A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence
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Copyright © 2021 by American Scientific Publishers Journal of Nanofluids
All rights reserved. Vol. 10, pp. 98–105, 2021
Printed in the United States of America (www.aspbs.com/jon)
A New Derivation of Exact Solutions for
Incompressible Magnetohydrodynamic
Plasma Turbulence
Adel M. Morad1, 2, ∗ , S. M. A. Maize3 , A. A. Nowaya3 , and Y. S. Rammah3
1
Department of Mathematics and Computer Science, Menoufia University, Shebin-Elkoom, 32511 Egypt
2
Department of Computational Mathematics and Mathematical Physics, Institute of Mathematics,
Mechanics and Computer Science, Southern Federal University, Rostov on Don, 344090 Russia
3
Physics Department, Faculty of Science, Menoufia University, Shebin Elkom, 32511 Egypt
The objective of this paper is to study the propagation of nonlinear, quasi-parallel, magnetohydrodynamic waves
of small-amplitude in a cold Hall plasma of constant density. Magnetohydrodynamic equations, along with the
cold plasma were expanded using the reductive perturbation method, which leads to a nonlinear partial differ-
ential equation that complies with a modified form of the derivative nonlinear evolution Schrödinger equation.
Exact solutions of the turbulent magnetohydrodynamic model in plasma were formulated for the physical quan-
tities that describe the problem completely by using the complex ansatz method. In addition, the complete
ARTICLE
set of equations was used taking into account the magnetic field, which can be considered to be constant in
the x-direction to create stable vortex waves. Vortex solutions of the modified nonlinear Schrödinger equation
(MNLS) were compared with the solutions of incompressible magnetohydrodynamic equations. The obtained
IP: 192.168.39.151
equations differ from the standard NLS equationOn:by Wed, 13 Oct 2021
one additional term10:20:36
that describes the interaction of the
nonlinear waves with the constant Copyright: American
density. The behaviorScientific Publishers
of both the velocity profile and the waveform of pres-
sure were studied. The results showed that thereDelivered by Ingenta
are clear disturbances in the identity of the velocity and thus
pressure. The identity of both velocity and pressure results from that a magnetic field is formed.
KEYWORDS: Plasma Turbulence, Incompressible Magnetohydrodynamics, Reductive Perturbation Method, Modified
Nonlinear Schrödinger Equation, The Complex Ansatz Method.
1. INTRODUCTION Re is considered the primary responsibility for determining
Fluid and plasma flows at large Reynolds numbers display that the fluid is in a laminar or turbulent flow. Re , defined
dynamic random behavior; this phenomenon is called tur- as vL/ since it is assumed that it takes values less than
bulence. Turbulence is a common phenomenon in which or equal to 2000 to become a laminar flow, taking values
seemingly spontaneous, unsteady, erratic, and disorderly greater than or equal to 2000 in the turbulent flow.3
movements describe a fluid flow.1 Many of the astrophys- An ionized gas, called plasma, is the most common
ical models in the universe are turbulent. The Solar wind, electrically conducting medium. Plasmas comprise 99%
the convective region of stars, interstellar plasma, accre- of the visible material in the universe. If the dynamical
tion disk, etc. are some examples. But in the atmosphere time and length scales of interest are large enough com-
and in the water of rivers and seas, turbulent flows can pared to those of the microscopic components of plasma,
be found. In the flow, structures formed are called eddies electrically-conducting media may be represented as flu-
where the eddies motion is unforeseeable. A flow that is ids. The single-fluid approximation is used in the frame-
disordered in time and space is a turbulent flow. It may be work of this work for studying turbulent systems.2
three-dimensional or two-dimensional at times, and may or Hydrodynamic (HD) turbulence in Navier-Stokes (NS)
may not have well-organized structures.2 Reynolds number model involving the velocity field, magnetohydrodynamic
(MHD) turbulence, and pressure, which also involves the
∗
magnetic field, will be considered in the following funda-
Author to whom correspondence should be addressed.
mental mode of turbulence. A set of differential equations
Emails: dr_adel_morad@yahoo.com,
adel.mourad@science.menofia.edu.eg must be determined in the turbulence study to understand
Received: 12 April 2021 the flow. The probability of adapting fluid models for
Accepted: 3 June 2021 compressible and incompressible cases of viscous fluids
98 J. Nanofluids 2021, Vol. 10, No. 1 2169-432X/2021/10/098/008 doi:10.1166/jon.2021.1765
Morad et al. A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence
in collisional plasma has been the focus of many exper- equations. Nonlinear dynamical processes assist us in com-
iments over the last few decades. In particular, when prehending the nature and the structure formation of the
modeling plasma turbulence, the NS equations coupled turbulent. By using the complex ansatz method, the solv-
with Maxwell equations are important. However, MHD is ability condition can be obtained as the model equation,
known to be the appropriate technique in the collisionless the system can be solved to get the four cases.
