the shock wave structure in a dense electronegative gas containing conductive particles

ISSN 0018�151X, High Temperature, 2013, Vol. 51, No. 5, pp. 575–582. © Pleiades Publishing, Ltd., 2013.Original Russian Text © V.A. Bityurin, A.C. Dobrovol’skaya, N.I. Klyuchnikov, 2013, published in Teplofizika Vysokikh Temperatur, 2013, Vol. 51, No. 5, pp. 643–651.

575

INTRODUCTION

When chemical reactions occur, the relaxationzones behind the shock wave front in a multicompo�nent gas mixture are characterized by a larger variety,considerably differing from that of a single�compo�nent gas [1]. In this work, a special kind analogue ofthe ionizing shock wave is considered, in which theionization arises not due to high energy atomic colli�sions, but owing to thermal electron emission fromsubmicrometer solid particles that are suspended inthe gas.

The study of such systems is of interest for a varietyof areas, such as combustion processes, cosmic gasdynamics, development of gas�cleaning systems, etc.During study of the impact of electric discharge onhigh velocity air�gas flows, intensive electrical fieldswere observed due to the presence of macroparticles[2, 3]. Earlier, the question on the structure of theshock wave was considered in [4, 5]. However, in theseworks, the chemical composition of the mixture wasassumed to be constant, while the macroparticles werecharacterized by a constant charge. In this work, theserestrictions have been removed and the correspondingquantities are calculated on the basis of equations ofchemical kinetics.

In the problem under consideration, the oncomingflow is represented as a neutral gas of density n with asuspension of macroscopic particles of density ng. Letus assume that the particles have a spherical form ofthe same radius a and ng n. We set the density of theparticle material to be 3 to 4 g/cm3, and the mass frac�tion of a neutral molecule and a particle to be ~10–6.This corresponds to the particle radius a = 1.5 × 10–8 m.As the neutral gas, we mean the binary air mixture that

�

contains oxygen and nitrogen in the proportionO2 : N2 = 1 : 4. The variables corresponding to theoxygen and nitrogen are single primed and doubleprimed, so that the density of air is given by n = n' + n''.The concentration of particles is ng = 1017 m–3, and thegas concentration is n = 1025 m–3.

ELECTRON STICKING REACTION

Consider the temperature range where the ioniza�tion (and even dissociation) of the main gas can becompletely neglected. In this case, the only source ofelectrons is thermal electron emission, whereas theonly source of ions is the electron sticking to the mol�ecules of oxygen. Molecular rotational and vibrationaldegrees of freedom are not taken into consideration.In describing the sticking reaction, we follow [6]. Sup�pose that at some initial time, in a homogeneousbinary mixture of molecules of oxygen and nitrogen,electrons are added with a homogeneous density n0.Since the carrying gas is rather dense, the most likelyevent is the formation of negative ions in triple colli�sions. The following two reactions occur:

(reaction 1),

(reaction 2),

depending on the molecule type that is the third parti�cle in the reaction.

We introduce the “ionization degree” (assumingthat sticking is the equivalent of ionization)

(1)

e O2 O2+ + O2– O2+=

e O2 N2+ + O2–

N2+=

α t( )ni t( )

ne t( ) ni t( )+��,=

PLASMA INVESTIGATIONS

The Shock Wave Structure in a Dense Electronegative Gas Containing Conductive Particles

V. A. Bityurin, A. C. Dobrovol’skaya, and N. I. KlyuchnikovJoint Institute for High Temperatures, Russian Academy of Sciences, Moscow, Russia

Received June 21, 2012

Abstract—The structure of a flat shock wave in an electronegative gas containing a rarified suspension ofsmall metallic particles (macroparticles) is considered. When heated due to thermal electron emission, theparticles are ionized in the shock compression that occurs in the carrying gas. The emitted electrons adhereto the electronegative neutral molecules, which gives rise to the formation of a multicomponent gas mixturecomposed of ions, electrons, neutrals, and positively charged macroparticles. The profiles of the correspond�ing variables are calculated within the Navier–Stokes approximation. The reaction of sticking is analyzedwith the use of chemical kinetics equations. The calculations were performed for the case of air, whichassumed by binary mixture of molecules O2 and N2. The initial temperature and the Mach number of theoncoming flow are T = 500 K and M = 3.

