Quaternionic Unification of Electromagnetism and Fluid Dynamics.pdf

A quaternionic unication of electromagnetism and

hydrodynamics

Arbab I. Arbab

Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321,

Khartoum 11115, Sudan,

Department of Physics and Applied Mathematics, Faculty of Applied Sciences and

Computer, Omdurman Ahlia University, P.O. Box 786, Omdurman, Sudan

E-mail: aiarbab@uofk.edu

Abstract. We have derived energy conservation equations from the quaternionic

Newton's law that is compatible with Lorentz transformation. This Newton's law

yields directly the Euler equation and other relations governing the uid motion.

With this formalism, the pressure contributes positively to the dynamics of the system

in the same way mass does. Hydrodynamic equations are derived from Maxwell's

equations by adopting an electromagnetohydrodynamics analogy. In this analogy the

hydroelectric eld is related to the local acceleration of the uid and the Lorentz

gauge is related to the incompressible uid condition. An analogous Lorentz gauge in

hydrodynamics is proposed. We have shown that the vorticity of the uid is developed

whenever the particle local acceleration of the uid deviates from the velocity direction.

We have shown that Lorentz force in electromagnetism corresponds to Euler force for

uids. Moreover, we have obtained a Faraday-like law and Ampere's -like law in

Hydrodynamics.

PACS numbers: 41.20.Jb; 47.10.-g; 47.10.A-

A quaternionic unication of electromagnetism and hydrodynamics

1. Introduction

We have recently formulated Maxwell equations using quaternion (Arbab and Satti,

2009 & Arbab, 2009). In this formalism, we have shown that Maxwell equations

have not included the magnetic eld produced by the charged particle. Consequently,

we have shown that this magnetic eld is given by an equation equivalent to Biot-

Savart law. Moreover, we have shown that the magnetic eld created by the charged

particle is always perpendicular to the particle direction of motion. We have also

introduced the quaternionic and compared it with the ordinary continuity equation.

Consequently, we have found that Maxwell equations predict that the electromagnetic

elds propagate in vacuum or a charged medium with the same speed of light in

vacuum. The present paper explores the idea of formulating the uid dynamics

equations using Quaternion description. Supported by the previous analysis done with

electromagnetism, we would like to further establish an analogy between quaternion

hydrodynamic and quaternion Maxwell's equations. We have shown that the quaternion

Newton's equation gives directly the non-relativistic limit of the energy momentum

conservation equation. These equations are the Euler equation, the continuity equation,

the energy momentum conservation equation. We have found an analogy between

hydrodynamics and electrodynamics. This analogy guarantees that one can derive the

equation of the former from the latter or vice versa. This analogy is quite impressing

since it allows us to visualize the ow of the electromagnetic eld that resembles the ow

of a uid. A Gauss-like equation in hydrodynamics is derived. This shows that these

GCEs must be satised for any uid ow. We have shown here that the generalized

Newton's second law yields directly the non-relativistic equations governing the motion

of uids. Moreover, we have shown earlier that the generalized continuity equations are

Lorentz invariant. The uid vorticity arises whenever the local acceleration of the uid

particles deviates from the velocity direction. The physical properties and the equations

governing the hydrodynamics are derived from Newton's second law. From this law Euler

equation is then derived, together with the equation of motion of the vorticity of the

uid. We remark also that Faraday-like law and Ampere's -like law in hydrodynamics

is obtained. Moreover, we have found that the diusion equation is compatible with

the GCEs. We therefore, emphasize that the GCEs should append any model dealing

with uid motion. According to this analogy and since the electrodynamics is written

in terms of the and , we should write the hydrodynamics equations in terms of the

hydroelectric eld and the vorticity in the same way Maxwell equations are written.

This is legitimate because of the analogy that exists between the two paradigms, viz.,

E h , E e and ! h , B, v , A and v 2

2 , '.

2. The Continuity equation

We have recently explored the application of quaternions to Maxwell's equations (Arbab

and Satti, 2009). We have found that the quaternionic Maxwell's equation reduces to

A quaternionic unication of electromagnetism and hydrodynamics

the ordinary Maxwell's equations but predicts the existence of a scalar wave competing

with the electromagnetic eld traveling at the speed of light in vacuum. Quaternions are

found to have interesting properties. The multiplication rule for the two quaternions,

A = (a 0 ; A) and B = (b 0 ; B) is given by (Tait, 1904)

AB = a 0 b 0 A B ;a 0 B Ab 0 + A B :

(1)

Therefore, the ordinary continuity equation is transformed into a quaternionic continuity

equation as

@ J

r J + @

; i

rJ =

@t + rc 2

+ r J

= 0;

(2)

where

r =

@t ; r

; J =

ic; J

(3)

The scalar and vector parts of Eq.(2) imply that

r J + @

@t = 0 ;

(4)

r +

c 2

@ J

@t = 0 ;

(5)

and

r J = 0 :

(6)

We call Eqs.(4)-(6) the generalized continuity equations (GCEs). In a covariant form,

Eqs.(5)-(6) read

@ J = 0 ; @ J @ J = 0 :

(7)

Hence the GCEs are Lorentz invariant. We remark that the GCEs are applicable to any

ow whether created by charged particles or neutral ones.

