Next Article in Journal
Risk Management Assessment in Oil and Gas Construction Projects Using Structural Equation Modeling (PLS-SEM)
Previous Article in Journal
Adsorption Factors in Enhanced Coal Bed Methane Recovery: A Review
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Electrostatic Shock Structures in a Magnetized Plasma Having Non-Thermal Particles

1
Department of Physics, Jahangirnagar University, Dhaka 1342, Bangladesh
2
Atomic Energy Centre, Health Physics Division, Dhaka 1000, Bangladesh
3
Atomic Energy Centre, Plasma Physics Division, Dhaka 1000, Bangladesh
*
Author to whom correspondence should be addressed.
Gases 2022, 2(2), 22-32; https://doi.org/10.3390/gases2020002
Submission received: 17 January 2022 / Revised: 18 March 2022 / Accepted: 22 March 2022 / Published: 25 March 2022

Abstract

:
A rigorous theoretical investigation has been made on the nonlinear propagation of dust-ion-acoustic shock waves in a multi-component magnetized pair-ion plasma (PIP) having inertial warm positive and negative ions, inertialess non-thermal electrons and positrons, and static negatively charged massive dust grains. The Burgers’ equation is derived by employing the reductive perturbation method. The plasma model supports both positive and negative shock structures in the presence of static negatively charged massive dust grains. It is found that the steepness of both positive and negative shock profiles declines with the increase of ion kinematic viscosity without affecting the height, and the increment of negative (positive) ion mass in the PIP system declines (enhances) the amplitude of the shock profile. It is also observed that the increase in oblique angle raises the height of the positive shock profile, and the height of the positive shock wave increases with the number density of positron. The applications of the findings from the present investigation are briefly discussed.

1. Introduction

Positive ions are produced by the electron impact ionization while negative ions are produced due to the attachment of electron with an atom [1], and the existence of both positive and negative ions or pair-ion (PI) can be observed in space plasmas, viz., cometary comae [2], upper regions of Titan’s atmosphere [3,4,5], plasmas in the D- and F-regions of Earth’s ionosphere [4,5,6]), and also laboratory plasmas, viz., ( K + , S F 6 ) plasma [7,8], ( A r + , F ) plasma [9], plasma processing reactors [10], plasma etching [11], combustion products [11], ( X e + , F ) plasma [12], neutral beam sources [13], ( A r + , S F 6 ) plasma [14,15,16], ( A r + , O 2 ) plasma, Fullerene ( C 60 + , C 60 ) plasma [17,18], etc. The dynamics of the plasma system and associated nonlinear electrostatic structures have been rigorously changed by the presence of massive dust grains in the PI plasma (PIP) [19,20,21,22]. Yasmin et al. [23] studied the nonlinear propagation of dust-ion-acoustic (DIA) waves (DIAWs) in a multi-component plasma and found that the shock profile associated with DIAWs is significantly modified by the existence of dust grains. A number of authors also examined the effects of the positron to the formation of solitary profile associated with electrostatic waves [24,25]. Rahman et al. [24] studied the electrostatic solitary waves in electron-positron-ion plasma, and observed that the amplitude of the solitary profile increases with increasing the number density of positron. Abdelsalam [25] investigated ion-acoustic (IA) solitary waves in a dense plasma, and demonstrated that the presence of the positron can cause an increase in the amplitude of the solitary profile.
Cairns et al. [26] first demonstrated the non-thermal distribution to investigate the effect of energetic particles on the formation of IA shock profile, and introduced the parameter α in the non-thermal distribution for measuring the amount of deviation of non-thermal plasma species from Maxellian–Boltzmann distribution. The non-thermal plasma species are regularly seen in the cometary comae [2], Earth’s ionosphere [4], the upper region of the Titans [3], etc. Haider et al. [27] investigated the IA solitary waves in the presence of non-thermal particles, and observed that the width of the solitary profile decreases with increasing of ions’ non-thermality. Pakzad and Javidan [28] studied the dust-acoustic (DA) solitary and shock waves in a dusty plasma having non-thermal ions, and reported that the amplitude of the wave increases with the decrease of the non-thermality of ions. Ghai et al. [29] studied the DA solitary waves in the presence of non-thermal ions, and found that the height of the solitary wave decreases with the increase of α .
Landau damping and the kinematic viscosity among the plasma species are the primary reasons for the formation of shock profile associated with electrostatic waves [30,31,32]. The existence of the external magnetic field is considered to be responsible for changing the configuration of the shock profile [33]. Sabetkar and Dorranian [30] examined the effects of external magnetic field on the formation of the DA solitary waves in the presence of non-thermal plasma species, and found that the amplitude of solitary wave increases with the increase in the value of oblique angle. Shahmansouri and Mamun [31] analyzed the DA shock waves in a magnetized non-thermal dusty plasma and demonstrated that the amplitude of shock wave increases with increasing the oblique angle. Malik et al. [32] studied the small-amplitude DA wave in magnetized plasma, and reported that the height of the shock wave enhances with oblique angle. Bedi et al. [34] studied DA solitary waves in a four-component magnetized dusty plasma, and highlighted that both compressive and rarefactive solitons can exist in the presence of an external magnetic field. To the best knowledge of the authors, no attempt has been made to study the DIA shock waves (DIASHWs) in a magnetized PIP by considering kinematic viscosity of both inertial warm positive and negative ion species, and inertialess non-thermal electrons and positrons in the presence of static negatively charged dust grains. The aim of our present investigation is, therefore, to derive Burgers’ equation and investigate DIASHWs in a magnetized PIP, and to observe the effects of various plasma parameters (e.g., mass, charge, temperature, kinematic viscosity, obliqueness, etc.) on the configuration of DIASHWs.
The layout of the paper is as follows: The basic equations are displayed in Section 2. The well known Burgers’ equation is derived in Section 3. Numerical analysis and discussion are presented in Section 4. A brief conclusion is given in Section 5.

