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Self-similar expansion of a warm dusty plasma—I. Unmagnetized case
Pkznrt. Spcr~rSG., Vol. 43, No. 9. pp. 1079-1085. 1995
Pergamon
Copyright I(‘) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 003’-0633195 $9.50+0.00 0032-0633(95)00021-6
Self-similar expansion of a warm dusty plasma-I.
Unmagnetized case
R. Bharuthram” and N. N. Raot Department
of Physics, University
of Durban-Westville,
Durban
4000. South Africa
Received 4 October 1994 : revised 13 January 1995 ; accepted 13 January 1995
expansion into vacuum of a II;semi-i&&e half-space is investigat.A by ir&tding the adiabatic equation of state for the da& fluid. Given the complex nature of city the anafysis is restricted to expansion of a collisionless plasma have constant charge. Exact b&i& for the ease of the as weil as negative charging or the warm case, numerical the relevant self-simiisothermal negatively a larger distance than an adi&&c one before the density drops to nearly zero. The &ects of several pksma parameters such as the density and the temperature ratios on the expansion profIles are d&XWed.
1. Introduction Plasmas consisting, in addition to the usual electrons and ions. of finite-size, charged particulate matter in the micron or sub-micron range have attracted much attention in recent years. These so-called dusty plasmas exist naturally in astrophysical and space environments such as cometary tails, planetary ring systems, interstellar and circumstellar clouds, etc. (Goertz, 1989). Nearer the Earth, the enhanced radar backscatter from the noctilucent clouds often observed in the polar regions during the summer seasons has been attributed to the presence of dust in the Earth’s lower ionospheric regions (Havnes et al., 1990). On the other hand, understanding of the
*Also at: Plasma
Physics Research Institute, University of Natal, Durban 4001, South Africa. TPermanent address : Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India. Co~re.spon&c~c to : R. Bharuthram
trapping as well as the movement of the dust grains is important in the fabrication of semiconductors using plasma-aided processes. The effect of the dust component on the collective behaviour of plasmas has been extensively studied in the last few years by various authors. In particular, various types of waves and instabilities have been investigated (de Angelis et al., 1988 : D’Angelo, 1990. I993 ; Shukla ct ~1.. 1991; Shukla, 1992 ; D’Angelo and Song. 1990; Rosenberg, 1993 ; Rao, 1993a ; Rawat and Rao, 1993). Because of the large mass of the dust particles compared to the electron and the ion mass. dusty plasmas support new kinds of waves in the very low-frequency regime. By considering thermal electrons and ions having Boltzmann distributions, Rao et al. (1990) predicted the existence of a novel kind of low-frequency electrostatic mode which they called the dust-acoustic wave. The inertial contribution to this mode comes from the dust fluid. Recently, the electromagnetic generalization of this mode to include compressional magnetic field and plasma density perturbations in a magnetized dusty plasma has been obtained by Rao (1993b). The expansion of a dusty plasma is a fundamental process which is important in many practical situations. For example. the dusty plasmas found in space environments have typically free boundaries and therefore can expand into vacuum. On the other hand, in laboratory devices the dust particles in the form of impurities are produced in the source regions and can subsequently expand into the central plasma region. In this regard, Lonngren ( 1990) examined the self-similar one-dimensional expansion of an unmagnetized cold dusty plasma into a vacuum using the model of Rao et al. (1990). The analytical solutions presented by Lonngren (1990) are, however, strictly valid for the case of identically zero electron density. The work of Lonngren has been extended by Y u and Luo (1992) who provide a more detailed analytical study as well as numerical solutions for the density profiles and the flow velocities. A study of the expansion process via kinetic theory (Luo and Y u, 1992a,b) produces results which are in agreement with those from the fluid model. In this
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma--l
1080
paper, we extend the work of Lonngren (1990) and Yu and Luo (1992) by allowing for a finite temperature for the dust particles by including the adiabatic equation of state. In Section 2, we derive the basic governing equations for the self-similar expansion. Exact analytical solutions for the cold case are also obtained both for the positively as well as the negatively charged dust grains. For the general case, numerical solutions are presented in Section 3. A detailed study is made of the effect of plasma parameters such as temperature and densities on the expansion profile. Finally, Section 4 contains a summary of our results and conclusions.
expansion into vacuum in the x-direction at the later times. The equations governing the self-similar expansion of the plasma are obtained by letting II = n(i) and L’= ~(5) where 5 = .vij.t, and i is a normalizing velocity to be specified a posteriori. Then, in terms of the self-similar variable (2). equations (3)~-( 5) become. respectively
(._&!!+ndr-_() 1
d<
-
nm(l?-it)G
dr
(7)
de -
= -Zenz
d$
-z
du
and 2. Theory We consider a three-component plasma consisting of electrons, ions and dust grains, and describe its dynamics by means of the fluid equations wherein the electrons and the ions are in thermal equilibrium at their respective temperatures in the presence of the self-consistent electrostatic field and the inertia is provided by the dust fluid (Rao et al., 1990). Accordingly, the densities of the electrons (n,) and the ions (n,) are given by the Boltzmann distributions, namely
From equation
(2), we find
d4 _=__dt while upon obtain
solving
T, drl,
(10)
en, d<
for 4 from equations
(1) and (2) we
B
(11)
n, = x 4
and
(2) where C$is the electrostatic potential, e the magnitude of the electron charge and T, (T,) the temperature of the ions (electrons). The subscript “0” indicates equilibrium quantities. The dynamics of the dust fluid is governed by the equation of continuity g + the momentum
and the adiabatic
;_(nz!) =0
conservation
equation
The system of equations condition
where b = n& and c( = T,/Te. Upon using equations (9), (11) and (6) to evaluate dp/d<, and then combining with equations (8) and (10) we arrive at
(12) Similarly
equations
(7) and (9) become,
respectively (13)
(3) and
equation
dp -=-
yp(a/?+n:“)
d<
(P-n:+‘)
1 dfz, (14) PC d<
At this point we wish to point out that for cold dust fluid (p = 0), equations (7) and (12) become
of state
is closed with the quasi-neutrality and II, = II, + zn.
