Dust acoustic shock waves in two-component dusty plasma

The effect of electron–dust collision on small-amplitude nonlinear dust acoustic (DA) waves in two-component thermal dusty plasma consisting of positively charged (due to thermionic emission) dust grains and electrons has been investigated incorporating the nonadiabaticity of dust-charge variation arising due to delays in the dust charging, i.e. due to small nonzero values of ωpd/νch, where ωpd is the dust-plasma frequency and νch is the dust-charging frequency. The propagation of small-amplitude DA waves is governed by a modified Korteweg–de Vries–Burger equation in which the Burger term arising due to the charge delay induced dissipation. Numerical investigations reveal that this equation has a shock-wave solution. Numerical investigations also reveal that in the absence of collision-induced dissipation the charge-delay-induced dissipation also causes the generation of a DA shock wave in two-component dusty plasma.


Introduction
It is well known that low-frequency oscillations in so-called three-component 'complex' plasmas consisting of electrons, ions and negatively charged dust grains give rise to some new lowfrequency eigenmodes, e.g. 'dust ion acoustic' (DIA) [1] and 'dust acoustic' (DA) [2]. The dust grains immersed in a plasma are electrically charged. In a laboratory plasma system, the dust grains are usually negatively charged. However, in space as well as in the laboratory, dust grains can be charged positively by UV irradiation [3,4] and by thermionic emission [5]- [7]. The charge on the dust grains is not fixed but fluctuates due to the fluctuating electric field. In a three-component dusty plasma with fluctuating dust charges and high dust densities, there also exists another wave mode called the 'dust Coulomb (DC) wave' or the 'dust electro-acoustic (DEA) wave' [8]. The damping or instabilities of the DA wave due to the effects of dust-charge variations and collisions have been studied both theoretically and experimentally [9]- [15].
The charge q d on the dust grain is an extra dynamical variable which controls grain motion but itself must be determined from the orbit motion limited (OML) grain charging equation where I is the total current flowing to the dust grain surface, ω pd (= (q 2 d n d0 )/( 0 m d )) is the dust-plasma frequency and ν ch [∼∂ I /∂q d ] is the dust-charging frequency. For nonzero values of ω pd /ν ch the linear DA wave mode in weakly coupled dusty plasma becomes damped [9]- [11]. Experimental observations [15] reveals that in inhomogeneous strongly coupled dusty plasma the DA wave becomes unstable due to the delay (ω pd /ν ch 1 but is nonzero; in the experiment in [15], it is O(10 −4 )) in the charging of dust grains. The nonlinear analysis of DA, DIA and DC in a weakly coupled three-component dusty plasma shows that this delay in the charging (which is usually called nonadiabatic dust-charge variations) causes a dissipation in dusty plasma. This dissipation leads to the formation of both DA shock, DC(DAE) shock waves [16] and DIA shock waves [17]- [19]. Another experimental observation [20] reveals that the DIA shock is generated due to ion-dust collision through ion viscosity [21,22] [29] for electron-ion plasma. On the other hand, in the absence of electron-dust collision, this mKdVB equation reduces to the well known KdV-Burger equation [16]. Numerical investigations show that this equation possesses a compressional shock-wave solution and depending on the magnitude of the electron-dust collision and ω pd /ν ch , it exhibits both monotonic and oscillatory shock.
The paper is organized as follows. Formulation of the problem is given in section 2. Section 3 deals with the nonlinear evolution equations describing nonlinear DA waves in twocomponent electron-dust plasma. The shock-wave solution of the mKdVB equation is given in section 4. Section 5 contains the numerical results and discussions. A summary of the result is given in section 6.