2. Governing Equations

We consider a multi-component PIP having inertial positively charged warm ions (mass m 1 ; charge e Z 1 ; temperature T 1 ; number density n 1 ˜ ), negatively charged warm ions (mass m 2 ; charge e Z 2 ; temperature T 2 ; number density n 2 ˜ ), inertialess electrons featuring non-thermal distribution (mass m e ; charge e ; temperature T e ; number density n e ˜ ), inertialess positrons obeying non-thermal distribution (mass m p ; charge e; temperature T p ; number density n p ˜ ), and static negatively charged massive dust grains (charge e Z d ; number density n d ); where Z 1 ( Z 2 ) is the charge state of the positive (negative) ion, and Z d is the charge state of the negative dust grains, and e is the magnitude of the charge of an electron. An external magnetic field B 0 is considered in the system directed along the z-axis defining B 0 = B 0 z ^ , where B 0 and z ^ denote the strength of the external magnetic field and unit vector directed along the z-axis, respectively. The dynamics of the PIP system is governed by the following set of equations [35,36,37,38,39]:
n ˜ 1 t ˜ + ´ · ( n ˜ 1 u ˜ 1 ) = 0 ,
u ˜ 1 t ˜ + ( u ˜ 1 · ´ ) u ˜ 1 = Z 1 e m 1 ´ ψ ˜ + Z 1 e B 0 m 1 ( u ˜ 1 × z ^ ) 1 m 1 n ˜ 1 ´ P 1 + η ˜ 1 ´ 2 u ˜ 1 ,
n ˜ 2 t ˜ + ´ · ( n ˜ 2 u ˜ 2 ) = 0 ,
u ˜ 2 t ˜ + ( u ˜ 2 · ´ ) u ˜ 2 = Z 2 e m 2 ´ ψ ˜ Z 2 e B 0 m 2 ( u ˜ 2 × z ^ ) 1 m 2 n ˜ 2 ´ P 2 + η ˜ 2 ´ 2 u ˜ 2 ,
´ 2 ψ ˜ = 4 π e [ n ˜ e + Z d n ˜ d + Z 2 n ˜ 2 Z 1 n ˜ 1 n ˜ p ] ,
where u ˜ 1 ( u ˜ 2 ) is the positive (negative) ion fluid velocity; η ˜ 1 = μ 1 / m 1 n 1 ( η ˜ 2 = μ 2 / m 2 n 2 ) is the kinematic viscosity of the positive (negative) ion; P 1 ( P 2 ) is the pressure of positive (negative) ion, and ψ ˜ represents the electrostatic wave potential. Now, we are introducing normalized variables, namely, n 1 n ˜ 1 / n 10 , n 2 n ˜ 2 / n 20 , n e n ˜ e / n e 0 , n p n ˜ p / n p 0 , n d n ˜ d / n d 0 , u 1 u ˜ 1 / C 2 , u 2 u ˜ 2 / C 2 (where C 2 = ( Z 2 k B T e / m 2 ) 1 / 2 , k B being the Boltzmann constant); ψ ψ ˜ e / k B T e ; t = t ˜ / ω P 2 1 [where ω P 2 1 = ( m 2 / 4 π e 2 Z 2 2 n 20 ) 1 / 2 ]; ´ = / λ D (where λ D = ( k B T e / 4 π e 2 Z 2 n 20 ) 1 / 2 ). The pressure term of the positive ion can be recognized as P 1 = P 10 ( n ˜ 1 / n 10 ) γ with P 10 = n 10 k B T 1 being the equilibrium pressure of the positive ion, and the pressure term of the negative ion can be recognized as P 2 = P 20 ( n ˜ 2 / n 20 ) γ with P 20 = n 20 k B T 2 being the equilibrium pressure of the negative ion, and γ = ( N + 2 ) / N (where N is the degree of freedom, and for a three-dimensional case, N = 3 , then γ = 5 / 3 ). For simplicity, we consider ( η ˜ 1 η ˜ 2 = η ), and η is normalized by ω P 2 λ D 2 . The quasi-neutrality condition at equilibrium for our plasma model can be written as Z 1 n 10 + n p 0 Z 2 n 20 + Z d n d 0 + n e 0 . Equations (1)–(5) can be expressed in the normalized form as [40]:
n 1 t + · ( n 1 u 1 ) = 0 ,
u 1 t + ( u 1 · ) u 1 = α 1 ψ + α 1 Ω c ( u 1 × z ^ ) α 2 n 1 γ 1 + η 2 u 1 ,
n 2 t + · ( n 2 u 2 ) = 0 ,
u 2 t + ( u 2 · ) u 2 = ψ Ω c ( u 2 × z ^ ) α 3 n 2 γ 1 + η 2 u 2 ,
2 ψ = λ e n e λ p n p + λ d n d ( 1 + λ e + λ d λ p ) n 1 + n 2 .
Other plasma parameters can be recognized as α 1 = Z 1 m 2 / Z 2 m 1 , α 2 = 5 T 1 m 2 / 2 Z 2 T e m 1 , α 3 = 5 T 2 / 2 Z 2 T e , λ e = n e 0 / Z 2 n 20 , λ d = Z d n d 0 / Z 2 n 20 , λ p = n p 0 / Z 2 n 20 , and Ω c = ω c / ω P 2 (where ω c = Z 2 e B 0 / m 2 ). Now, the expression for the number density of electrons [26,41,42,43] and positrons [26,44] following non-thermal distribution can be, respectively, written as
n e = ( 1 β ψ + β ψ 2 ) exp ( ψ ) ,
n p = ( 1 + β α 4 ψ + β α 4 2 ψ 2 ) exp ( α 4 ψ ) ,
where β = 4 α / ( 1 + 3 α ) ( α represents the number of non-thermal populations in our considered model) and α 4 = T e / T p . Now, by substituting Equations (11) and (12) into Equation (10), and expanding up to the third order in ψ , we can write
2 ψ = λ e λ p + n d λ d + n 2 Λ n 1 + σ 1 ψ + σ 2 ψ 2 + · · · ,
where Λ = 1 + λ e + λ d λ p , σ 1 = λ e ( 1 β ) λ p α 4 ( β 1 ) , and σ 2 = λ e / 2 λ p α 4 2 / 2 . We note that the terms containing σ 1 and σ 2 are the contribution of non-thermal distributed electrons and positrons.

