The modified Burgers’ equation of nonlinear propagation of Dust-Ion Acoustic Shock Waves (DIASWs) for multicomponent unmagnetized dusty plasma consisting of adiabatic ion fluid, Boltzmann distributed electrons and positrons and static negatively charged dust fluid has been derived using the standard reductive perturbation method. The solution of modified Burgers’ equation in nonplanar geometry is numerically analyzed and it has been found that, the nonplanar geometry effects have a very significant role in the formation of shock waves. Further more; it is found that the planar geometry shock structure with higher amplitude is the strongest, followed by cylindrical and spherical shock waves amplitudes respectively. It is also observed that; an increase in positron concentration decreases the amplitude of the DIASWs. Keywords: Nonplanar Geometry multicomponent, Dust- ion-Acoustic Shocks Adiabatic Dusty Plasm

Akpabio, Louis E.

Antia, Akaninyene D.

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.51, 2016  56  Nonplanar Geometry in Multi-Component Dust-Ion-Acoustic Shock Waves for Adiabatic Dusty Plasma  Louis E. Akpabio   and Akaninyene D. Antia Theoretical Physics Group, Department of Physics, University of Uyo, Uyo, Nigeria.  Abstract The modified Burgers’ equation of nonlinear propagation of Dust-Ion Acoustic Shock Waves (DIASWs) for multicomponent unmagnetized dusty plasma consisting of adiabatic ion fluid, Boltzmann distributed electrons and positrons and static negatively charged dust fluid has been derived using the standard reductive perturbation method. The solution of modified Burgers’ equation in nonplanar geometry is numerically analyzed and it has been found that, the nonplanar geometry effects have a very significant role in the formation of shock waves. Further more; it is found that the planar geometry shock structure with higher amplitude is the strongest, followed by cylindrical and spherical shock waves amplitudes respectively. It is also observed that; an increase in positron concentration decreases the amplitude of the DIASWs. Keywords:  Nonplanar Geometry multicomponent, Dust- ion-Acoustic Shocks Adiabatic Dusty Plasma    1. INTRODUCTION  Plasmas, in general, consisting of electrons and positrons of equal masses and ion are usually characterized as electron-positron-ion plasma [1, 2]. The presence of ions brings about the existence of several low frequency waves which other wise do not propagate in electron-positron plasmas.  There are electron-positron in astrophysical plasmas such as in magnetosphere of plasmas, in active galactive nuclei, in early universe and in the region of the accretion disks surrounding the central black holes [3-8]. Due to impressive developments in plasma, there has been considerable interest in different type of linear and nonlinear wave structure such as solitons, double layers, vortices etc, in electron-positron plasmas [9-15] as well as in multi-component electron-positron-ion plasmas [16-18]. Most of the astrophysical plasmas usually contain highly charged (negative/positive) impurities or dust particulates in addition to the electrons, positrons and ions. It is a well known fact that, the presence of static charged dust grains modifies the existing plasma wave spectra [19-21]. Dust ion acoustic shocks in an unmagnetized dusty plasma many arise when there is a balance between the nonlinearity (associated with the harmonic generation) and the kinetic viscosity introduced by the dust ion drag. The formation of Dust Ion Acoustic Shock waves (DIASWs) was observed by Nakamura et al [22].  They found out that, both monotonic and oscillatory shock waves exist and the dust density has vital role on the shock waves and phase velocity of the wave. A number of studies have been made on the propagation of DIASWs in dusty plasma by several investigators [23-26]. Recently, several theoretical investigations [27-31] on the properties of dust-ion acoustic and dust acoustic solitary waves in nonplanar geometry had been carried out. Sahu [32] of recent, have carried out theoretical investigation on the effect of nonplanar DIASWs in an Adiabatic Dusty Plasma. Since, the properties of wave motions-electron-positron-ion-dust plasma should be different from those in three component electron-ion-dust plasma.  