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Nonthermal effects on the cylindrical dusty ion shocks in nonthermal viscous space plasma
Journal Pre-proofs Nonthermal effects on the cylindrical dusty ion shocks in nonthermal viscous space plasma Abdelwahed, E.K. El-Shewy, A.A. El-Rahman, N.F. Abdo PII: DOI: Reference:
S0273-1177(19)30729-X https://doi.org/10.1016/j.asr.2019.09.051 JASR 14476
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Advances in Space Research
Received Date: Accepted Date:
19 December 2018 28 September 2019
Please cite this article as: Abdelwahed, El-Shewy, E.K., El-Rahman, A.A., Abdo, N.F., Nonthermal effects on the cylindrical dusty ion shocks in nonthermal viscous space plasma, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.09.051
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Nonthermal eects on the cylindrical dusty ion shocks in nonthermal viscous space plasma , E. K. El-ShewyW2>3 , A. A. El-Rahman4 , N. F. Abdo 2>3 1 College of science and Humanitarian Studies, Physics Department, Prince Sattam Bin Abdul Aziz University, Kingdom of Saudi Arabia 2 Taibah University, Al-Madinah Al-Munawarrah, Kingdom of Saudi Arabia. 3 Theoretical Physics Group, Faculty of Science, Mansoura University, Mansoura, Egypt. 4 Department of Physics, Faculty of Science, Assiut University, New Valley Branch, New Valley, Egypt. Abdelwahed
1> 3
June 22, 2019
Abstract The propagating cylindrical shock dust ion wave (CDISW) in dusty four component plasma with three viscous component (ion and two polarity charged grains) has been introduced. The three dimensional (3D) Cylindrical Burgers (CB) equation is derived. The propagating cylindrical shock characteristics are established to becomes a very signicantly improved by the supports of electron nonthermality, ion and negative (positive) kinematics viscosity coecients. Furthermore, the shock strength depends on cylindrical directions. The obtained results may be protable in understanding both the laboratory and space applications of plasmas.. Keywords: Dust plasmas; Cylindrical Burgers; dust cylindrical ion acoustic shocks; nonthermal electrons.
W E_Mail:
[email protected]. [email protected]
1
1
Introduction
The study of nonlinear excitations has received a worldwide attention in the ineld of laboratory and space plasmas (Sahu et al., 2013; Barkan et al, 1996; Saini et al., 2013; Rao et al., 1990; Saha and Chatterjee, 2014; Sijo et al., 2015; Sreekala et al., 2017; Horanyi et al., 1993; Adnan et al., 2018). Merlino et al. (1998) presented a Laboratory study for wave forms and instabilities regions in dusty plasmas. The charging and wave progress in dust space plasmas investigations reported the importantance of dusty plasma applications in the last decades (Barkan and Merlino, 1995; Horanyi, M., Mendis, 1986 ; Nakamura et al. 1997; Nejoh 1997;Klumov et al., 2000; Mamun, 2008; Zaghbeer et al, 2018). The instability eects in nonextensive two polarities dusty plasmas have been examined (Zaghbeer et al 2018). After theoretical forecasting and laboratory observation of DIAWS existence (Shukla and Siline 1992; Barkan and Merlino, 1995) extensive investigations have been developed to discussing the features and applications of these wave types (Moslem et al. 2005; Banerjee and Maitra, 2016; Samanta et al, 2013; Elwakil et al. 2011; Baluku et al. 2010). Elwakil et al. (2011) introduced the non adiabatic eect of charge uctuations on the wave features in inhomogeneous dusty plasmas. It was noted that the dissipation variable coecient Burger shock term is obtained. Ion acoustic propagation in normal and dust Plasmas have been observed by many authors (Luo et al., 1999; Sarma and Nakamura, 2009; Popel et al., 2005; Nakamura et al., 1999). Luo et al. (1999) deliberated experimentally the shock produced in the subsistence of grain particles. In another experiment, Sarma and Nakamura (2009) studied the IASW characteristics in the dusty plasma containing ions of negative charge. Also, The Dissipative propagation in in Q machine devices of IA shocks and its properties have bed reported (Popel et al 2005). The Contribution of dressing nonlinear DIA properties of inhomogeneous dusty mesospheric plasma has been investigated (Attia et al 2010). Furthermore, shock characteristics of dust media were studied for dierent physical parameters and distributions (El-Shewy et al., 2011; Popel and Tsytovich, 1999; Abdelwahed, 2015; Hussain et al., 2013; El-Hanbaly et al., 2015; El-Hanbaly et al. 2016; Abdelwahed 2015). Since Cairns et al. (1995) introduced the fast nonthermal population distribution, many publications in many environmental shocks have been investigated (El-Taibany et al. 2005; Elwakil et al. 2014; Saha et al. 2014; El-Shewy et al. 2017; Hossen et al. 2017). On the other hand, the shock generation in many dierent geometries have been studied (Mannan and Mamun, 2011; Hossen et al., 2014; El-Shewy et al., 2015; Jannat et al., 2015; Zaman et al., 2017; Dev et. al., 2015a; Ghai and
Saini, 2017). Ghai and Saini investigated shock acoustic formation in dusty plasmas with two ions temperature and depletion of electrons corresponding to Earth magnetotail. The inuences of superthermal two 2
ions and dust viscosity on the DA characteristics of forming shock at critical nonlinearity coecient have been examined. It was noted that the wave strength for Burgers equation in normal case decreases whereas it increases in critical case with superthermal indexes (Ghai and Saini, 2017). Dev et. al studied dusty model having dierent ion temperatures and uctuated charge of dusty particles. The shock forming solution has been derived via tanh and its modied method. They constructed that, in higher nonlinearity, the shock extent reduces forever and constant width is produced for all parameters that affect the domain of that investigation (Dev et al., 2015b). Sahu and Tribeche (2012) studied the dependence of nonextensive shock excitations in nonplanar form. The cylindrical time damping superthermal solitons plasmas has been reported (El-Shewy et al., 2018 a). The non-planar form of acoustic nucleus shock structures in degenerate plasma systems has been examined (Zaman et al., 2017). In this study, we aim to examine the electrostatic nonlinear cylindrical dust ionic shock (CDIS) in four dusty component plasma having three viscous components namely, ion and two polarities charged grains.