3. Derivation of Burgers’ Equation

To derive Burgers’ equation [45,46,47] for the DIASHWs propagating in a PIP, we first introduce the stretched co-ordinates [48,49] as
ξ = ϵ ( l x x + l y y + l z z V p t ) ,
τ = ϵ 2 t ,
where V p is the phase speed and ϵ is a smallness parameter denoting the weakness of the dissipation ( 0 < ϵ < 1 ). It is noted that l x , l y , and l z (i.e., l x 2 + l y 2 + l z 2 = 1 ) are the directional cosines of the wave vector of k along x, y, and z-axes, respectively. Then, the dependent variables can be expressed in power series of ϵ as [49]
n 1 = 1 + ϵ n 1 ( 1 ) + ϵ 2 n 1 ( 2 ) + ϵ 3 n 1 ( 3 ) + · · · ,
n 2 = 1 + ϵ n 2 ( 1 ) + ϵ 2 n 2 ( 2 ) + ϵ 3 n 2 ( 3 ) + · · · ,
u 1 x , y = ϵ 2 u 1 x , y ( 1 ) + ϵ 3 u 1 x , y ( 2 ) + · · · ,
u 2 x , y = ϵ 2 u 2 x , y ( 1 ) + ϵ 3 u 2 x , y ( 2 ) + · · · ,
u 1 z = ϵ u 1 z ( 1 ) + ϵ 2 u 1 z ( 2 ) + · · · ,
u 2 z = ϵ u 2 z ( 1 ) + ϵ 2 u 2 z ( 2 ) + · · · ,
ψ = ϵ ψ ( 1 ) + ϵ 2 ψ ( 2 ) + · · · .
Now, by substituting Equations (14)–(22) into Equations (6)–(9) and (13), and collecting the terms containing ϵ , the first-order equations become
n 1 ( 1 ) = 3 α 1 l z 2 3 V p 2 2 α 2 l z 2 ψ ( 1 ) ,
u 1 z ( 1 ) = 3 V p α 1 l z 3 V p 2 2 α 2 l z 2 ψ ( 1 ) ,
n 2 ( 1 ) = 3 l z 2 3 V p 2 2 α 3 l z 2 ψ ( 1 ) ,
u 2 z ( 1 ) = 3 V p l z 3 V p 2 2 α 3 l z 2 ψ ( 1 ) .
Now, the phase speed of DIASHWs can be read as
V p V p + = l z a 1 + a 1 2 36 σ 1 a 2 18 σ 1 ,
V p V p = l z a 1 a 1 2 36 σ 1 a 2 18 σ 1 ,
where a 1 = ( 6 σ 1 α 2 + 6 σ 1 α 3 + 9 + Λ n 1 ) and a 2 = 4 σ 1 α 2 α 3 + 6 α 2 + 6 α 1 α 3 Λ . The x and y-components of the first-order momentum equations can be written as
u 1 x ( 1 ) = 3 l y V p 2 Ω c ( 3 V p 2 2 α 2 l z 2 ) ψ ( 1 ) ξ ,
u 1 y ( 1 ) = 3 l x V p 2 Ω c ( 3 V p 2 2 α 2 l z 2 ) ψ ( 1 ) ξ ,
u 2 x ( 1 ) = 3 l y V p 2 Ω c ( 3 V p 2 2 α 3 l z 2 ) ψ ( 1 ) ξ ,
u 2 y ( 1 ) = 3 l x V p 2 Ω c ( 3 V p 2 2 α 3 l z 2 ) ψ ( 1 ) ξ .
Now, by following the next higher-order terms, the equation of continuity, momentum equation, and Poisson’s equation can be written as
n 1 ( 1 ) τ V p n 1 ( 2 ) ξ + l x u 1 x ( 1 ) ξ + l y u 1 y ( 1 ) ξ + l z u 1 z ( 2 ) ξ + l z ξ n 1 ( 1 ) u + z ( 1 ) = 0 , u 1 z ( 1 ) τ V p u 1 z ( 2 ) ξ + l z u 1 z ( 1 ) u 1 z ( 1 ) ξ + α 1 l z ψ ( 2 ) ξ
+ α 2 l z ξ 2 3 n 1 ( 2 ) 1 9 ( n 1 ( 1 ) ) 2 η 2 u 1 z ( 1 ) ξ 2 = 0 ,
n 2 ( 1 ) τ V p n 2 ( 2 ) ξ + l x u 2 x ( 1 ) ξ + l y u 2 y ( 1 ) ξ + l z u 2 z ( 2 ) ξ + l z ξ n 2 ( 1 ) u 2 z ( 1 ) = 0 , u 2 z ( 1 ) τ V p u 2 z ( 2 ) ξ + l z u 2 z ( 1 ) u 2 z ( 1 ) ξ l z ψ ( 2 ) ξ
+ α 3 l z ξ 2 3 n 2 ( 2 ) 1 9 ( n 2 ( 1 ) ) 2 η 2 u 2 z ( 1 ) ξ 2 = 0 ,
σ 1 ψ ( 2 ) + σ 2 [ ψ ( 1 ) ] 2 + n 2 ( 2 ) Λ n 1 ( 2 ) = 0 .
Finally, the next higher-order terms of Equations (6)–(9) and (13), with the help of Equations (23)–(37), can provide Burgers’ equation as
Ψ τ + A Ψ Ψ ξ = C 2 Ψ ξ 2 ,
where Ψ = ψ ( 1 ) is used for simplicity. In Equation (38), the nonlinear coefficient (A) and dissipative coefficient (C) are given by the following expression:
A = ( M 1 S 1 3 M 2 S 2 3 2 σ 2 S 1 3 S 2 3 ) M 3 S 1 S 2 , and C = η 2 ,
where M 1 = Λ ( 81 α 1 2 V p 2 l z 4 6 α 2 α 1 2 l z 6 ) , M 2 = 81 V p 2 l z 4 6 α 3 l z 6 , and M 3 = 18 V p l z 2 ( 1 + α 1 Λ ) . It is well established that the ion fluids are viscous, and in realistic situations, ion fluids which occur in both space [50] and laboratory [50,51] plasmas, where the effect of viscous force cannot be neglected. It is important to note here that Burger’s equation, defined by Equation (38), is derived for our multi-ion dusty plasma system, when the effect of dispersion is negligible compared to that of dissipation.
Now, we look forward to the stationary shock wave solution of this Burgers’ equation by taking ζ = ξ U 0 τ and τ = τ , where U 0 is the speed of the shock waves in the reference frame. These allow us to represent the stationary shock wave solution as [49,52,53]
Ψ = Ψ m 1 tanh ζ Δ ,
where Ψ m is the amplitude and Δ is the width. The expression of the amplitude and width can be given by the following equations:
Ψ m = U 0 A , and Δ = 2 C U 0 .