(6)
In the above equations. n is the dust number density, I’ the dust fluid velocity, p the dust pressure, y the ratio of specific heats and Z the charge number of the dust particles. Note that Z > 0 (CO) for the positively (negatively) charged dust particles. We assume the dusty plasma to occupy at the initial time the semi-infinite half-space .Y < 0 and consider its
(16) If we normalize the velocity with respect to i = J( 7’Jm), then equations (15) and (16) are just equations (13) and (14) of Lonngren (1990). These equations are also used by Yu and Luo (1992). In these papers. the dust particles are negatively charged, thus from equation (6) our Z is equivalent to their -Z.
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma--I
1081
Returning to our system obtain from equation (14)
(35)
of equations
(12)-(14),
we
(17) where C is a constant conditions. Combining we have
to be determined by the initial this equation with equation (12).
we obtain the analytical solutions cases of dust particle charging : (A) Negatively
charged
for the following
dust particles
two
(Z < 0)
I .‘Z
(18) which when used in equation
(13) yields
where
I‘ = ,iWJH
(1%
$_
where
Vz:+‘-B)_~ (ZI (ap+?z:+
Here. in order that requires, a < $+ ’ Upon obtaining dr/dj from equation with equation (I 8) to obtain
(19), we combine
(B) Positively
solution
charged
’)
(36) is well behaved.
dust particles
(27) one
(Z > 0)
where ZT,
(;=_+-__-___
171
ZT, fl( I +r)‘nf
”
where
2m (,-@_tn;+‘)2
(fl-n:-‘,
Ei(y+ + -z;r
From equation we have
(B-C’) Zn:
(2% )
(p-ng+‘)’
condition
(6)
(23)
Equations (1 I ), ( 17), (19), (21) and (23) constitute a complete set to describe the self-similar expansion of a dusty plasma with warm dust particles. For the general case, we may numerically integrate equation (21) to obtain n,(l). Then, the solutions for ?z,(<), n(c), p(t) and o(4) follow from the other equations in the above set. Before proceeding to such solutions. we give below analytical solutions for the case of cold dust species. For the cold dust fluid, we set p = 0 in expressions (20) and (22) for H and G, respectively. Upon defining N = II:+ ’ , equation (21) is rewritten as
(24) Defining
z(cg+rl:+
(22)
( 11) and the quasi-neutrality
n _
* = __---
1)p (afl+nf+‘)n:
In this case, one requires /I > 17:_’ so that the solution (28) is well behaved. The above transcendental relations yield the implicit solutions for n,(t). The constants C. and C, are determined by the initial conditions. We wish to point out that in the work ofYu and Luo (1992), where the dust particles are negatively charged, a combination of their equations (5) and (9) will yield a solution which is the same as solution (26) obtained above.
3. Numerical results We begin by presenting the governing equations in the normalized form. If p0 and n,, are the initial dust fluid pressure and number density, respectively, then one can use the equation of state (5) to show that the constant C in equation (17) is given by C = p,)/(n,,Z):‘. The dust fluid pressure p is normalized by p,, and the ion density n, by its initial value ~1,~.Writing 2 = p /ZJ where p = + 1 (-. 1) for Z > 0 (CO), and normalizing the velocity by E. = \i( T,/m), equations (19)-(23) become
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma-I
1082 Ax
(6-E;“)
yApr?:
(cX6+$+‘)
1.2
(31)
+ (6-vlpl”)
-
ion
density dust dsadty
-
alsctron
&
*+
(l+r’)S$+’
+:‘(i’fl)
2(&+$-t’)’
2
A@(aste+‘) (6-ny+‘)’
I
1
-
dsnalty
_I n1
and
fi=
(6 -$‘I) Ziii
(33)
where 6 = n,,/n,, and A = p”/(T,ni”). In addition, the normalized dust pressure and the electron number density are given, respectively, by (34)
6
I& = fi;
(35)
where A and ii, are normalized with respect to Q. In the above equations, tildes indicate normalized quantities. For real solutions and for Z > 0 (< 0). that is, for ,U = + 1 (- I), it is necessary that the parameter 6 is given suchthat6>1 (@, respectively. For chosen values of 6, A and a (determined by the initial conditions), we present below some numerical solutions of the set of equations (30)-(35). In each case, the equations are solved using a numerical code which employs the Runge-Kutta-Fehlberg algorithm (Burden and Faires, 1985). By suitably combining the fourth- and fifth-order modified Runge-Kutta formulae, this algorithm provides an automatic variation of the stepsize so that the local truncation error is minimized for a given tolerance on the solutions to be computed. As pointed out below, the code is tested using the exact analytical solutions obtained for the case of the cold dust fluid. Figure 1 shows the expansion of the plasma from the point 5 = 0 for parameters 6 = 0.1, 2 = 1.O and Z = - 1 (singly negatively charged dust particles). Here A = 0, corresponding to the cold dust fluid. We wish to point out that for the curves in Fig. 1 the numerical solutions of equation (32) for n’, are identical to the results obtained from the transcendental equation (26). It is seen that C1and fi decrease with 5. In order to satisfy the quasi-neutrality condition, ri, increases with < until about < = 3.0 when the dust density n’ zy 0 and ri, = ri,. In Fig. 2a and b we examine the effect of a finite dust temperature. We observe that for A = 0.5 an isothermal dust fluid (y = 1) expands over a larger distance than an adiabatic one (y = 3), before its density drops to zero. The corresponding behaviour of ti, and ric are shown in Fig. 2b. In Fig. 3 are shown plots of the dust velocity B (equation (30)) for the parameters corresponding to Figs 1 and 2. The variation of the dust fluid pressure 9 with 4 is given in Fig. 4 for the isothermal and adiabatic cases. For 2 = - 100, the curves show similar behaviour, although the typical expansion distance increases significantly, as is presented for 5 in Fig. 5.