case.4–6 The goal of this research is to study the properties
The aerodynamic and thermal dynamics of a New- of nonlinear turbulent dynamics in incompressible one-
tonian incompressible fluid which has a constant prop- dimensional hydrodynamic and MHD turbulence by using
erty turbulent flowing through a high-performance two- an approximation method that gives the ability to save the
dimensional channel of a horizontal heat transfer with nonlinear behaviors of equations and provide the solutions
a rectangular cross-section were studied using computa- of small amplitude to describe long-range effects in the
7 8
tional fluid dynamics (CFD). A numerical investigation system. Through that, a framework has been evaluated,
of a fluid’s dynamic and thermal dynamics with a con- and results represented in the figures, which have shown
stant property turbulent flowing through a two-dimensional very good agreement of the governing parameters in the
horizontal rectangular channel was carried out by Menni problem within the researched field.
et al. 9 10
In this study, the upper surface was kept at a This work introduces an accurate analytical solution
constant temperature, while the lower surface was kept for analyzing nonlinear turbulent systems by driving the
warm. 11 12
The internal structure of the channel’s thermal MNLS equation for the model. The novel model solu-
and dynamic analyses has been meticulously examined. In tions are represented by assuming that the medium can
be treated as an MHD multi-fluid system. In the case of
Refs. [13–18], the authors encompassed diverse domains
the MHD equations, to generate a single evolution equa-
of speed and heat, as well as distinct profiles of friction
tion, the special assumptions and holding low order terms
and heat exchange coefficients in a channel with chang-
must be chosen. The novelty of this study, that for the
ARTICLE
ing walls. Abo-Dahab and Abdelhafez19 investigated the
first time we explored the effect of the intrinsic magnetic
effects of magnetic field and thermal radiation on molyb-
field on velocity and pressure fluctuations lately. And we
denum disulfide nanofluid combined. Mebarek and others
extended that by applying RPM to achieve the examina-
studied an inclined ring filled with molten potassium, the
IP: 192.168.39.151 On: Wed, tion13ofOctsystem
2021parameters
10:20:36 associated with nonlinear terms
stability of natural convection, under the influence of
Copyright: radial Scientific
American in the Publishers
differential equations of NS turbulence and MHD
magnetism.19–21 The natural convection of MHD flow of a byturbulence.
Delivered Ingenta This technique determining the velocity fluc-
Newtonian nanofluid in a baffled enclosure of U-shaped is tuation and studying the fluid pressure behavior.
investigated using a homogeneous single-phase nanofluid
model.22 23
Transport phenomena in viscous and incompressible 2. INCOMPRESSIBLE NAVIER-STOKES
fluid regimes adjacent to a spinning vertical cone of ther- EQUATION
mal radiation and transpiration effects are studied using For the simplest case, the equations reflect the bulk fluid
a mathematical model.24–26 Chamkha et al. studied the velocity evolution v and the magnetic field B in an
incompressible steady flow properties such as viscosity, incompressible flow with constant mass density may be
thermal radiation, conductivity, and viscous dissipation in expressed as:2
order to treat a wide range of scientific and technologi- v 1
cal problems.27–31 They also investigated the mass transfer + v · v = − p + B · B + 2 v (1)
t 4
of an electrically conducting fluid and simultaneous heat B
by combining stagnation convection flow on a flat plate + v · B = B · v + 2 B (2)
t
embedded in the presence of magnetic field effects, fluid
these equations are restricted because of the incompress-
wall blowing, and heat generation or absorption effects
ibility condition and the magnetic field has solenoidal
which temperature-dependent, and in the presence of a
character,
magnetic field, the flow of unsteady mixed convection over
·v = ·B = 0 (3)
a vertical cone moving in a fluid layer with an angu-
lar velocity time-dependent was investigated. Dogonchi Considering the limits in Eq. (3). for velocity and mag-
et al.32 33 were the first to use a hybrid approach to inves- netic field, p can measure easily. Therefore, beyond the
tigate the influence of a magnetic field on natural con- incompressible limit, p is a passive quantity. Working with
vection in a cavity with an inclined elliptical heater with a non-dimensional form of the MHD Eqs. set (1), (2) and
a nanoparticle form factor Significant factors’ impact on (3) is beneficial. If the mean magnetic field assumed t be
flow and heat transfer characteristics are studied. 34 35 zero, all variables depend on two spatial parameter x and
The “Reductive Perturbation Method” (RPM) 36 37
that is time t, and Eq. (1)–(3) take the following form:
a beneficial approach, that preserving the effects of non- v 1 2
+ v· v = − p+ B· B + 2 v (4)
linear, dispersive, and dissipative in relevant differential t x x 4 x x
J. Nanofluids, 10, 98–105, 2021 99
A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence Morad et al.