4. Numerical Analysis and Discussion

Now, we would like to observe the basic properties of DIASHWs in a magnetized PIP having inertial pair-ions, inertialess non-thermal distributed electrons and positrons, and static negatively charged massive dust grains by changing the various plasma parameters, viz., ion kinematic viscosity, oblique angle, non-thermality of electrons and positrons, mass, charge, and number density of the plasma species. Note that the viscous force acting in positive and negative ion fluid of the plasma model under consideration is the source of dissipation, and is responsible for the formation of shock structures [50]. It can be seen from the literature that the PIP system can support these conditions: m 2 > m 1 (i.e., H + O 2 [8,10,13,14,15,16], A r + S F 6 [14,15,16,54], and X e + S F 6 [14,15,16,54]), m 2 = m 1 (i.e., H + H [8,10,13,14,15,16] and C 60 + C 60 [17,18,55]), and m 2 < m 1 (i.e., A r + F [10,13]). Equation (41) shows that under consideration of U 0 > 0 and C > 0 , no shock wave will exist if A = 0 as the amplitude of the wave becomes infinite, which clearly violates the reductive perturbation method [56,57,58,59]. Thus, A can be positive (i.e., A > 0 ) or negative (i.e., A < 0 ), according to the value of other plasma parameters. The left panel of Figure 1 illustrates the variation of A with α 4 , and it is obvious from this figure that A can be negative, zero, and positive according to the values of α 4 when other plasma parameters are α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , λ p = 1.5 , λ e = 1.7 , λ d = 0.05 , δ = 30 , and α = 0.5 . The point at which A becomes zero for the value of α 4 is known as the critical value of α 4 (i.e., α 4 c ). In our present analysis, the critical value of α 4 is α 4 c 01 . Thus, the negative (positive) shock profile can exist for the value of α 4 < α 4 c ( α 4 > α 4 c ). The right panel of Figure 1 describes the effects of the external magnetic field on the formation of the positive shock profile. The increase in oblique angle raises the height of the positive shock profile, and this result is analogous to the result of Ref. [31].
The left and right panels of Figure 2 represent the variation of the positive and negative shock profiles with ion kinematic viscosity (via η ), respectively, when other plasma parameters remain constant. It is really interesting that the steepness of both positive and negative shock profiles declines with the increase of η without affecting the height.
The height of the positive shock profile is very sensitive to the change of non-thermality of the electrons and positrons, which can be seen in the left panel of Figure 3. There is a decrease in the amplitude of positive shock profile when electrons and positrons deviate from thermodynamic equilibrium, and this result is compatible with the result of Ref. [60]. The variation of the DIASHWs with negative ion charge state, negative ion, and positron number densities (via λ p ) can be observed in the right panel of Figure 3. It is clear from the right panel of Figure 3 that as we increase the positron (negative ion) number density, the height of the positive shock wave increases (decreases) when the charge of the negative ion remains constant, or the the height of the positive shock wave decreases with the charge of the negative ion for a fixed value of the number density of positron and negative ion.
The charge and mass of the positive and negative ions are vigorously responsible for changing the height of the positive shock profile. Figure 4 describes the nature of DIASHWs with the variation values of α 1 , i.e., α 1 > 1 (left panel) and α 1 , i.e., α 1 < 1 (right panel), respectively, under consideration of α 4 > α 4 c . it is obvious from this figure that (a) due to the α 4 > α 4 c , we have observed positive shock potential of DIASWs; (b) the height of the positive shock profile increases (decreases) with rising value of positive (negative) ion mass for a fixed value of the their charge state, but as we increase the charge state of the positive (negative) ion, then the amplitude of the positive shock profile diminishes (enhances) when their masses are constant; (c) the height of the amplitude of the shock profile is taller (shorter) under the assumption of m 2 < m 1 ( m 2 > m 1 ). Thus, the dynamics of positive shock profile of DIASWs in PIP system rigorously change with these conditions m 2 < m 1 (i.e., α 1 < 1 ) and m 2 > m 1 (i.e., α 1 > 1 ).
We can also observe the effects of mass and charge of positive and negative ions in the formation of the negative shock profile from Figure 5 under consideration of α 4 < α 4 c . The left panel represents the variation of potential with ξ under consideration of m 2 > m 1 , i.e., α > 1 , while the right panel represents the variation of potential with ξ under consideration of m 2 < m 1 , i.e., α < 1 . It can be seen from this figure that the magnitude of the amplitude decreases (increases) with the mass of negative (positive) ion when their charge remains constant. Physically, the order of the variation of the potential, as well as the dynamics of the plasma medium, is invariant under the conditions m 2 > m 1 or m 2 < m 1 .