In this present work, we have considered an unmagnetized multi-component adiabatic dusty plasma; and have studied the basic properties of DIASWs in such a dusty plasma which was not considered in the earlier investigation [32]. The paper is organized in the following manner. The basic equations governing the adiabatic dusty plasma system under consideration are given in section 2. The nonplanar DIASWs are investigated by the reduction perturbation method (RPM) in section 3. In section 4 we present the numerical results and discussion. Section 5 is the conclusion.  2. GOVERNING EQUATIONS A four-component unmagnetized plasma comprising of static negatively charged dust fluid, Boltzmann distributed electrons and positrons, and adiabatic ion fluid. The dynamics of the DIAWs – nonplanar geometry for such a dusty plasma, is governed by the following normalized fluid equations:       ( ) )1(01 =∂∂+∂∂iii unrrrtn νν    +∂∂−∂∂−=+∂∂νηαφrnrprnuntun iiliiiii1     Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.51, 2016  57    )2(2 −∂∂∂∂rururrii νν   ( ) )3(013 =∂∂+∂∂+∂∂iiiii urrrprputp νν   ( ) ( ) )4(011expexp =+++−−−µδφδφµ pip n  In the above equation, in  is the ion number density normalized by its equilibrium value iio un ,  is the ion fluid speed normalized by ( ) φ,21ieBi mTKC =  is the wave potential normalized by ieB peTK , is the ion thermal pressure normalized by eiiBio TTTKn =α, . The time and space variable are normalized by reciprocal plasma frequency ( ) 2121 4 enmw ioipi pi=−  and the Debye length ( ) 2124 enTK ioeBD piλ =  respectively.  Where o=ν , for one-dimensional geometry and 2,1=ν  for cylindrical and spherical geometry respectively. iη  is the viscosity coefficient of ion fluid normalized by ioeoDpi nnw =µλ ,2  and iopop nn=δ . We note that in equation (4), we have assumed the quasi-neutrality condition.  3. DERIVATION OF NONPLANAR BURGERS’ EQUATION We derive the Burgers’ equation (1)-(4) by employing the reductive perturbation method (RMP) and the stretched coordinates [33] ( )tVr o−=∈ 21ξ  and ,21 t=∈τ  where ε is a smallness parameter measuring the weakness of the nonlinearity and 0V  is the phase speed of DIASWs normalized by iC .  Equations (1)-(4) can be expressed in terms of ξ  and τ as follows:      ( ) ( ) ( ) ( ) )5(0123212321=∈+∈+∂+∂∈−∂∈ iiooiiioi unVVunnVnτξτνε ξξτ    1021212123 φξξξτ ∂∈−=∂∈+∂∈−∂∈ iiiiiiii nuununVun        [ +∂∈∈+∂∈− iii unp 2202121 ξξ ηα      )6(11202203002 ∈+∈−∂ ∈+∈ii uVVuVVτξτντξτνξ    iiiii ppupVp 32121230 +∂∈+∂∈−∂∈ ξξτ    )7(01002321= ∈+∈+∂∈ ii uVVuτξτνξ    ++−−++ ...211...211 22 φφδφφµ p       ( ) )8(01 =−++ µδ p  where 021 ηη =∈  is assumed [34]. We can expand the dependent variables iii pun ,,  and φ  in power series of ∈  as follows:      ...1 221 +∈+∈+= iii nnn  Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.51, 2016  58       2210 iii uuu ∈+∈+=       )9(1 221 iii ppp ∈+∈+=       2210 φφφ ∈+∈+=  We obtain the lowest order of the coefficient ∈  as follows by substituting equation (9) into equation (5)-(8):            ( ) )10(11 φδµ pin +=        ( ) )11(101 φδµ pi Vu +=       ( ) )12(3 11 φδµ pip +=           ( ) ( )13130pV δµα ++=  The next higher order in ∈ , is given as the following set of equations,    ( ) )14(001211201 =+∂+∂+∂−∂ τνξξξτVuuunnVn iiiiii    211011201 iiiiiii punVuuuVu ξξξξτ α∂+∂−∂+∂−∂    )15(120112 ii un ξξξ ηφφ ∂+∂−−∂=    )16(03330111211201 =+∂+∂+∂+∂−∂ τνξξξξτVuupupupVp iiiiiiii    ( ) ( ) )17(21 2122 φδµφδµ ppin −++=  Now, using equations (10)-(17) and eliminating 222 ,, iii pun  and 2φ , we finally obtain a modified Burgers’ equation      )18(02121111 =∂−∂++∂ φφφφτνφ ξξτ CA  where           ( ) ( ) ( )( )( )( ) )19(132312223pppppAδµδµαδµδµδµα+++−−+++=      )20(20η=C   4. NUMERICAL RESULTS AND DISCUSSION  The modified Burgers’ equation describing the nonlinear propagation of the DIASWs for multicomponent unmagnetized dusty plasma consisting of adiabatic ion fluid, Boltzmann distributed electrons and positrons and static negatively charged dust fluid is given in Equation (18). The stationary DIASWs of this modified Burgers’ equation for planar geometry ( )0=ν  is   ( ))21(2tanh11  −−=CVVAV τξφ  where V is a constant velocity normalized by iC . For nonplanar geometry, an exact analytical solution of equation (18) is not possible, hence it is solved numerically. The results are shown in Figs. 1-5.  