2
Model Equations and shock solutions
Consider unmagnetized collisionless dusty four component plasma with viscous ion, dust uids of negatively and positively charge and nonthermal electrons. Basic uid equations given as: Cql + =(ql xl ) = 0> Cw Cq + =(q x ) = 0> Cw Cq+ + =(q+ x+ ) = 0= Cw
(1)
Momentum equations read: Cxl + (xl =) xl + ! = l 2 xl > Cw
Cx + (x =) x ! = 2 x > Cw Cx+ + (x+ =) x+ + ! = + 2 x+ = Cw
(2)
Poisson equation coupled last equations as: q l ql + q+ + qh = 0>
3
(3)
and qh in nonthermal form reads: qh
¢ ¡ 1 ! + !2 exp (!) > = 4( )= 1 + 3
=
(4)
In equations(1-4), the quantities ql > q > q+ > qh (xl > x > x+ ) are the densities (velocities) of ions, negative grains, positive grains that normalized by ³ ´1@2 ¢ ¡ (0) (0) (0) E Wh ql > q > q+ > qh0 Np , potential ! is normalized with NEhWh > wlph and l ¢1@2 ¢1@2 ¡ ¡ 2 vsdfh are normalized using $ 1 and g = NE Wh @4 h2 qh0 > sh = pl @4 h qh0 respectively, = ] pl @p and = ]+ pl @p+ = Here, NE and Wh are the Boltzmann constant and the temperature of electron, h is electro charge,Also, pm are the masses with p+ ¿ p , is measured of electron nonthermality. Neutrality charge conditions read, + 1 = l + +> (0)
(0)
(0)
(0)
(5) (0)
(0)
with l = ql @qh > = ] q @qh and + = ]+ q+ @qh where ] and ]+ are charge numbers of grains, respectively. Also, l > ( + ) are the ion, negative (positive) kinematics viscosity coecients of positively and negatively charged dusty grain.
2.1
Nonlinear shock analysis
By the method of reductive-perturbation , one can insert the nonplanar stretched co-ordinates with small dimensionless parameter : U = 1@2 (u w) >
= 1@2 >
] = }>
and
W = 3@2 w> (6)
where is phase speed and determined later. All the physical quantities in this model is expanded in as: (1)
(2)
ql>>+ = 1 + ql>>+ + 2 ql>>+ + ===> (1)
(2)
xm = (xl>>+ + xl>>+ + ===)> (1)
(2)
ym = 3@2 (yl>>+ + yl>>+ + ===)> (1)
(2)
zm = 3@2 (zl>>+ + zl>>+ + ===)> ! = (!(1) + !(2) + === )>
(7)
where xl>>+ > yl>>+ and zl>>+ are the velocities in directions of U> > dqg ]. The value m (m = l> > +) is assumed to be very small values, so, its values 1 given as m = 2 m0 , with m0 is denite values. To deduce Burgers equation, the following conditions must be satised, || > qm = 1> xm = ym = zm = 0> ! = 0= 4
(8)
Substitution of equations (6) and (7) in equations (1-4), equating the coefcients of the same powers. The lowest-order relations for ion can be written as: 1 1 (1) (1) xl = !(1) > (9.a) ql = 2 !(1) > (1)
(1)
Cyl 1 C!(1) Czl 1 C!(1) = > = > 2 CU CU C] W C whereas for negative dust are given by (1)
q =
(1) ! > 2
(1)
x =
(1)
Cy C!(1) = > CU W 2 C
(1) ! >
(9.b)
(10.a)
(1)
Cz C!(1) = > CU C]
(10.b)
(1) ! >
(11.a)
and for (m = +): (1)
q+ =
(1) ! > 2
(1)
x+ =
(1)
Cy+ C!(1) = > CU W 2 C
(1)
Cz+ C!(1) = = CU C]
(11.b)
Linear dispersion relation reads l + + + 2 + 2 = 0=
(12)
By using the R(2 ) equations, one gets: (1) (2) (1) (1) (1) ´ C ³ (1) (1) x Czl 1 Cyl Cql Cq (2) + + = 0> l + l + ql xl + xl CW CU CU W C] W C (2) (1) (1) (1) Cx Cx C!(2) C 2 xl (1) Cxl l + l + xl + = l0 > CU CW CU CUÃ CU2 ! (2) (1) (1) (1) Cyl 1 U C!(1) Cyl C 2 yl C!(2) (1) Cyl + + xl + > = l0 CU CW CU W C W C CU2
(2)
(1)
Czl Czl + CU CW
(1) (1) Czl
+ xl
CU
+
(1)
C 2 zl C!(2) = l0 > C] CU2
5
(13)
(2) (1) (1) (1) ´ Cq(1) x Cz Cq C ³ (1) (1) 1 Cy (2) + + + = 0> q x + x + CU CU CW W W C C] (1) (2) (1) (1) Cx Cx C 2 x C!(2) (1) Cx + + x = 0 > CW CU CU CU CU2 Ã ! (2) (1) (1) (1) Cy Cy C 2 y U C!(1) C!(2) (1) Cy + + x = > + 0 CU CW CU W C W C CU2
(2)
(1)
(1)
(1)
Cz Cz C 2 z C!(2) (1) Cz + + x = 0 > CU CW CU C] CU2
(14)
(2) (1) (1) (1) ´ x(1) Cq+ Cz+ Cq+ C ³ (1) (1) 1 Cy+ (2) + + + = 0> q+ x+ + x+ + + + CU CW CU W W C C] (2) (1) (1) (1) Cx Cx C 2 x+ C!(2) (1) Cx+ + + x+ + + + = +0 > CU CU CU CW CU2 Ã ! (2) (1) (1) (1) Cy+ Cy+ C 2 y+ U C!(1) C!(2) (1) Cy+ + + x+ + > = +0 CU CW CU W C W C CU2
(1)
(2)
(1)
(1)
Cz C 2 z+ Cz+ C!(2) (1) Cz+ + + x+ + = +0 > CW CU CU C] CU2
(15)
And 1 ³ (1) ´2 =0 (16) ! 2 With aid of the obtained equation (9-12) and eliminating second order quantities in equations (13-16), we can obtain the 3D-Cylindrical Burgers equation: (2)
(2)
q l ql
C CU
