A new model for the reformulation of compressible fluid equations

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A new model for the reformulation of compressible fluid equations S. Demir a,∗, A. Uymaz b, M. Tanıs¸ lı a a b

Faculty of Science, Department of Physics 26470 Eskis¸ ehir, Anadolu University, Turkey Graduate School of Sciences 26470 Eskis¸ ehir, Anadolu University, Turkey

a r t i c l e

i n f o

Article history: Received 27 March 2016 Revised 1 July 2016 Accepted 6 October 2016 Available online xxx Keywords: Biquaternion Fluid equations Maxwell equations Matrices

a b s t r a c t Stimulating the biquaternionic generalization of electric and magnetic fields in electromagnetism, a new model has been proposed to present the Maxwell type equations of compressible fluids. The fluid wave equation has been expressed in a compact and elegant manner. Similarly, the generalized Pointing theorem for fluids has been derived analogous to electromagnetism and linear gravity. Moreover, a brief survey on biquaternionic matrices and the corresponding matrix representations of fluid equations have been given. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

1. Introduction Quaternions were invented by Hamilton [1] in 1843 in order to extend complex numbers to the three dimensions. This basic mathematical entity forms an associative division algebra and composed of four components, i.e, one real and three imaginary. A biquaternion is a complex combination of two quaternions, and therefore it also named as complex quaternion. In other words, biquaternions can be seen as the quaternions with complex coefficients. Because of having eight real elements, they have capability to express until eight component mathematical structures. One of the purposes of this paper is to propose new model based on quaternions for the Maxwell type equations of compressible fluids stimulating the biquaternionic generalization of electric and magnetic fields in electromagnetism. Actually, the researches related to quaternions in electromagnetic theory can be traced back to Maxwell’s own studies. Although today, 3-dimensional vector representation is used for formulating electromagnetism, in his famous book Treatise on Electricity and Magnetism [2], he also gave their quaternionic forms in a number of places [3]. On the other hand, in the presence of non-zero photon rest mass, this deviation leads us straightforwardly through the Proca equation which is one of the generalization of Maxwells equations [4]. Similarly, in contrast to the classical Maxwell equations, by assuming the existence of magnetic monopoles according to Dirac’s theory [5], Maxwell’s equations show more symmetry between the electric and magnetic fields. However, by using biquaternions, all of these equations can be rewritten in the simple and compact form [6–19]. Moreover, biquaternions can also be employed to generalize the Maxwell equations in presence of electric, magnetic sources and massive photons [20– 26]. There also are perfect analogies between linear gravity and electromagnetism. Similarly to Dirac’s magnetic monopole theory, the existence of gravitomagnetic mass for the linear gravity can be proposed. Furthermore, with the addition of ∗

Corresponding author. E-mail address: [email protected] (S. Demir).

http://dx.doi.org/10.1016/j.cjph.2016.10.011 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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the terms due to the finite mass of graviton, the usual Maxwell’s equations for electromagnetism may be transformed into Proca-type gravitational field equations [27]. As realized previously for electromagnetic theory, some biquaternionic models have also been proposed to formulate the generalization of gravity both covering the Proca and gravitomagnetic monopole terms. In relevant literature, using the structural symmetry between the fields equations of electromagnetism and linear gravity, the gravitational field and generalized wave equations have also been reformulated in terms of biquaternions [28– 30]. On the other hand, biquaternion formalism has been used to unify the theory of linear gravity and electromagnetism in the presence of both electric, magnetic, gravitoelectric and gravitomagnetic charges [31–39]. As pointed before, the analogy between the field equations of electromagnetism and linear gravity has led the scientist to deriving Maxwell like equations for gravitation. Stimulating from these efforts, an analogy between the structure of electrodynamics and fluid dynamics has been explored in the literature for incompressible and compressible flows, and also plasma. Although there were several attempts in the past, these generalizations were defined under restricted conditions. The first attempt has been done by Logan [40] to develop a set Maxwell equations for fluids, but it was restricted to the case of the one-dimensional Rayleigh problem. On the other hand, analogous equations for incompressible fluids have been first presented by Troshkin [41]. Similarly, initiating a new theory of turbulence, the analogy between the Navier–Stokes and Maxwell equations has been investigated by Marmanis [42] in order to describe the motion of an incompressible fluid at high Reynolds numbers. A system of equations of compressible fluid has been reformulated by Kambe [43,44] in a form analogous to electromagnetism governed by Maxwell equations with source terms. According to this approach, vorticity behaves like the magnetic field while the velocity field performs as the part of a vector potential and the enthalpy plays the part of a scalar potential in electromagnetism. Similarly, Scofield and Huq [45,46] have derived the fluid dynamical analogs of the electrodynamical Lorentz force law and Poynting theorem. Since Marmanis [42] has developed the analogous Maxwell type formalism for an incompressible fluid without considering dissipation terms, in the recent paper [47], Abreu et al. have carried out a Maxwell type electromagnetic model for a compressible fluid with dissipation term. A correlation function and dispersion relation have also been analyzed as functions of the Reynolds number. In spite of the fact that all of the relationships have been introduced for incompressible turbulent flow and compressible flow, the application of these analogies have not been investigated in case of plasma. But in their paper [48], Thompson and Moeller have extended these derivations in order to determine the Maxwell type equations for a plasma. Although Maxwell’s equations of electromagnetism have been expressed in many mathematical forms in relevant literature, the same is not true for their analogous equations in fluid mechanics. On the other hand, the Maxwell type fluid equations are rewritten in terms of octons [49]. Associative octons are the eight-component values defined by Mironov and Mironov [50–55]. This mathematical formalism can be seen as an alternative structure to the eight component octonions [56]. Non-associative and non-commutative octonions form the widest normed division algebra after the algebras of real numbers, complex numbers, and quaternions. They have many attractive mathematical properties to reformulate and generalize the field equations of electromagnetism and linear gravity [57–69]. Contrary to expectations, in the recent paper [70], it was proven that the non-associative nature of hyperbolic octonions is not a obstacle to derive a compact and elegant formulation for fluids. On the other hand, the corresponding matrix representation of octons [52] is not useful. Similarly, different multiplication rules of the basis elements of various types of octonions in literature lead to some difficulties in the mathematical operations related to vectors. However, Hamilton quaternions with four components is composed of a scalar and a vector. Naturally, this structure allows one to perform easily combined operations simultaneously with scalars and vectors. Furthermore, mathematically an exact isomorphism exists between biquaternions and their corresponding matrix representations. Therefore, the quaternionic reformulation of the fluid Maxwell equations has been proposed in this paper. Stimulating the similarities between Maxwell equations of electromagnetism and linear gravity, the field and wave equations for fluids have been generalized in a compact and simple way. The organization of the paper is as follows: biquaternion algebra and the corresponding matrix representations are given in Section 2. The biquaternionic formulations of fluid Maxwell equations, and derivation of the generalized wave equations are in Section 3. Energy relations and Pointing theorem for fluids analogous to electromagnetic theory are proposed in Section 4. Finally, discussions and perspective of our paper are given in the last Section 5. 2. Preliminaries A quaternion is a hypercomplex number with four real components