B 2
2
x t, respectively. It is necessary to apply the following
+ v· B = B· v+ B (5) partial relations,37
t x x x
t = t − vgr + 2
(12)
·v = ·B = 0 (6)
x x x = x + (13)
In this work, we assumed that the magnetic field acting in xx = xx + 2 x + 2
(14)
the x direction takes a constant value Bx = constant, so the
derivative of the magnetic field in this direction is always
3.1. First Order Approximation
equal to zero, B/x = 0.
Using the RPM method by substituting Eq. (9)–(10) and
(11) into Eqs. (4)–(5) and (6), using the properties (12)–
3. THE REDUCTIVE (13) and (14), a set of equations for all values of n and l
PERTURBATION METHOD can be obtained. The different powers terms are collected:
The reductive perturbation theory is presented as a solu- For a harmonic at l = 1, the following results can be:
tion method used to reduce a system of general nonlinear ik2 + v11
hyperbolic to a solvable nonlinear single equation which p11 = B11 = 0 (15)
k
defines a wide system.36 37 The approach of RPM used
with more general systems, like dispersion or dissipation, The importance of a harmonic at l = 1, comes in having
demonstrates that they can be reduced to the Korteweg- a basic relation, which is the dispersion relation, which
de Vries equation (KdV) or the Burgers equation for long links the wave number with the wave frequency,
waves. It is also seen that the Schrödinger equation gov- = −ik2 (16)
erns a wide field for the slow modulation propagation of an
ARTICLE
infinitesimal amplitude plane wave and that a general wave where and are unity. Eq. (16) shows that can be
system reduced to a nonlinear Schrödinger form equation positive and this happens with the growth of time to cause
for small and finite amplitudes, which are known as the instability of the system while the can take negative
generalized nonlinear Schrödinger equation. values so that the system becomes stable with the decay
IP: 192.168.39.151 On: Wed, of 13 Oct 2021 10:20:36
Now, the F is a state vector that is implied
Copyright:
time.
to be Scientific
American Publishers
ui Bi pi in space x and time t. The small deviations The dispersion relation for the waveguide is a plot relat-
Delivered by Ingenta
from the condition of equilibrium that are known as F 0 = ing the characteristic wave vector of each k to the fre-
T
0 1 1 . The state vector F expanded in a power series quency of that wave. The phase velocity vph = /k,
of small parameter and has an expansion as, which used to measure how fast a constant phase point is
moving, and the group velocity vgr = −2ik measures how
n n fast the wave energy is moving generally. In other words,
F =F0+ F (7)
n=1
the dispersion relation introduces stability properties of the
MHD waves from the relation between vph and vgr , which
And, F F etc.: are functions of slow variables that
0 1 provide three cases of dispersion character. Moreover, the
can be defined as: dispersion relation can be normal at vph /vgr > 1, while it
is anomalous at vph /vgr < 1 and finally it is non-dispersive
= x − vgr t = 2t (8) at vph = vgr . The dispersion relation shows us a basic per-
ception of the system’s behavior if it is stable or not.
vgr is the group velocity, that used to determine the veloc- As shown in Figure 1, the dispersion relation represents
ity along the modulation direction of the wave. The depen- normal dispersion at vph > vgr , which gives good agree-
dent variable can be expanded as F n = n ilkx− t
l=− Fl e , ment with the dispersion properties of the famous KdV
then the problem physical quantities take the form, equation in nonlinear hydrodynamic dispersive media.
