Open Journal of Acoustics, 2011, 1, 7075 doi:10.4236/oja.2011.13009 Published Online December 2011 (http://www.SciRP.org/journal/oja) Copyright © 2011 SciRes. OJA TimeDependent Nonplanar DustIonAcoustic Gardner Double Layers Farah Deeba*, A. A. Mamun Department of Physi c s , Jahangirnagar University, Dhaka, Bangladesh Email: *deeba.ju35@yahoo.com Received September 24, 2011; revised October 30, 2011; accepted Novembe r 10, 2011 Abstract A theoretical investigation has been made on the nonplanar (cylindrical and spherical) dustionacoustic (DIA) double layers (DLs) in a dusty plasma system, containing inertial ions, Boltzmann electrons, and negatively charged stationary dust. In this investigation, in order to analyze the time dependent nonplanar DIA DLs, we have used the modified Gardner equation, which has been obtained by employing the reductive perturbation method. It has been found that the behaviors of DIA DLs have been significantly modified by the time period and the nonplanar geometry. The nonplanar DIA DLs has been found to be similar with planar DIA DLs only at large time scale and the cylindrical DIA DLs have been found to be smaller than the spherical DLs, but larger than the planar DLs. Keywords: DustIonAcoustic, Double Layers, Nonplanar 1. Introduction The existence of novel dustionacoustic (DIA) waves was first predicted by Shukla and Silin [1] about twenty years ago. Nearly four years later, the prediction of Shu kla and Silin [1] was conclusively verified by a labo ratory experiment of Barkan et al. [2]. Then the linear features of the DIA waves have been rigorously investi gated by a number of authors [1,3,4]. The linear proper ties of the DIA waves in dusty plasmas are now well understood from both theoretical and experimental points of view [15]. The nonlinear structures (viz., solitary waves, shock structure, and double layers) associated with the DIA waves have also received a great deal of interest because they have a great impact in understanding the basic pro perties of the localized electrostatic perturbations not only in space [69], but also in laboratory dusty plasmas [1014]. A number of investigations have been made on these nonlinear structures [612], particularly DIA soli tary waves (SWs) [11,1519], shock waves [12,13,2024], and double layers (DLs) [2528]. All of these work [11, 12,1519] are confined in planar geometry. Since the waves observed in laboratory devices are certainly not bounded in onedimension, the investigations made on 1D (planar) nonlinear DIA waves, may not be appropri ate for realistic space or laboratory dusty plasma situa tions. Recently, a few investigations have been made on finite amplitude nonplanar DIA solitary and shock struc tures [2931]. But in all of these investigations [2931], authors have used the KdV or Burgers equations, which are not valid for a parametric regime corresponding to A = 0 or A ~ 0 (where A is the coefficient of the nonlinear term of the KdV or Burger equation [2931], and A ~ 0 means here that A is not equal 0, but A is around 0). This is because, the latter gives rise to infinitely large ampli tude structures which break down the validity of the re ductive perturbation method [32]. On the other hand, to the best of our knowledge, no attempt has been made on nonplanar DIA DLs. There fore, in our present work we are going to analyze the nonplanar DIA DLs in dusty plasma system by deriving e modified Gardner (mG) equation. th 2. Basic Equations We consider the nonlinear propagation of the DIA waves in an unmagnetized nonplanar (cylindrical and spherical) dusty plasma, consisting of inertial ions, Boltzmann electrons, and negatively charged stationary dust. Thus, at equilibrium we have 00 0 , where d = iedd nnZn is the number of electrons residing onto the dust grain surface, and 0d(0i n) is the dust (ion) number density at equilibrium. The nonlinear dynamics of the DIA waves, n