5. Conclusions

In our present investigation, we considered a multi-component magnetized PIP having static dust grains, non-thermal electrons, and positrons. Burgers’ equation was derived by employing the reductive perturbation method for studying DIASHWs. The results that we have found from this investigation can be summarized as follows:
  • The negative (positive) shock profile can exist for the value of α 4 < α 4 c ( α 4 > α 4 c ).
  • The steepness of both positive and negative shock profiles declines with the increase of η without affecting the height.
  • The increase in oblique angle raises the height of the positive shock profile.
  • The height of the positive shock wave increases with the number density of positron.
  • The amplitude of the positive shock profile increases (decreases) with increasing the value of positive (negative) ion mass.
The results are applicable in understanding the criteria for the formation of DIASHWs in astrophysical plasmas, viz., cometary comae [2], upper regions of Titan’s atmosphere [3,4,5], plasmas in the D- and F-regions of Earth’s ionosphere [4,5,6], and also in laboratory environments, viz., H + O 2 [8,10,13,14,15,16], ( K + , S F 6 ) plasma [7,8], X e + S F 6 [14,15,16,54], H + H [8,10,13,14,15,16], ( A r + , F ) plasma [9], plasma processing reactors [10], plasma etching [11], combustion products [11], ( X e + , F ) plasma [12], neutral beam sources [13], ( A r + , S F 6 ) plasma [14,15,16], ( A r + , O 2 ) plasma, Fullerene ( C 60 + , C 60 ) plasma [17,18], etc.

Author Contributions

All authors contributed equally to complete this work. All authors have read and agreed to the published version of the manuscript.

