Self-Similar Expansion of a Dusty Plasma

J. Plasma Physics (1997), vol. 58, part 3, pp. 467–474. Printed in the United Kingdom

� 1997 Cambridge University Press

467

Self-similar expansion of dusty plasmas

S. R. P I L L A Y,1 S. V. S I N G H,1 R. B H A R U T H R A M1†and M. Y. Y U2

1Department of Physics, University of Durban–Westville, Durban 4000, South Africa2Theoretische Physik I, Ruhr-Universitat Bochum, D-44780 Bochum, Germany

(Received 27 September 1996)

The radially symmetric self-similar expansion of a dusty plasma is investigated incylindrical and spherical geometries. The electrons and ions are assumed to be inBoltzmann equilibria, while the dynamics of the dust particles is governed by thefluid equations. The effects of finite dust pressure as well as dust-charge variationare included.

1. IntroductionDusty plasmas are of interest because of their importance in studies of space en-vironments, such as planetary rings, cometary tails, and the Earth’s ionosphere,as well as in many industrial and laboratory devices. The dust grains are usuallymassive and negatively charged particles of micrometre or submicrometre size.They can significantly alter the dynamics of a plasma by introducing new physicaltime and space scales. Of particular interest is the problem of transport of impurityspecies from their source regions, such as near the tails of comets and spacecraft.

There have been several studies of the quasineutral self-similar planar expansionof dusty plasmas (Andersson et al. 1980; Liberman and Velikovich 1986; Lonngren1990; Yu and Luo 1992). Although mathematically self-similar solutions are a par-ticular class of solutions, experience has shown that they are very often useful indescribing the long-time behaviour of fluid systems (Zel’dovich and Raizer 1967)regardless of their earlier histories. Lonngren (1990) investigated the problem ofone-dimensional self-similar expansion of a dusty plasma into a vacuum. An exactanalytical solution was obtained for a plasma with zero electron density. Yu andLuo (1992) extended this problem to include electrons, and showed that during theexpansion the dust density can vanish. The results were shown to agree well withthose obtained from collisionless kinetic theory (Luo and Yu 1992a). It was alsoshown (Luo and Yu 1992b) that, in the presence of two or more species of dustparticles, the densities of the various dust species can vanish one by one, dependingupon their charge-to-mass ratios (the smaller ones vanish first). For most practi-cal applications, however, the expansion occurs in spherical or cylindrical geometry.Therefore, Yu and Bharuthram (1994) studied the self-similar expansion of dust par-ticles in cylindrical geometry. Three different self-similar variables were considered,and the results were shown to be similar to those for planar geometry, except that

† Also at Plasma Physics Research Institute, University of Natal, Durban 4001, SouthAfrica.

468 S. R. Pillay et al.

the dust density vanished only asymptotically. The electron density can also vanishrapidly, with the dust and ion charges balancing each other. The self-similar expan-sion of a warm dusty plasma was investigated for an unmagnetized (Bharuthramand Rao 1995) and a magnetized (Rao and Bharuthram 1995) plasma. In all of theabove studies, the dust particles were assumed to have uniform charge and mass.

However, in an actual situation the charge on a dust grain can vary in accordancewith the local (varying) electrostatic potential. Therefore Melandsø et al. (1993)studied dust acoustic waves in planetary rings, taking account of dust-charge vari-ation. Yu and Luo (1995) investigated the self-similar expansion of a warm dustyplasma containing variable-charge dust grains. It was found that the grain chargecan vary significantly during the expansion because of the rapid change of the elec-trostatic potential. In the present paper, we study the expansion of a dusty plasmain cylindrical and spherical geometries, taking into account the charge-variationeffects.

2. Cylindrical expansionSince the electrons and ions are lighter than the dust particles, on the time scaleof the dust motion they can be considered to be in Boltzmann equilibria. Theirdensities are thus given by

ne = ne0 exp(eφ

Te

), (1)

ni = ni0 exp(−eφTi

), (2)

where nj0 and Tj (j = e, i) are the unperturbed densities and temperatures of theelectrons and ions, e is the electron charge and φ is the electrostatic potential.

The dust particles with density n, radial velocity v, charge q, massm and pressurep are assumed to obey the warm-fluid equations. Their dynamics is governed bythe set of radial equations

∂n

∂t+∂(nv)∂r

+nv

r= 0, (3)

∂v

∂t+ v

∂v

∂r= − q

m

∂φ

∂r− 1mn

∂p

∂r, (4)

where the pressure is governed by the adiabatic equation of state(∂

∂t+ v

∂

∂r

)p

n3 = 0. (5)

We note that the pressure force can compete with the electrostatic force in theearlier stages of the self-similar expansion. In fact, the cooling effect arising fromthe pressure variation is an important feature of the expansion.

The charge of a dust grain is assumed to be determined only by the microscopicelectron and ion currents flowing into the grain. Charging through photo or impactemission is excluded. In the probe model, the grain-charging process is governedby the charge-conservation equation

∂q

∂t+ v

∂q

∂r= Ie + Ii, (6)

where the average electron and ion currents Ie and Ii flowing into the grains are

Self-similar expansion of dusty plasmas 469

determined by the local electron and ion densities and the potential differencebetween the grain surface and the local plasma.