Ã
(2)
+ q+ + (1 ) !(2) +
!(1) C 2 !(1) C!(1) C!(1) + + d !(1) e CW 2W CU CU2
1 C 2 !(1) C 2 !(1) + = 0> 2 C] 2 2W 2 C2
! + (17)
where
d = e =
3( l 2 + + 2 4 @3) > 23 (1 ) ¢ ¡ l l0 + 0 + + +0 22 (1 )
6
(18.a) (18.b)
By the transformation: p = ] 1 O2u + U Ou = > ¢ ¡ 2 O2u + 1 O2u = = 2Ou
W
(19)
Equation (19) can be transformed to C 2 !(1) C!(1) C!(1) + D!(1) E = 0> C C C 2 D = d Ou > E = e O2u =
(20)
Where Ou is the direction cosine in U direction. Equation (20) can X0 be solved under the transformation " = 2E ( X0 ) (Ghai and Saini, 2017) and nally by using the Tanh analytical method (Maliet and Hereman, 1996), the cylindrical shock is obtained in the form: !(1) =
X0 X0 (1 tanh ( ( X 0 ))= d Ou 2e O2u
(21)
Where X0 is the speed of shock.
3
Results and Discussions
Nonlinear cylindrical dust ion shock (CDIS) in dusty four component plasma with viscous ion has been introduced. The 3D-Cylindrical Burgers equation is obtained and solved to give the propagating cylindrical shocks. The physical eects of ionic and two types grains kinematic viscosity coecients on the properties of the cylindrical shocks are studied. To perform our results, numerical investigation was performed according to mesosphere parameters (Popel et al., 2005; Zaghbeer et al., 2018; El-Shewy et al., 2018a). To show the contribution of nonthermality on the shock formation, the shock amplitude and width against the parameter of nonthermal electron are plotted in Figs. 1-2. Generally, It was conrmed that this model supports two shock waves namely negative and positive cylindrical shocks. Fig. 1 show that the positive shock amplitude raises with parameter of nonthermality up to a critical point and it began to be reduced with for negative shocks. The width of shock decreases with as in Fig. 2. However, one of our goals was to examine the uid viscosity and cylindrical coordinate eects on the shock formation in this model. We have examined the eects of some parameters such the ion, negative (positive) charged grain kinematics viscosity 7
coecients l > (+ )> the direction cosine Ou and the dependence parameter on the cylindrical shock features. For example, Figs. 3 concern the eect of l on the shock prole. It was noted that, l increases the shock steepness while the strength of the shock still unvaried. In Fig. 4, the shock amplitude variation against + and Ou is depicted for X0 = 0=55> = 0=85> = 0=002, = 0=005 in the two regions of shock polarity i.e. = 0=2 for positive shock and = 0=38 for negative shock. It was illustrated that, both + and Ou decrease the amplitude and shock strength of positive shock while Ou decrease the amplitude and shock strength of negative shock and + increase the amplitude and shock strength of negative shock. On the other hand, + and Ou increase the width of both positive and negative shocks as depicted in Fig. 5. Also, in Fig. 6 the positive (negative) kinematics viscosity coecients + and increase the width and steepness of each polarity shocks. On the other hand, the two types of mesospheric cylindrical shock formation with all coordinate information are depicted in Figs. 7. In apsence of some parameters, this investigation concedes with other studies in interpret the nonlinear wave pictures in the opposite polarities plasmas (Mendis and Rosenberg, 1994; El-Shewy and El-Rahman, 2018b; Zaghbeer et al., 2014; Rahman et al., 20008). In conclusions, It was conrmed that all model parameters such as the nonthermality of electron > ion, negative (positive) kinematics viscosity coecients of positively and negatively charged grain l > (+ )> the directions Ou and parameters aected the mesosphere cylindrical shock characteristics. Our returns from this study may be of numerous importance in distinguishing the structure properties of shock formation mechanism and the comparison between nonlinear formations of waves in mesosphere space (Popel et al., 2005; Zaghbeer et al., 2018; El-Shewy et al., 2018a).
8
Acknowledgment This project was supported by the deanship of scientic research at Prince Sattam Bin Abdulaziz University under the research project No. 10259/01/2019.