F. DEEBA ET AL.71 whose phase speed is much smaller (larger) than the electron (ion) thermal speed, in a nonplanar geometry is governed by 1=0, v iii nrnu tr r (1) = ii i uu u tr , r (2) 1=r rr r , (3) =1exp() i n, (4) where =0 for 1D planar geometry, and =1(2) for a nonplanar cylindrical (spherical) geometry; i is the ion number density normalized by its equilibrium value ; is the ion fluid speed normalized by n 0i ni u = iBei ; CkTm is the electrostatic wave potential normalized by Be kT e, is the surface charge den sity normalized by Be kT e; 00 =dd i nn , is the ratio of dust number density to the ion number density (the quantity is often used to measure the quantity of dust particles present in the plasma system); i is the mass of the ion (electron); is the ion (ele ctron) temperature, d m() e m () ie TT is the number of electrons resid ing on a dust grain surface; k is the Boltzmann con stant, and e is the magnitude of the electron charge. The time and space variables are in units of the ion plasma period 12 1 0 =4e pii i mn 2 and the Debye radius 12 2 0 =4πe DmB ei kT n , respectively. 3. Derivation of mG Equation To study finite amplitude DIA DLs by employing the reductive perturbation method [27,33,34], we first in troduce the stretched coordinates: = p rVt , , (5) 3 =t (6) where is a smallness parameter (0<<1 ) measur ing the weakness of the dispersion, and V i n (normalized by i) is the phase speed of the perturbation mode. We then expand all the dependent variables (viz. , , C i u , and ) in power series of : (1)2 (2)3 (3) =1 , iiii nnnn (7) (1)2 (2)3 (3) =0 , iiii uuuu (8) (1)2 (2)3 (3) =0 , (9) (1)2(2)3 (3) =0 , (10) Now, Expressings (1)(4) in terms of and , and substituting (7)(10) into the resulting [Equations (1)(4) expressed in terms of and ], one can easily deve lop different sets of equations in various powers of . To the lowest order in , one obtains (1) (1) 2 =,= , ii p un (11) VV (1)2 = p 1 =0, , e V (12) where (1) = and =1 e . The expression for 2 V in (12) represents the linear dispersion relation for the DIA waves propagating in a dusty plasma under consideration. This equation clearly indicates that the DIA wave phase speed ( V) increases with the increase of the dust charge density (0dd n). To the next higher order in , we obtain a set of equations, which, after using (11) and (12), can be simplified as 2 ( V 2)2 (2) (2) ( 342 3 =,=, 22 ii p ppp un VVV 2) (13) (2) 2 1 0,123. 2AA=== (14) It is obvious from (14) that since =0A0 . Now by solving for =0A , we found can be zero for both =1 and =32 . But when =1 , 0dd n n (dust number density at equilibrium) will be equal to 0i (ion number density at equilibrium), which means that there is no electron present in the system, i.e., all electrons will be captured by dust particles. As our model contains all of electron, ion, and dust, this is an invalid condition for our present model. For our system <1 . Therefore the only valid solution of for =0A is given by ==23 c. (15) For around its critical value (c ), i.e. for = c corresponding to 0 = A, we can express 0 as 0 = = c c . A ss (16) where c is a small and dimensionless parameter, and can be taken as the expansion parameter , i.e. c , and for =1s<c and =1s for >c . So, (2) can be expressed as (2) 2 1, 2s (17) This means that for c , (2) must be included in the third order Poisson’s equation. To the next higher order in , we obtain a set of equations: Copyright © 2011 SciRes. OJA
F. DEEBA ET AL. 72 (3) (3)3 32 3 (2) 2 2 = 0, ii pp p p nu VVV V (18) (3) 3 2( 3 (3) 11 2 =0, i pp p p u VVV V 2) (19) 2 (3) (3)(2)3 2 2 1 6 =0. 2 ie ee n s (20) Now, using (11)(14) and (18)(20), we finally obtain a nonlinear dynamical equation of the form: 3 2 3=0, 2s (21) where 2 15 1 =11 21 , 5 (22) 3 2 1 =1 2 . (23) Equation (21) is known as modified Gardner (mG) equation. The modification is due to the extra term, 2 , which arises due to the effects of the nonplanar geometry. Because of the existence of both 2 and 3 term, this equation supports both Sws and DLs solutions. We have already mentioned that =0 corresponds to a 1D planar geometry which reduces (21) to a standard Gardner (sG) equation. We are now going to numerically analyze the mG equation. However, for a better under standing, before going to numerical solutions of mG equation, we first briefly discuss the stationary double layer (DL) solution of this standard Gardner equation [i.e. (21) with =0 ]. The stationary DL solution of the sG equation [i.e. (21) with =0 ] is obtained by con sidering a moving frame (moving with speed 0 U) 0 =U , and imposing all the appropriate boundary conditions for DL solution, including 0 , dd 0 , 22 dd 0 at . These boun dary conditions for the stationary DL solution allow us to express the sG equation as 2 1d ()=0, 2d V (24) where the pseudopotential ()V is 23 0 ()= . 