n
n 3.2. Second Order Approximation
v= vl eilkx− t
(9)
n=1 l=− At this point we need four-wave components, for O 2 ,
the long-wave component of the orders l = 0 1 2 3 are
n
n
p = 1+ pl eilkx− t
(10) obtained, in the form:
n=1 l=− 10
iv11
n
n v12 =
B = 1+ Bl eilkx− t
(11) k
n=1 l=− B12 = 0
10 10
The physical variables are considered real and be func- ivgr v11 2i v11
tions of the slow and fast independent variables and p12 = − +
k k2
100 J. Nanofluids, 10, 98–105, 2021
Morad et al. A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence
Dispersion relation partial differential equations that are being studied and
0 ← (KdV) vph
KdV vgr → constructed. The articles present a range of interesting
techniques, including inverse scattering theory, sin-cosine
method, expanded tanh-function method, and other.38–40
The wave frequency, Ω
–5
← (MHD) vph
4. THE EXACT SOLUTIONS OF NLS
– 10 (MHD) vgr →
EQUATION
In this section, the complex ansatz method is proposed to
obtain the exact complex traveling wave solutions of NLS
without converting them into real and imaginary parts. The
– 15
partial differential equation is given by Refs. [37, 38];
0 1 2 3 4
f v vx vt vxx = 0
The wavenumber, k
(20)
Fig. 1. The relation between v and k of our MHD model in comparison
with the KdV dispersion properties. By using the complex transformation,
v22 = 0 = iK − (21)
kv11 2 Equation (21) can convert the partial differential Eq. (20)
b22 =
−ik2 − 2 into ordinary differential equation (ODE)
kv11 2
p22 = gv iKv −iv − K 2 v = 0 (22)
−ik2 − 2
ARTICLE
v32 = 0 b32 = 0 p32 = 0 (17)
The suggested solutions could be;
The second-order of O and l = 1; that is an impor- 2
tant step in the solution by the method of reduction per- v = li=0 ai F i + li=1 bi F i−1 Gi (23)
IP: 192.168.39.151 On: Wed, 13 Oct 2021 10:20:36
turbation, as it results in the compatibility condition,
Copyright: and Scientific Publishers
American By making the balance of the highest order linear term,
this is an important role in calculating the group Delivered
velocity. by Ingenta
the constants ai bi , and the integer l can be determined.
The coupled Riccati equations will be,
vgr = −2ik = (18)
k
F = −F G (24)
vgr is a characteristic of a dispersive medium, it used to
determine how the medium will affect a wave traveling And,
through it. G = 1 − G2 (25)
3.3. Third Order Approximation The two types of solution are admitted as:
By using derived equations in the order O 3 with
first harmonic mode l = 1, the solvability condition was F = ±sech G = tanh (26)
obtained by the model equation,
The relation between F and G is G2 = 1 − F 2 , and,
i v11 + P v11 + Qv11 2 v11 = 0
(19) F = ±csch G = coth (27)
which is the well-known envelope MNLS equation with
The exact solutions for Eq. (20) can be produced by
real coefficients P and Q; it describes the evolution of
substituting Eq. (23) into Eq. (22), using Eq. (25) repeat-
the envelope of the modulated wave group. The dispersion
edly with Eqs. (26) or (27) and by setting each coefficient
coefficients P and Q are satisfied by;
of F i and GF i to zero which results in a set of linear equa-
2kvgr − 3 2 k3 tions for each K ai bi , these equations are solved to
P= Q=
k2 −k4 2 + ik2 +2 2 give us the ODE form of Eq. (20).
Equation (19) can be considered as the modified non- −iv + P K 2 v + Qv2 v = 0 (28)
linear Schrördinger equation (MNLS), because of its sym-
metry of NLS in quantum mechanics theory which is The balancing between v and v2 v yields l = 1, as this
nonlinear equation and has great interest. This equation indicates the following form of the solution
can be used in plasma waves and nonlinear optics. Here,
exact solutions are carried to both linear and nonlinear v = a0 + a1 F + b1 G (29)
J. Nanofluids, 10, 98–105, 2021 101
A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence Morad et al.