Figure caption: Fig. 1. The amplitude variation against for X0 = 0=55> = 0=85> + = 1=6> = 0=02, = 0=005> Ou = 0=5 and = 0=5= Fig. 2. The width variation against for X0 = 0=55> = 0=85> + = 1=6> = 0=02, = 0=005> Ou = 0=5, = 0=5> l0 = 0=05> 0 = 0=4 and +0 = 0=15= Fig. 3. Variation of solution (21) with for various values of l0 and for X0 = 0=55> = 0=02, = 0=005> = 0=85> + = 1=5> Ou = 0=5> = 0=5> ] = 0=5> U = 0=5> +0 = 0=5> 0 = 0=7 and (a) = 0=2, (b) = 0=38= Fig.4. The amplitude variation against Ou for various values of + and X0 = 0=55> = 0=85> = 0=002, = 0=005 and = 0=2 in positive region and = 0=38 in negative region= Fig.5. The width variation against Ou for various values of + and X0 = 0=55> = 0=85> = 0=002, = 0=005 and = 0=2 in positive region and = 0=38 in negative region= Fig. 6. The width variation against +0 for various values of 0 and for X0 = 0=55> = 0=85> + = 1=8> = 0=002, = 0=005> Ou = 0=5> = 0=5> l0 = 0=05 and = 0=2. Fig. 7. Three dimensional wave prole !(1) (21) against U and at = 1=0 and ] = 0=5 for X0 = 0=55> Ou = 0=5> = 0=002> = 0=005> = 0=85> + = 1=5> +0 = 0=5> 0 = 0=7> l0 = 0=2 and (a) = 0=2, (b) = 0=4=
9
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[46] Samanta, U. K., Saha, A., Chatterjee, P., 2013. Bifurcations of dust ion acoustic travelling waves in a magnetized quantum dusty plasma. Phys. Plasma 20, 052111. [47] Sarma, A., Nakamura, Y., 2009. Ion-acoustic shock waves with negative ions in presence of dust particulates. Phys. Lett. A 373, 4174. [48] Shukla, P., Siline, V., 1992, Dust ion-acoustic wave, Phys. Scr. 3, 508. [49] Sijo,S., Michael, M., Sreekala, G., Neethu, T., Renuka, G., Venugopal, C., 2015. Phys. Plasmas, Oblique solitary waves in a ve component plasma 22, 123704. [50] Sreekala, G. , Varghese, A., Jayakumar, N., Michael, M., Sebastian, S., Venugopal, C., 2017. Stability of the magnetosonic wave in a cometary multi-ion plasma, Advances in Space Research 59, 2679. [51] Zaghbeer, S.K., El-Shewy, E.K., El-Hanbaly, Salah, H.H., H.Sheta, N., ElShewy, E.K., Elgarayhi, A., 2018. On the growth rate instability of nonextensively opposite polarity dusty plasmas. Advances in Space Research 62, 1728. [52] Zaman, D.M.S., Amina, M, Dip, P.R., Mamun, A.A., 2017. Planar and non-planar nucleus-acoustic shock structures in self-gravitating degenerate quantum plasma systems, Eur. Phys. J. Plus 132, 457.
13
Amplitude
U
۴ܑǤ Ǥ ߩܷ ൌ ͲǤͷͷǡ ߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤǡ ߙ ൌ ͲǤͲʹǡߤ ൌ ͲǤͲͲͷǡ ܮ ൌ ͲǤͷܽ݊݀ȣ ൌ െͲǤͷǤ
Width
U
۴ܑǤ Ǥ ߩߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤǡ ߙ ൌ ͲǤͲʹǡ ܷ ൌ ͲǤͷͷǡ ߤ ൌ ͲǤͲͲͷǡ ܮ ൌ ͲǤͷǡ ȣ ൌ െͲǤͷǡ ߟ ൌ ͲǤͲͷǡ ߟି ൌ ͲǤͶߟା ൌ ͲǤͳͷǤ
I +1/
KL KL KL
W
Fig. 3.a.
KL KL KL
I +1/
W Fig. 3.b. ۴ܑǤ Ǥ ሺʹͳሻɒߟ ߙ ൌ ͲǤͲͲʹǡ ߤ ൌ ͲǤͲͲͷǡ ߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤͷǡ ܮ ൌ ͲǤͷǡ ȣ ൌ െͲǤͷǡ ܼ ൌ ͲǤͷǡ ܴ ൌ ͲǤͷǡ ߟା ൌ ͲǤͷǡ ߟି ൌ ͲǤǡ ܷ ൌ ͲǤͷͷሺሻߩ ൌ ͲǤʹǡ ሺܾሻߩ ൌ ͲǤͶǤ
Amplitude
G G G G G G
Lr
۴ܑǤ Ǥ ܮ ߜା ߜି ൌ ͲǤͺͷǡ ܷ ൌ ͲǤͷͷǡ ߙ ൌ ͲǤͲͲʹǡ ߤ ൌ ͲǤͲͲͷߩ ൌ ͲǤʹ݅݊݀݊ܽ݊݅݃݁ݎ݁ݒ݅ݐ݅ݏ ߩ ൌ ͲǤ͵ͺ݅݊݊݁݃ܽ݊݅݃݁ݎ݁ݒ݅ݐǤ
G G
G
Width
Lr ۴ܑǤ Ǥ ܮ ߜା ߜି ൌ ͲǤͺͷǡ ܷ ൌ ͲǤͷͷǡ ߙ ൌ ͲǤͲͲʹǡ ߤ ൌ ͲǤͲͲͷሺሻߩ ൌ ͲǤʹߩ ൌ ͲǤ͵ͺ݅݊݊݁݃ܽ݊݅݃݁ݎ݁ݒ݅ݐǤ
Width
K
K
K
K 0
۴ܑǤ Ǥ ߟା ߟି ߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤͺǡ ߙ ൌ ͲǤͲͲʹǡ ܷ ൌ ͲǤͷͷǡ ߤ ൌ ͲǤͲͲͷǡ ܮ ൌ ͲǤͷǡ ȣ ൌ െͲǤͷǡ ߟ ൌ ͲǤͲͷߩ ൌ ͲǤʹǤ
Fig. 7.a. 7
Fig. 7.b. 7 ۴ܑǤ ૠǤ Ԅሺଵሻ ሺʹ ʹͳሻ ܴȣɒ ൌ ͳǤͲ ܮ ൌ ͲǤͷǡ ߙ ൌ ͲǤͲͲʹ ʹǡ ߤ ൌ ͲǤͲͲ Ͳͷǡ ߜି ൌ ͲǤͺ ͺͷǡ ߜା ൌ ͳǤͷ ͷǡ ൌ ͲǤͷ ߟା ൌ ͲǤͷǡ ߟି ൌ ͲǤǡ ߟ ൌ ͲǤʹǡ ܷ ൌ ͲǤͷͷ ሺሻߩ ൌ ͲǤʹ ʹǡ ሺܾሻߩ ൌ ͲǤͶǤ Ͳ
S0273-1177(19)30729-X https://doi.org/10.1016/j.asr.2019.09.051 JASR 14476
To appear in:
Advances in Space Research
Received Date: Accepted Date:
19 December 2018 28 September 2019
Please cite this article as: Abdelwahed, El-Shewy, E.K., El-Rahman, A.A., Abdo, N.F., Nonthermal effects on the cylindrical dusty ion shocks in nonthermal viscous space plasma, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.09.051
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Nonthermal eects on the cylindrical dusty ion shocks in nonthermal viscous space plasma , E. K. El-ShewyW2>3 , A. A. El-Rahman4 , N. F. Abdo 2>3 1 College of science and Humanitarian Studies, Physics Department, Prince Sattam Bin Abdul Aziz University, Kingdom of Saudi Arabia 2 Taibah University, Al-Madinah Al-Munawarrah, Kingdom of Saudi Arabia. 3 Theoretical Physics Group, Faculty of Science, Mansoura University, Mansoura, Egypt. 4 Department of Physics, Faculty of Science, Assiut University, New Valley Branch, New Valley, Egypt. Abdelwahed
1> 3
June 22, 2019