DOI: 10.1134/S0018151X13050039

576

HIGH TEMPERATURE Vol. 51 No. 5 2013

BITYURIN et al.

where ne(t) and ni(t) are the electron and ion densitiesat the moment t. For the reactions under consider�ation, the chemical kinetics equation has the form

(2)

where

is the effective frequency for the detachment of the

electron. Here is the rate of the electron detach�ment in the rth reaction (r = 1, 2). The quantity αeq

denotes the ionization degree at equilibrium

where K(T) is the equilibrium constant for the givenreaction,

Here me is the electron mass, kB and ћ are the Bolt�zmann and Planck constants, and A is the electronaffinity energy in eV. It is worth noting that the densityof ions and electrons is many orders of magnitudelower than the density of the neutral component. So,the variation of n' can be completely neglected and wecan assume this variable to be the constant in all of theformulas.

With allowance for ne(t) + ni(t) = n0, from (1) wedirectly obtain the expression for the density of ions

(3)

and for the rate of the density variation

(4)

The explicit temporal dependence of the ion den�sity is obtained using the solution to (2)

(5)

and substituting (2) into (3). From (3), we also obtainthe equilibrium densities of ions and electrons

(6)

In what follows, nonuniform analogs of equations (6)are used when writing the hydrodynamics equations.

HYDRODYNAMICS EQUATIONS

In this work, the shock wave structure is analyzed inthe Navier–Stokes approximation. Though from aformal point of view the hydrodynamics equations arenot applicable for such small space intervals like thewidth of the shock wave transition, it was noted a long

dα t( )dt

�� νdα t( ) αeq–αeq 1–

��,=

νd kd1( )n' kd

2( )n ''+=

( )rdk

αeqn'K t( )

1 n'K T( )+��,=

K T( ) 12�� 2π�

2

mekBT��⎝ ⎠⎛ ⎞

3/2 11604.5AT

��⎝ ⎠⎛ ⎞ .exp=

ni t( ) α t( )n0=

dni t( )dt

�� n0dα t( )

dt�� νd

α t( ) αeq–αeq 1–

��n0.= =

α t( ) αeq 1νdt

1 αeq–��–⎝ ⎠

⎛ ⎞exp–=

nieq αeqn0

n 'K T( )1 n'K T( )+��n0,= =

neeq 1 αeq–( )n0

11 n'K T( )+��n0.= =

time ago that hydrodynamics leads to quite fair resultsfor the shock wave structure [7]. The calculations fordense argon, which are based on the use of moleculardynamics, demonstrate a good agreement with theNavier–Stokes approximation [8]. Therefore, thehydrodynamic approach proves to be an effectivemethod in solving such problems.

Study of the structure of the shock wave is based onthe equations of one�dimensional (along the x axis)stationary gas motion. In essence, these equations rep�resent the equations of mass balance (or the number ofparticles), momentum, and energy written for eachcomponent of the mixture

(7)

(8)

(9)

Index a here takes the values g, e, i, and n denotesthe quantities corresponding to macroparticles, ions,and neutrals: ua is the hydrodynamic velocity; ρa =mana is the mass density; εa = 3kBT/ma is the internalenergy per unit mass (the temperature of all compo�nents is assumed to be the same); Pa is the xx�compo�nent of the tensor of pressure; qa is the heat flux den�sity; and Γa is the rate of the density variation of thecomponent a due to chemical reactions (obviously,Γg = Γn = 0). We also assume that both components ofthe neutral subsystem have the same velocity

(10)

So, equations (7)–(9) at a = n represent a systemequations for the neutral components provided condi�tion (10) holds with allowance for the definitions

The quantity Fa (the force of friction) is determinedby the expression

where

(11)

ddx��na x( )ua x( ) Γa x( ),=

ddx�� ρa x( )ua

2 x( ) Pa x( )+[ ] ea x( )na x( )E x( ) Fa x( ),+=

ddx�� ρa x( )εa x( ) 1

2��ρa x( )ua

2 x( ) Pa x( )+ +⎝ ⎠⎛ ⎞ ua x( )