I Fig. 1. Plot of number densities as a function of the normalized self-similar variable 5, for Z = - 1.O and A = 0.0. with C(= 1.O and 6 = 0. I 1 .o
,
r
1
I
I
5
6
I
-
7’3.0
-
- 7=1.0
R
0
1
2
1
I
(4 1
I
-
0.0
1 U
I
I
I
1
2
3
I
- deotron dandty ion donalty
I
I
5
J 6
b) Fig. 2. (a) Plot of dust density E versus < for 7 = I .O and 3.0. (b) Plot of ion and electron densities versus < for 7 = 1.0 and 3.0. In both (a) and (b) A = 0.5 and Z = - 1.0. with Y = 1.O and 6 = 0.1
and N. N. Rao : Self-similar
R. Bharuthram I
I
I
expansion
1
of a warm dusty plasma0.010
I
I
1083 I
I
1
I
1
A-O.0
-
- - A-O.6 and y-3.0 A-O.6 and y-1.0
-
A-O.0 A-O.6 and y-1.0 - - A-0.6 and y'3.0
\ \ \ \ \ \ \ \ \ \
\ \ I 0
J.
1
0
2
3
4
5
10
20
\; 30
8
A
60
t
Fig. 5. Plot of the dust density ri versus i for the cabe when Z = - 100. Other fixed parameters are as in Fig. I
Fig. 3. Plot of the dust fluid velocity i; versus < for the parameters corresponding to Figs 1 and 2
I
I 50
6
c
1.2
I. 40
I
I
I
A-0.6 and
y-1.0 - - A-O.6 and y-3.0 -
p
1.0
a much further counterpart.
0.6
4. Discussion
0.6
0.4
0.2
-1 0
1
2
c3
4
5
6
Fig. 4. Plot of dust fluid pressure J? versus < for the parameters corresponding to Fig. 2
We next consider results for positively charged dust particles (recall Z > 0 requires 6 = n,,/n,, > 1). For Z = + 1. the relevant curves are shown in Fig. 6ad, while for 2 = + 100 the corresponding behaviour of n” is presented in Fig. 7. From comparison of Figs 2-5 one observes that a positively charged dust fluid expands over
distance
than
its negatively
charged
In this paper, we have investigated the self-similar expansion of an unmagnetized, collisionless dusty plasma consisting of Boltzmann distributed electrons and ions. and a thermal dust component whose dynamics is governed by the fluid equations, including the adiabatic equation of state. For both negatively and positively charged dust particles. it is found that the dust component expands over a larger distance when its initial pressure is non-zero as compared to the expansion of a cold fluid. For the latter case, the numerical solutions of the set of governing equations are in exact agreement with the corresponding analytical solutions which are in agreement with Lonngren (1990) and Yu and Luo (1992). It is found that an isothermal (I$ = 1) dusty plasma expands much further than one which is adiabatic (7 = 3). For the chosen set of parameters, a positively charged dusty plasma expands further than a negatively charged one. A possible explanation for this is that for positively charged dust particles the density of the oppositely charged particles, i.e. electrons, is larger at i = 0 than that of the positive ions for the negatively charged dust particles. Consequently, the attractive electrostatic forces due to the more dense electrons (at < = 0) are able to drag the positive dust particles further out than the expansion experienced by the negative dust particles due to the positive ions. In the Appendix
1084
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma-l
E
0.5 -
no.4
. duet ion density den&y - electron dsmity
-
fp
-
_
-
@‘I
(aI I 1.10 :.