Use Eq. (26) with F ∗ = F and G∗ = −G, substitute number and is the ratio of the velocity of system particles
Eq. (29) into Eq. (28) and evaluate the coefficients of both to the speed of sound in the fluid.41
F i and GF i to zero, we obtain
v2 1/2
Qa30 − Qa0 b12 = 0 −a1 + 2Qa0 a1 b1 = 0 Ms = (30)
Cs
3Qa0 a21 + b1 + Qa0 b12 = 0
In incompressible flow,42 we can say that plasma moving at
−K 2
P a1 + 3Qa20 a1 − Qa1 b12
= 0 low Mach numbers, so the material derivative is qual zero
2K P a1 + Qa1 + Qa1 b1 = 0 Qa0 b1 − Qb1 = 0
2 3 2 2 3 and the plasma density remains constant with the applica-
tion of pressure on the system. When it takes values less
a2K 2 P b1 + Qa21 b1 + Qb13 = 0 than 1, the bulk particle’s velocity is small in compari-
The following given cases are produced by solving these son to the speed of sound. In fluid dynamics, when the
algebraic equations; divergence of the flow velocity is zero . v = 0, a flow is
√ √ considered incompressible. Solutions involving small per-
i 2K P turbation in density and pressure fields are agreed by the
case 1 a0 = − √ a1 = 0
Q differing density set and may allow pressure stratification
√ √ in the domain.
i 2K P
b1 = − √ = 2K 2 P As seen in Figure 2, the system appears homogeneous
Q horizontally, so that it becomes clear that the velocity is
√ √
i 2K P constant in both x and t, so that the velocity is induced to
case 2 a0 = √ a1 = 0 a sudden rise until it reaches a maximum value by increas-
Q
√ √ ing time and this disturbance is noticeable around the axis
i 2K P of space x. This increase in the velocity occurs due to
b1 = − √ = −2K P
ARTICLE
2
Q the formation of vortex waves. The velocity in hydrody-
√ √ namic turbulence consists of both compression and vortex
i 2K P
case 3 a0 = − √ a1 = 0 motions.
Q IP: 192.168.39.151 On: Wed, 13 Oct 2021 10:20:36 v = v⊥ + v
√ √
i 2K P Copyright: American Scientific Publishers
b1 = √ = −2K 2 P Delivered bywhere
Ingentav which is longitudinal, curl-free, and leads to
Q shocks, v⊥ is rotational, divergence-free, and is associ-
√ √
i 2K P ated with vortex structures. A solenoidal flow velocity field
case 4 a0 = √ a1 = 0 defines an incompressible flow. But besides having zero
Q
√ √ divergence, a solenoidal field also has the extra connota-
i 2K P tions of having a non-zero curl (i.e., rotational component),
b1 = √ = 2K P 2
Q meaning that the flow is mainly solenoidal at the subsonic
and incompressible limit.
Figures 2 and 3 show that the flow quantities v are
5. MACH NUMBER AND ITS homogeneous, where the upstream and downstream of
CONSEQUENCES these vortex waves should be considered stationary in sys-
The magnitude of many of the compressibility effects is tem reference transition layers. The flow is initialized with
determined by Mach number which is the dimensionless a constant mass-density and zero flow velocity. For large
Fig. 2. Space-time plot of the velocity real and imaginary parts in the case of our mode at K = 01 m, k = 01 m, = 10−4 m2 /s, and = 102 m2 /s.
102 J. Nanofluids, 10, 98–105, 2021
Morad et al. A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence
distinctive feature of incompressible flow. By looking at
the behavior of the velocity inside the system, it becomes
clear to us that the magnetic field applied to the system
is constant, and this led to the homogeneity of the veloc-
ity field until an intrinsic magnetic field occurred within
the system which lead to sudden fluctuations of the veloc-
ity waves. This intrinsic magnetic field is responsible for
many changes that occur within the system, including the
behavior of pressure as well.
6. PRESSURE DETERMINATIONS
In nature, there is no incompressible fluid. In nanofluids,
this is frequently the case. When the pressure variation
is small compared to the absolute pressure, we can con-
sider gases incompressible. It is easy to obtain solutions
for pressure values, which of course are identical to the
boundary solutions for velocity in the same declared sys-
Fig. 3. The graph shows the velocity solution release due to the vortex tem. The total pressure may satisfy either the normal com-
waves in plasma in incompressible case.
ponent at the walls, or the tangential component of it, but
not both. The thing that is worth pointing out, the incom-
scale, the solenoid force infuses energy and course the pressible fluid means that it has a constant density and
turbulent flow after dynamic times and statistically sta- has a low Mach. The first means that the density does not
ARTICLE
tionary. The transition to turbulence is taken into account change at all, but the second means that the field veloc-
and reveals significance in time. This hydrodynamic tur- ity is small compared to the speed of sound, but it allows
bulence behavior can be explained by the sudden change the density to change as a function of temperature and not
in the dissipation of kinetic energy IP: induced by the vortex
192.168.39.151 pressure.