��+ qa x( ) ea x( )na x( )ua x( )E x( )=

+ Qa x( ) Fa x( )ua x( ).+

u' x( ) u'' x( ) un x( ),= =

ρn x( ) ρ' x( ) ρ'' x( ),+=

nn x( ) n' x( ) n'' x( ),+=

Pn x( ) P ' x( ) P '' x( ),+=

qn x( ) q' x( ) q'' x( ),+=

ρ' x( )ε' x( ) ρ'' x( )ε'' x( )+ ρn x( )εn x( ).=

Fa x( ) Fab x( ),b∑=

Fab x( ) 163

��Ωab1 1,( )mabna x( )nb x( ) ua x( ) ub x( )–[ ].–=


THE SHOCK WAVE STRUCTURE IN A DENSE ELECTRONEGATIVE GAS 577

Here is the known Chapman–Cowling inte�gral [9] and mab = mamb/(ma + mb) is the reduced mass.Equation (11) for the friction force is obtained fromthe solution to a system of Boltzmann equations cor�responding to the first Chapman–Enskog approxima�tion. Provided the isothermal conditions hold, thequantity Qa(x) = (x) (the heat of friction) is

given by

(12)

The system equations must be supplemented by theMaxwell equation to determine the self�consistentelectric field E(x)

(13)

where eg(x) = Zg(x), ee(x) = –e, ei(x) = –e, en(x) = 0 arethe charges of the particles of the components (e is theelementary charge); ε0 is the dielectric permittivity ofthe vacuum.

Now, we concretize this common system of equa�tions as applied to our problem. In the expanded form,the mass balance equations corresponding to (7) aregiven by

(14)

(15)

(16)

(17)

The first term in the right hand side of (16)describes the rate of the electron production due tothermal electron emission. By summing (15) and (16),we obtain the law for the conservation of the electriccurrent

(18)

where

In writing dynamics equations (8) for charged par�ticles, one should bear in mind that their density has avalue of the order ~10–8 of the density of neutrals.Thus, we can neglect the collisions between chargedparticles and take into account only collisions withneutrals. In addition, to a first approximation, we mayneglect the thermal motion of macroparticles due totheir large mass, assuming that

(1,1)abΩ

Qabb∑

Qab x( )mb

ma mb+��Fab x( ) ua x( ) ub x( )–[ ].–=

ddx��E x( ) 1

ε0

�� ea x( )na x( ),a∑=

ddx��ng x( )ug x( ) 0,=

ddx��ni x( )ui x( ) Γ x( ),=

ddx��ne x( )ue x( ) d

dx��Zg x( )ng x( )ug x( ) Γ x( ),–=

ddx��nn x( )un x( ) 0.=

dj x( )dx

�� 0,=

j x( ) ea x( )na x( )ua x( ).a∑=

Pg x( ) 0, qg x( ) 0.= =

Furthermore, by the assumption about collisions,we have for electrons and ions

where pa(x) = na(x)kBT(x) (a = i, e, n). Integral in(11) and the friction force Fab(x) in (12) are associatedwith the parameters of the oncoming flow (the corre�sponding values are marked by subscript “1”); i.e., thetemperature dependence of the integrals is ignored. Itis convenient to pass from integrals to the diffusioncoefficients in the expression for the force of frictionon the neutral component, using the known formulasof the kinetic theory [9]. With all these assumptions,the equations of motion of charged particles take theform

(19)

(20)

(21)

where Da is the diffusion coefficient of the a�type par�ticle in air (a = g, i, e). Moreover, in Eq. (21), the iner�tial contribution of electrons is omitted due to thesmall mass.