. . . ion -
80
I
I
dermity denmity
ebatroo
*
I
. . . A=O.S and 74.0 - - A-O.6 and 74.;/
16 --
A-0.0
00
6
,
’
/
10 F
16
20
(4 1.2
I
, -
1.0 -
-
-
I
A-O.6 and 793.0 A-0.6 and y-1.0
-
Fig. 6. (a) Plot of number densities versus [ for A = 0.0. (b) Plot of dust density fi versus t: for 7 = 1.O and 3.0, with A = O.S. (c) Plot of ion and electron densities versus < for the parameters of(b). (d) Plot of dust fluid velocity 2; versus [ for the parameters of (a) and (b). (e) Plot of dust fluid pressure p versus 5 for the parameters of (b). In all the above plots 2 = + 1.O, CI= 1.O and 6 = 1.1
we point out the differences that arise when, instead of the full adiabatic equation of state, one uses the usual simple model, namely, Vp = yTVn in the dust fluid momentum equation. Our model is restricted to a one-dimensional expansion in order to obtain some insight into a physical process that is mathematically quite complex. In reality, it would be more correct to investigate a three-dimensional cylindrical or spherical expansion. This would require a numerical solution of the complex set of nonlinear equations which govern the dynamics of the plasma expansion. Furthermore, we have neglected the charge variation effects of the dust particles which, as shown in several recent studies, could play an important role.
The present paper considers the self-similar expansion of an unmagnetized dusty plasma. However, many of the plasmas of practical interest either in the space environment or in the laboratory are embedded in an external magnetic field. Hence, it is necessary to investigate the effect of an external magnetic field on the expansion process. This problem is investigated in the next paper (Rao and Bharuthram, 199.5).
Acknowledgements. One of us (N.N.R.) thanks the FRD (South Africa) for supporting his visit to the University of DurbanWestville. He is grateful to the members of the Physics Department for their warm hospitality.
R. Bharuthram
and N. N. Rao: Self-stmilar
0.0012
I
I
expansion I
A-O.6 -
y-1.0
and y-3.0
A-O.0
II \
I
\
0.0006
\ \
0.0005
0.0004
I
IOXS
R., Self-similar expansion of a warm dusty plasmaII. Magnetized case. Phrtwr. Sptrcc SC,;.43. 1087-1093. 1995. Rao, N. N., Shukla, P. K. and Yu, M. Y’., Dust-acoustic w:~ves in dusty plasmas. Plrrtwt. Sptrw SC;. 34, 543. lY90. Rawat, S. P. S. and Rao. N. N., Kelvin ~Helmholtz Instability driven by sheared dust flow. Plmet. .Spu~c ki. 41, 137. 1993. Rosenberg, M., Ion- and dust-acoustic instabilities in dusty plamas. Ph7(‘/. .~pu~cJ9;. 41, ‘20. 1993. Shukla, P. K., Low-frequency modes In dust> plasmas. I’/II~,s~c~~I .S‘L.l.il)tl/45, 504. 1992. Shukla, P. K., Y’u, M. Y. and Bharuthram. R.. Linear and nonlinear dust drift waves. J. ,~Yc,o@,~.Rc.\. 96. 2 1343. 195,I Yu, M. Y. and Luo. H., Self-similar motion of ;I dusty plasma Phr.\. f_c~r. A161, 506. 1992. Rae, N. N. and Bharuthram,
- - A-O.6and
n
of a warm dusty plasma-
\
0.0003
Appendix
0.0002
\
0.0001 I
\
I
\
‘--_,
0.0000
0
50
I
100
150
Here. we point out the differences in the analysis if one uses the visual simple model, namely. Vp = ;‘TVtt in the momentum equation in order to allow for the pressure term. IF one disregards the adiabatic equation of state (5) and rewrites equation (4) as
200
c Fig. 7. Plot of the dust density ri versus ; for the case when Z = + 100. Other hxed parameters are as in Fig. 6
then equation
(2 I ) is replaced by
$4_ References
d< -
Burden, R. L. and Faires, J. D., .Yurwtkal
Anul~~sis, 3rd Edn. Chap. 5. p. ??O. Prindle. Weber & Schmidt, Boston, 1985. D’Angelo, N., Low-frequency electrostatic waves in dusty plasmas. Pltrrtrr. .Spc~ce Sci. 38, 1143. 1990. D’Angelo, N., The Rayleigh-Taylor instability in dusty plasmas. Plurwt. Sprrce Sc,i. 41, 469. 1993. D’Angelo, N. and Song, B., The Kelvin-Helmholtz instability in dusty plasmas. Plutwt. Spuw Sci. 38, 1577, 1990. de Angelis, U.. Formisano, V. and Giordano, M., Ion plasma waves in dusty plasmas : Halley’s comet. J. Plastna P1t.w. 40, 399. lYS8. Goertz, C. K., Dusty plasmas in the solar system. Rer. Geopk~x 27, 171. 1989. Havnes, 0.. de Angelis, U., Bingham, R., Goertz, C. K., Mortil, G. E. and ‘fsytovich, V., On the role of dust in the summer
J. A tm1.r. Tr~rr.P/IJ~.s.52, 637, 1990. of a dusty plasma into a vacuum. Plmet. *Sp:p’IL’c SC,;. 38, 1457, 1990. Luo, H. and Yu, M. Y., Kinetic theory of self-similar motion of a dusty plasma. Ph~xics F/uids B4, I 122, 1992~ Luo, H. and Yu, M. Y., Self-similar expansion of dusts in a plasma. Pkr.sics Fluirk B4, 3066. 1992b. Rao, N. N.. Low-frequency waves in magnetized dusty plasmas. .I. P/u.\ttrtr Plr.r.v. 49, 375. lY93a. Rao, N. N.. Dust-magnetoacoustic waves in magnetized dusty plasmas. Pht,.\ictr Suipttr 48, 363. 1993b. mesopause.