q = q0 e0 + q1 e1 + q2 e2 + q3 e3

(1)

where q0 , q1 , q2 , q3 are real numbers, and e0 , e1 , e2 , e3 are quaternion basis elements that satisfy the multiplication rules in Table 1. Because basis elements can be seen as orthogonal unit spatial vectors, a quaternion q can be expressed as a linear combination of a scalar q0 and a spatial vector q = q1 e1 + q2 e2 + q3 e3 :

q = q 0 + q.

(2)

Therefore, scalars and spatial vectors are in the subspace of quaternions. Please cite this article as: S. Demir et al., A new model for the reformulation of compressible fluid equations, Chinese Journal of Physics (2016), http://dx.doi.org/10.1016/j.cjph.2016.10.011

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Table 1 Multiplication rules of quaternion basis elements.

e0 e1 e2 e3

e0

e1

e2

e3

1 e1 e2 e3

e1 −1 −e3 e2

e2 e3 −1 −e1

e3 −e2 e1 −1

Biquaternions are the quaternions with complex components and actually originally proposed Hamilton in 1853. In order to obtain a biquaternion Q, two quaternions q = q0 e0 + q1 e1 + q2 e2 + q3 e3 and q = q0 e0 + q1 e1 + q2 e2 + q3 e3 should be combined as

Q = q + iq = (q0 + iq0 )e0 + (q1 + iq1 )e1 + (q2 + iq2 )e2 + (q3 + iq3 )e3 . Here i is the most acknowledged complex unit(i =

√

(3a)

−1). Biquaternion Q is also expressed as

Q = Q0 e0 + Q1 e1 + Q2 e2 + Q3 e3

(3b)

where Q0 , Q1 , Q2 , Q3 are complex numbers. Similarly to a quaternion, a biquaternion can also be expressed as

P = P0 + P

(4)

where P0 is its scalar part and P = P1 e1 + P2 e2 + P3 e3 is its vector part. The product of two biquaternions P and Q is the same as the real quaternions product,

PQ = (P0 + P )(Q0 + Q ) = P0 Q0 + P0 Q+Q0 P − P · Q + P × Q

(5)

P · Q = P1 Q1 + P2 Q2 + P3 Q3

(6)

P × Q = (P2 Q3 − Q3 P2 )e1 + (P3 Q1 − Q1 P3 )e2 + (P1 Q2 − Q2 P1 )e3

(7)

where

and

are the well known scalar and vector products, respectively. Explicitly, the biquaternion product is associative but not commutative. The conjugation of Q is obtained by changing the sign of the components of the imaginary basis elements,

¯ = Q0 − Q = Q0 e0 − Q1 e1 − Q2 e2 − Q3 e3 . Q

(8)

The conjugation of product of two biquaternions satisfies the following relation

¯ )=Q ¯ P¯ . (PQ

(9)

Similarly to complex numbers, the complex conjugate of Q is also defined as

Q∗ = Q0∗ e0 + Q1∗ e1 + Q2∗ e2 + Q3∗ e3 = (q0 − iq0 )e0 + (q1 − iq1 )e1 + (q2 − iq2 )e2 + (q3 − iq3 )e3 .