Instead of the balance equations of momentum andenergy for neutrals, it is expedient to use the corre�sponding equations for the mixture by summing (8)and (9) over all components with allowance for a smallmolar part of charged particles. As a result, we obtain

(22)

(23)

where

and η and κ are the viscosity and thermal conductivityof the neutral component, respectively. In deriving

Pi e, x( ) pi e, x( ), qi e, x( ) 0,= =(1,1)abΩ

ug x( ) ddx��ug x( )

Zg x( )eE x( )mg

��kBT1

mgnn1Dg

��–=

× nn x( ) ug x( ) un x( )–[ ],

ddx�� ρi x( )ui

2 x( ) pi x( )+[ ] eni x( )E x( )–=

–kBT1

nn1Di

��ni x( )nn x( ) ui x( ) un x( )–[ ],

ddx��pe x( ) ene x( )E x( )–=

–kBT1

nn1De

��ne x( )nn x( ) ue x( ) un x( )–[ ],

ddx�� ρn x( )un

2 x( ) ρg x( )ug2 x( ) Pn x( ) ��+ +

– 12��ε0E2 x( ) 0,=

ddx�� ρn x( )εn x( ) 1

2��ρn x( )un

2 x( ) Pn x( )+ + un x( )⎩⎨⎧

+ qn x( ) 12��ρg x( )ug

3 x( )+⎭⎬⎫

0,=

Pn x( ) pn x( ) 43��η d

dx��un x( ), qn x( )– κ d

dx��T x( ),–= =

578


BITYURIN et al.

Eq. (23) by summing the components in Eq. (9), theterms in the right�hand side of (9), corresponding tothe friction forces and the heat of friction, are summedwith a zero resultant value, but at a glance the remain�ing term j(x)E(x) is not equal to zero. However, withallowance for the electric current conservation Eq. (18),j(x) = const, since at x = ∞ (behind the shock wavefront) the current is zero. This is true at all x. So, theaforementioned term is also equal to zero.

The system of hydrodynamics equations (13)–(17)and (19)–(23) lies at the foundation of our next con�sideration.

THE SHOCK WAVE STRUCTURE

Assume that a flow of dirty air moves along the x�axis from –∞ to +∞. We seek a nonuniform solution tothe hydrodynamics equations that corresponds to ashock wave. The initial and final states are electricallyneutral, and the hydrodynamic velocities of all thecomponents are the same. Equations (14), (17), (22),and (23) are integrated explicitly with the result

(24)

(25)

(26)

(27)

where the constants of integration are expressedthrough the parameters of the oncoming flow. Hereβ = ρg1/ρn1 is the mass fraction of macroparticles andhn1 = εn1 + pn1/ρn1 is the specific enthalpy of the neu�tral component. The electric field energy density canbe neglected as compared to the neutral gas pressure.

In writing the expression for the rate of ion produc�tion Γ(x) in (15), we assume that at each point of theflow, the chemical reaction can be described by theformulas for the homogeneous system which arerelated to the temperature and the density of neutralsand macroparticles at the given point. The value Γ(x)is obtained using (4), (5) and has the form

where

ng x( )ug x( ) ng1u1,=

nn x( )un x( ) nn1u1,=

ρn x( )un2 x( ) ρg x( )ug

2 x( ) Pn x( )+ +

= pn1 ρn1 1 β+( )u12,+

ρn x( )εn x( ) 12��ρn x( )un

2 x( ) Pn x( )+ + un x( ) qn x( )+⎩⎨⎧

+ 12��ρg x( )ug

3 x( )⎭⎬⎫

ρn1 hn112��u1

2+⎝ ⎠⎛ ⎞ u1

12��βu1

3+ ,=

Γ x( ) Zg x( )ng x( )νd x( )α x( ) αeq x( )–

1 αeq x( )–��,=

α x( ) αeq x( ) 1νd x( )

1 αeq x( )–( )un x( )��–exp–

⎩ ⎭⎨ ⎬⎧ ⎫

,=

The effective frequency of detachment νd(x) wasconsidered in [6], and its explicit expression is given by

where d ' = 3.12 × 10–8 cm and d '' = 3.29 × 10–8 cm arethe effective diameters of nitrogen and oxygen mole�cules, respectively, that were fitted using the viscositydata corresponding to their binary mixture [10].