Lonngten,
K. E., Expansion
k’
H ‘itt, C;;
(,47)
where
We observe H’ = Hand
from equations C;’ = G provided
(20). (23). (.A?) and
(‘43) that
and (Ah) where we have used expression (23) for the dust density tt. Using the ideal gas law p = tzT for the dust component. one tinds that equation (A6) is satisfied only if;’ = I. Thus. the results obtained from the simple model employing V/I = ;‘TVtt in the dust fluid momentum equation do differ from those obtained using the exact equation of state (5) when the adiabatic expansion is considered.
Pergamon
Copyright I(‘) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 003’-0633195 $9.50+0.00 0032-0633(95)00021-6
Self-similar expansion of a warm dusty plasma-I.
Unmagnetized case
R. Bharuthram” and N. N. Raot Department
of Physics, University
of Durban-Westville,
Durban
4000. South Africa
Received 4 October 1994 : revised 13 January 1995 ; accepted 13 January 1995
expansion into vacuum of a II;semi-i&&e half-space is investigat.A by ir&tding the adiabatic equation of state for the da& fluid. Given the complex nature of city the anafysis is restricted to expansion of a collisionless plasma have constant charge. Exact b&i& for the ease of the as weil as negative charging or the warm case, numerical the relevant self-simiisothermal negatively a larger distance than an adi&&c one before the density drops to nearly zero. The &ects of several pksma parameters such as the density and the temperature ratios on the expansion profIles are d&XWed.
1. Introduction Plasmas consisting, in addition to the usual electrons and ions. of finite-size, charged particulate matter in the micron or sub-micron range have attracted much attention in recent years. These so-called dusty plasmas exist naturally in astrophysical and space environments such as cometary tails, planetary ring systems, interstellar and circumstellar clouds, etc. (Goertz, 1989). Nearer the Earth, the enhanced radar backscatter from the noctilucent clouds often observed in the polar regions during the summer seasons has been attributed to the presence of dust in the Earth’s lower ionospheric regions (Havnes et al., 1990). On the other hand, understanding of the
*Also at: Plasma
Physics Research Institute, University of Natal, Durban 4001, South Africa. TPermanent address : Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India. Co~re.spon&c~c to : R. Bharuthram
trapping as well as the movement of the dust grains is important in the fabrication of semiconductors using plasma-aided processes. The effect of the dust component on the collective behaviour of plasmas has been extensively studied in the last few years by various authors. In particular, various types of waves and instabilities have been investigated (de Angelis et al., 1988 : D’Angelo, 1990. I993 ; Shukla ct ~1.. 1991; Shukla, 1992 ; D’Angelo and Song. 1990; Rosenberg, 1993 ; Rao, 1993a ; Rawat and Rao, 1993). Because of the large mass of the dust particles compared to the electron and the ion mass. dusty plasmas support new kinds of waves in the very low-frequency regime. By considering thermal electrons and ions having Boltzmann distributions, Rao et al. (1990) predicted the existence of a novel kind of low-frequency electrostatic mode which they called the dust-acoustic wave. The inertial contribution to this mode comes from the dust fluid. Recently, the electromagnetic generalization of this mode to include compressional magnetic field and plasma density perturbations in a magnetized dusty plasma has been obtained by Rao (1993b). The expansion of a dusty plasma is a fundamental process which is important in many practical situations. For example. the dusty plasmas found in space environments have typically free boundaries and therefore can expand into vacuum. On the other hand, in laboratory devices the dust particles in the form of impurities are produced in the source regions and can subsequently expand into the central plasma region. In this regard, Lonngren ( 1990) examined the self-similar one-dimensional expansion of an unmagnetized cold dusty plasma into a vacuum using the model of Rao et al. (1990). The analytical solutions presented by Lonngren (1990) are, however, strictly valid for the case of identically zero electron density. The work of Lonngren has been extended by Y u and Luo (1992) who provide a more detailed analytical study as well as numerical solutions for the density profiles and the flow velocities. A study of the expansion process via kinetic theory (Luo and Y u, 1992a,b) produces results which are in agreement with those from the fluid model. In this
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma--l
1080
paper, we extend the work of Lonngren (1990) and Yu and Luo (1992) by allowing for a finite temperature for the dust particles by including the adiabatic equation of state. In Section 2, we derive the basic governing equations for the self-similar expansion. Exact analytical solutions for the cold case are also obtained both for the positively as well as the negatively charged dust grains. For the general case, numerical solutions are presented in Section 3. A detailed study is made of the effect of plasma parameters such as temperature and densities on the expansion profile. Finally, Section 4 contains a summary of our results and conclusions.