(10)

Generally, the norm of a biquaternion Q is a complex scalar and determined by

¯ =Q ¯ Q = Q02 + Q12 + Q22 + Q32 . NQ = QQ

(11)

If a biquaternion has unit norm, it is named as unit biquaternion. Finally, on condition that its norm is non-zero, the inverse of a biquaternion Q is given as

Q−1 =

¯ Q 1 = (Q0 e0 − Q1 e1 − Q2 e2 − Q3 e3 ). NQ NQ

(12)

Since the norm of a biquaternion may be zero, unlike the quaternion algebra the biquaternion algebra is not a division algebra. 2.1. Matrix representations of biquaternions In general, corresponding matrix representations of physical quantities can be used in order to simplify equation manipulations. Mathematically, an exact isomorphism exists between biquaternions and their corresponding matrix representations. Similarly to real quaternionic matrices [71], the column matrix representation of an arbitrary biquaternion Q = Q0 e0 + Q1 e1 + Q2 e2 + Q3 e3 is defined as Q=



Q0 ,

Q1 ,

Q2 ,

Q3

T

=



Q0 ,

QT

T

(13)

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where superscript T indicates transpose of a matrix. Likewise, Q0 also denotes a scalar part of biquaternion while Q symbolizes its vector part. The product of biquaternions P and Q is defined in terms of matrices [72] as



−P T P0 I3 + P˜

P0 P

PQ =

  Q0 Q

⎡

P0 ⎢P1 =⎣ P2 P3

or



−Q T Q0 I3 − Q˜

Q0 Q

PQ =

  P0 P

⎡

Q0 ⎢Q 1 =⎣ Q2 Q3

−P1 P0 P3 −P2

−P2 −P3 P0 P1

−Q1 Q0 −Q3 Q2

⎤⎡ ⎤

−P3 Q0 P2 ⎥⎢Q1 ⎥ −P1 ⎦⎣Q2 ⎦ P0 Q3

(14)

⎤⎡ ⎤

−Q2 Q3 Q0 −Q1

−Q3 P0 −Q2 ⎥⎢P1 ⎥ . Q1 ⎦⎣P2 ⎦ Q0 P3

(15)

Here, P˜ and Q˜ are special matrices expressed as



−P3 0 P1

0 P3 −P2

P˜ =

Q˜ =

P2 −P1 0

−Q3 0 Q1

0 Q3 −Q2

(16)

Q2 −Q1 0

(17)

and I3 also denotes 3 × 3 identity matrix. Although the quaternion product is associative and distributive respect to addition and subtraction, it is not commutative. On the other hand, by introduction the following matrices +



P=

⎡



P0 ⎢P1 −P T =⎣ P2 P0 I3 + P˜ P3

P0 P

and

 ˘ = Q

T

−Q Q0 I3 − Q˜

Q0 Q



⎡

Q0 ⎢Q1 =⎣ Q2 Q3

−P1 P0 P3 −P2 −Q1 Q0 −Q3 Q2

−P2 −P3 P0 P1

⎤

−P3 P2 ⎥ −P1 ⎦ P0

−Q2 Q3 Q0 −Q1

(18)

⎤

−Q3 −Q2 ⎥ Q1 ⎦ Q0

(19)

then the commutative property of biquaternion product can be obtained. Eqs. (14) and (15) prove that the product of biquaternions P and Q defined in Eq. (5) can commute simply with a sign change +

˘ P. PQ = Q

(20)

Considering the triple product of biquaternions PQR in matrix form, this operation can also be realized easily by means of the above matrix property [71] ++

+

˘Q PQR = PR

(21)

or ++

+

˘ PQ. PQR = R

(22)

It is explicitly seen that the special matrices of biquaternions emerge an effective tool to overcome their algebraic difficulties. 3. Biquaternionic form of fluid maxwell equations The dynamics of an ideal fluid can be constructed on Euler’s equation of motion, continuity equation, entropy equation and vorticity equation. Because the Maxwell type equations of compressible fluid has been previously derived by Kambe [43,44] in terms of traditional vector algebra, in this section, the biquaternionic reformulation of these equations has been given. In order to reformulate fluid Maxwell equations in terms of biquaternions, let’s firstly assume that fluid is ideal and isentropic. The four dimensional differential operator in terms of biquaternion variables can be defined as

=i

1 ∂ 1 ∂ +∇ =i e0 + a0 ∂ t a0 ∂ t

∂ ∂ ∂ e + e + e ∂x 1 ∂y 2 ∂z 3

(23)

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where its conjugate is

∗ = −i

1 ∂ 1 ∂ + ∇ = −i e0 + a0 ∂ t a0 ∂ t

∂ ∂ ∂ e + e + e . ∂x 1 ∂y 2 ∂z 3

(24)

Here a0 denotes the speed of sound in a uniform state of fluid at rest and ∇ describes the vector part of biquaternionic differential operator given by

∇=

∂ ∂ ∂ e + e + e . ∂x 1 ∂y 2 ∂z 3

(25)

The biquaternionic fluid potential can be introduced to generalize field variables h and v analogous to scalar and vector potentials in electromagnetism

 = −ih + a0 v = −ihe0 + a0 [v1 e1 + v2 e2 + v3 e3 ] or, in matrix form



Ψ = −ih,

a0 v1 ,

a0 v2 ,

a0 v3

T



= −ih,

(26)

a0 vT

T

.