Formulation of the problem
We consider two-component, unmagnetized, nonrelativistic thermal dusty plasma consisting of positively charged dust grains and electrons. The positively charged dust grains undergo elastic collisions with the electrons. The charges on the dust grains fluctuate according to the electric field E. In this situation at equilibrium, E = 0, the overall charge neutrality condition becomes where n e0 (n d0 ) is the electron (dust) equilibrium number density and z d0 e is the (positive) charge residing on the dust grains.
To incorporate the effect of the friction-dominated dust fluid momentum loss through electron-dust elastic collisions on the DA wave, we assume that the electron-dust collision frequency is much larger than the dust-plasma frequency.
and R e = m e n e ν ed (v e − v d ), are the momentum losses of dust grains and electrons per unit volume due to electron-dust elastic collisions, where m d (m e ) is the dust (electron) mass, n d (n e ) is the dust (electron) number density, v d (v e ) is the dust (electron) velocity, ν ed and ν de are the electron-dust and dust-electron collision frequency, then Krook's model suggests that where the electron-dust collision frequency [30] is given by In the above expression = ln[ √ 5λ De /za] is the Coulomb logarithm, where λ De = ( 0 T e )/(n e0 e 2 ) is the electron plasma Debye length, a is the dust grain radius, z = (z d0 e 2 )/(4π 0 aT e ) is the nondimensional dusty plasma parameter and V te = √ T e /m e is the electron thermal velocity.
The one-dimensional behaviour of the thermal plasma under consideration may be described by the following normalized equations for the two-fluid model: To close this system, instead of using an electron continuity equation we consider the following current displacement equation for electrons and dust fluids: In the above set of equations and q d is the total charge on the dust grains. The timescale T is normalized in . The normalized electron-dust collision frequency is ν ed = ν ed /ω pd . The space scale X and dust velocity V d , electron velocity V e are normalized in units of electron Debye length λ De (= ( 0 T e )/(n e0 e 2 )) and DA speed The electric field E is normalized by e/(λ De T e ). In the dust momentum conservation equation (5), the momentum loss due to viscous drag arising from dust-dust and dust-electron collisions through the dust viscous stress η de ∂ 2 v d /∂ x 2 is ignored compared to that from the frictional force ν de v d . This is because the ratio of the friction force to viscous stress is of the order The inequality holds good for (z d0 T e )/T d 1 (even though for thermal dusty plasma T d = T e as z d0 1) and ν de /ω pd 1 (our assumption). Relation (2) has been used to derive the dust momentum conservation equation (5). The momentum loss due to electron-dust collisions has been considered in both the dust and electron momentum conservation equations (5) and (6). It is assumed that the electron-dust collision frequency ν ed is much larger than the dust-plasma frequency ω pd , i.e. ν ed /ω pd 1 and δ = ( . Hence by neglecting the term O(ω 2 pd /ω 2 pe ), equation (6) can be rewritten as The normalized charge, Q d , on the dust grain is determined by the following OML dustcharging equation: where I Th e is the thermionic emission current [5]- [7] and I e is the electron plasma current. The normalized expressions of the I Th e and I e currents for spherical dust grains of radius a are as follows: 142.5 and where W e is the work function, z = z d0 e 2 /4π 0 aT e is the nondimensional dusty plasma parameter and 4π 0 a is the capacitance of the spherical dust grains of radius a.
The dust-charging frequency ν ch is given by where ω pe is the electron plasma frequency and V te is the electron thermal velocity.

Nonlinear evolution equations
Our aim is to find the equations for a single fluid (dust), which describe the behaviour of nonlinear DA waves. By eliminating N e and V e from equations (5), (7), (8) and (9)-(12), we obtain The grain-charging equation (10) together with (11) and (12), becomes where β d = 1/(z(z + 1)) and Q is the perturbed part of the normalized dust charge Q d , i.e. Q d = 1 + Q.
In order to study the small-amplitude nonlinear DA wave in electron-dust two-component plasma, the reductive perturbation technique has been employed and the following stretched coordinate introduced: where λ is the phase velocity of the linear DA wave normalized by the DA speed and measures the order of the smallness of the perturbations. The dynamical variables N d , V d , Q and the electric field E are expanded as where To employ the reductive perturbation technique, we first consider the dust grain charging equation (17). Using stretching (18), perturbation expansion (19) for all the variables and expressions (11) and (12) we arrive at To include the effect of charge delay in nonlinear evolution equations and to make the nonlinear perturbation consistent with that of (18) and (19), we chose the following scaling [16]: where ν d ≈ O(1). Using the above scaling in equation (20) and equating the terms in lowest powers of , i.e. the term O( ) and the terms O( 2 ), we get the following equations: The expression for the normalized electron-dust-collision frequency in terms of the dust-charging frequency can be rewritten as To make the nonlinear perturbation consistent with that of (19) and (21), we assume the following scaling: where ν c ≈ O(1). The justification of scalings (21) and (25) is discussed in section 5.
Applying equations (19) and (25) to dynamical equations (14)- (16) and equating the terms at the lowest powers of i.e. O( Equating the O( 5 2 ) terms, we get from (14)-(16) 142.7 For DA waves the value of λ follows from equations (22) and (26) Here β d arises due to the dust-charge variation. The above expression thus shows that the linear DA wave phase velocity is modified due to the dust-charge variation. Eliminating ∂ N (2) d /∂ξ from equations (29) and (30) and then using (22), the following relationship can be obtained: Adding equation (30) with the resulting equation obtained by the elimination of ∂ V (2) d /∂ξ from equations (28) and (32), we arrive at the following equation: Finally using equations (22), (23) in (33) and then eliminating E (1) with the help of equation (27), we get the following mKdVB equation: where The above expressions show that the dust-charge variation modifies both the coefficient of nonlinearity α and the coefficient of dispersion β, as the term β d arising due to dust-charge  [29] for two-component electron-ion plasma, and the dissipation is proportional to the electron-dust collision frequency ν ed . On the other hand, in the absence of electron-dust collisions (ν c ≈ 0 ⇒ ν ≈ 0), we recover the well known KdV-Burger equation and the dissipation is proportional to µ arising due to the delay (ν d = 0) in the charging. Hence the charge delay plays a dissipative role in dusty plasma and this dissipation leads to the formation of a shock wave (figure 3 with ν = 0, dotted curve) in two-component dusty plasma. Again, with the help of relation (25) and by virtue of equation (24) the expression for ν can be rewritten as Similarly with the help of (21) the expression for µ can be rewritten as with β d = 1/(z(z + 1)). From (39) and (40), we have (41) Using the numerical values given in table 1, we see that Thus the dissipation arising due to the delay in the charging is much greater than the dissipation arising due to electron-dust collisions under the assumption that the electron-dust collision frequency is much greater than the dust-plasma frequency in two-component dusty plasma.