expansion into vacuum in the x-direction at the later times. The equations governing the self-similar expansion of the plasma are obtained by letting II = n(i) and L’= ~(5) where 5 = .vij.t, and i is a normalizing velocity to be specified a posteriori. Then, in terms of the self-similar variable (2). equations (3)~-( 5) become. respectively
(._&!!+ndr-_() 1
d<
-
nm(l?-it)G
dr
(7)
de -
= -Zenz
d$
-z
du
and 2. Theory We consider a three-component plasma consisting of electrons, ions and dust grains, and describe its dynamics by means of the fluid equations wherein the electrons and the ions are in thermal equilibrium at their respective temperatures in the presence of the self-consistent electrostatic field and the inertia is provided by the dust fluid (Rao et al., 1990). Accordingly, the densities of the electrons (n,) and the ions (n,) are given by the Boltzmann distributions, namely
From equation
(2), we find
d4 _=__dt while upon obtain
solving
T, drl,
(10)
en, d<
for 4 from equations
(1) and (2) we
B
(11)
n, = x 4
and
(2) where C$is the electrostatic potential, e the magnitude of the electron charge and T, (T,) the temperature of the ions (electrons). The subscript “0” indicates equilibrium quantities. The dynamics of the dust fluid is governed by the equation of continuity g + the momentum
and the adiabatic
;_(nz!) =0
conservation
equation
The system of equations condition
where b = n& and c( = T,/Te. Upon using equations (9), (11) and (6) to evaluate dp/d<, and then combining with equations (8) and (10) we arrive at
(12) Similarly
equations
(7) and (9) become,
respectively (13)
(3) and
equation
dp -=-
yp(a/?+n:“)
d<
(P-n:+‘)
1 dfz, (14) PC d<
At this point we wish to point out that for cold dust fluid (p = 0), equations (7) and (12) become
of state
is closed with the quasi-neutrality and II, = II, + zn.
(6)
In the above equations. n is the dust number density, I’ the dust fluid velocity, p the dust pressure, y the ratio of specific heats and Z the charge number of the dust particles. Note that Z > 0 (CO) for the positively (negatively) charged dust particles. We assume the dusty plasma to occupy at the initial time the semi-infinite half-space .Y < 0 and consider its
(16) If we normalize the velocity with respect to i = J( 7’Jm), then equations (15) and (16) are just equations (13) and (14) of Lonngren (1990). These equations are also used by Yu and Luo (1992). In these papers. the dust particles are negatively charged, thus from equation (6) our Z is equivalent to their -Z.
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma--I
1081
Returning to our system obtain from equation (14)
(35)
of equations
(12)-(14),
we
(17) where C is a constant conditions. Combining we have
to be determined by the initial this equation with equation (12).
we obtain the analytical solutions cases of dust particle charging : (A) Negatively
charged
for the following
dust particles
two
(Z < 0)
I .‘Z
(18) which when used in equation
(13) yields
where
I‘ = ,iWJH
(1%
$_
where
Vz:+‘-B)_~ (ZI (ap+?z:+
Here. in order that requires, a < $+ ’ Upon obtaining dr/dj from equation with equation (I 8) to obtain
(19), we combine
(B) Positively
solution
charged
’)
(36) is well behaved.
dust particles
(27) one
(Z > 0)
where ZT,
(;=_+-__-___
171
ZT, fl( I +r)‘nf
”
where
2m (,-@_tn;+‘)2
(fl-n:-‘,
Ei(y+ + -z;r
From equation we have
(B-C’) Zn:
(2% )
(p-ng+‘)’
condition
(6)
(23)
Equations (1 I ), ( 17), (19), (21) and (23) constitute a complete set to describe the self-similar expansion of a dusty plasma with warm dust particles. For the general case, we may numerically integrate equation (21) to obtain n,(l). Then, the solutions for ?z,(<), n(c), p(t) and o(4) follow from the other equations in the above set. Before proceeding to such solutions. we give below analytical solutions for the case of cold dust species. For the cold dust fluid, we set p = 0 in expressions (20) and (22) for H and G, respectively. Upon defining N = II:+ ’ , equation (21) is rewritten as
(24) Defining
z(cg+rl:+
(22)
( 11) and the quasi-neutrality
n _
* = __---
1)p (afl+nf+‘)n:
In this case, one requires /I > 17:_’ so that the solution (28) is well behaved. The above transcendental relations yield the implicit solutions for n,(t). The constants C. and C, are determined by the initial conditions. We wish to point out that in the work ofYu and Luo (1992), where the dust particles are negatively charged, a combination of their equations (5) and (9) will yield a solution which is the same as solution (26) obtained above.