Funding

The research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

Sharmin Jahan would like to acknowledge “NST Fellowship” for their financial support to complete this work.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Shukla, P.K.; Mamun, A.A. Introdustion to Dusty Plasma Physics; Institute of Physics: Bristol, UK, 2002. [Google Scholar]
  2. Chaizy, P.; Reme, H.; Sauvaud, J.A.; D’Uston, C.; Lin, R.P.; Larson, D.E.; Mitchell, D.L.; Anderson, K.A.; Carlson, C.W.; Korth, A.; et al. Negative ions in the coma of comet Halley. Nature 1991, 349, 393. [Google Scholar] [CrossRef]
  3. Coates, A.J.; Crary, F.J.; Lewis, G.R.; Young, D.T.; Waite, J.H.; Sittler, E.C. Discovery of heavy negative ions in Titan’s ionosphere. Geophys. Res. Lett. 2007, 34, L22103. [Google Scholar] [CrossRef] [Green Version]
  4. Massey, H. Negative Ions, 3rd ed.; Cambridge University Press: Cambridge, UK, 1976. [Google Scholar]
  5. Sabry, R.; Moslem, W.M.; Shukla, P.K. Fully nonlinear ion-acoustic solitary waves in a plasma with positive-negative ions and nonthermal electrons. Phys. Plasmas 2009, 16, 032302. [Google Scholar] [CrossRef]
  6. Abdelwahed, H.G.; El-Shewy, E.K.; Zahran, M.A.; Elwakil, S.A. On the rogue wave propagation in ion pair superthermal plasma. Phys. Plasmas 2016, 23, 022102. [Google Scholar] [CrossRef]
  7. Song, B.; D’Angelo, N.; Merlino, R.L. Ion-acoustic waves in a plasma with negative ions. Phys. Fluids B 1991, 3, 284. [Google Scholar] [CrossRef] [Green Version]
  8. Sato, N. Production of negative ion plasmas in a Q machine. Plasma Sources Sci. Technol. 1994, 3, 395. [Google Scholar] [CrossRef]
  9. Nakamura, Y.; Tsukabayashi, I. Observation of modified Korteweg-de Vries solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 1984, 52, 2356. [Google Scholar] [CrossRef]
  10. Gottscho, R.A.; Gaebe, C.E. Negative ion kinetics in rf glow discharges. IEEE Trans. Plasma Sci. 1986, 14, 92. [Google Scholar] [CrossRef]
  11. Sheehan, D.P.; Rynn, N. Negative-ion plasma sources. Rev. Sci. Lnstrum. 1988, 59, 8. [Google Scholar] [CrossRef]
  12. Ichiki, R.; Yoshimura, S.; Watanabe, T.; Nakamura, Y.; Kawai, Y. Negative-ion plasma sources. Phys. Plasmas 2002, 9, 4481. [Google Scholar] [CrossRef]
  13. Bacal, M.; Hamilton, G.W. H and D production in plasmas. Phys. Rev. Lett. 1979, 42, 1538. [Google Scholar] [CrossRef]
  14. Wong, A.Y.; Mamas, D.L.; Arnush, D. Negative ion plasmas. Phys. Fluids 1975, 18, 1489. [Google Scholar] [CrossRef]
  15. Cooney, J.L.; Gavin, M.T.; Lonngren, K.E. Soliton propagation, collision, and reflection at a sheath in a positive ion-negative ion plasma. Phys. Fluids B 1991, 3, 2758. [Google Scholar] [CrossRef]
  16. Nakamura, Y.; Odagiri, T.; Tsukabayashi, I. Ion-acoustic waves in a multicomponent plasma with negative ions. Plasma Phys. Control. Fusion 1997, 39, 105. [Google Scholar] [CrossRef]
  17. Oohara, W.; Hatakeyama, R. Pair-ion plasma generation using fullerenes. Phys. Rev. Lett. 2003, 91, 205005. [Google Scholar] [CrossRef]
  18. Hatakeyama, R.; Oohara, W. Properties of pair-ion plasmas using fullerenes. Phys. Scr. 2005, 116, 101. [Google Scholar] [CrossRef]
  19. Shukla, P.K.; Silin, V.P. Dust ion-acoustic wave. Phys. Scr. 1992, 45, 508. [Google Scholar] [CrossRef]
  20. Shukla, P.K. Dust ion-acoustic shocks and holes. Phys. Plasmas 2000, 7, 1044. [Google Scholar] [CrossRef]
  21. Baluku, T.K.; Hellberg, M.A.; Kourakis, I.; Saini, N.S. Dust ion acoustic solitons in a plasma with kappa-distributed electrons. Phys. Plasmas 2010, 17, 053702. [Google Scholar] [CrossRef] [Green Version]
  22. Sultana, S.; Islam, S.; Mamun, A.A. Envelope solitons and their modulational instability in dusty plasmas with two-temperature superthermal electrons. Astrophys. Space Sci. 2014, 351, 581. [Google Scholar] [CrossRef]
  23. Yasmin, S.; Asaduzzaman, M.; Mamun, A.A. Dust ion-acoustic shock waves in nonextensive dusty plasma. Astrophys. Space Sci. 2013, 343, 245. [Google Scholar] [CrossRef]
  24. Rahman, A.; Kourakis, I.; Qamar, A. Electrostatic solitary waves in relativistic degenerate electron-positron-ion plasma. IEEE Trans. Plasma Sci. 2015, 43, 974. [Google Scholar] [CrossRef]
  25. Abdelsalam, U.M.; Moslem, W.M.; Shukla, P.K. Ion-acoustic solitary waves in a dense pair-ion plasma containing degenerate electrons and positrons. Phys. Lett. A 2008, 372, 4057. [Google Scholar] [CrossRef]
  26. Carins, R.A.; Mamun, A.A.; Bingham, R.; Dendy, R.O.; Nairn, C.M.C.; Shukla, P.K. Electrostatic solitary structures in non-thermal plasmas. Geophys. Res. Lett. 1995, 22, 2709. [Google Scholar] [CrossRef]
  27. Haider, M.M.; Sultana, I.; Khatun, S.J.; Nasrin, J.; Tasniml, N.; Rahman, O.; Akter, S. Nonthermal distribution of electro-negative light-ions in heavy-ion-acoustic solitary and shock structures. Theor. Phys. 2019, 4, 124. [Google Scholar] [CrossRef]
  28. Pakzad, H.R.; Javidan, K. Dust acoustic solitary and shock waves in strongly coupled dusty plasmas with nonthermal ions. Pramana 2009, 73, 913. [Google Scholar] [CrossRef]