2612 Us V 4 (25) We note here that 0 U and are always positive since 1 is always valid. It is obvious from (25) that =0 =0 d() () ==0, d V V (26) 2 2 =0 d() <0. d V (27) The conditions (26) and (27) imply that DL solution of (24) exist if and only if = = d() () ==0. d m m V V (28) where m is the amplitude of the DLs. The latter can be expressed as 2 0= 6 s U, (29) 0 6 =. m U (30) Now, using (25) and (30) in (24) we have 2 2 2 d=0, dm (31) where =6 . Now, integrating (31) the stationary DL solution of sG equation [i.e. (21) with =0 ] can be directly given by =1tanh 2 m , (32) where is the DL width of the DLs, and is given 2 =. m (33) From (32) and (33), it is clear that DLs can be formed in the dusty plasma system if and only if <0 , i.e. << U , where =0.74 L (=1 U ), obtained from =0 , is the lower (upper) limit of above (below) which DLs exist. In Figures 1 and 2, the variations of m with 0 U (Figure 1) and e (Figure 2) have been graphically shown. On the other hand, since >0 and >0 , (32) and (33) indicate that the DLs are associated with positive potential if , i.e. =1s<c =1 , and associated with negative potential if , i.e. c s> . It is obvious from Figures 13 that > c which indicates that our DLs are associated only with negative potential. The parametric regimes for the existence of negative Copyright © 2011 SciRes. OJA
F. DEEBA ET AL.73 Figure 1. Showing the variation of the amplitude of DIA DLs with U0 at μ = 0.75. Figure 2. Showing the variation of the amplitude of DIA DLs with μ at U0 = 0.1. Figure 3. Showing the parametric regime for the existence of DLs. The upper (lower) surface plot represents the lower (upper) limit of μ below (above) which DLs exist. DLs are bounded by the lower and upper surface plot of Figure 3, and DLs exist for parameters corresponding to any point in between two (=0 ) surface plots. The point to be noted here that if we would keep only the lower order nonlinear term of (21) (viz. the third term of (21) or the term containing 2 ) instead of the higher order nonlinear term (viz. the fourth term of (21) or the term containing 3 ), we would obtain the solitary structures. On the other hand, in our present work, we have kept both the terms containing 2 and 3 , and have obtained the DL structures (associated with 2 and 3 ). 4. Numerical Analysis Now in order to analyze the nonplanar DIA DLs, we turn to (21) with the term 2 , which is due to the effects of the nonplanar (cylindrical or spherical) geo metry. An exact analytic solution of (21) is not possible. Therefore, we have numerically solved (21), and have studied the effects of cylindrical and spherical geome tries on timedependent DIA DLs. The results are de picted in Figures 4 and 5. The initial condition, which we have used in numerical analysis, is in the form of the stationary solution of (21) without the term 2 . Figure 4 shows how the effects of a cylindrical geometry modify the DIA DLs, and Figure 5 shows how the effects of a spherical geometry modify the DIA DLs. From the numerical solutions of (21) (displayed in Figures 4 and 5) we may conclude that for a large value of (e.g. =40 ), the cylindrical (=1 ) and sphe rical (=2 ) DLs are almost similar to 1D planar (=0 ) structures. This is because when the value of is large, the term 2 , which is due to the effects of the cylindrical or spherical geometry, is no longer dominant. However, as the value of decreases, the term 2 becomes dominant, and spherical and cylin drical DL structures differ from 1D (planar) ones. It has been observed that as the value of decreases, the amplitude of these localized pulses increases. It is also found that the amplitude of cylindrical DIA DL stru ctures is larger than those of 1D planar ones, but smaller than that of the spherical ones. 5. Discussion and Conclusions In this paper we have investigated timedependent non planar dustionacoustic Gardner double layers in a dusty plasma system (composed of inertial ions, Boltzmann electrons, and negatively charged stationary dust), by deriving modified Gardner (sG) equation. The outcomes, which have been obtained from this investigation can be pinpointed as follows: 1) The dusty plasma system under consideration su pports both finite amplitude planar and nonplanar DLs, whose basic features (polarity, amplitude, width, etc.) de pend on the ion and dust number densities. 2) The DLs having large width exist for 0.74<<1 and only have negative potential, i.e., no positive DLs Copyright © 2011 SciRes. OJA