The initial condition that we have considered for all the numerical results, is the form of the stationary solution of equation (21) without the geometry term ( )τν 2  at 10−=τ .  The shock wave structure for different geometries with respect to the considered parameters is presented in Fig.1. It is obvious that, the developed shock amplitude in the different geometry are different from each other, while the shock steepness is the same for all geometry. The planar geometry shock structure, with higher amplitude is the strongest.  The cylindrical shock wave amplitudes are bigger than that of the spherical shock wave, but smaller than that of the planar geometry shock wave Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.51, 2016  59  structure.  -0.100.10.20.30.40.50.60.7-30 -20 -10 0 10 20 30ν=0ν=1ν=2 Fig. 1: Showing how the shock profile  ( ξφ sV1  curves) varies in different geometries for 3.0,5.0,5.0,5 0 ===−= pδαητ  and 4.0=µ . In Figures 2 and 3, we present the variation of cylindrical shock structures and spherical shock structures for different values of τ  respectively. It can be observed that as time increases ( )τ , the amplitude of the cylindrical shock waves also increases. It can also be observed that, the shock wave profile for the nonplanar (Fig. 2 and Fig.3) are similar to what is obtained for the planar geometry of fig. 1 in terms of the amplitude for larger value of τ . This result confirms the fact that, large value of the nonplanar geometrical effect ( )τν 2  is no longer dominant.  -0.100.10.20.30.40.50.60.7-30 -20 -10 0 10 20 30τ=-3τ=-5τ=-7τ=-10 Fig. 2: Variation of 1φ  with respect to ξ  at different values of τ for the cylindrical geometry ( )1=v  and the other parameters being the same as figure 1.   Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.51, 2016  60  -0.100.10.20.30.40.50.6-30 -20 -10 0 10 20 30τ=-3τ=-5τ=-7τ=-10 Fig. 3: Variation of 1φ  with respect to ξ  at different values of τ for the spherical geometry ( )2=v  and the other parameters being the same as figure 1. -0.200.20.40.60.811.2-30 -20 -10 0 10 20 30δp=0δp=0.1δp=0.18δp=0.3 Fig. 4: Variation of 1φ  with respect to ξ  at different values of the positron concentration for the cylindrical geometry ( )1=v  and the other parameters being the same as figure 1.     Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.51, 2016  61  -0.200.20.40.60.811.2-30 -20 -10 0 10 20 30δp=0δp=0.1δp=0.18δp=0.3 Fig. 5: Variation of 1φ  with respect to ξ  at different values of the positron concentration for the spherical geometry ( )2=v  and the other parameters being the same as figure 1. The effect of positron concentration on the shock wave structures for cylindrical and spherical geometries are shown in figures 4 and 5 respectively. From these two figures (Fig.4 and Fig.5), it is found that increase in positron concentration decreases the amplitude of the DIASWs for nonplanar geometry. It is obvious that; in physical situation, any increase in positron concentration decreases the ion concentration.  Since, we are dealing with DIASWs; the amplitude of the shock structure will decrease for increase in positron concentration.  5.  CONCLUSION  We have derived the nonplanar Burgers’ equation for Dust-ion-acoustic shock waves in a four component unmagnetized plasma comprising of static negatively charged dust fluid, Boltzmann distributed electrons and positrons, and adiabatic ion fluid by using the standard reductive perturbation method. We have found out that; the developed shock amplitude are different in the different geometries. With the planar geometry presently a higher amplitude in the shock structure, followed by the cylindrical and spherical shock structure respectively. For large negative values of τ , it is observed that the nonplanar geometries approaches the planar geometry. Finally, increase in positron concentration decreases the amplitude of the DIASWs. 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Nonplanar Geometry in Multi-Component Dust-Ion-Acoustic Shock Waves for Adiabatic Dusty Plasma

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Nonplanar Geometry in Multi-Component Dust-Ion-Acoustic Shock Waves for Adiabatic Dusty Plasma

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