(27)

Here h is enthalpy per unit mass, and v1 , v2 , v3 are the components of continuous velocity field v along the x−, y− and z−direction, respectively. Operation of the complex conjugate of differential operator ∗ on the fluid generalized potential  gives



 = ∗

 =



1 ∂ −i + ∇ [−ih + a0 v] a0 ∂ t





1 ∂h − − a0 ∇ ·v + a0 ∇ × v + i −∇ h − a0 ∂ t

 ∂v . ∂t

(28)

According to fluid Lorenz gauge conditions derived by Kambe [43],

∂h + a20 ∇ ·v = 0 ∂t

(29)

the scalar terms in the first parenthesis vanish. If two different vector fields for fluids are introduced as

E ≡ −∇ h −

∂v ∂t

(30)

and

H ≡∇ ×v

(31)

then the following compact equation can easily be written

∗  = F .

(32)

Here F is the generalized fields of fluids in biquaternion form

F = a0 H + iE = a0 [H1 e1 + H2 e2 + H3 e3 ] + i[E1 e1 + E2 e2 + E3 e3 ]

(33)

= (a0 H1 + iE1 )e1 + (a0 H2 + iE2 )e2 + (a0 H3 + iE3 )e3 and it combines the fluid electric E and fluid magnetic fields H in a single expression. Here E1 , H1 , E2 , H2 , E3 , H3 are the components of the E and H vector fields along the x−, y−and z−direction, respectively. In order to express the matrix form of Eq. (32), the multiplication relation given in Eq. (14) will be useful

⎡ + ∗

D

−i a10 ∂∂t

=⎣ ∇

−∇

⎤

T

−i a10 ∂∂t I3

˜⎦ +∇

−ih a0 v

⎡ F=

∂vy ∂vz ⎤ x − a10 ∂∂ht − a0 ( ∂v ∂x + ∂y + ∂z ) ⎢ a0 ( ∂vz − ∂vy ) − i( ∂ h + ∂vx )⎥ ∂y ∂z ∂x ∂t ⎥ ⎢

⎡

⎢ ⎢ =⎢ ⎢ ⎣

−i a10 ∂∂t

∂ ∂x ∂ ∂y ∂ ∂z

− ∂∂x

−i a10 ∂∂t

∂ ∂z ∂ − ∂y

− ∂∂y − ∂∂z

−i a10 ∂∂t

∂ ∂x

⎤ ⎡ −ih⎤ ∂ ⎥ ∂ y ⎥⎢a0 vx ⎥ ⎢ ⎥ − ∂∂x ⎥ ⎥⎣a0 vy ⎦ 1 ∂ ⎦ a0 vz −i − ∂∂z

a0 ∂ t

⎡ 0⎤

⎢F1 ⎥ ⎢ a ( ∂vx − ∂vz ) − i( ∂ h + ∂vy )⎥ = ⎢F2 ⎥. 0 ⎢ ⎣ ⎦ ∂z ∂x ∂y ∂t ⎥ F3 ⎣ a ( ∂vy − ∂vx ) − i( ∂ h + ∂vz )⎦ 0 ∂x ∂y ∂z ∂t

(34)

Elements of the final matrix F give the following components of the fields related to compressible fluids

1 ∂h − − a0 a0 ∂ t



∂vx ∂vy ∂vz + + ∂x ∂y ∂z



= 0,

(35)

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 a0

 a0

 a0

∂vz ∂vy − ∂y ∂z ∂vx ∂vz − ∂z ∂x ∂vy ∂vx − ∂x ∂y

 



∂ h ∂vx −i + ∂x ∂t 

∂ h ∂vy −i + ∂y ∂t



 −i

∂ h ∂vz + ∂z ∂t

 = a0 H1 + iE1 ,

(36)

= a0 H2 + iE2 ,

(37)

= a0 H3 + iE3 .

(38)

 

Explicitly, while the first element of F is the fluid Lorenz gauge conditions in Eq. (29), its other elements are the x−, y− and z− components of vector fields given in Eqs. (30) and (31), respectively. Similarly, operation of the biquaternionic differential operator on F exposes another relation for fluids





1 ∂ i + ∇ [a0 H + iE ] a0 ∂ t

F =



−a0 ∇ ·H + a0 ∇ × H −

=

(39)

1 ∂E a0 ∂ t







+ i −∇ ·E + ∇ × E +

∂H . ∂t

Here, let’s define the biquaternionic generalized fluid source density

J = −iq +

1 1 J = −iqe0 + [J1 e1 + J2 e2 + J3 e3 ] a0 a0

which is represented by the following matrix



J = −iq,

1 J , a0 1

1 J , a0 2



T 1 J a0 3



= −iq,

(40)



1 T T J . a0

(41)

Here ϱ is the fluid source density

q=−

∂ (∇ · v ) − ∇ 2h ∂t

(42)

and J represents the fluid current source density given as

J=

∂ 2v ∂h +∇ + a20 ∇ × (∇ × v ). ∂t ∂t2

(43)

It can easily be proved that these quantities satisfy the following relation

∂q + ∇ ·J = 0 ∂t

(44)

which is named as fluid continuity equation analogously to classical electromagnetic theory [43]. Using these definitions, then Eq. (39) can be expressed by the following compact form

F = J.

