J. Plasma Fusion Res. SERIES, Vol.2 (1999) 177-179

Stability Analysis of Wave Propagation in 2-Fluid Dusty
Plasma SystemTAKAISHI Takeshi* and KUSANO Kanya'Hiroshima-DENKI Institute of Technology, Hiroshima 739-0321, Japan
Hiroshima University, Higashi-Hiroshima 7 39-8 526, J apan
1

(Received: 25 December 1998 / Accepted: 6 May 1999)

Abstract
The stability of electro-static waves in 2-fluid dusty plasma system is investigated. We study thefluid dynamics of the dust particles and ions with the electrons following the Boltzmann distribution. We
find some branches in its linear dispersion that are approximated to the dust-ion-acoustic wave when 466+0. (qao is the equilibrium charge on the dust particles.) We show the stability of these waves in I
dimensional system,and demonstrate the growth of these waves.

Keywords:
dust, plasma, dispersion relation

1. Introduc'tion
We study the corporate dynamics of dust particles
and ions. We investigate the 2-fluid dusty plasma!dt n, *!,n^ v,) = oa =-rUAi
aadQ
asystem,where 2 species fluid dynamics with the charge
evolution process of the dust particles.In this paperwe
show the results of the linear dispersion relationin thehomogeneous dusty plasma.We find the acoustic-likemode,which is approximated to dust-ion-acoustic
mode,of the linear dispersion relationin this system witharuo*'oA;'oaaaa,aa^ Q^tV^=-Qt_" "ctx." = I"(ea,Q)+1"(qo,nr)
dta.lniv i) =Q
n'* 6;

homogeneous medium.

,dQ

tu'*'tar't

mi dx

- L*,

2. Linear Dispersion Relation
We assume that the dynamics of electrons behavesas the massless motion with Boltzmann distribution, andions and dust particles as the fluid motion, because of
the large mass ratio between electrons and ions (dust= 4n (-en.(Q) + en,+ e an),whereparticles). In this paper we study 1-dimensional
unmagnetized cold dusty plasma system for simplicity,and eliminate the decreament of ni in our 2-fluid modelfrom the condition that n", n;)) rt6. We consider thevariation of the charge of the dust particles as thecollision current [].
*Present address: Hiroshima Kokusai Gakuin University,| '",n ^,1[r'r, =-(T)t R2en"(d)exp ,#,/sor,\j1(ao, n' ) = l----------r I R'e n,\^'J,-,%lpr,)@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchHiroshima 739-0321, Japan, E-mail: ttks@ g.hkg.acipt'17

Takaishi r . et al., Stability Analysis of wave propagation in 2-Fluid Dusty plasma System(qo < 0).

n, ni, and n d are the number densities of elec-trons, ions, and dust particles, v1 and v6 are the velocity
of ions and dust particles, Ti and T" are the temperatureof ions and dust particles, respectively. 46 is the localaverage charge of dust particles, R is the size of dust

0.001

1. ,10grain.

6We study the propagation of the electro-static
waves in homogeneous medium.

1, . 10-ei)4ao=o
In the case of qoo - 0 (no dust charge in theequilibrium), the equilibrium conditions are ft=Land nis = n9. However, from milm" = 1836, the first

i. ,10 "equation is not realistic in this plasma.
ii) q6s+ 0 (4ao < 0)

1, , 10-15

In the case of qdo+ 0 (existence of the dust charge

ln-1750 -1500 -12s0 -1000 -750 -500

the equilibrium), the equilibrium conditions

arel-,\ I\tt2J/Fig. 1 n,o, ndo versus eoo/ewhen D"o = 10-4, R = lg-a, moo= 10-15. Solid line describe the no, dotted line
describe n,, and hashed line describe n,.

\n,^=[!!LlIt -9ltsl . 'ol'aoolng'"Pr,) Rr"/\r,/m,l \-250 0\andnoo=dlo(n! -n,01
r.t5

under the condition that n66 > 0. Figure I shows nio, nao

for n! = 10-6. When we treat this system oider n"= n,)) k4, wa may set qasle = 1700.

We derive the following linear dispersion relation
with the Fourier mode exp (i (kx - at)),L.250= a)2?i1t + k:D)a3 +lC,(tt + k?r.)+ 4En do C rl ro2 + i ( rrfu + of,i)kz a

0. 7s- t@'or+al)cr *? $,c.ltl
where0,25['.;5-,,=-fi"[1*,,RT")\C, =-]

tl"o,Fig.2 Real frequency ar, normalized to ar., versus

cr=- L I"o,wavenumber k, normalized to ko", when eaole = 1738.

ftio,."=-(T)t R2e n2expPgt\' ^\Rr"/the dust-ion-acoustic mode o)ore = ({;)"(a2ra + a2o)tt2 [2). Figure 3 shows the growth rate ofandthis mode. This mode is slowly growing for ft/ftp" > l 8.

al^=(4n,.o'noolt''\m"l3. Conclusions

Figure 2 shows the frequency ofthese electro-static
waves when eaole = -1783. These are approximated toIn this paper we have presented the 2-fluid dusty
plasma system for the condition that n., ni )) tra.
178

Takaishi T. et al., Stability Analysis of Wave Propagation in 2-Fluid Dusty Plasma System

Treating the dynamics of ions as the fluid motion, we
7.

can analyze the wave propagation that the mode is basedon that of the plasma without dust particles. We mainly

5.i0

consider the case 4oo < 0. We find the modified ionacoustic mode, which is approximated to the dust-ionacoustic mode. For the case that t is much larger than

5.10

2.5,10ftp", electrons distribute homogeneously in space. Whenthe density of ions locally decreases, the negative charge-2.5.10of dust particles increases, and it makes the electro-static
attractive force between ions and dust particles larger.

-5.10

Then the electro-static waves grow so that the-7.5'10

fluctuations of the density of ions and the charge of dust
particles are in the out-of-phase.

Referencestll L. Spitzer, Jr., Physical Process in the Intersteller

Fig. 3 Growth rate, normalized to al", versus k/kD" whenqaol e = -1738.

tzl

r79

Medium, John Wiley & Sons, New York (1978).
P.K. Shukla, and V.P. Silin, P/rys. Scr. 45, 508(1992\.