  29. Ghai, Y.; Saini, N.S.; Eliasson, B. Landau damping of dust acoustic solitary waves in nonthermal plasmas. Phys. Plasmas 2018, 25, 013704. [Google Scholar] [CrossRef]
  30. Sabetar, A.; Dorranian, D. Effect of obliqueness and external magnetic field on the characteristics of dust acoustic solitary waves in dusty plasma with two-temperature nonthermal ions. J. Theor. Appl. Phys. 2015, 9, 150. [Google Scholar]
  31. Shahmansouri, M.; Mamun, A.A. Dust-acoustic shock waves in a magnetized non-thermal dusty plasma. J. Plasma Phys. 2014, 80, 593. [Google Scholar] [CrossRef]
  32. Malik, H.K.; Srivastava, R.; Kumar, S.; Singh, D. Small amplitude dust acoustic solitary wave in magnetized two ion temperature plasma. J. Taibah Univ. Sci. 2020, 14, 417. [Google Scholar] [CrossRef] [Green Version]
  33. Islam, M.K.; Biswas, S.; Chowdhury, N.A.; Mannan, A.; Salahuddin, M.; Mamun, A.A.D. Obliquely propagating ion-acoustic shock waves in a degenerate quantum plasma. Contrib. Plasma Phys. 2022, 62, e202100073. [Google Scholar] [CrossRef]
  34. Bedi, C.; Gill, T.S.; Bains, A.S. Four component magnetized dusty plasma containing non-thermal electrons. J. Phys. Conf. Ser. 2010, 208, 012037. [Google Scholar] [CrossRef]
  35. Adhikary, N.C. Effect of viscosity on dust-ion acoustic shock wave in dusty plasma with negative ions. Phys. Lett. A 2012, 376, 1460. [Google Scholar] [CrossRef]
  36. Sahu, B.; Sinha, A.; Roychoudhury, R. Nonlinear features of ion acoustic shock waves in dissipative magnetized dusty plasma. Phys. Plasmas 2014, 21, 103701. [Google Scholar] [CrossRef]
  37. Atteya, A.; Sultana, S.; Schlickeiser, R. Dust-ion-acoustic solitary waves in magnetized plasmas with positive and negative ions: The role of electrons superthermality. Chin. J. Phys. 2018, 56, 1931. [Google Scholar] [CrossRef]
  38. El-Nabulsi, R.A. Time-nonlocal kinetic equations, jerk and hyperjerk in plasmas and solar physics. Adv. Space Phys. 2018, 61, 2914. [Google Scholar] [CrossRef]
  39. El-Nabulsi, R.A. Non-standard magnetohydrodynamics equations and their implications in sunspots. Proc. R. Soc. A 2020, 476. [Google Scholar] [CrossRef]
  40. El-Labany, S.K.; Behery, E.E.; El-Razek, H.N.A.; Abdelrazek, L.A. Shock waves in magnetized electronegative plasma with nonextensive electrons. Eur. Phys. J. D 2020, 74, 104. [Google Scholar] [CrossRef]
  41. Ourabah, K.; Gougam, L.A.; Tribeche, M. Nonthermal and suprathermal distributions as a consequence of superstatistics. Phys. Rev. E 2015, 91, 012133. [Google Scholar] [CrossRef]
  42. Davis, S.; Avaria, G.; Bora, B.; Jain, J.; Morcao, J.; Pavez, C.; Soto, L. Single-particle velocity distributions of collisionless, steady-state plasmas must follow superstatistics. Phys. Rev. E 2019, 100, 023205. [Google Scholar] [CrossRef] [Green Version]
  43. Ourabah, K. Demystifying the success of empirical distributions in space plasmas. Phys. Rev. Res. 2020, 2, 023121. [Google Scholar] [CrossRef]
  44. Beck, C.; Cohen, E.G.D. Superstatistics. Phys. A Stat. Mech. Its Appl. 2003, 322, 267. [Google Scholar] [CrossRef] [Green Version]
  45. Hussain, S.; Mahmood, S. Korteweg-de Vries Burgers equation for magnetosonic wave in plasma. Phys. Plasmas 2011, 18, 052308. [Google Scholar] [CrossRef]
  46. Michael, M.; Willington, N.T.; Jayakumar, N.; Sebastian, S.; Sreekala, G.; Venugopal, C. Korteweg–deVries–Burgers (KdVB) equation in a five component cometary plasma with kappa described electrons and ions. J. Theor. Appl. Phys. 2016, 10, 289. [Google Scholar] [CrossRef] [Green Version]
  47. Lu, D.; Seadawy, A.; Yaro, D. Analytical wave solutions for the nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov and two-dimensional Kadomtsev-Petviashvili-Burgers equations. Res. Phys. 2019, 12, 2164. [Google Scholar] [CrossRef]
  48. Washimi, H.; Taniuti, T. Propagation of Ion-Acoustic Solitary Waves of Small Amplitude. Phys. Rev. Lett. 1966, 17, 996. [Google Scholar] [CrossRef]
  49. Hossen, M.M.; Nahar, L.; Alam, M.S.; Sultana, S.; Mamun, A.A. Electrostatic shock waves in a nonthermal dusty plasma with oppositely charged dust. High Energy Density Phys. 2017, 24, 9. [Google Scholar] [CrossRef]
  50. Shikha, R.K.; Orani, M.M.; Mamun, A.A. Roles of positively charged dust, ion fluid temperature, and nonthermal electrons in the formation of modified-ion-acoustic solitary and shock waves. Results Phys. 2021, 27, 104507. [Google Scholar] [CrossRef]
  51. Nakamura, Y.; Bailung, H.; Shukla, P.K. Observation of Ion-Acoustic Shocks in a Dusty Plasma. Phys. Rev. Lett. 1999, 83, 1602. [Google Scholar] [CrossRef]
  52. Karpman, V.I. Nonlinear Waves in Dispersive Media; Pergamon Press: Oxford, UK, 1975. [Google Scholar]
  53. Hasegawa, A. Plasma Instabilities and Nonlinear Effects; Springer: Berlin, Germany, 1975. [Google Scholar]
  54. Nakamura, Y.; Bailung, H.; Lonngren, K.E. Oblique collision of modified Korteweg–de Vries ion-acoustic solitons. Phys. Plasmas 1999, 6, 3466. [Google Scholar] [CrossRef]