On: Wed, 13 Oct 2021 10:20:36
waves. Copyright: American Scientific
After aPublishers
period of time stage in which the fluid is able to
The intrinsic magnetic field generated by Delivered moving byslide
Ingenta
at the wall, the velocity field presented in Figure 2
nanoparticles is responsible for producing steepen vortex does not contribute to one that the Navier boundary con-
waves. These vortex waves are fluctuated and have a finite dition obeys. The relation between the rate of shear at the
amplitude and can steepen into being a jump and be a wall and the slip velocity can be measured if the veloc-
layer. This layer is a thin transition and these waves move ity field is advanced in time with the pressure, after the
faster than the related to the linear group velocity of the initial time step, change with x sinusoidally. The behav-
underlying waves and showed a local plane. One can con- ior of the pressure, as shown in Figure 4, is very similar
sider upstream and downstream of these vortex planes for to the behavior of the velocity field, and this is due to
the flow quantities v to be homogeneous and can also be the pressure being greatly affected by the velocity of the
stationary in transition layers of system reference. system particles, where the pressure appears homogeneous
In the other words, the speed with which the particles in both time t and direction x until it has clear distur-
move is the Alfvén velocity, and this velocity is a basic and bances at a certain time that the intrinsic magnetic field
Fig. 4. Space-time plot of the pressure real and imaginary parts in the case of our mode at K = 01 m, k = 01 m, = 10−4 m2 /s, and = 102 m2 /s.
J. Nanofluids, 10, 98–105, 2021 103
A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence Morad et al.
which the solutions of that model represent rising in the
velocity due to the occurrence of the vortex wave. Initially,
the system remains horizontally homogeneous and occurs
physically sudden rising in the velocity occurs and growth
until it reaches its maximum value with increase in the
time at t > 0 001s and perturbed about the space axis.
There is good agreement of the velocity behavior between
our results and the previous measurements under the effect
of a constant magnetic field. Therefore, one can say that
the velocity field has an identity of great importance and
a clear influence on the characteristics of the studied
system.42
(iii) The disturbances in both velocity and pressure of
the system appeared at t > 0 001 s and 0.5 m < x <
0.5 m, which occur as a result of configuring the intrinsic
magnetic field is generated by moving particles. More-
Fig. 5. The graph shows the 2D pressure plasma solution in both time
over, this intrinsic magnetic field controls various sys-
t and space x for incompressible case.
tem properties for spreading i.e., viscosity, pressure, and
nonlinearity.
controls these disturbances. Figures 4 and 5 clarify that (iv) In the turbulent incompressible system, their 2D
ARTICLE
the pressure profile is a process that has the characteristics plane-like spatial cover all scales of spatial length. There-
of random homogeneity. It can be assumed that the term fore, vortex coupled with velocity fluctuation on all spatial
for describing pressure fluctuations when using the MNLS scales. Moreover, the small-amplitude waves propagate in
equation is small if compared to IP:the other terms ofOn:
192.168.39.151 the13cold
the Wed, Oct plasma are governed by the KdV equation and
2021 10:20:36
MNLS equation. Using this assumption, Copyright:
one can American
consider Scientific
waves are Publishers
vortex waves of the MNLS equation that can
the quasi-adiabatic evolution of these vortex waves Delivered
and bybehave
Ingentaas KdV solutions. Finally, the drawn figures have
derive the equation to help us in this evolution description shown very good agreement within the investigated area of
which gives a good agreement with the results introduced the governing parameters in the problem. We recommend
in Ref. [43].
that there be a lot of studies to clarify the extent of the
Figure 5 presents the vortex waves of the pressure
stability of this system and the factors affecting its stabil-
propagation with homogeneous random processes and
also shows that the distribution of pressure after a short ity. It is also possible to study the effect of the magnetic
period of the fluctuations which have steepened into the Prandtl number and the relationship of this number to the
vortex. stability of the system. And there could be future works
that will focus on the other phenomena in nanofluids
flow.