(45)

If the scalar and vector terms in the left and right sides of this equation are matched reciprocally,



F =

−a0 ∇ ·H + a0 ∇

= −iq +

1 ∂E ×H− a0 ∂ t





∂H + i −∇ ·E + ∇ × E + ∂t



1 J a0

(46)

then the following equations can be written,

∇ · E = q,

(imaginary scalar term)

(47a)

∇ · H = 0,

(scalar term )

(47b)

∂H + ∇ × E = 0, ∂t −

∂E + a20 [∇ × H ] = J . ∂t

(imaginary vector term) (vector term )

(47c)

(47d)

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Explicitly, these are the Maxwell type equations for ideal fluids in terms of biquaternion basis and coincide derived vectorial expressions by Kambe [43,44]. Furthermore, the compact formulation in Eq. (45) proves that by means of biquaternions, the Maxwell equations for ideal fluid can be expressed in a single equation analogously to compact equations for electromagnetic theory [6–18] and linear gravity [30]. Let us consider direct application of Eq. (45) to the matrix form of biquaternions

⎡

i a10 ∂∂t

+

−∇

DF = ⎣ ∇

i a10 ∂∂t I3

T

⎢ ⎢ =⎢ ⎢ ⎣

0 ˜⎦ F +∇

⎡ J=

⎡

⎤

i a10 ∂∂t

∂ ∂x ∂ ∂y ∂ ∂z

− ∂∂x

− ∂∂y − ∂∂z

i a10 ∂∂t

∂ ∂z ∂ − ∂y

i a10 ∂∂t

∂ ∂x

⎤

⎡

−a0 ( ∂∂Hx1 + ∂∂Hy2 + ∂∂Hz3 ) − i( ∂∂Ex1 + ∂∂Ey2 + ∂∂Ez3 ) ⎢− 1 ∂ E1 + i ∂ H1 + a0 ( ∂ H3 − ∂ H2 ) + i( ∂ E3 − ∂ E2 )⎥ ∂t ∂y ∂z ∂y ∂z ⎥ ⎢ a0 ∂ t

⎢− 1 ∂ E2 + i ∂ H2 + a0 ( ∂ H1 ⎢ a0 ∂ t ∂t ∂z ⎣− 1 ∂ E3 + i ∂ H3 + a0 ( ∂ H2 a0 ∂ t ∂t ∂x

⎤ ⎡ ⎤ 0 ∂ ⎥ ∂ y ⎥⎢a0 H1 + iE1 ⎥ ⎢ ⎥ − ∂∂x ⎥ ⎥⎣a0 H2 + iE2 ⎦ 1 ∂ ⎦ a0 H3 + iE3 i − ∂∂z

a0 ∂ t

⎤

−iq ⎢ a10 J1 ⎥ ⎢1J ⎥ 2 ⎥. − ∂∂Hx3 ) + i( ∂∂Ez1 − ∂∂Ex3 )⎥ ⎥=⎢ ⎣ a10 J ⎦ ∂ H1 ∂ E2 ∂ E1 ⎦ − ∂ y ) + i( ∂ x − ∂ y ) a0 3

(48)

As expected, the elements of the resulting matrix describe the eight components of the fluid-Maxwell equations in Eq. (47),



−a0

∂ H1 ∂ H2 ∂ H3 + + ∂x ∂y ∂z

−

1 ∂ E1 + a0 a0 ∂ t

−

1 ∂ E2 + a0 a0 ∂ t

1 ∂ E3 − + a0 a0 ∂ t

  



∂ H3 ∂ H2 − ∂y ∂z ∂ H1 ∂ H3 − ∂z ∂x ∂ H2 ∂ H1 − ∂x ∂y



∂ E1 ∂ E2 ∂ E3 −i + + ∂x ∂y ∂z 

 +i



 +i





= −iq,

∂ H1 ∂ E3 ∂ E2 + − ∂t ∂y ∂z ∂ H2 ∂ E1 ∂ E3 + − ∂t ∂z ∂x



∂ H3 ∂ E2 ∂ E1 +i + − ∂t ∂x ∂y



(49)

=

1 J1 , a0

(50)

=

1 J2 , a0

(51)

=

1 J3 . a0

(52)

 

The first relation is a complex combination of scalar equations related to compressible fluids, while the others are respectively x−, y− and z− components of vectorial expression in Eq. (47). On the other hand, re-operation of biquaternionic differential operator  on the Eq. (32) gives

∗  = F.