  55. Oohara, W.; Date, D.; Hatakeyama, R. Electrostatic waves in a paired Fullerene-ion plasma. Phys. Rev. Lett. 2005, 95, 175003. [Google Scholar] [CrossRef] [PubMed]
  56. Jahan, S.; Sharmin, B.E.; Chowdhury, N.A.; Mannan, A.; Roy, T.S.; Mamun, A.A. Electrostatic ion-acoustic shock waves in a magnetized degenerate quantum plasma. Plasma 2021, 4, 426–434. [Google Scholar] [CrossRef]
  57. Akter, J.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Dust-acoustic envelope solitons in an electron-depleted plasma. Plasma Phys. Rep. 2021, 47, 725. [Google Scholar] [CrossRef]
  58. Islam, M.K.; Noman, A.A.; Akter, J.; Chowdhury, N.A.; Mannan, A.; Roy, T.S.; Salahuddin, M.; Mamun, A.A. Modulational instability of dust-ion-acoustic waves in pair-ion plasma having non-thermal non-extensive electrons. Contrib. Plasma Phys. 2021, 61, e202000214. [Google Scholar] [CrossRef]
  59. Banik, S.; Heera, N.M.; Yeashna, T.; Hassan, M.; Shikha, R.K.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Modulational instability of ion-acoustic waves and associated envelope solitons in a multi-component plasma. Gases 2021, 1, 148–155. [Google Scholar] [CrossRef]
  60. Alinejad, H. Dust ion-acoustic solitary and shock waves in a dusty plasma with non-thermal electrons. Astrophys. Space Sci. 2010, 327, 131. [Google Scholar] [CrossRef]
Figure 1. The variation of nonlinear coefficient A with α 4 (left panel) and Ψ with ζ for different values of δ (right panel) along with α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , λ p = 1.5 , λ e = 1.7 , λ d = 0.05 , α = 0.5 , and V p V p + .
Figure 1. The variation of nonlinear coefficient A with α 4 (left panel) and Ψ with ζ for different values of δ (right panel) along with α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , λ p = 1.5 , λ e = 1.7 , λ d = 0.05 , α = 0.5 , and V p V p + .
Gases 02 00002 g001
Figure 2. The variation of Ψ with ζ for different values of η under consideration of α 4 > α 4 c (left panel) and α 4 < α 4 c (right panel) along with α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , λ p = 1.5 , λ e = 1.7 , λ d = 0.05 , δ = 30 , α = 0.5 , U 0 = 0.01 , and V p V p + .
Figure 2. The variation of Ψ with ζ for different values of η under consideration of α 4 > α 4 c (left panel) and α 4 < α 4 c (right panel) along with α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , λ p = 1.5 , λ e = 1.7 , λ d = 0.05 , δ = 30 , α = 0.5 , U 0 = 0.01 , and V p V p + .
Gases 02 00002 g002
Figure 3. The variation of Ψ with ζ for different values of α (left panel) and λ p (right panel) under consideration of α 4 > α 4 c along with α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , α 4 = 1.5 , λ e = 1.7 , λ d = 0.05 , δ = 30 , η = 0.3 , U 0 = 0.01 , and V p V p + .
Figure 3. The variation of Ψ with ζ for different values of α (left panel) and λ p (right panel) under consideration of α 4 > α 4 c along with α 1 = 1.5 , α 2 = 0.05 , α 3 = 0.03 , α 4 = 1.5 , λ e = 1.7 , λ d = 0.05 , δ = 30 , η = 0.3 , U 0 = 0.01 , and V p V p + .
Gases 02 00002 g003
Figure 4. The variation of Ψ with ζ for different values of α 1 , i.e., α 1 > 1 (left panel) and α 1 , i.e., α 1 < 1 (right panel) under consideration of α 4 > α 4 c along with α 3 = 0.03 , α 4 = 1.5 , λ e = 1.7 , λ p = 1.5 , λ d = 0.05 , δ = 30 , η = 0.3 , α = 0.5 , U 0 = 0.01 , and V p V p + .
Figure 4. The variation of Ψ with ζ for different values of α 1 , i.e., α 1 > 1 (left panel) and α 1 , i.e., α 1 < 1 (right panel) under consideration of α 4 > α 4 c along with α 3 = 0.03 , α 4 = 1.5 , λ e = 1.7 , λ p = 1.5 , λ d = 0.05 , δ = 30 , η = 0.3 , α = 0.5 , U 0 = 0.01 , and V p V p + .
Gases 02 00002 g004
Figure 5. The variation of Ψ with ζ for different values of α 1 , i.e., α 1 > 1 (left panel) and α 1 , i.e., α 1 < 1 (right panel) under consideration of α 4 < α 4 c along with α 3 = 0.03 , α 4 = 1.5 , λ e = 1.7 , λ p = 1.5 , λ d = 0.05 , δ = 30 , η = 0.3 , α = 0.5 , U 0 = 0.01 , and V p V p + .
Figure 5. The variation of Ψ with ζ for different values of α 1 , i.e., α 1 > 1 (left panel) and α 1 , i.e., α 1 < 1 (right panel) under consideration of α 4 < α 4 c along with α 3 = 0.03 , α 4 = 1.5 , λ e = 1.7 , λ p = 1.5 , λ d = 0.05 , δ = 30 , η = 0.3 , α = 0.5 , U 0 = 0.01 , and V p V p + .
Gases 02 00002 g005
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Jahan, S.; Banik, S.; Chowdhury, N.A.; Mannan, A.; Mamun, A.A. Electrostatic Shock Structures in a Magnetized Plasma Having Non-Thermal Particles. Gases 2022, 2, 22-32. https://doi.org/10.3390/gases2020002

AMA Style

Jahan S, Banik S, Chowdhury NA, Mannan A, Mamun AA. Electrostatic Shock Structures in a Magnetized Plasma Having Non-Thermal Particles. Gases. 2022; 2(2):22-32. https://doi.org/10.3390/gases2020002

Chicago/Turabian Style

Jahan, Sharmin, Subrata Banik, Nure Alam Chowdhury, Abdul Mannan, and A A Mamun. 2022. "Electrostatic Shock Structures in a Magnetized Plasma Having Non-Thermal Particles" Gases 2, no. 2: 22-32. https://doi.org/10.3390/gases2020002

Article Metrics

Back to TopTop