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In this paper, the solutions of three dimensional incompressible magnetohydrodynamics (MHD) equations are obtained by usin g sin method and Riccati auxiliary equation. This paper obtains the soliton solutions by the aid of software Mathematica.

The word MHD is made up of three terms magneto indicating magnetic field, hydro referring liquid, and dynamics meaning movement. The field of MHD is a fascinatingly rich field of physics and applied mathematics that considers the behavior of an electrically conducting fluid in the presence of an external electromagnetic field. Although inspiring in its own right, MHD also has numerous engineering and science applications. These range from the pursuit of reliable energy sources such as nuclear fusion [

The MHD description governs the large-scale dynamics of plasmas, and applies to many laboratory as well as astrophysical configurations. Incompressible MHD has traditionally focused on topics like MHD turbulence, dynamo aspects, and singular structure formation. We inspect what kind of waves that can exist through linearization of the MHD equations in parallel with applying Fourier transforms. As a reminder, the incompressible MHD equations are

∇ ⋅ V _ = ∇ ⋅ B _ = 0 , (1)

V _ t + ( V _ ⋅ ∇ ) V _ − ( B _ ⋅ ∇ ) B _ + ∇ ( P _ + 12 | B _ | 2 ) (2)

− ν 1 V _ x x − ν 2 V _ y y − ν 3 V _ z z = 0 ,

B _ t + ( V _ ⋅ ∇ ) B _ − ( B _ ⋅ ∇ ) V _ (3)

− η 1 B _ x x − η 2 B _ y y − η 3 B _ z z = 0,

where

V _ = ( V 1 ( x , y , z , t ) , V 2 ( x , y , z , t ) , V 3 ( x , y , z , t ) ) T ,

B _ = ( B 1 ( x , y , z , t ) , B 2 ( x , y , z , t ) , B 3 ( x , y , z , t ) ) T and P _ = P _ ( x , y , z , t )

represent the unknown velocity field, the magnetic field and the pressure of the flow, respectively, and ν 1 , ν 2 , ν 3 , η 1 , η 2 and η 3 are the viscosity coefficients of the flow. The field of incompressible MHD is a particularly rich subset of physics and applied mathematics. The challenges inherent in the equations provide a plethora of research opportunities. Aside from purely academic pursuits, MHD also plays an important role in the development of engineering technologies. Designing suitable engineering systems using electrically conducting fluids requires using computational techniques.

One of the most prominent reasons for this difficulty is the phenomenon of fluid turbulence which again rears its head in MHD [

This paper is organized as follows. In Section 1 we review the main governing equations of incompressible MHD. In Section 2, the s i n ( k ξ ) − c o s ( k ξ ) method and the exact solutions for the incompressible MHD problem are presented. Finally, Section 3 contains the conclusion.

The three dimensional incompressible MHD system (1)-(3) in the scalar form

V 1 , x + V 2 , y + V 3 , z = B 1 , x + B 2 , y + B 3 , z = 0 , (4)

V 1 , t + V 1 V 1 , x + V 2 V 1 , y + V 3 V 1 , z + P x + B 2 ( B 2 , x − B 1 , y ) + B 3 ( B 3 , x − B 1 , z ) − ν 1 V 1 , x x − ν 2 V 1 , y y − ν 3 V 1 , z z = 0 , (5)

V 2 , t + V 1 V 2 , x + V 2 V 2 , y + V 3 V 2 , z + P y + B 1 ( B 1 , y − B 2 , x ) + B 3 ( B 3 , y − B 2 , z ) − ν 1 V 2 , x x − ν 2 V 2 , y y − ν 3 V 2 , z z = 0 , (6)

V 3 , t + V 1 V 3 , x + V 2 V 3 , y + V 3 V 3 , z + P z + B 1 ( B 1 , z − B 3 , x ) + B 2 ( B 2 , z − B 3 , y ) − ν 1 V 3 , x x − ν 2 V 3 , y y − ν 3 V 3 , z z = 0 , (7)