Shock-wave solution
On transforming to the wave frame the modified KdVB equation (34) reduces to On introducing the transformations equation (44) is rewritten as a set of simultaneous equations for u and v du dη This system of equations has two singular points at (u, v) = (0, 0) and (2V /α, 0). The first one, i.e. (u, v) = (0, 0), is a saddle point, whereas (u, v) = (2V /α, 0) is a stable node or a stable focus according to (µ + νV ) 2 > or < 4βV.
A stable node corresponds to a monotonic shock (dissipation dominant) front (figure 4) while a stable focus implies that the shock structure is oscillatory (dispersion dominant) ( figure 3).

Numerical results and discussions
In our numerical analysis, we considered the following dusty plasma parameters: plasma temperature T e = T d = 5 eV, 0.5 eV; dust number density n d0 ≈ 5 × 10 13 m −3 ; average dust grain radius a ≈ 1 µm; dust mass density ρ d ≈ 10 3 kg m −3 so that the dust mass m d ≈ 4 3 πρ d a 3 ≈ 4.19 × 10 −15 kg; electron mass m e ≈ 9.1094 × 10 −31 kg implies m e /m d ≈ 2.17 × 10 −16 . The values of the ratio of the dust-charging frequency (ν ch ) to the dust-plasma frequency (ω pd ) and the values of the normalized electron-dust collision frequency (ν ed ) (equation (24)) are shown in table 1 for different values of the plasma temperature, T e , and the work function, W e . It is found that for T e = T d = 5 eV, ω pd /ν ch ≈ O(10 −3 ) and for T e = T d = 0.5 eV, it is O(10 −4 ). Hence, to include the effects of charge delay, these values justify the scaling ω pd /ν ch ≈ ν d 1/2 (equation (21)) on the basis of which charging equation (equation (18)) is approximated. This table also shows that for T e = T d = 5 eV, the normalized (normalized by the dust-plasma frequency, ω pd ) electron-dust collision frequency is 1/ν ed ≈ O(10 −4 )) and for  Thus the numerical values justify our assumption that the electron-dust collision frequency is greater than the dust-plasma frequency and also justify the scaling ν ed ≈ ν c −1/2 (equation (25)). This table also suggests that with the increased work function W e , the dust-charging frequency decreases, whereas with the increase in plasma temperature, the charging frequency increases. Figures 1 and 2 show the variations in the coefficient of the nonlinear term, α, and the coefficient of the dispersive term, β, with the work function, W e , for different plasma temperatures. Figure 2 shows that the nonlinearity of the DA wave decreases with W e and increases with the plasma temperature. Figure 3 shows that β behaves in a qualitatively similar way to α.
The system of equations (46) have been solved numerically by the usual Runge-Kutta-Felhberg method with the help of the values of the above-mentioned parameters. Figures 3 and 4 are drawn for W e = 2, T d = T e = 5 eV and Mach number M = 2. In figure 3 the solid curve represents the oscillatory shock structure for the dissipation arising due to both the electron-dust collision and nonadiabatic dust-charge variations. The dotted curve represents the same for the dissipation arising only due to the nonadiabatic dust-charge variation, i.e. dissipation arising due to a delay in the charging. This figure shows that the DA wave in a two-component electrondust plasma exhibits an oscillatory shock wave. Figure 4 is drawn for T e = T d = 5 eV and W e = 2 eV and shows the monotonic nature of the DA shock wave. These two figures also show the transition of an oscillatory shock to a monotonic shock when the dissipation increases. Figure 5 shows that the shock strength given by   increases as the work function, W e , increases. This figure also shows that as the plasma temperature increases, the shock strength decreases. Thus the DA shock speed increases as the work function increases and the plasma temperature decreases. 142.13

Summary
We summarize the results as follows.
(1) In this paper the nonlinear characteristics of a DA wave in two-component electron-dust (positively charged due to thermionic emission) dusty plasma has been studied incorporating nonadiabatic dust-charge variations (i.e. the charge delay has been included) under the assumption that the electron-dust collision frequency is greater than the dust-plasma frequency (by equation (25) and table 1). It is seen that the nonlinear DA wave is governed by a mKdVB equation (34). The dissipation arising due to the delay in the charging leads to the formation of a shock wave in two-component dusty plasma.  2). (4) The dissipation arising due to a delay in the charging has larger effect on the DA wave than the electron-dust collision-induced dissipation under the assumption ν ed = ν ed /ω pd 1 in two-component electron-dust dusty plasma (equation (42)).