(53)

Using the compact fluid Maxwell equation in (45), the following expression can be written

 = J

(54)

where  is the d’Alembertian operator which is provided by



1 ∂  =  =   = i +∇ a0 ∂ t ∗

∗



1 ∂ −i +∇ a0 ∂ t





=



1 ∂2 − ∇2 . a20 ∂ t 2

(55)

Actually, Eq. (54) is the compact form of fluid wave equations in terms of potential like variables. It hides the relation





1 ∂2 1 − ∇ 2 [−iheo + a0 [v1 e1 + v2 e2 + v3 e3 ]] = −iqe0 + [J1 e1 + J2 e2 + J3 e3 ] a0 a20 ∂ t 2

and also contains the following components of the wave equation of fluids:

1 a20

∂ 2h − ∇ 2 h = q, ∂t2

(56a)

1 a20

∂ 2 v1 ∂ 2 v1 ∂ 2 v1 ∂ 2 v1 1 − − − = 2 J1 , ∂t2 ∂ x2 ∂ y2 ∂ z2 a0

(56b)

1 a20

∂ 2 v2 ∂ 2 v2 ∂ 2 v2 ∂ 2 v2 1 − − − = 2 J2 , ∂t2 ∂ x2 ∂ y2 ∂ z2 a0

(56c)

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1 a20

∂ 2 v3 ∂ 2 v3 ∂ 2 v3 ∂ 2 v3 1 − − − = 2 J3 . ∂t2 ∂ x2 ∂ y2 ∂ z2 a0

(56d)

However, the fluid wave equations can also be obtained in terms of fields. For this aim, the product of Eq. (45) by ∗ from the left gives

∗ F = ∗ J, and

 

(57)





1 ∂2 1 ∂ − ∇ 2 [a0 H + iE ] = −i +∇ a0 ∂ t a20 ∂ t 2 1 a0



−iq +

1 J a0



       1 ∂q ∂ 2H 1 ∂ 2E 1 1 ∂J 2 2 − a ∇ H + i − ∇ E = − + ∇ · J + [ ∇ × J ] − i q + . ∇ 0 a0 ∂ t a0 ∂t2 a20 ∂ t 2 a20 ∂ t

Equating real and imaginary scalar and vector terms, three different equations can be written. The first equations of them is the fluid continuity equation previously defined in Eq. (44), and goes to zero. The other equations

1 a20

∂ 2H 1 − ∇ 2 H = 2 [∇ × J ] ∂t2 a0

(58)

1 a20

1 ∂J ∂ 2E − ∇ 2 E = −∇ q − 2 ∂t2 a0 ∂ t

(59)

and

also are the wave equations in terms of fluid magnetic H and fluid electric E fields analogous to the fields in electromagnetic theory, respectively. 4. Energy relations for fluids The biquaternionic formalism presented in this paper allows us a convenient tool to obtain energy relations and derive the generalized Pointing theorem for fluids analogous to electromagnetism. For this aim, let’s multiply Eq. (45) from left by the complex conjugation of F

F∗ F = F∗ J where

F∗

(60)

is defined as

∗

F = a0 H − iE = a0 [H1 e1 + H2 e2 + H3 e3 ] − i[E1 e1 + E2 e2 + E3 e3 ]

(61)

= (a0 H1 − iE1 )e1 + (a0 H2 − iE2 )e2 + (a0 H3 − iE3 )e3 . The relation in Eq. (60)



[a 0 H − i E ] i





1 ∂ 1 + ∇ [a0 H + iE ] = [a0 H − iE ] −iq + J a0 ∂ t a0

implies to the following biquaternionic operations



[a 0 H − i E ] −



1 ∂E − a0 ∇ · H + a0 ∇ × H + i a0 ∂ t = [−H · J + H × J − qE ] + i

and

 −E ·



1

a0





∂H −∇ ·E+∇ ×E ∂t

E·J−

∂H ∂E +H· − E · (∇ × E ) − a20 H · (∇ × H ) ∂t ∂t



1 E × J − qa0 H a0





∂H ∂E −H× − E (∇ · E ) − a20 H (∇ · H ) + E × (∇ × E ) + a20 H × (∇ × H ) ∂t ∂t       1 ∂E ∂H +i − E· − a0 H · + a0 E · ( ∇ × H ) − a0 H · ( ∇ × E ) a0 ∂t ∂t      1 ∂E ∂H +i E× + a0 H × − a0 H ( ∇ · E ) + a0 E ( ∇ · H ) a0 ∂t ∂t



+ E×

(62)

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 − a0 E × ( ∇ × H ) + a0 H × ( ∇ × E ) = [−H · J + H × J − qE ] + i

1 a0

E·J−

1 E × J − qa0 H a0



If the similar components on the left and right sides of this equation are reciprocally equalized, the imaginary scalar part of Eq. (62) will be

−

1 a0



E·

∂E ∂t





− a0 H ·

∂H ∂t



+ a0 E · ( ∇ × H ) − a0 H · ( ∇ × E ) =

1 E · J. a0

(63)

According to the following identities

E·

∂E 1 ∂ 1 ∂ 2 = (E · E ) = E , ∂t 2 ∂t 2 ∂t

(64)

H·

∂H 1 ∂ 1 ∂ 2 = (H · H ) = H , ∂t 2 ∂t 2 ∂t

(65)

and

∇ · ( E × H ) = H · ( ∇ × E ) − E · ( ∇ × H ),

(66)

then the Eq. (63) can be re-expressed as

  ∂ E 2 + a20 H 2 + a20 ∇ · (E × H ) + E · J = 0. ∂t 2

(67)

Now, if the fluid energy density

E 2 + a20 H 2 2

U=

(68)

and the fluid Poynting vector are defined in terms of biquaternions,

S = a20 E × H

(69)

Eq. (67) becomes

∂U + ∇ · S + E · J = 0. ∂t

(70)

As in electrodynamics, this equation expresses the fluid Poynting theorem in terms of biquaternions and states the transfer of energy-momentum. Actually, these equations are similar to the energy relation derived by Scofield and Huq [45] on tensor notation. The analogous term between the electromagnetic field and the fluid case is (E · J). In electromagnetic theory, this term is the density of electrical power dissipated by Lorenz force acting on the electrical current and also defines the (resistive) energy losses. In fluid case, this energy loss arises from the frictional forces. The more detailed discussion about the fluid Poynting theorem can be found in the papers by Scofield and Huq [45,46]. On the other hand, the second equation is the real vectorial part of Eq. (62)

∂H ∂E −H× − E (∇ · E ) − a20 H (∇ · H ) + E × (∇ × E ) + a20 H × (∇ × H ) ∂t ∂t

E×

= −qE + H × J .