B 1 , t + V 1 B 1 , x + V 2 B 1 , y + V 3 B 1 , z − B 1 V 1 , x − B 2 V 1 , y − B 3 V 1 , z − η 1 B 1 , x x − η 2 B 1 , y y − η 3 B 1 , z z = 0 , (8)

B 2 , t + V 1 B 2 , x + V 2 B 2 , y + V 3 B 2 , z − B 1 V 2 , x − B 2 V 2 , y − B 3 V 2 , z − η 1 B 2 , x x − η 2 B 2 , y y − η 3 B 2 , z z = 0 , (9)

B 3 , t + V 1 B 3 , x + V 2 B 3 , y + V 3 B 3 , z − B 1 V 3 , x − B 2 V 3 , y − B 3 V 3 , z − η 1 B 3 , x x − η 2 B 3 , y y − η 3 B 3 , z z = 0. (10)

To find the travelling wave solution for Equations (4)-(10), we take the transformation

V i ( x , y , z , t ) = v i ( ξ ) , B i ( x , y , z , t ) = b i ( ξ ) , P ( x , y , z , t ) = p ( ξ ) , i = 1 , 2 , 3 (11)

where ξ = x + y + α z + β t , and change the Equations (4)-(10) into the following ordinary differential equations

v ′ 1 + v ′ 2 + α v ′ 3 = b ′ 1 + b ′ 2 + α b ′ 3 = 0 , (12)

( β + v 1 + v 2 + α v 3 ) v ′ 1 + p ′ + b 2 ( b ′ 2 − b ′ 1 ) + b 3 ( b ′ 3 − α b ′ 1 ) − ( ν 1 + ν 2 + α 2 ν 3 ) v ″ 1 = 0, (13)

( β + v 1 + v 2 + α v 3 ) v ′ 2 + p ' + b 1 ( b ′ 1 − b ′ 2 ) + b 3 ( b ′ 3 − α b ′ 2 ) − ( ν 1 + ν 2 + α 2 ν 3 ) v ″ 2 = 0, (14)

( β + v 1 + v 2 + α v 3 ) v ′ 3 + α p ′ + b 1 ( α b ′ 1 − b ′ 3 ) + b 2 ( α b ′ 2 − b ′ 3 ) − ( ν 1 + ν 2 + α 2 ν 3 ) v ″ 3 = 0, (15)

( β + v 1 + v 2 + α v 3 ) b ′ 1 − ( b 1 + b 2 + α b 3 ) v ′ 1 − ( η 1 + η 2 + α 2 η 3 ) b ″ 1 = 0 , (16)

( β + v 1 + v 2 + α v 3 ) b ′ 2 − ( b 1 + b 2 + α b 3 ) v ′ 2 − ( η 1 + η 2 + α 2 η 3 ) b ″ 2 = 0 , (17)

( β + v 1 + v 2 + α v 3 ) b ′ 3 − ( b 1 + b 2 + α b 3 ) v ′ 3 − ( η 1 + η 2 + α 2 η 3 ) b ″ 3 = 0 , (18)

where ' = d d ξ balancing the highest order of linear terms with nonlinear

terms in the system (12)-(18) suggests the following ansatz

v 1 = γ 1 + γ 2 ψ , v 2 = γ 3 + γ 4 ψ , v 3 = γ 5 + γ 6 ψ , (19)

b 1 = δ 1 + δ 2 ψ , b 2 = δ 3 + δ 4 ψ , b 3 = δ 5 + δ 6 ψ , (20)

p = ρ 1 + ρ 2 ψ + ρ 3 ψ 2 , (21)

where γ i , δ i , ρ i , i = 1 , 2 , 3 , α and β are constants to be determined, and the function ψ satisfying a Riccati equation