Now we pass to the dimensionless variables. As thelength scale, we take

where η is the viscosity of the neutral component attemperature T1 of the oncoming flow. This value isclose to the free path length of neutrals and reflects theviscous nature of the shock compression. Under thegiven initial conditions, L = 5.6 × 10–6 cm. Below weuse the following dimensionless variables

where E0 = kBT1/eL. After passing to the dimension�less variables, equations (26) and (27) take the form

(28)

(29)

where k = is the ratio of the heat capacities and Pr =

is the Prandtl number. Since the molar fractions of

macroparticles, electrons, and ions are small, it isassumed that the thermodynamical functions of themixture coincide with those for the pure neutral com�ponent. Hereafter, the densities of neutrals and mac�roparticles are excluded with the use of (24) and (25).

αeq x( ) n' x( )K T x( )( )1 n' x( )K T x( )( )+��.=

νd x( ) π 0.8d'2 0.4 2 d ' d''+( )2 m' m''+m''

��+=

× nn x( )kBT x( )

m'�� 11604.5A

T x( )��–⎝ ⎠

⎛ ⎞ ,exp

L 43�� ηρn1u1

��,=

ξ xL��, W

un

u1

��, Viui

u1

��, Veue

u1

��, Uug

u1

��,= = = = =

θ TT1

��, G EE0

��, Nini

nn1

��, Nene

nn1

��,= = = =

ddξ��W ξ( ) β U ξ( ) 1–[ ] W ξ( ) 1–+=

+ θ ξ( ) W ξ( )–

kM12W ξ( )

��,

34�� 1

PrM12 k 1–( )

�� ddξ��θ ξ( ) β

2�� U ξ( ) 1–[ ]=

× U ξ( ) 2W ξ( ) 1+–[ ]

+ θ ξ( ) W ξ( ) k 1–( ) k–+

M12k k 1–( )

�� 12�� W ξ( ) 1–[ ]2

,–

pc

cv

pc

ηκ



The equations for charged particles and Maxwell’sequation are written in the form

(30)

(31)

(32)

(33)

(34)

(35)

where Sca = is the Schmidt number for the a�type

particle, K = μ = mn/mg = 10–6, ζ = ng1/nn1 = 10–8,

and δ = m'/mn = 1.11. The value Ld in (35) is given by

.

In form, the latter formula coincides with the expres�sion for the Debye radius, being dependent on thedensity of particles, but it depends on the density ofneutral particles. In our calculations, we assumed thatthe interaction of electrons and ions with neutral mol�ecules is described by the Sutherland potential [9]. Itsattractive part has the distance dependence ~1/r4,while its repulsive part is taken in the approximation ofhard spheres. The corresponding Ω integrals were cal�culated numerically. To a great accuracy, the interac�tion between the macroparticles and molecules is cal�culated by the model of hard spheres. We used the fol�lowing (calculated) values of Schmidt’s number: Scg =2239.01, Sci = 495.78, and Sce = 0.0171. The relativelyhigh value of Sci is due to the resonant chargeexchange that was considered similarly to [11, 12].

At last, it is necessary to obtain the equation fordetermining the charge Zg(ξ)e of the macroparticle.

U ξ( ) ddξ��U ξ( )

= μ

kM12

�� Zg ξ( )G ξ( ) 43��Scg

U ξ( ) W ξ( )–W ξ( )

��– ,

ddξ��Ni ξ( )Vi ξ( ) K Zg ξ( ) U ξ( ) W ξ( ) θ ξ( ), , ,( ),=

ddξ��Ni ξ( )Vi

2 ξ( ) 1

kM12δ

�� ddξ��Ni ξ( )θ ξ( )– Ni ξ( )G ξ( )–

⎩⎨⎧

=

– 43��Sci

Ni ξ( ) Vi ξ( ) W ξ( )–[ ]W ξ( )

��⎭⎬⎫

,

ddξ��Ne ξ( )Ve ξ( )

= ζ ddξ��Zg ξ( ) K Zg ξ( ) U ξ( ) W ξ( ) θ ξ( ), , ,( ),–

ddξ��Ne ξ( )θ ξ( ) Ne ξ( )G ξ( )+

= 43��Sce

Ne ξ( ) Ve ξ( ) W ξ( )–[ ]W ξ( )