3. Numerical results We begin by presenting the governing equations in the normalized form. If p0 and n,, are the initial dust fluid pressure and number density, respectively, then one can use the equation of state (5) to show that the constant C in equation (17) is given by C = p,)/(n,,Z):‘. The dust fluid pressure p is normalized by p,, and the ion density n, by its initial value ~1,~.Writing 2 = p /ZJ where p = + 1 (-. 1) for Z > 0 (CO), and normalizing the velocity by E. = \i( T,/m), equations (19)-(23) become
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma-I
1082 Ax
(6-E;“)
yApr?:
(cX6+$+‘)
1.2
(31)
+ (6-vlpl”)
-
ion
density dust dsadty
-
alsctron
&
*+
(l+r’)S$+’
+:‘(i’fl)
2(&+$-t’)’
2
A@(aste+‘) (6-ny+‘)’
I
1
-
dsnalty
_I n1
and
fi=
(6 -$‘I) Ziii
(33)
where 6 = n,,/n,, and A = p”/(T,ni”). In addition, the normalized dust pressure and the electron number density are given, respectively, by (34)
6
I& = fi;
(35)
where A and ii, are normalized with respect to Q. In the above equations, tildes indicate normalized quantities. For real solutions and for Z > 0 (< 0). that is, for ,U = + 1 (- I), it is necessary that the parameter 6 is given suchthat6>1 (
I Fig. 1. Plot of number densities as a function of the normalized self-similar variable 5, for Z = - 1.O and A = 0.0. with C(= 1.O and 6 = 0. I 1 .o
,
r
1
I
I
5
6
I
-
7’3.0
-
- 7=1.0
R
0
1
2
1
I
(4 1
I
-
0.0
1 U
I
I
I
1
2
3
I
- deotron dandty ion donalty
I
I
5
J 6
b) Fig. 2. (a) Plot of dust density E versus < for 7 = I .O and 3.0. (b) Plot of ion and electron densities versus < for 7 = 1.0 and 3.0. In both (a) and (b) A = 0.5 and Z = - 1.0. with Y = 1.O and 6 = 0.1
and N. N. Rao : Self-similar
R. Bharuthram I
I
I
expansion
1
of a warm dusty plasma0.010
I
I
1083 I
I
1
I
1
A-O.0
-
- - A-O.6 and y-3.0 A-O.6 and y-1.0
-
A-O.0 A-O.6 and y-1.0 - - A-0.6 and y'3.0
\ \ \ \ \ \ \ \ \ \
\ \ I 0
J.
1
0
2
3
4
5
10
20
\; 30
8
A
60
t
Fig. 5. Plot of the dust density ri versus i for the cabe when Z = - 100. Other fixed parameters are as in Fig. I
Fig. 3. Plot of the dust fluid velocity i; versus < for the parameters corresponding to Figs 1 and 2
I
I 50
6
c
1.2
I. 40
I
I
I
A-0.6 and
y-1.0 - - A-O.6 and y-3.0 -
p
1.0
a much further counterpart.
0.6
4. Discussion
0.6
0.4
0.2
-1 0
1
2
c3
4
5
6
Fig. 4. Plot of dust fluid pressure J? versus < for the parameters corresponding to Fig. 2
We next consider results for positively charged dust particles (recall Z > 0 requires 6 = n,,/n,, > 1). For Z = + 1. the relevant curves are shown in Fig. 6ad, while for 2 = + 100 the corresponding behaviour of n” is presented in Fig. 7. From comparison of Figs 2-5 one observes that a positively charged dust fluid expands over
distance
than
its negatively
charged
In this paper, we have investigated the self-similar expansion of an unmagnetized, collisionless dusty plasma consisting of Boltzmann distributed electrons and ions. and a thermal dust component whose dynamics is governed by the fluid equations, including the adiabatic equation of state. For both negatively and positively charged dust particles. it is found that the dust component expands over a larger distance when its initial pressure is non-zero as compared to the expansion of a cold fluid. For the latter case, the numerical solutions of the set of governing equations are in exact agreement with the corresponding analytical solutions which are in agreement with Lonngren (1990) and Yu and Luo (1992). It is found that an isothermal (I$ = 1) dusty plasma expands much further than one which is adiabatic (7 = 3). For the chosen set of parameters, a positively charged dusty plasma expands further than a negatively charged one. A possible explanation for this is that for positively charged dust particles the density of the oppositely charged particles, i.e. electrons, is larger at i = 0 than that of the positive ions for the negatively charged dust particles. Consequently, the attractive electrostatic forces due to the more dense electrons (at < = 0) are able to drag the positive dust particles further out than the expansion experienced by the negative dust particles due to the positive ions. In the Appendix
1084
R. Bharuthram and N. N. Rao : Self-similar expansion of a warm dusty plasma-l
E
0.5 -
no.4
. duet ion density den&y - electron dsmity
-
fp
-
_
-
@‘I
(aI I 1.10 :.
. . . ion -
80
I
I
dermity denmity
ebatroo
*
I
. . . A=O.S and 74.0 - - A-O.6 and 74.;/
16 --
A-0.0
00
6
,
’
/
10 F
16
20
(4 1.2
I
, -
1.0 -
-
-
I
A-O.6 and 793.0 A-0.6 and y-1.0
-
Fig. 6. (a) Plot of number densities versus [ for A = 0.0. (b) Plot of dust density fi versus t: for 7 = 1.O and 3.0, with A = O.S. (c) Plot of ion and electron densities versus < for the parameters of(b). (d) Plot of dust fluid velocity 2; versus [ for the parameters of (a) and (b). (e) Plot of dust fluid pressure p versus 5 for the parameters of (b). In all the above plots 2 = + 1.O, CI= 1.O and 6 = 1.1
we point out the differences that arise when, instead of the full adiabatic equation of state, one uses the usual simple model, namely, Vp = yTVn in the dust fluid momentum equation. Our model is restricted to a one-dimensional expansion in order to obtain some insight into a physical process that is mathematically quite complex. In reality, it would be more correct to investigate a three-dimensional cylindrical or spherical expansion. This would require a numerical solution of the complex set of nonlinear equations which govern the dynamics of the plasma expansion. Furthermore, we have neglected the charge variation effects of the dust particles which, as shown in several recent studies, could play an important role.