ψ ′ 2 + ε k 2 ψ 2 = ϵ k 4 , k ≥ 0 , ϵ = ± 1 (22)

then we obtain three kinds of general solutions [

ψ = k sin k ξ or ψ = k cos k ξ , when ϵ = 1 , (23)

ψ = constant , when k = 0 , (24)

ψ = k c o s h k ξ , when ϵ = − 1. (25)

Substituting (19)-(21) into system (12)-(18) and using the Riccati equation (22), we obtain [

( γ 2 + γ 4 + α γ 6 ) ψ ′ = ( δ 2 + δ 4 + α δ 6 ) ψ ′ = 0, (26)

( γ 2 β ) ψ ′ + ( γ 1 + γ 2 ψ ) ( γ 2 ψ ′ ) + ( γ 3 + γ 4 ψ ) ( γ 2 ψ ′ ) + α ( γ 5 + γ 6 ψ ) ( γ 2 ψ ′ ) + ( ρ 2 ψ ′ ) + ( 2 ρ 3 ψ ψ ′ ) + ( δ 3 + δ 4 ψ ) ( δ 4 − δ 2 ) ψ ′ + ( δ 5 + δ 6 ψ ) ( δ 6 − α δ 2 ) ψ ′ − ( ν 1 + ν 2 + α 2 ν 3 ) ( γ 2 ψ ″ ) = 0 , (27)

( β + γ 1 + γ 2 ψ + γ 3 + γ 4 ψ + α γ 5 + α γ 6 ψ ) ( γ 4 ψ ′ ) + ( ρ 2 ψ ′ ) + ( 2 ρ 3 ψ ψ ′ ) + ( δ 1 + δ 2 ψ ) ( δ 2 − δ 4 ) ψ ′ + ( δ 5 + δ 6 ψ ) ( δ 6 − α δ 4 ) ψ ′ − ( ν 1 + ν 2 + α 2 ν 3 ) ( γ 4 ψ ″ ) = 0 , (28)

( β + γ 1 + γ 2 ψ + γ 3 + γ 4 ψ + α γ 5 + α γ 6 ψ ) ( γ 6 ψ ′ ) + ( α ρ 2 ψ ′ ) + ( 2 α ρ 3 ψ ψ ′ ) + ( δ 1 + δ 2 ψ ) ( α δ 2 − δ 6 ) ψ ′ + ( δ 3 + δ 4 ψ ) ( α δ 4 − δ 6 ) ψ ′ − ( ν 1 + ν 2 + α 2 ν 3 ) ( γ 6 ψ ″ ) = 0 , (29)

( β + γ 1 + γ 2 ψ + γ 3 + γ 4 ψ + α γ 5 + α γ 6 ψ ) ( δ 2 ψ ′ ) − ( δ 1 + δ 2 ψ + δ 3 + δ 4 ψ + α δ 5 + α δ 6 ψ ) ( γ 2 ψ ′ ) − ( η 1 + η 2 + α 2 η 3 ) ( δ 2 ψ ″ ) = 0 , (30)

( β + γ 1 + γ 2 ψ + γ 3 + γ 4 ψ + α γ 5 + α γ 6 ψ ) ( δ 4 ψ ′ ) − ( δ 1 + δ 2 ψ + δ 3 + δ 4 ψ + α δ 5 + α δ 6 ψ ) ( γ 4 ψ ′ ) − ( η 1 + η 2 + α 2 η 3 ) ( δ 4 ψ ″ ) = 0, (31)

( β + γ 1 + γ 2 ψ + γ 3 + γ 4 ψ + α γ 5 + α γ 6 ψ ) ( δ 6 ψ ′ ) − ( δ 1 + δ 2 ψ + δ 3 + δ 4 ψ + α δ 5 + α δ 6 ψ ) ( γ 6 ψ ′ ) − ( η 1 + η 2 + α 2 η 3 ) ( δ 6 ψ ″ ) = 0. (32)