��,–

ddξ��G ξ( ) L2

Ld2

�� ζZg ξ( )U ξ( )�� Ne ξ( )– Ni ξ( )– ,=

1n aD

η

ρ

1 1

,n

Ln u

Γ

Ldε0kBT1

nn1e2��=

Following [6], we assume that the charge rapidlyreaches the local equilibrium value that is determinedby the condition of the zero value of the resultant elec�tric current (of electrons) on the surface of the macro�particle:

(36)

where the resultant current I is determined by the for�mulas

Here nth is the density of the emitted electrons on

the surface of the macroparticle, = is the

thermal velocity of the electron, is the velocity cor�responding to the first Bohr orbit, a0 is the Bohr radius,Ry is the Rydberg constant, and X is the electron workfunction of the particle material (in eV). The value ofω is given by

Owing to the complexity of separate parts of theseequations and due to the rigid nature of the system,direct integration of system (28)–(36) is difficult. As isknown, the latter factor gives rise to additional diffi�culties [13]. Thus, while solving this problem, it isworthwhile to use simplifying assumptions.

For the initial parameters in use, the mass fractionβ is equal to 0.01. Due to the smallness of this value, inEqs. (28) and (29) we take β = 0. In this case, theseequations are reduced to the equations for the pureneutral component. Physically, this means thatcharged particles move on the background of the givenmacroscopic motion of the carrying gas. At Pr = 3/4,that is close to the Prandtl number of air, and equa�tions (28) and (29) were solved exactly by Bekker [14].This solution is used in the present work. Bekkerobtained the dependence ξ = ξ(W) whose inversedependence has only the approximate form

I Zg ξ( ) U ξ( ) W ξ( ) θ ξ( ), , ,( ) 0,=

I 3πa2nth ξ( )vT e, θ ξ( )2πaωDeZg ξ( )

θ ξ( )ωZg ξ( )θ ξ( )

��⎝ ⎠⎛ ⎞exp 1–

��+=

× nth ξ( )ζnn1

ωZg ξ( )θ ξ( )

��⎝ ⎠⎛ ⎞exp

U ξ( ) 10.2nn1

W ξ( )��K θ ξ( )( )+

��– ,

nth ξ( ) 1

8π2a03

��v0

vT e,

��kBT1

2

Ry2

��θ ξ( )3/2 1ωZg ξ( )θ ξ( )

��+⎝ ⎠⎛ ⎞=

× 1θ ξ( )�� ωZg ξ( ) 11604.5

T1

��X+⎝ ⎠⎛ ⎞– .exp

,T ev18

e

k T

mπB

0v

ω e2

4πε0akBT1

�� .=

W ξ( )W2 W2 ξ k 1+( )/2k–[ ]exp+

1 W2 ξ k 1+( )/2k–[ ]exp+��,=

580


BITYURIN et al.

where W2 = is the velocity behind the

shock wave front at ξ = +∞ (the values in this state aremarked with the subscript “2”). At ξ = 0 we have W =

(in both the approximate and exact solutions);i.e., the velocity of the flow equals the critical velocityof sound at this point. The temperature Θ(ξ) in Bek�ker’s solution is related to W(ξ) by the relationship

(Θ2 = 2.68, which corresponds to T2 = 1339.5 K).On the right hand side of the dynamics equation of

macroparticles (Eq. (30)), the ratio of the Coulombic

force to the force of friction has the order Due to this, the Coulombic force can be neglected. Inthis case, the dynamics equation of macroparticles canbe separated from other equations of the system. It canbe solved directly, since the velocity W(ξ) is known.The obtained values allow one to calculate the chargedistribution of the macroparticle behind the shockwave front. This distribution is shown in Fig. 1.