7. CONCLUSIONS AND FUTURE WORK
In the present study, we studied the small-amplitude per-
turbations behavior in cold plasma. MHD equations, along Highlights
with the state equation was developed by using the RPM. • The governing equations that describe plasma turbu-
Therefore, we can say that the medium can be thought of lence term were solved exactly.
as a system of multifluid MHD. As a consequence, a non- • The solution can be applied to understand several flow
linear wave equation with a modified form of the deriva- systems.
tive nonlinear Schrödinger equation is obtained. The key • The average velocity of the turbulent fluctuations is
remarks of the present work are as follows: smaller than the velocity of sound in the medium, which
(i) Analytical studies of isotropic incompressible tur-
leads to disturbances in velocity waves.
bulence in the non-magnetic case are presented to
• The intrinsic magnetic field was generated by moving
describe the properties of hydrodynamic turbulence such
as the velocity, pressure, and Mach number of the particles while lead to the disturbances.
flow. • The fluctuations produce changes in the properties of
(ii) In incompressible hydrodynamic turbulence, the rel- the turbulence to a large extent.
ative importance of vortex-related motions over shock • These properties were studied and presented in the
motions is likely to depend on the velocity field are present manuscript.
104 J. Nanofluids, 10, 98–105, 2021
Morad et al. A New Derivation of Exact Solutions for Incompressible Magnetohydrodynamic Plasma Turbulence
NOMENCLATURE 11. Y. Menni, A. Azzi, and A. J. Chamkha, J. Brazil. Soc. Mech. Sci.
v Velocity component along x axe m/s Eng. 12, 3888 (2018).
12. Y. Menni, A. Chamkha, C. Zidani, and B. Benyoucef, Math. Model.
t Time (s)
Eng. Probl. 6, 52 (2019).
B Applied magnetic field (Wb/m2 ) 13. Y. Menni, A. J. Chamkha, A. Azzi, and C. Zidani, International
p Fluid pressure P a Journal of Fluid Mechanics Research 47, 23 (2020).
Re Reynolds number – 14. Y. Menni, A. Azzi, and A. J. Chamkha, Heat Transfer Research 50,
Bx Magnetic field acting in the x direction (Wb/m2 ) 1781 (2019).
15. Y. Menni, A. J. Chamkha, A. Azzi, and B. Benyoucef, World J.
k Wavenumber m Model Simul. 15, 213 (2019).
L Length scale m 16. Y. Menni, A. Chamkha, G. Lorenzini, and B. Benyoucef, Mathemat-
Ms Mach number – ical Modelling of Engineering Problems 6, 170 (2019).
Cs Speed of sound in the fluid (m/s) 17. Y. Menni, A. J. Chamkha, and A. Azzi, J. Comput. Appl. Mech. 6,
vgr Group velocity (m/s) 741 (2020).
18. J. Raza, F. Mebarek-Oudina, and A. J. Chamkha, Multidiscipline
P Dispersion coefficient in NLS − Modeling in Materials and Structures 15, 737 (2019).
Q Nonlinearity coefficient in NLS − 19. M. Abo-Dahab, M. A. Abdelhafez, F. Mebarek-Oudina, and S. M.
ai , bi Constants are defined in Eq. (23) – Bilal, Indian Journal of Physics 95, 1 (2021).
K Wavenumber of the new model equation (m) 20. F. Mebarek-Oudina, R. Bessaih, B. Mahanthesh, A. J. Chamkha, and
J. Raza, Int. J. Numer. Methods Heat Fluid Flow 31, 1172 (2020).
21. A. K. Abu-Nab, E. S. Selima, and A. M. Morad, Physica Scripta
Greek Symbols 96, 035222 (2021).
Magnetic resistivity (m2 /s) 22. A. Zaim, A. Aissa, F. Mebarek-Oudina, B. Mahanthesh, G.
Lorenzini, M. Sahnoun, and M. El Ganaoui, Propulsion and Power
Kinematic viscosity (m2 /s)
Research 9, 383 (2020).
Wave frequency (1/s) 23. R. Slimani, A. Aissa, F. Mebarek-Oudina, U. Khan, M. Sahnoun,
Angular frequency of the new model equation (rad/s) A. J. Chamkha, and M. A. Medebber, Eur. Phys. J. Appl. Phys. 92,
ARTICLE
10904 (2020).
24. M. A. Ismael, T. Armaghani, and A. J. Chamkha, J. Taiwan Inst.
ABBREVIATIONS Chem. Eng. 59, 138 (2016).
RPM—Reductive perturbation method; NS—Navier- 25. M. Veera Krishna, and A. J. Chamkha, Results in Physics 15, 102652
IP: 192.168.39.151
Stokes equations; MHD—Magnetohydrodynamic; KdV—On: Wed, 13 Oct 2021 10:20:36
(2019).