Then setting the coefficients of all powers of ψ , ψ ′ and ψ ψ ′ to zero, we will get a set of algebraic system with respect to variables γ i , δ i , ρ i , i = 1 , 2 , 3 , α and β

( γ 2 + γ 4 + α γ 6 ) = 0 , ( δ 2 + δ 4 + α δ 6 ) = 0 , (33)

( ν 1 + ν 2 + α 2 ν 3 ) ( γ 2 k 2 ) = 0 , (34)

γ 2 β + γ 1 γ 2 + γ 3 γ 2 + α γ 5 γ 2 + ρ 2 + δ 3 δ 4 − δ 3 δ 2 + δ 5 δ 6 − α δ 5 δ 2 = 0 , (35)

γ 2 2 + γ 2 γ 4 + α γ 6 γ 2 + 2 ρ 3 + δ 4 2 − δ 4 δ 2 + δ 6 2 − α δ 6 δ 2 = 0 , (36)

( ν 1 + ν 2 + α 2 ν 3 ) ( γ 4 k 2 ) = 0 , (37)

γ 4 β + γ 1 γ 4 + γ 3 γ 4 + α γ 5 γ 4 + ρ 2 + δ 1 δ 2 − δ 1 δ 4 + δ 5 δ 6 − α δ 5 δ 4 = 0 , (38)

γ 4 2 + γ 2 γ 4 + α γ 4 γ 6 + 2 ρ 3 + δ 2 2 − δ 4 δ 2 + δ 6 2 − α δ 6 δ 4 = 0 , (39)

( ν 1 + ν 2 + α 2 ν 3 ) ( γ 6 k 2 ) = 0 , (40)

γ 6 β + γ 1 γ 6 + γ 3 γ 6 + α γ 5 γ 6 + α ρ 2 + α δ 1 δ 2 − δ 1 δ 6 + α δ 3 δ 4 − δ 3 δ 6 = 0 , (41)

α γ 6 2 + γ 2 γ 6 + γ 4 γ 6 + 2 α ρ 3 + α δ 2 2 − δ 6 δ 2 + α δ 4 2 − δ 6 δ 4 = 0 , (42)

( η 1 + η 2 + α 2 η 3 ) ( δ 2 k 2 ) = 0 , (43)

δ 2 β + γ 1 δ 2 + γ 3 δ 2 + α γ 5 δ 2 − γ 2 δ 1 − γ 2 δ 3 − α δ 5 γ 2 = 0 , (44)

γ 2 δ 2 + γ 4 δ 2 + α γ 6 δ 2 − γ 2 δ 2 − δ 4 γ 2 − α δ 6 γ 2 = 0 , (45)

( η 1 + η 2 + α 2 η 3 ) ( δ 4 k 2 ) = 0 , (46)

δ 4 β + γ 1 δ 4 + γ 3 δ 4 + α γ 5 δ 4 − γ 4 δ 1 − γ 4 δ 3 − α δ 5 γ 4 = 0 , (47)

γ 2 δ 4 + γ 4 δ 4 + α γ 6 δ 4 − γ 4 δ 2 − δ 4 γ 4 − α δ 6 γ 4 = 0 , (48)

( η 1 + η 2 + α 2 η 3 ) ( δ 6 k 2 ) = 0 , (49)

δ 6 β + γ 1 δ 6 + γ 3 δ 6 + α γ 5 δ 6 − γ 6 δ 1 − γ 6 δ 3 − α δ 5 γ 6 = 0 , (50)

γ 2 δ 6 + γ 4 δ 6 + α γ 6 δ 6 − γ 6 δ 2 − δ 4 γ 6 − α δ 6 γ 6 = 0. (51)

From the output of symbolic computation software Mathematica, we obtain a solution, namely,

