above term is positive we can say that it decreases or the function stabilizes.

If the above term goes to 0 it gives us the root of the equation .i.e., either the collapsing

growth rate of Saiph (Kappa Orion) is positive or negative sign of growth rate.

( )2/1

222/1

2

2/1

2

++=

ss

CG h

Here the growth rate provides us the frequency of system for the thrust in it if we go for

the inverse of growth rate that gives us the time module for the Hall thruster DANISHA


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[ ] ( )2/1

222/112/1

2

2

++=

ss

CGt h

One can see in the above equation we one extra term square of a constant which is

playing crucial role here as the time increases by that factor than the gravitational

collapse in simple derivation. Skin depth and viscosity also plays vital role as these terms

associates with it the Hazarika constant term in it as this term enhances the thrust by10.85 times, the time of flight enhances to 86800 hrs from 8000 hrs. of SPT-100 for

DANISHA.as application it can be used for rockets which hovers around the earth forsatellites etc.,it can be used for mars rover or so in future.

Here we see that the function which is dependent on current diffusivity and skin depth ispositive i.e., stabilizes the growth rate of star .if current diffusivity is more than 1000 the

potential collapses, similarly the skin depth values (1-10) shows the collapse of the

potential of Kappa Orion(Saiph).It stabilizes for >1 always for if

and s remain constant, roots lies between 30 and 40.If we plot growth against the

current diffusivity and skin depth we can see the implosion and collapse of potential.

OnGravitationalredshiftpatternFor mass of the star including in the stellar spectra with those in spectra obtained in the

laboratory, the ratio M/R is too small for a gravitational redshift to be apparent. A white

dwarf is an old star whose interior consists of atoms whose electron structure havecollapsed and so it is very small

Rc

GMsaiph

=

2


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12

Rc

GMsaiphWith this condition no photons can even leave the star as it cannot radiate

i.e., a black hole.

111

152 RcGMsun

Schwarzschildsradius of Saiph star (Kappa Orionis)

2215

2

c

GM

c

GMR sun

saiph

s ==

ss RR 15=

Schwarzschilds radius of Saiph star (Kappa Orionis) is 15 times the Schwarzschilds

radius of sun.

Power is given by

hpRCP=

Let us assume that the power of SPT-100 is 1500000 Nm, and then the power of

DANISHA will be 10.85 times that ofSPT-100 which comes out to be 16275000Nm

As the plasma grows 15 times or more the implosion process starts thus giving the thrustto the system .On the pattern of gravitational collapse in Saiph star (Orion kappa) the

collapse occurs much faster with Fuzzy differential inclusion (FDI) by Inverse Square of

a constant term.


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In this fig. we can see that the gravitational potential collapses to zero when plottedagainst current diffusivity.

Intheaboveplotoneobservethatthecurrentdiffusivityenhancesthegrowthrate.


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Inthisfig. densityalsoenhancesthegrowth rate of the system.

REFERENCES

1. Bhatia, P.K and Hazarika, A.B.Rajib: Physica Scripta 51,775(1995)

2. Bhatia, P.K and Hazarika, A.B.Rajib: Physica Scripta 53, 57(1996)3. Bhatia, P.K and Hazarika, A.B.Rajib Ind. Acad. Math 29(1), 141(2007)

4. Jeans, J.H: Philosophical transaction of royal society of London, 199, 1(1902)

5. El-Sayed, M.F and Mohamed, R.A: ISRN Mechanical engineering, 2011, art.ID597172 (2011)

6. Chandrasekhar, S: Hydrodynamic and Hyromagnetic stability, Dover publication, USA

(1981)

fdi in danisha hall thruster using gravitational collapse pattern in saiph star(kappa orion)

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