Abstract The propagating cylindrical shock dust ion wave (CDISW) in dusty four component plasma with three viscous component (ion and two polarity charged grains) has been introduced. The three dimensional (3D) Cylindrical Burgers (CB) equation is derived. The propagating cylindrical shock characteristics are established to becomes a very signicantly improved by the supports of electron nonthermality, ion and negative (positive) kinematics viscosity coecients. Furthermore, the shock strength depends on cylindrical directions. The obtained results may be protable in understanding both the laboratory and space applications of plasmas.. Keywords: Dust plasmas; Cylindrical Burgers; dust cylindrical ion acoustic shocks; nonthermal electrons.
W E_Mail:
[email protected]. [email protected]
1
1
Introduction
The study of nonlinear excitations has received a worldwide attention in the ineld of laboratory and space plasmas (Sahu et al., 2013; Barkan et al, 1996; Saini et al., 2013; Rao et al., 1990; Saha and Chatterjee, 2014; Sijo et al., 2015; Sreekala et al., 2017; Horanyi et al., 1993; Adnan et al., 2018). Merlino et al. (1998) presented a Laboratory study for wave forms and instabilities regions in dusty plasmas. The charging and wave progress in dust space plasmas investigations reported the importantance of dusty plasma applications in the last decades (Barkan and Merlino, 1995; Horanyi, M., Mendis, 1986 ; Nakamura et al. 1997; Nejoh 1997;Klumov et al., 2000; Mamun, 2008; Zaghbeer et al, 2018). The instability eects in nonextensive two polarities dusty plasmas have been examined (Zaghbeer et al 2018). After theoretical forecasting and laboratory observation of DIAWS existence (Shukla and Siline 1992; Barkan and Merlino, 1995) extensive investigations have been developed to discussing the features and applications of these wave types (Moslem et al. 2005; Banerjee and Maitra, 2016; Samanta et al, 2013; Elwakil et al. 2011; Baluku et al. 2010). Elwakil et al. (2011) introduced the non adiabatic eect of charge uctuations on the wave features in inhomogeneous dusty plasmas. It was noted that the dissipation variable coecient Burger shock term is obtained. Ion acoustic propagation in normal and dust Plasmas have been observed by many authors (Luo et al., 1999; Sarma and Nakamura, 2009; Popel et al., 2005; Nakamura et al., 1999). Luo et al. (1999) deliberated experimentally the shock produced in the subsistence of grain particles. In another experiment, Sarma and Nakamura (2009) studied the IASW characteristics in the dusty plasma containing ions of negative charge. Also, The Dissipative propagation in in Q machine devices of IA shocks and its properties have bed reported (Popel et al 2005). The Contribution of dressing nonlinear DIA properties of inhomogeneous dusty mesospheric plasma has been investigated (Attia et al 2010). Furthermore, shock characteristics of dust media were studied for dierent physical parameters and distributions (El-Shewy et al., 2011; Popel and Tsytovich, 1999; Abdelwahed, 2015; Hussain et al., 2013; El-Hanbaly et al., 2015; El-Hanbaly et al. 2016; Abdelwahed 2015). Since Cairns et al. (1995) introduced the fast nonthermal population distribution, many publications in many environmental shocks have been investigated (El-Taibany et al. 2005; Elwakil et al. 2014; Saha et al. 2014; El-Shewy et al. 2017; Hossen et al. 2017). On the other hand, the shock generation in many dierent geometries have been studied (Mannan and Mamun, 2011; Hossen et al., 2014; El-Shewy et al., 2015; Jannat et al., 2015; Zaman et al., 2017; Dev et. al., 2015a; Ghai and
Saini, 2017). Ghai and Saini investigated shock acoustic formation in dusty plasmas with two ions temperature and depletion of electrons corresponding to Earth magnetotail. The inuences of superthermal two 2
ions and dust viscosity on the DA characteristics of forming shock at critical nonlinearity coecient have been examined. It was noted that the wave strength for Burgers equation in normal case decreases whereas it increases in critical case with superthermal indexes (Ghai and Saini, 2017). Dev et. al studied dusty model having dierent ion temperatures and uctuated charge of dusty particles. The shock forming solution has been derived via tanh and its modied method. They constructed that, in higher nonlinearity, the shock extent reduces forever and constant width is produced for all parameters that affect the domain of that investigation (Dev et al., 2015b). Sahu and Tribeche (2012) studied the dependence of nonextensive shock excitations in nonplanar form. The cylindrical time damping superthermal solitons plasmas has been reported (El-Shewy et al., 2018 a). The non-planar form of acoustic nucleus shock structures in degenerate plasma systems has been examined (Zaman et al., 2017). In this study, we aim to examine the electrostatic nonlinear cylindrical dust ionic shock (CDIS) in four dusty component plasma having three viscous components namely, ion and two polarities charged grains.