ν 1 = ν 2 = − 2 ν 3 , η 1 = η 2 = − 2 η 3 , α = 2 , γ 2 = γ 4 = δ 2 = δ 4 = − γ 6 = − δ 6 = d 0 , ρ 3 = − 3 2 d 0 2 , ρ 2 = d 0 ( 3 − c 0 ) , β = c 0 − a 0 , (52)

where

a 0 = γ 1 + γ 3 + 2 γ 5 , c 0 = δ 1 + δ 3 + 2 δ 5 ,

where a 0 , c 0 , ρ 2 , ρ 3 and k are arbitrary constants. Since k is a arbitrary parameter, according to (19)-(21), (23)-(25) and (52), we obtain three kinds of travelling wave solutions for the new coupled MHD system (1)-(3), namely

1) a periodic solution with ϵ = 1

V 1 = γ 1 + k d 0 sin k ξ , V 2 = γ 3 + k d 0 sin k ξ , V 3 = γ 5 − k d 0 sin k ξ , (53)

B 1 = δ 1 + k d 0 sin k ξ , B 2 = δ 3 + k d 0 sin k ξ , B 3 = δ 5 − k d 0 sin k ξ , (54)

P = ρ 1 + k d 0 ( 3 − c 0 ) sin k ξ − 3 2 d 0 2 k 2 sin 2 k ξ , (55)

2) a soliton solution with ϵ = − 1

V 1 = γ 1 + k d 0 cosh k ξ , V 2 = γ 3 + k d 0 cosh k ξ , V 3 = γ 5 − k d 0 cosh k ξ , (56)

B 1 = δ 1 + k d 0 cosh k ξ , B 2 = δ 3 + k d 0 cosh k ξ , B 3 = δ 5 − k d 0 cosh k ξ , (57)

P = ρ 1 + k d 0 ( 3 − c 0 ) cosh k ξ − 3 2 d 0 2 k 2 cosh 2 k ξ , (58)

3) a constant solution with k = 0

V 1 = γ 1 , V 2 = γ 3 , V 3 = γ 5 , (59)

B 1 = δ 1 , B 2 = δ 3 , B 3 = δ 5 , (60)

P = ρ 1 , (61)

where ξ = x + y + 2 z + ( c 0 − a 0 ) t . The MHD equations govern the dynamics of the velocity and the magnetic field in electrically-conducting fluids and reflect the basic physics laws of conservation. These equations can be implemented to study various problems in plasma, liquid metals, saltwater as well as astrophysics. The MHD equations involve coupling between the incompressible Navier-Stokes equations (when the magnetic field B _ is identically equal to 0) governing the fluid and incompressible Euler equations for B _ = 0 , ν 1 , ν 2 , ν 3 = 0 . This paper examines the soliton solutions for the three-dimensional incompressible MHD equations with only magnetic diffusion (without velocity dissipation). MHD deals with the dynamics of an electrically conducting fluid under the influence of magnetic field. The magnetic field, which is present everywhere in the universe, generates magnetic force and this force influences the dynamics of moving fluid, potentially changing the geometry or strength of magnetic field itself. It has been found that the difference in the phase may occur between speed and fluctuations of the magnetic field when the kinetic and magnetic Reynolds numbers are very large. Since the speed and fluctuations of the magnetic field in a circular polarized, the phase difference makes them no longer parallel or anti-parallel like that in the incompressible MHD.

This paper presents stabilized exact soliton solutions for the incompressible MHD equations. These stabilized soliton solutions are focused at incompressible fluids and the main technological applications in mind are those related with material processing techniques. The flow considered here is incompressible and parallel to the magnetic filed. Several classes of soliton solutions are obtained in three-dimensional Cartesian coordinates. Previously, Neukirch [

Aldhabani, M. and Sayed, S.M. (2018) Travelling Wave Solutions for Three Dimensional Incompressible MHD Equations. Journal of Applied Mathematics and Physics, 6, 114-121. https://doi.org/10.4236/jamp.2018.61011