Copyright: American Scientific
The Korteweg-de Vries equation; HD—Hydrodynamic; Publishers
26. Y. Menni, A. J. Chamkha, and A. Azzi, Special Topics and Reviews
ODE—Ordinary differential equation; PDF—Partial dif- Delivered by Ingenta
in Porous Media—An International Journal 10, 49 (2019).
27. M. Ghalambaz, A. Doostani, E. Izadpanahi, and A. J. Chamkha, J.
ferential equation; NLS—Nonlinear Schrördinger equa- Therm. Anal. Calorim. 139, 2321 (2020).
tion; MNLS—Modified nonlinear Schrördinger equation. 28. B. Kumar, G. S. Seth, R. Nandkeolyar, and A. J. Chamkha, Int. J.
Therm. Sci. 146, 106101 (2019).
Conflicts of Interest 29. A. J. Chamkha, M. A. Mansour, A. M. Rashad, H. Kargarsharifabad,
and T. Armaghani, J. Thermophys. Heat Transfer. 34, 836 (2020).
There are no conflicts to declare.
30. A. J. Chamkha and A. A. Khaled, Int. J. Numer. Methods Heat Fluid
Flow 10, 94 (2000).
Acknowledgments: The authors would like to thank 31. H. S. Takhar, A. J. Chamkha, and G. Nath, Heat Mass Transfer 39,
the editor and anonymous referees. 297 (2003).
32. A. S. Dogonchi, T. Armaghani, A. J. Chamkha, and D. D. Ganji,
Arab. J. Sci. Eng. 44, 7919 (2019).
References and Notes 33. A. S. Dogonchi and D. D. Ganji, Journal of the Taiwan Institute of
1. M. Lesieur, Turbulence in Fluids. Kluwer Academic Publishers, Dor- Chemical Engineers 69, 1 (2016).
drecht (1997). 34. Y. Menni, A. J. Chamkha, and A. Azzi, J. Nanofluids 8, 893
2. D. Biskamp, Magnetohydrodynamical Turbulence. Cambridge Uni- (2018).
versity Press, Cambridge, (2003). 35. Y. Menni, A. Chamkha, G. Lorenzini, N. Kaid, H. Ameur, and M.
3. G. Belmont, L. Rezeau, C. Riconda, and A. Zaslavsky, Introduction Bensafi, Mathematical Modelling of Engineering Problems 6, 415
to Plasma Physics. Elsevier, ISTE Press, United Kingdom (2019). (2019).
4. T. Z. Abdel Wahid and A. M. Morad, Applied Advances in Mathe- 36. A. M. Morad, M. Abu-Shady, and G. I. Elsawy, Physica Scripta 95,
matical Physics, 2020, 1289316 (2020). 025205 (2020).
5. E. M. Elsaid, T. Z. Abdel Wahid, and A. M. Morad, Results in 37. E. S. Selima, Y. Mao, Y. Xiaohua, A. M. Morad, T. Abdelhamid,
Physics 19, 103554 (2020). and B. I. Selim, Appl. Math. Model 57, 376 (2018).
6. T. Z. Abdel Wahid and A. M. Morad, IEEE Access 8, 76423 (2020). 38. A. M. Abourabia and A. M. Morad, Physica A. 437, 333 (2015).
7. Y. Menni, A. Chamkha, C. Zidani, and B. Benyoucef, Int. J. Numer. 39. A. M. Abourabia, T. S. El-Danaf, and A. M. Morad, Chaos, Solitons
Methods Heat Fluid Flow 30, 3027 (2019). and Fractals 41, 716 (2009).
8. Y. Menni, A. Chamkha, G. Lorenzini, N. Kaid, H. Ameur, and M. 40. A. M. Morad, E. S. Selima, and A. K. Abu-Nab, European Physical
Bensafi, Mathematical Modelling of Engineering Problems 6, 15 Journal Plus 136, 306 (2021).
(2019). 41. G. Gogoberidze, Phys. Plasmas 14, 022304 (2007).
9. Y. Menni, A. Azzi, A. J. Chamkha, and S. Harmand, Int. J. Numer. 42. F. H. Shu, Physics of Astrophysics, University Science Books, Vol-
Methods Heat Fluid Flow 29, 3908 (2019). ume II: Gas Dynamics (A Series of Books in Astronomy). Mill
10. Y. Menni, A. Azzi, A. J. Chamkha, and S. Harmand, J. Comput. Vallye, CA. ISBN: 0935702652 (1992).
Appl. Mech. 5, 231 (2019). 43. B. T. Kress and D. C. Montgomery, J. Plasma Phys. 64, 371 (2000).
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