2
Model Equations and shock solutions
Consider unmagnetized collisionless dusty four component plasma with viscous ion, dust uids of negatively and positively charge and nonthermal electrons. Basic uid equations given as: Cql + =(ql xl ) = 0> Cw Cq + =(q x ) = 0> Cw Cq+ + =(q+ x+ ) = 0= Cw
(1)
Momentum equations read: Cxl + (xl =) xl + ! = l 2 xl > Cw
Cx + (x =) x ! = 2 x > Cw Cx+ + (x+ =) x+ + ! = + 2 x+ = Cw
(2)
Poisson equation coupled last equations as: q l ql + q+ + qh = 0>
3
(3)
and qh in nonthermal form reads: qh
¢ ¡ 1 ! + !2 exp (!) > = 4( )= 1 + 3
=
(4)
In equations(1-4), the quantities ql > q > q+ > qh (xl > x > x+ ) are the densities (velocities) of ions, negative grains, positive grains that normalized by ³ ´1@2 ¢ ¡ (0) (0) (0) E Wh ql > q > q+ > qh0 Np , potential ! is normalized with NEhWh > wlph and l ¢1@2 ¢1@2 ¡ ¡ 2 vsdfh are normalized using $ 1 and g = NE Wh @4 h2 qh0 > sh = pl @4 h qh0 respectively, = ] pl @p and = ]+ pl @p+ = Here, NE and Wh are the Boltzmann constant and the temperature of electron, h is electro charge,Also, pm are the masses with p+ ¿ p , is measured of electron nonthermality. Neutrality charge conditions read, + 1 = l + +> (0)
(0)
(0)
(0)
(5) (0)
(0)
with l = ql @qh > = ] q @qh and + = ]+ q+ @qh where ] and ]+ are charge numbers of grains, respectively. Also, l > ( + ) are the ion, negative (positive) kinematics viscosity coecients of positively and negatively charged dusty grain.
2.1
Nonlinear shock analysis
By the method of reductive-perturbation , one can insert the nonplanar stretched co-ordinates with small dimensionless parameter : U = 1@2 (u w) >
= 1@2 >
] = }>
and
W = 3@2 w> (6)
where is phase speed and determined later. All the physical quantities in this model is expanded in as: (1)
(2)
ql>>+ = 1 + ql>>+ + 2 ql>>+ + ===> (1)
(2)
xm = (xl>>+ + xl>>+ + ===)> (1)
(2)
ym = 3@2 (yl>>+ + yl>>+ + ===)> (1)
(2)
zm = 3@2 (zl>>+ + zl>>+ + ===)> ! = (!(1) + !(2) + === )>
(7)
where xl>>+ > yl>>+ and zl>>+ are the velocities in directions of U> > dqg ]. The value m (m = l> > +) is assumed to be very small values, so, its values 1 given as m = 2 m0 , with m0 is denite values. To deduce Burgers equation, the following conditions must be satised, || > qm = 1> xm = ym = zm = 0> ! = 0= 4
(8)
Substitution of equations (6) and (7) in equations (1-4), equating the coefcients of the same powers. The lowest-order relations for ion can be written as: 1 1 (1) (1) xl = !(1) > (9.a) ql = 2 !(1) > (1)
(1)
Cyl 1 C!(1) Czl 1 C!(1) = > = > 2 CU CU C] W C whereas for negative dust are given by (1)
q =
(1) ! > 2
(1)
x =
(1)
Cy C!(1) = > CU W 2 C
(1) ! >
(9.b)
(10.a)
(1)
Cz C!(1) = > CU C]
(10.b)
(1) ! >
(11.a)
and for (m = +): (1)
q+ =
(1) ! > 2
(1)
x+ =
(1)
Cy+ C!(1) = > CU W 2 C
(1)
Cz+ C!(1) = = CU C]
(11.b)
Linear dispersion relation reads l + + + 2 + 2 = 0=
(12)
By using the R(2 ) equations, one gets: (1) (2) (1) (1) (1) ´ C ³ (1) (1) x Czl 1 Cyl Cql Cq (2) + + = 0> l + l + ql xl + xl CW CU CU W C] W C (2) (1) (1) (1) Cx Cx C!(2) C 2 xl (1) Cxl l + l + xl + = l0 > CU CW CU CUÃ CU2 ! (2) (1) (1) (1) Cyl 1 U C!(1) Cyl C 2 yl C!(2) (1) Cyl + + xl + > = l0 CU CW CU W C W C CU2
(2)
(1)
Czl Czl + CU CW
(1) (1) Czl
+ xl
CU
+
(1)
C 2 zl C!(2) = l0 > C] CU2
5
(13)
(2) (1) (1) (1) ´ Cq(1) x Cz Cq C ³ (1) (1) 1 Cy (2) + + + = 0> q x + x + CU CU CW W W C C] (1) (2) (1) (1) Cx Cx C 2 x C!(2) (1) Cx + + x = 0 > CW CU CU CU CU2 Ã ! (2) (1) (1) (1) Cy Cy C 2 y U C!(1) C!(2) (1) Cy + + x = > + 0 CU CW CU W C W C CU2
(2)
(1)
(1)
(1)
Cz Cz C 2 z C!(2) (1) Cz + + x = 0 > CU CW CU C] CU2
(14)
(2) (1) (1) (1) ´ x(1) Cq+ Cz+ Cq+ C ³ (1) (1) 1 Cy+ (2) + + + = 0> q+ x+ + x+ + + + CU CW CU W W C C] (2) (1) (1) (1) Cx Cx C 2 x+ C!(2) (1) Cx+ + + x+ + + + = +0 > CU CU CU CW CU2 Ã ! (2) (1) (1) (1) Cy+ Cy+ C 2 y+ U C!(1) C!(2) (1) Cy+ + + x+ + > = +0 CU CW CU W C W C CU2
(1)
(2)
(1)
(1)
Cz C 2 z+ Cz+ C!(2) (1) Cz+ + + x+ + = +0 > CW CU CU C] CU2
(15)
And 1 ³ (1) ´2 =0 (16) ! 2 With aid of the obtained equation (9-12) and eliminating second order quantities in equations (13-16), we can obtain the 3D-Cylindrical Burgers equation: (2)