3. Newton's second law of motion:

The motion of the mass (m) is governed by the Newton's second law. The quaternionic

Newton force reads (Arbab and Satti, 2009)

F = mV (rV ) ;

(8)

where

F =

c P ; F

; V = (ic ; v) :

(9)

The vector part of Eq.(8) yields the two equations

v 2

F = m

@t + r

v rv

;

(10)

A quaternionic unication of electromagnetism and hydrodynamics

and

rv = v

c 2 @v

@t :

(11)

The scalar part of Eq.(8) yields the two equations

P = mc 2

rv + v

c 2 @v

;

(12)

and

v rv = 0 :

(13)

For a continuous medium (uid) containing a volume V , one can write Eq.(10) as

@t + r( v 2

2 ) v (rv)

= f ;

(14)

where m = V and f = V . Using the vector identity

2 r(vv) = v(rv) + (vr)v,

Eq.(14) becomes

@t + (v r)v

= f :

(15)

This is the familiar Euler equation describing the motion of a uid. For a uid moving

under pressure (P r ) one can write the pressure force density as

f P = rP r ;

(16)

so that Eq.(15) becomes

@t + (v r)v

= rP r :

(17)

Using Eq.(5), Eq.(12) can be written as

@t + r S = f v;

S = (c 2 )v; u = v 2 ; P = f v: (18)

Eq.(18) is an energy conservation equation, where S is the energy ux and u is the

energy density of the moving uid. With pressure term only, Eq.(18) yields

@t + r (c 2 + P r )v = P r rv:

(19)

The source term on the right hand side in the above equation is related to the work

needed to expand the uid. It is shown by Lima et al. (1997) that such a term has

to be added to the usual equation of uid dynamics to account for the work related

to the local expansion of the uid. It is thus remarkable we derive the fundamental

hydrodynamics equations from just two simple quaternionic equations, the continuity

and Newton's equation. For incompressible uids rv = 0 so that Eq.(19) becomes

@t + r (c 2 + P r )v = 0 ;

(20)

which states that the pressure contributes positively to the energy ow. This means the

total energy ow of the moving uid is

S total = (c 2 + P r )v;

(21)

A quaternionic unication of electromagnetism and hydrodynamics

and the total momentum density of the ow is given by

~ p = ( + P r

c 2 )v;

(22)

which must be conserved. This is analogous to the general theory of relativity where

the pressure and mass are sources of gravitation. Equation (20) states also that there

is no loss or gain of energy. However, when viscous terms considered loss of energy

into friction will arise. In standard cosmology the general trend of introducing the bulk

viscosity () is by replacing the pressure term P r by the eective pressure (Weinberg,

1972)

P e: = P r rv: (23)

Substituting this in Eq.(17) and dening the vorticity of the uid by ! = rv, we get

@t + (v r)v

= rP r + r 2 v + r! ;

(16a)

which reduces to the Navier-Stokes equation for irrotational ow (! = 0). Equation

(20) can be put in a covariant form as

@ T = 0 ; T = ( + P r

c 2 ) v v P r g :

(24)

where T is the energy momentum tensor of a perfect uid, v is its velocity and g is

the metric tensor with signature (+++-). It is interesting to remark that we pass from

quaternion Newton's law to relativity without any osetting. This is unlike the case of

ordinary Newton's law where relativistic eects can't be included directly.

Using the vector identity, r(f A) = f(r A) A(rf), with J = v, Eq.(6)

can be written as

r J = r (v) = rv v (r) = 0 ;

(25)

which upon using Eq.(4) transforms into

rv = v

c 2 @v

@t :

(26)

Thus, Eq.(10) derived from Newton's second law is equivalent to one of the continuity

equations, viz., Eq(5). Taking the dot product of Eq.(5) with a constant velocity v, we

get

c 2 v @ J

v r +

@t = 0 ;

which yields

dt = @

@t (1 v 2

c 2 ) :

According to Lorentz transformation, if the density in the rest frame is , it will be

0 = (1 v 2

c 2 ) in the moving inertial frame. Thus, taking the total derivative, is

The viscous pressure can be obtained from the viscous force, F = A d~v dr ) P v = F A = rv

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