The electron grain current can be written as (Barnes et al. 1992)

Ie = −πa2e

(8Teπme

)1/2

ne exp(eq

CTe

), (7)

where a is the average grain radius and me is the electron mass. The potentialdifference ∆φ between the grain and the local plasma is expressed in terms of thedust charge q by introducing the constant grain capacitance C = 4πε0a such thatq = C∆φ. The average ion grain current is (Barnes et al. 1992)

Ii = πa2e

(8Tiπmi

)1/2

ni

(1− eq

CTi

), (8)

where mi is the mass of the ion. When dtq = 0, the constant grain charge is de-termined by the balance of the electron and ion grain currents, or Ii + Ie = 0,corresponding to the so-called floating potential at the grain surface. The corre-sponding charge-neutrality condition is given by

e(ni − ne) + nq = 0. (9)

We now introduce the dimensionless self-similar variable ξ = x/cdst and makethe Ansatze

nj =ni0Nj(ξ)ωpdt

, n =ni0N (ξ)ωpdt

, q = eZ(ξ),

v = cdsV (ξ), φ =TiΦ(ξ)e

,

where ωpd = (4πni0 e2/m)1/2 is the reduced ion plasma frequency, cds = (Ti/m)1/2

is the dust sound speed and Z is the variable dust-grain charge number. Equations(1) and (2) can then be written as

Ne = Ne0 exp (δΦ), (10)

Ni = exp (−Φ), (11)

where δ = Ti/Te and Ne0 = ne0/ni0. Following the procedure of Lonngren (1990)and Yu and Luo (1992), we obtain a system of coupled nonlinear ordinary differ-ential equations describing the self-similar expansion process of a three-componentplasma in cylindrical geometry as

dN

dξ=ξN 3ZαA−NB(Ni + δNe)

ξ(V − ξ)Dc, (12)

dV

dξ=−[ξN 2ZαA− PC−N 2Z2(V − ξ)

]ξDc

, (13)

dP

dξ=PN

[3ξNZαA− (V − ξ)2C + 2ξNZ2

]ξ(V − ξ)Dc

(14)

dNidξ

=−NiN

{ξαA[N (V − ξ)2 − 3P ]− ZB

}ξ(V − ξ)Dc

, (15)

dZ

dξ=

αA

V − ξ , (16)


where

α =a2 ni0 (8πTi/mi)

1/2

ωpd, β =

e2

C Ti, γ =

(miTemeTi

)1/2

, Ne = Ni + ZN.

We have also defined

A = Ni(1− βZ)− γNe exp(βδZ),

B = 2ξP +N (V − ξ)3,

C = (Ni + δNe)[3V − ξ],Dc = (Ni + δNe)[N (V − ξ)2 − 3P ]−N 2Z2.

Equations (12)–(16) are highly nonlinear inhomogeneous ordinary differentialequations, and it is difficult to find analytically the condition for the existenceof meaningful solutions. However, we note that if all the derivatives are treatedas independent variables and the equations are considered as algebraic equationsthen, for the solutions (more exactly, the derivatives of N , V , P , Ni and Z) to exist,the determinant Dc of the coefficients must not vanish (Yu and Luo 1995). This isin contrast to the constant-charge case where the equations are homogeneous (andthe corresponding determinant must vanish) and can be integrated analytically.

We have integrated (12)–(16) numerically. Depending on the values of variousparameters such as α, β and δ and the initial conditions, different types of solutionscan be found. Figures 1 and 2 show two typical results for cylindrical expansion. Itshould be pointed out that the initial values are not arbitrary, instead they mustsatisfy the quasineutrality condition Ni + ZN = Ne and Dc� 0. The parametersand initial conditions for Fig. 1 are α = 0.001, β = 0.1, δ = 0.3, N0 = 0.3, Ni0 = 1,V0 = 10, Z0 = −2 and P0 = 1 for a hydrogen plasma. It can be seen from Fig. 1 thatfor the initial stage (ξ < 3) of the expansion the pressure force is dominant in theexpansion. Note that the dust density decreases rapidly. The dust expansion endsat ξ ≈ 10, where the determinant Dc vanishes and the densities of ions, electronsand dust particles become constant.

Figure 2 shows the expansion of a dusty plasma for the parameters α = 0.01,β = 0.1, δ = 0.01, N0 = 0.3, Ni0 = 1, V0 = 10, Z0 = −4 and P0 = 0.5. Herethe dust density decreases slowly compared with the case shown in Fig. 1, and theintegration stops at ξ ≈ 3.8. Further integration gives unphysical results. Solutionsdo not exist for Dc < 0.