F. DEEBA ET AL. 74 Figure 4. Showing the effects of cylindrical geometry on DIA DLs at μ = 0.75. Figure 5. Showing the effect of spherical geometry on DIA DLs at μ = 0.75. have been formed. 3) The magnitude of the amplitude of the DLs in creases with the increase of , but decreases with the increase of 0 U . 4) The magnitude of the amplitudes of both cylindrical and spherical DLs increase with decrease of . 5) The spherical DLs have larger amplitude and poten tial than the cylindrical and planar DLs. We, finally hope that our results may be useful in un derstanding the localized electrostatic disturbances in both space environments [69], and laboratory devices [1114]. 6. Acknowledgements The Third World Academy of Science (TWAS) Research Grant for research equipment is gratefully acknowled ged. 7. References [1] P. K. Shukla and V. P. Silin, “Dust IonAcoustic Wave,” Physica Scripta, Vol. 45, No. 5, 1992, p. 508. doi:10.1088/00318949/45/5/015 [2] A. Barkan, N. D'Angelo and R. L. Merlino, “Experiments on IonAcustic Waves in Dusty Plasmas,” Planetary and Space Science, Vol. 44, No. 1, 1996, pp. 239242. doi:10.1016/00320633(95)001093 [3] P. K. Shukla and M. Rosenberg, “Boundary Effects on DustIonAcoustic and DustAcoustic Waves in Collisio nal Dusty Plasmas,” Physics of Plasmas, Vol. 6, No. 3, 1999, pp. 10381040. doi:10.1063/1.873345 [4] P. K. Shukla and A. A. Mamun, “Introduction to Dusty Plasma Physics,” Institute of Physics Publishing Ltd., Bris tol, 2002. doi:10.1887/075030653X [5] R. L. Merlino and J. Goree, “Dusty plasmas in the labo ratory, Industry, and Space,” Physics Today, Vol. 57, No. 7, 2004, pp. 3239. doi:10.1063/1.1784300 [6] V. E. Fortov, et al., “Complex (Dusty) Plasmas: Current Status, Open Issues, Perspectives,” Physics Report, Vol. 421, No. 12, 2005, pp. 1103. doi:10.1016/j.physrep.2005.08.007 [7] O. Ishihara, “Complex Plasma: Dusts in plasma,” Journal of Physics D, Vol. 40, No. 8, 2007, p. R121. doi:10.1088/00223727/40/8/R01 [8] P. K. Shukla and B. Eliasson, “Colloquium: Fundamentals of DustPlasma Interactions,” Reviews of Modern Physics, Vol. 81, No. 1, 2009, pp. 2544. doi:10.1103/RevModPhys.81.25 [9] G. E. Morfill and A. V. Ivlev, “Complex plasmas: An In terdisciplinary Research Field,” Reviews of Modern Phy sics, Vol. 81, No. 4, 2009, pp. 13531404. doi:10.1103/RevModPhys.81.1353 [10] K. E. Lonngren, “Soliton Experiments in Plasmas,” Plas ma Physics, Vol. 25, No. 9, 1983, p. 943. doi:10.1088/00321028/25/9/001 [11] Y. Nakamura and A. Sharma, “Observation of IonAcoustic Solitary Waves in a Dusty Plasma,” Physics of Plasmas, Vol. 8, No. 9, 2001, pp. 39213926. doi:10.1063/1.1387472 [12] Y. Nakamura, H. Bailung and P. K. Shukla, “Observation of IonAcoustic Shocks in a Dusty Plasma,” Physical Re views Letters, Vol. 83, No. 8, 1999, pp. 16021605. doi:10.1103/PhysRevLett.83.1602 [13] Q. Z. Luo, N. D'Angelo and R. L. Merlino, “Experimen tal Study of Shock Formation in a Dusty Plasma,” Phys ics of Plasmas, Vol. 6, No. 9, 1999, pp. 34553458. doi:10.1063/1.873605 [14] C. Charles, “A Review of Recent Laboratory Double La yer Experiments,” Plasma Sources Science and Technol ogy, Vol. 16, No. 4, 2007, pp. R1R25. doi:10.1088/09630252/16/4/R01 [15] R. Bharuthram and P. K. Shukla, “Large Amplitude Ion Acoustic Solitons in a Dusty Plasma,” Planetary and Space Science, Vol. 40, No. 7, 1992, pp. 973977. doi:10.1016/00320633(92)90137D [16] S. I. Popel and M. Y. Yu, “Ion Acoustic Solitons in Impu rityContaining Plasmas,” Contributions to Plasma Phys ics, Vol. 35, No. 2, 1995, pp. 103108. Copyright © 2011 SciRes. OJA
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