The present paper considers the self-similar expansion of an unmagnetized dusty plasma. However, many of the plasmas of practical interest either in the space environment or in the laboratory are embedded in an external magnetic field. Hence, it is necessary to investigate the effect of an external magnetic field on the expansion process. This problem is investigated in the next paper (Rao and Bharuthram, 199.5).
Acknowledgements. One of us (N.N.R.) thanks the FRD (South Africa) for supporting his visit to the University of DurbanWestville. He is grateful to the members of the Physics Department for their warm hospitality.
R. Bharuthram
and N. N. Rao: Self-stmilar
0.0012
I
I
expansion I
A-O.6 -
y-1.0
and y-3.0
A-O.0
II \
I
\
0.0006
\ \
0.0005
0.0004
I
IOXS
R., Self-similar expansion of a warm dusty plasmaII. Magnetized case. Phrtwr. Sptrcc SC,;.43. 1087-1093. 1995. Rao, N. N., Shukla, P. K. and Yu, M. Y’., Dust-acoustic w:~ves in dusty plasmas. Plrrtwt. Sptrw SC;. 34, 543. lY90. Rawat, S. P. S. and Rao. N. N., Kelvin ~Helmholtz Instability driven by sheared dust flow. Plmet. .Spu~c ki. 41, 137. 1993. Rosenberg, M., Ion- and dust-acoustic instabilities in dusty plamas. Ph7(‘/. .~pu~cJ9;. 41, ‘20. 1993. Shukla, P. K., Low-frequency modes In dust> plasmas. I’/II~,s~c~~I .S‘L.l.il)tl/45, 504. 1992. Shukla, P. K., Y’u, M. Y. and Bharuthram. R.. Linear and nonlinear dust drift waves. J. ,~Yc,o@,~.Rc.\. 96. 2 1343. 195,I Yu, M. Y. and Luo. H., Self-similar motion of ;I dusty plasma Phr.\. f_c~r. A161, 506. 1992. Rae, N. N. and Bharuthram,
- - A-O.6and
n
of a warm dusty plasma-
\
0.0003
Appendix
0.0002
\
0.0001 I
\
I
\
‘--_,
0.0000
0
50
I
100
150
Here. we point out the differences in the analysis if one uses the visual simple model, namely. Vp = ;‘TVtt in the momentum equation in order to allow for the pressure term. IF one disregards the adiabatic equation of state (5) and rewrites equation (4) as
200
c Fig. 7. Plot of the dust density ri versus ; for the case when Z = + 100. Other hxed parameters are as in Fig. 6
then equation
(2 I ) is replaced by
$4_ References
d< -
Burden, R. L. and Faires, J. D., .Yurwtkal
Anul~~sis, 3rd Edn. Chap. 5. p. ??O. Prindle. Weber & Schmidt, Boston, 1985. D’Angelo, N., Low-frequency electrostatic waves in dusty plasmas. Pltrrtrr. .Spc~ce Sci. 38, 1143. 1990. D’Angelo, N., The Rayleigh-Taylor instability in dusty plasmas. Plurwt. Sprrce Sc,i. 41, 469. 1993. D’Angelo, N. and Song, B., The Kelvin-Helmholtz instability in dusty plasmas. Plutwt. Spuw Sci. 38, 1577, 1990. de Angelis, U.. Formisano, V. and Giordano, M., Ion plasma waves in dusty plasmas : Halley’s comet. J. Plastna P1t.w. 40, 399. lYS8. Goertz, C. K., Dusty plasmas in the solar system. Rer. Geopk~x 27, 171. 1989. Havnes, 0.. de Angelis, U., Bingham, R., Goertz, C. K., Mortil, G. E. and ‘fsytovich, V., On the role of dust in the summer
J. A tm1.r. Tr~rr.P/IJ~.s.52, 637, 1990. of a dusty plasma into a vacuum. Plmet. *Sp:p’IL’c SC,;. 38, 1457, 1990. Luo, H. and Yu, M. Y., Kinetic theory of self-similar motion of a dusty plasma. Ph~xics F/uids B4, I 122, 1992~ Luo, H. and Yu, M. Y., Self-similar expansion of dusts in a plasma. Pkr.sics Fluirk B4, 3066. 1992b. Rao, N. N.. Low-frequency waves in magnetized dusty plasmas. .I. P/u.\ttrtr Plr.r.v. 49, 375. lY93a. Rao, N. N.. Dust-magnetoacoustic waves in magnetized dusty plasmas. Pht,.\ictr Suipttr 48, 363. 1993b. mesopause.
Lonngten,
K. E., Expansion
k’
H ‘itt, C;;
(,47)
where
We observe H’ = Hand
from equations C;’ = G provided
(20). (23). (.A?) and
(‘43) that
and (Ah) where we have used expression (23) for the dust density tt. Using the ideal gas law p = tzT for the dust component. one tinds that equation (A6) is satisfied only if;’ = I. Thus. the results obtained from the simple model employing V/I = ;‘TVtt in the dust fluid momentum equation do differ from those obtained using the exact equation of state (5) when the adiabatic expansion is considered.