3. Spherical expansionThe continuity and momentum equations for the dust fluid in spherical geometryin the cold-plasma limit are

∂ n

∂t+∂(nv)∂r

+2nvr

= 0, (17)

∂v

∂t+ v

∂v

∂r= − q

m

∂φ

∂r. (18)

Using the charge-balance equation (6) and the same self-similar variables andnormalizations as in the last section, we obtain the following set of nonlinear


1.4

1.2

1.0

0.8

0.6

0.4

0.2

01 2 3 4 5 6 7 8 9 10

f

Figure 1. The self-similar parameters for cylindrical expansion. Here and in the Fig. 2,the solid, dash-dotted, dotted, dashed, +++++ and ××××× lines represent N/N0, Ne/Ne0,Ni/Ni0, V/V0, Z/Z0, and P/P0 respectively. We have used α = 0.001, β = 0.1 and δ = 0.3,and the initial conditions are N0 = 0.3, Ni0 = 1, V0 = 10, Z0 = −2 and P0 = 1.

1.4

1.2

1.0

0.8

0.6

0.4

0.2

01.0 1.5 2.0 2.5 3.0 3.5 4.0

f

Figure 2. Cylindrical expansion for α = 0.01, β = 0.1, δ = 0.01, N0 = 0.3, Ni0 = 1, V0 = 10,Z0 = −4 and P0 = 0.5.


1.2

1.0

0.8

0.6

0.4

0.2

01 2 3 4 5 6 7 8 9 10�

f

Figure 3. Spherical expansion for α = 0.001, β = 0.1, δ = 0.3, N0 = 0.1, Ni0 = 1, V0 = 10and Z0 = −3. Here and in Fig. 4 circles represent Z/Z0; the rest of the key is as in Fig. 1.

inhomogeneous ordinary differential equations for the spherical expansion of adusty plasma:

dN

dξ= −N (V − ξ)2(2V − ξ)(Ni + δNe)− αZξN 2A

ξ(V − ξ)Ds, (19)

dNidξ

=NNi(V − ξ) [Z(2V − ξ)− αξA]

ξDs, (20)

dV

dξ=NZ2(2V − ξ)− αξNZA

ξDs, (21)

where Ds= (V − ξ)2(Ni + δNe)−NZ2 is the determinant of the coefficients of thederivatives.

Equations (19)–(21) along with (16) are solved numerically. For solutions to exist,the determinant Ds again must not vanish. Figure 3 shows the self-similar solutionsfor the spherical expansion of a dusty plasma. The plasma parameters and initialconditions are α = 0.001, β = 0.1, δ = 0.3, N0 = 0.1, Ni0 = 1, V0 = 10 and Z0 = −3,where the subscript 0 denotes the (initial) value of the corresponding variable atξ = 1. It can be seen from Fig. 3 that the dust density falls off more rapidly than inthe cylindrical case. As the determinant Ds vanishes at ξ ≈ 10 the dust expansionends. Interestingly, the dust density vanishes with the determinant. At this pointthe electron and ion densities become equal. Figure 4 shows the spherical expansionfor the parameters and initial conditions α = 0.001, β = 0.1, δ = 0.1, N0 = 0.8,Ni0 = 1, V0 = 10 and Z0 = −1. Here the dust density shows the same behaviour asis in Fig. 3, but the ion density decreases rapidly. As the dust density approaches


1.2

1.0

0.8

0.6

0.4

0.2

01 2 3 4 5 6 7 8 9 10�

f

Figure 4. Spherical expansion for α = 0.001, β = 0.1, δ = 0.1, N0 = 0.8, Ni0 = 1, V0 = 10and Z0 = −1.

zero, there is a slight increase in the electron density and the dust charge Z. Theincrease in the electron density can be explained by the quasineutrality conditionNi + ZN = Ne. In order to maintain quasineutrality at all times, the electrondensity must increase. Again, solutions do not exist for Ds < 0.

4. DiscussionUsing the multi-fluid model, we have investigated self-similar cylindrical and spher-ical expansions of dusty plasmas by taking into account the effect of dust-chargevariation. The ordinary differential equations governing the self-similar variablesare highly nonlinear, and it is not possible to obtain analytically the existenceconditions for solutions. The equations were therefore solved numerically.

In the case of cylindrical expansion, the dust density does not need to vanishcompletely, although the pressure can vanish during the early stages of the self-similar expansion. For spherical expansion, the dust density can approach zero.Solutions do not exist or cease to exist when the determinant of the coefficientscorresponding to the set of nonlinear equations for the derivatives is or becomesnegative (Dc,s 6 0). The fall-off in the dust density is more rapid in the sphericalcase.

The results here can be useful for estimating the distributions of dust particlesor other impurities in the vicinities of rocket exhausts, asteroids, comets and otherdust-shedding bodies (Bernhardt et al. 1995). They can also be useful in predictingthe distribution of impurity particles in laboratory or industrial plasmas (Bohmeet al. 1994).


Acknowledgements

Dr S.V. Singh wishes to express his gratitude for the hospitality provided by thePhysics Department, University of Durban–Westville, Durban, South Africa. Thiswork has been supported by Foundation for Research and Development, SouthAfrica, and the Sonderforschungsbereich 191 Niedertemperatur Plasmen, Germany.

References

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Self-Similar Expansion of a Dusty Plasma