(71)

Using the following identities

∇ (E · E ) = 2[(E · ∇ )E + E × (∇ × E )]

(72)

∇ (H · H ) = 2[(H · ∇ )H + H × (∇ × H )]

(73)

and

Eq. (71) will be



∇

E 2 + a20 H 2 2

 +

∂ (E × H ) + qE − H × J = E (∇ · E ) + (E · ∇ )E + a20 H (∇ · H ) ∂t

+ a20 (H · ∇ )H

(74)

and it expresses the energy-momentum relation for fluids. Third equation can be obtained from real scalar part of Eq. (62)

−E ·

∂H ∂E +H· − E · (∇ × E ) − a20 H · (∇ × H ) = −H · J . ∂t ∂t

(75)

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and it is a trivial corollary. Finally, imaginary vectorial part of Eq. (62) will also be

1 (E × a20

∂E ∂H )+H× − H (∇ · E ) + E (∇ · H ) − E × (∇ × H ) + H × (∇ × E ) ∂t ∂t 1 E × J. a20

= −qH −

(76)

In order to verify the energy relation in the Eq. (60) by means of matrices, let us consider the compact equation + +

+

F∗ DF = F∗ J

(77)

which implies the following matrix product



0 F∗

−F ∗T F˜∗

⎡ 1 ∂ i a0 ∂ t ⎣ ∇

−F1∗ 0 F3∗ −F2∗

−F2∗ −F3∗ 0 F1∗

⎤ 0 0 ∗ ∂ I +∇ ⎦ ˜ F = F 3

−∇ i a10 ∂ t

or

⎡0 ∗ ⎢F1 ⎢F2∗ ⎣ ∗ F3

⎡0 ∗ ⎢F1 = ⎢F2∗ ⎣ ∗ F3

−F1∗ 0 F3∗ −F2∗

T

⎡

⎤ i a1 ∂∂t −F3∗ 0 ∂ ∗ ⎢ F2 ⎥⎢ ∂ x ∗ ⎥⎢ ∂ −F1 ⎢ ∂ y ⎦ ∂ 0 ⎣

− ∂∂x

i a10 ∂∂t

∂ ∂z ∂ − ∂y

∂z

⎡

−F ∗T F˜∗

− ∂∂y − ∂∂z



− ∂∂z

−iq 1 J a0

⎤

⎡0⎤

∂ ⎥ ∂ y ⎥⎢F1 ⎥ ⎢ ⎥ − ∂∂x ⎥ ⎥⎣F2 ⎦ 1 ∂ ⎦ F3 i a0 ∂ t

i a10 ∂∂t

∂ ∂x

⎤

⎤ −iq −F3∗ ∗ ⎢ 1 J1 ⎥ F2 ⎥ a0 ⎢ ⎥ −F1∗ ⎥⎢ a10 J2 ⎥. ⎦⎣ 1 ⎦ 0 J a0 3

−F2∗ −F3∗ 0 F1∗

(78)

Products of the last matrices on both sides of equation give the following relation

⎡0 ∗ ⎢F1 ⎢F2∗ ⎣ ∗

−F1∗ 0 F3∗ −F2∗

F3

−F2∗ −F3∗ 0 F1∗

⎡

⎤

⎡

⎤

⎤ −( ∂∂Fx1 + ∂∂Fy2 + ∂∂Fz3 ) − a10 (F1∗ J1 + F2∗ J2 + F3∗ J3 ) −F3∗ ∗ ⎢ i 1 ∂ F1 + ∂ F3 − ∂ F2 ⎥ ⎢−iqF1∗ + a10 (F2∗ J3 − F3∗ J2 )⎥ F2 ⎥⎢ a0 ∂ t ∂y ∂z ⎥ ⎢ ⎥ ∗ ⎥⎢ 1 ∂ F2 ∂ F ∂ F ⎥ 3 1 −F1 ⎢ i −iqF2∗ + a10 (F3∗ J1 − F1∗ J3 )⎥. ⎦ a0 ∂ t + ∂ z − ∂ x ⎥ = ⎢ ⎣ ⎦ 0 ⎣ i 1 ∂ F3 + ∂ F2 − ∂ F1 ⎦ −iqF3∗ + a10 (F1∗ J2 − F2∗ J1 ) a0 ∂ t ∂x ∂y

After performing rest of matrix product on the left side, the following equations can be written