Now, the right hand side of Eq. (31) can be calcu�lated (it is shown in Fig. 2). This makes it possible tointegrate Eq. (31) to obtain the expression for the ioncurrent Ji(ξ) = Ni(ξ)Vi(ξ). In this case, it is not neces�sary to solve Eq. (33), since we can use the above�men�tioned equality for the resultant current. From thisresult we obtain

Figure 3 shows the corresponding two quantities.Knowing the values of the currents makes it possible toeliminate the densities Ni and Ne from the remainingequations. In what follows, we assume that the pres�sure gradients of electrons and ions weakly influencethe velocity distribution, and, therefore, they can beneglected. This is justified below. After that, Eq. (34)reduces to the algebraic form and can be solved relativeto the velocity of electrons

(37)

After substituting (37) into (35), we obtain the sys�tem of two equations

( )

( )

+ −

+

M

M

21

21

2 1

1

k

K

2W

θ ξ( ) 1k 1–( )M1

2

2�� W2 ξ( ) 1–[ ]–=

− −

∼ ∼Sc 1 410 .g

Je ξ( ) Ne ξ( )Ve ξ( ) ζZg ξ( ) Ji ξ( ).–= =

Ve ξ( ) W ξ( ) 1 34Sce

��G ξ( )– .=

Vi ξ( ) ddξ��Ji ξ( )Vi ξ( )

= Ji ξ( )

δkM12

�� G ξ( )–4Sci

3��

Vi ξ( ) W ξ( )–W ξ( )

��– ,

ddξ��G ξ( )

= L2

Ld2

��ζZg ξ( )U ξ( )

��Ji ξ( )Vi ξ( )��–

Je ξ( )W ξ( ) 1 3/4Sce( )G ξ( )–[ ]��–

⎩ ⎭⎨ ⎬⎧ ⎫

.

2.0

100

2.4

2.8

5

2

1

0

–2 –1 10 ξ

lnξ

Zg

Fig. 1. Distribution of the macroparticle charge number inthe shock wave (the electron work function is X = 2.5 eV).

2

1000 50

4

0

K, 10–10

ξ

Fig. 2. The dimensionless production rate of negative ions(the electron affinity is A = 0.87 eV).

00 5

7

14

21

10lnξ

Je

Ji

J, 10–9

Fig. 3. The dimensionless densities of the ion and electroncurrents behind the shock wave front.



The solution to these equations together with (37)determines the velocity profiles and electric fieldbehind the shock wave front. The results of numericalintegration are shown in Figs. 4 and 5.

As we can see from Fig. 4, due to the large crosssection of resonant charge exchange, the ions arenearly “frozen” into the neutral component. On theother hand, owing to their large mass, the friction ofmacroparticles by neutrals is weak, giving rise to a sig�nificant separation of charge and creating the electricfield. Moreover, the contribution of the electrons tothe charge density is negligible. Thus, in essence, elec�trons do not take part in the formation of the electricfield. It is the ion freezing�in effect and the large elec�tric field acting on the electrons that determine thedynamics of ions and electrons and make the pressure

gradients to be unimportant. The electrons acquirelarge velocities in the electric field, returning to thefinal state much later than the other charged compo�nents. This determines the width of the relaxationzone, which is ~1 cm.

Shown in Fig. 6 is the distribution for the electricpotential

The jump of this potential at the front of the shockwave is about 570 V. Figure 7 shows the density ofcharged particles in the relaxation zone. It is worthnoting that the decrease in the density of electrons is

φ ξ( )kBT1

e�� G ξ( ) ξ.d

ξ0

ξ

∫–=

2

0 5 10

4

6

8

0 4 8

0.4

0.8

0

1

3

4

2

1

2

3

4

lnξ

Fig. 4. Distribution of the dimensionless velocities of thecomponents of the gas mixture behind the shock wavefront: 1, macroparticles; 2, neutrals; 3, ions; 4, electrons.

–60 5 10

–4

–2

0

lnξ

E, 105 V/m

Fig. 5. The electric field distribution.

0

0 4 8

300

600

lnξ

φ, V

Fig. 6. The electric potential distribution.

0

0 5 10

2

4

6

10–8

1

2

3

lnξ

Fig. 7. The dimensionless densities of charged particles behindthe shock wave front: 1, ions; 2, macroparticles; 3, electrons.

582


BITYURIN et al.

caused not only by the detachment, but is also due tothe sharp increase in the velocity.