(2)
q l ql
C CU
Ã
(2)
+ q+ + (1 ) !(2) +
!(1) C 2 !(1) C!(1) C!(1) + + d !(1) e CW 2W CU CU2
1 C 2 !(1) C 2 !(1) + = 0> 2 C] 2 2W 2 C2
! + (17)
where
d = e =
3( l 2 + + 2 4 @3) > 23 (1 ) ¢ ¡ l l0 + 0 + + +0 22 (1 )
6
(18.a) (18.b)
By the transformation: p = ] 1 O2u + U Ou = > ¢ ¡ 2 O2u + 1 O2u = = 2Ou
W
(19)
Equation (19) can be transformed to C 2 !(1) C!(1) C!(1) + D!(1) E = 0> C C C 2 D = d Ou > E = e O2u =
(20)
Where Ou is the direction cosine in U direction. Equation (20) can X0 be solved under the transformation " = 2E ( X0 ) (Ghai and Saini, 2017) and nally by using the Tanh analytical method (Maliet and Hereman, 1996), the cylindrical shock is obtained in the form: !(1) =
X0 X0 (1 tanh ( ( X 0 ))= d Ou 2e O2u
(21)
Where X0 is the speed of shock.
3
Results and Discussions
Nonlinear cylindrical dust ion shock (CDIS) in dusty four component plasma with viscous ion has been introduced. The 3D-Cylindrical Burgers equation is obtained and solved to give the propagating cylindrical shocks. The physical eects of ionic and two types grains kinematic viscosity coecients on the properties of the cylindrical shocks are studied. To perform our results, numerical investigation was performed according to mesosphere parameters (Popel et al., 2005; Zaghbeer et al., 2018; El-Shewy et al., 2018a). To show the contribution of nonthermality on the shock formation, the shock amplitude and width against the parameter of nonthermal electron are plotted in Figs. 1-2. Generally, It was conrmed that this model supports two shock waves namely negative and positive cylindrical shocks. Fig. 1 show that the positive shock amplitude raises with parameter of nonthermality up to a critical point and it began to be reduced with for negative shocks. The width of shock decreases with as in Fig. 2. However, one of our goals was to examine the uid viscosity and cylindrical coordinate eects on the shock formation in this model. We have examined the eects of some parameters such the ion, negative (positive) charged grain kinematics viscosity 7
coecients l > (+ )> the direction cosine Ou and the dependence parameter on the cylindrical shock features. For example, Figs. 3 concern the eect of l on the shock prole. It was noted that, l increases the shock steepness while the strength of the shock still unvaried. In Fig. 4, the shock amplitude variation against + and Ou is depicted for X0 = 0=55> = 0=85> = 0=002, = 0=005 in the two regions of shock polarity i.e. = 0=2 for positive shock and = 0=38 for negative shock. It was illustrated that, both + and Ou decrease the amplitude and shock strength of positive shock while Ou decrease the amplitude and shock strength of negative shock and + increase the amplitude and shock strength of negative shock. On the other hand, + and Ou increase the width of both positive and negative shocks as depicted in Fig. 5. Also, in Fig. 6 the positive (negative) kinematics viscosity coecients + and increase the width and steepness of each polarity shocks. On the other hand, the two types of mesospheric cylindrical shock formation with all coordinate information are depicted in Figs. 7. In apsence of some parameters, this investigation concedes with other studies in interpret the nonlinear wave pictures in the opposite polarities plasmas (Mendis and Rosenberg, 1994; El-Shewy and El-Rahman, 2018b; Zaghbeer et al., 2014; Rahman et al., 20008). In conclusions, It was conrmed that all model parameters such as the nonthermality of electron > ion, negative (positive) kinematics viscosity coecients of positively and negatively charged grain l > (+ )> the directions Ou and parameters aected the mesosphere cylindrical shock characteristics. Our returns from this study may be of numerous importance in distinguishing the structure properties of shock formation mechanism and the comparison between nonlinear formations of waves in mesosphere space (Popel et al., 2005; Zaghbeer et al., 2018; El-Shewy et al., 2018a).