1 ∂ F1 i + a0 ∂ t

−F1∗ =−

 −F1∗

∂ F3 ∂ F2 − ∂y ∂z





−

F2∗

1 ∂ F2 i + a0 ∂ t

∂ F1 ∂ F3 − ∂z ∂x





−

F3∗

1 ∂ F3 i + a0 ∂ t

1 ∗ (F J1 + F2∗ J2 + F3∗ J3 ) a0 1

∂ F1 ∂ F2 ∂ F3 + + ∂x ∂y ∂z



 −

F3∗



∂ F1 ∂ F2 ∂ F3 + + ∂x ∂y ∂z

= −iqF2∗ +

 −F3∗

1 ∂ F2 i + a0 ∂ t

∂ F1 ∂ F3 − ∂z ∂x



 +

F2∗

1 ∂ F3 i + a0 ∂ t

 +

1 ∂ F1 i + a0 ∂ t

F3∗

∂ F3 ∂ F2 − ∂y ∂z



 −

F1∗

∂ F2 ∂ F1 − ∂x ∂y



1 ∗ (F J3 − F3∗ J2 ) a0 2

1 ∂ F3 i + a0 ∂ t

∂ F2 ∂ F1 − ∂x ∂y



(81)

 − F2∗ i

1 ∗ (F J2 − F2∗ J1 ). a0 1

(80)



1 ∗ (F J1 − F1∗ J3 ) a0 3

∂ F1 ∂ F2 ∂ F3 + + ∂x ∂y ∂z

= −iqF3∗ +





(79)

= −iqF1∗ +

−F2∗

∂ F2 ∂ F1 − ∂x ∂y

1 ∂ F1 + a0 ∂ t

∂ F3 ∂ F2 − ∂y ∂z



 + F1∗ i

1 ∂ F2 + a0 ∂ t

∂ F1 ∂ F3 − ∂z ∂x



(82)

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11

If vectorial notation is used in order to simplify expressions, Eq. (79) will be

1 a0

−i



F∗ ·

∂F ∂t



− F ∗ · (∇ × F ) = −

1 ∗ F · J. a0

(83)

Similarly, Eqs. (80)–(82) respectively express x−, y− and z−components of the following equation,

1 −F (∇ · F ) + i a0 ∗



∂F F × ∂t



+ F ∗ × (∇ × F ) = −iqF ∗ +

∗

1 ∗ F × J. a0

(84)

Hence, Eqs. (83) and (84) can be combined as

1 i a0





∂F ∂F −F · + F∗ × ∂t ∂t



∗

− F ∗ (∇ · F ) +



(85)

−F ∗ · (∇ × F ) + F ∗ × (∇ × F ) = −iqF ∗ +

1 (−F ∗ · J + F ∗ × J ). a0

If the biquaternion product in Eq. (5) is taken into account, this equation can be re-written as



F and

∗

1 ∂F i −∇·F +∇×F a0 ∂ t

 F∗ i



= F ∗ (−iq +

1 J) a0



1 ∂ 1 + ∇ F = F ∗ (−iq + J ). a0 ∂ t a0

(86)

Explicitly, this form of above equation coincides biquaternionic energy relation previously given in Eq. (60). 5. Conclusions Although Maxwell’s equations of electromagnetism have been expressed in many mathematical forms, the same is not true for their analogous equations in fluid mechanics. In this paper, a suitable and elegant reformulation of fluid Maxwell equations has been presented. The biquaternionic generalization of fluid equations was absent in relevant literature. Therefore this paper fills a gap which was not formulated in similar studies before. Unfortunately the eight dimensional hypercomplex numbers do not have a consistent vector interpretation, and this situation leads to some difficulties for the description of vector fields. On the other hand, biquaternions with four complex components can easily be associated vectors and also eight dimensional physical quantities. Furthermore, since biquaternon algebra unifies the dot and cross products of vectors into a single product, Eq. (32) confirms that all Maxwell type field equations of fluids can be expressed as one biquaternionic equation. Similarly, the wave equations of fluids have been expressed by Eq. (54) in compact and elegant manner. Generally, these types of unification of the field equations of fluids could bring to significative advantages in case of extending the results from one discipline to other one. Although electromagnetism and fluid mechanics appear very different from each other, the proposed formalism reveals analogous results to the perviously derived biquaternionic formulations of Maxwell equations in electromagnetism [6–18]. On the other hand, stimulating from the analogy between linear gravity and electromagnetism formulated before [30], a formalism related to fluid-gravity equivalence may be proposed. The biquaternionic formalism presented in this paper allows us a convenient tool to obtain energy relations and derive the generalized Poynting theorem for fluids analogous to electromagnetism. On the other hand, the fluid Poynting equation in (70) is similar to the energy formula derived by Scofield and Huq [45] on tensor notation, and also octonic consideration [55]. On the other hand, one of the aims of this work is proposing an alternative mathematical tool to present the Maxwell type equations of compressible fluids. In contrast to the previous studies in relevant literature, it has been proven in this paper that corresponding matrix representations of biquaternionic fluid equations can be used in order to simplify equation manipulations. Mathematically, it is shown that an exact isomorphism exists between these biquaternionic fluid equations and their corresponding matrix representations. This isomorphism may lead to the emergence of a powerful tool for investigating various problems in fluid mechanics. References [1] [2] [3] [4] [5] [6] [7] [8]

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