CONCLUSIONS

In this work, we examined the shock wave structurein a gas that contains metallic macroparticles (nano�particles). An important physical factor considered isthe thermal emission of particles upon their heating inthe shock wave, which leads to the appearance of thecharge on a macroparticle, and formation of the mul�ticomponent structure of the mixture behind theshock wave front, consisting of electrons, negativeions, neutrals, and positively charged macroparticles.We obtained the density distributions of particles andthe charge of macroparticles along the shock layer.The second effect observed in our study is the separa�tion of positive and negative charges occurring due tothe macroscopic motion of charged components, con�trary to the usual situation in a plasma where such aseparation occurs due to thermal motion. Thisunusual polarization leads to the creation of largeelectric fields that have been observed experimentally[3]. Furthermore, in this work we obtain the distribu�tion of the electric field along the shock wave.

Our investigation holds at a small molar fraction ofmacroparticles (10–8) and the smallness of their massfraction (β = 0.01). Under these assumptions, themotion of neutrals can be thought of as unperturbedmotion, so that the distribution of the mass velocityand temperature is determined by the density shockcompression in the neutral gas. However, with anincrease in the mass fraction of macroparticles, theycan disturb the motion of the neutral gas and the dis�tribution of these parameters will be determined by theshock transition in the binary mixture of macroparti�cles and neutrals. This question was discussed earlierin the literature (see [15] and references therein).

REFERENCES

1. Zel’dovich, Ya.B. and Raizer, Yu.P., Physics of ShockWaves and High�Temperature Hydrodynamic Phenom�ena, New York: Dover, 1967.

2. Velikodniy, V.Yu. and Bityurin, V.A., in Proceedings ofthe Fifth International Workshop on MagnetoplasmaAerodynamics for Aerospace Applications, Moscow, Rus�sia, April 7–10, 2003, Moscow: Joint Institute for HighTemperatures of the Russian Academy of Sciences,2003, p. 429.

3. Bityurin, V.A., Velicodny, V.Yu., Vorotilin, V.P.,Grishin, V.G., Eremeev, A.V., Nikitenko, L.K., Timo�feev, I.B., Janovsky, J.G., and Van Wie, D., in Proceed�ings of the Sixth International Workshop on Magneto�plasma Aerodynamics for Aerospace Applications, Mos�cow, Russia, May 24–27, 2005, Moscow: Joint Institutefor High Temperatures of the Russian Academy of Sci�ences, 2005, vol. 2, p. 552.

4. Velikodnyi, V.Yu. and Bityurin, V.A., Dokl. Phys., 1998,vol. 43, no. 7, p. 389.

5. Bityurin, V.A. and Klyuchnikov, N.I., Fluid Dyn., 2005,vol. 40, no. 3, p. 494.

6. Bityurin, V.A. and Klyuchnikov, N.I., High Temp.,2011, vol. 49, no. 4, p. 466.

7. Serrin, J., in Series: Encyclopedia of Physics. Mathemati�cal Principles of Classical Fluid Mechanics, Truesdell, C.,Ed., Berlin: Springer�Verlag, 1959, vol. 3/8/1.

8. Hoover, W.G., Phys. Rev. Lett., 1979, vol. 42, no. 23,p. 1531.

9. Hirschfelder, J.O., Curtiss, Ch.F., and Bird, R.B.,Molecular Theory of Gases and Liquids, New York: JohnWiley and Sons, 1954.

10. Cole, W.A. and Wakeham, W.A., J. Phys. Chem. Ref.Data, 1985, vol. 14, no. 1, p. 209.

11. Mason, E.A., Vanderslice, J.T., and Yos, J.M., Phys.Fluids, 1959, vol. 2, no. 6, p. 688.

12. Rapp, D. and Francis, W.E., J. Chem. Phys., 1962, vol. 37,no. 11, p. 2631.

13. Hairer, E. and Wanner, G., Solving Ordinary DifferentialEquations: II. Stiff and Differential–Algebraic Problems,Berlin: Springer�Verlag, 1996.

14. Kochin, N.E., Kibel, I.A., and Roze, N.V., TheoreticalHydromechanics, New York: Wiley, 1964, part 2.

15. Fernandez�Feria, R. and Fernandez de la Mora, J.,J. Fluid Mech., 1987, vol. 179, p. 21.

Translated by G. Dedkov

the shock wave structure in a dense electronegative gas containing conductive particles

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