8
Acknowledgment This project was supported by the deanship of scientic research at Prince Sattam Bin Abdulaziz University under the research project No. 10259/01/2019.
Figure caption: Fig. 1. The amplitude variation against for X0 = 0=55> = 0=85> + = 1=6> = 0=02, = 0=005> Ou = 0=5 and = 0=5= Fig. 2. The width variation against for X0 = 0=55> = 0=85> + = 1=6> = 0=02, = 0=005> Ou = 0=5, = 0=5> l0 = 0=05> 0 = 0=4 and +0 = 0=15= Fig. 3. Variation of solution (21) with for various values of l0 and for X0 = 0=55> = 0=02, = 0=005> = 0=85> + = 1=5> Ou = 0=5> = 0=5> ] = 0=5> U = 0=5> +0 = 0=5> 0 = 0=7 and (a) = 0=2, (b) = 0=38= Fig.4. The amplitude variation against Ou for various values of + and X0 = 0=55> = 0=85> = 0=002, = 0=005 and = 0=2 in positive region and = 0=38 in negative region= Fig.5. The width variation against Ou for various values of + and X0 = 0=55> = 0=85> = 0=002, = 0=005 and = 0=2 in positive region and = 0=38 in negative region= Fig. 6. The width variation against +0 for various values of 0 and for X0 = 0=55> = 0=85> + = 1=8> = 0=002, = 0=005> Ou = 0=5> = 0=5> l0 = 0=05 and = 0=2. Fig. 7. Three dimensional wave prole !(1) (21) against U and at = 1=0 and ] = 0=5 for X0 = 0=55> Ou = 0=5> = 0=002> = 0=005> = 0=85> + = 1=5> +0 = 0=5> 0 = 0=7> l0 = 0=2 and (a) = 0=2, (b) = 0=4=
9
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13
Amplitude
U
۴ܑǤ Ǥ ߩܷ ൌ ͲǤͷͷǡ ߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤǡ ߙ ൌ ͲǤͲʹǡߤ ൌ ͲǤͲͲͷǡ ܮ ൌ ͲǤͷܽ݊݀ȣ ൌ െͲǤͷǤ
Width
U
۴ܑǤ Ǥ ߩߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤǡ ߙ ൌ ͲǤͲʹǡ ܷ ൌ ͲǤͷͷǡ ߤ ൌ ͲǤͲͲͷǡ ܮ ൌ ͲǤͷǡ ȣ ൌ െͲǤͷǡ ߟ ൌ ͲǤͲͷǡ ߟି ൌ ͲǤͶߟା ൌ ͲǤͳͷǤ
I +1/
KL KL KL
W
Fig. 3.a.
KL KL KL
I +1/
W Fig. 3.b. ۴ܑǤ Ǥ ሺʹͳሻɒߟ ߙ ൌ ͲǤͲͲʹǡ ߤ ൌ ͲǤͲͲͷǡ ߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤͷǡ ܮ ൌ ͲǤͷǡ ȣ ൌ െͲǤͷǡ ܼ ൌ ͲǤͷǡ ܴ ൌ ͲǤͷǡ ߟା ൌ ͲǤͷǡ ߟି ൌ ͲǤǡ ܷ ൌ ͲǤͷͷሺሻߩ ൌ ͲǤʹǡ ሺܾሻߩ ൌ ͲǤͶǤ
Amplitude
G G G G G G
Lr
۴ܑǤ Ǥ ܮ ߜା ߜି ൌ ͲǤͺͷǡ ܷ ൌ ͲǤͷͷǡ ߙ ൌ ͲǤͲͲʹǡ ߤ ൌ ͲǤͲͲͷߩ ൌ ͲǤʹ݅݊݀݊ܽ݊݅݃݁ݎ݁ݒ݅ݐ݅ݏ ߩ ൌ ͲǤ͵ͺ݅݊݊݁݃ܽ݊݅݃݁ݎ݁ݒ݅ݐǤ
G G
G
Width
Lr ۴ܑǤ Ǥ ܮ ߜା ߜି ൌ ͲǤͺͷǡ ܷ ൌ ͲǤͷͷǡ ߙ ൌ ͲǤͲͲʹǡ ߤ ൌ ͲǤͲͲͷሺሻߩ ൌ ͲǤʹߩ ൌ ͲǤ͵ͺ݅݊݊݁݃ܽ݊݅݃݁ݎ݁ݒ݅ݐǤ
Width
K
K
K
K 0
۴ܑǤ Ǥ ߟା ߟି ߜି ൌ ͲǤͺͷǡ ߜା ൌ ͳǤͺǡ ߙ ൌ ͲǤͲͲʹǡ ܷ ൌ ͲǤͷͷǡ ߤ ൌ ͲǤͲͲͷǡ ܮ ൌ ͲǤͷǡ ȣ ൌ െͲǤͷǡ ߟ ൌ ͲǤͲͷߩ ൌ ͲǤʹǤ
Fig. 7.a. 7
Fig. 7.b. 7 ۴ܑǤ ૠǤ Ԅሺଵሻ ሺʹ ʹͳሻ ܴȣɒ ൌ ͳǤͲ ܮ ൌ ͲǤͷǡ ߙ ൌ ͲǤͲͲʹ ʹǡ ߤ ൌ ͲǤͲͲ Ͳͷǡ ߜି ൌ ͲǤͺ ͺͷǡ ߜା ൌ ͳǤͷ ͷǡ ൌ ͲǤͷ ߟା ൌ ͲǤͷǡ ߟି ൌ ͲǤǡ ߟ ൌ ͲǤʹǡ ܷ ൌ ͲǤͷͷ ሺሻߩ ൌ ͲǤʹ ʹǡ ሺܾሻߩ ൌ ͲǤͶǤ Ͳ