# МАТЕМАТИЧЕСКИЙ АНАЛИЗ УРАВНЕНИЙ СОХРАНЕНИЯ ДВУХФАЗНЫХ СМЕСЕЙ

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• 532.525 DOI: 10.14529/mmp140202

.. , ..

-

. , . , -

. , , -

. , .

. -

-

. ,

-

. ,

. , . .. -

.

,

, .

: ; ; -

.

- , . - . , , [1, 2]. , [3]. [4, 5], [6].

, , - , , - . , , . .. , .. [7, 8] , , - . , - , . .

2014, 7, 2 29

• .. , ..

- . , . [9], . - .

1.

, (s) (g). , . . [9], ,

st

+ vssx

= svsx

ss

F, (1)

gt

+ vggx

= gvgx

gg

(vs vg)sx

+gg

F (1 gs

)Cgg

, (2)

ss

[vst

+ vsvsx

]= s

sx

+ (g s)sx

( +

Cs2

)(vs vg), (3)

gg

[vgt

+ vgvgx

]= g

gx

+( +

Cs2

)(vs vg), (4)

ss

[est

+ vsesx

]= ss

vsx

+

x

(ks

Tsx

) h(Ts Tg) (s s)F, (5)

gg

[egt

+ vgegx

]= gg

vgx

+

x

(kg

Tgx

)+ h(Ts Tg) +

+(s s)F (vs vg)pgsx

+ (vs vg)2 (es eg)Cs , (6)

st

+ vssx

= F +Css

, s = 1 g. (7)

, . . [9]. (g) (s) - ; i, vi, pi, Ti, ei, i, i, hi, i, ki , , ,, , , , , - i- (i = s, g) , -

, h , Cs ,

i = ii

(ii

)iTi

, c , F = sg[ps pg s]/c. (1), (2) ; (3), (4) -

; (5), (6) ; (7) .

, - D. :

vs = vs +D, (8)

vg = vg +D. (9)

30 .

• :

x = x+Dt. (10)

:

x=

x, (11)

( t

)=

( t

)+

( x

)D. (12)

- (1) (8) (12). (1) :

st

+sx

D + (vs D)sx

= s(vs D)

x s

sF

st

+sx

D + vssx

D1x

= svsx

ss

F.

:

st

+ vssx

= svsx

ss

F. (13)

, (2) (8) (12) :

gt

+ vggx

= gvgx

gg

(vs vg)sx

+gg

F (1 gs

)Csg

. (14)

- (3) - (8) (12). (3) :

ss

[vst

+Dvsx

+ vsvsx

Dvsx

]=

= ssx

+ (g s)sx

( +

Cs2

)(vs vg)

ss

[vst

+ vsvsx

]= s

sx

+ (pg ps)sx

( +

Cs2

)(vs vg). (15)

, (4) - :

gg

[vgt

+ vgvgx

]= g

gx

+( +

Cs2

)(vs vg). (16)

2014, 7, 2 31

• .. , ..

- (5). (8) (12), -:

ss

[est

+Desx

+ vsesx

D esx

]=

= ssvsx

+

x

(ks

Tsx

) h(Ts Tg) (ps s)F

ss

[est

+ vsesx

]= sps

vsx

+

x

(ks

Tsx

)+ h(Ts Tg) (ps s)F. (17)

, (6) :

gg

[egt

+Degx

+ vgegx

D egx

]= gpg

vgx

+

x

(kg

Tgx

)

h(Ts Tg) + (ps s)F (vs vg)pgsx

+ (vs vg)2 (es eg)Cs

gg

[egt

+ vgegx

]= gpg

vgx

+

x

(kg

Tgx

)+ h(Ts Tg) +

+(ps s)F (vs vg)pgsx

+ (vs vg)2 (es eg)Cs . (18)

, , (7). (8) (12) :

st

+Dsx

+ vssx

Dsx

= F +Css

,

:

st

+ vssx

= F +Css

. (19)

, , -, (13) (14) , - (15) (16) , (17) (18) , - (19). (13) (19) , , -, (1) (7) .

. - (3) (4) - vs vg .

ssvs

[vst

+ vsvsx

]= vs

[ s

psx

+ (pg ps)sx

( +

Cs2

)(vs vg)

]32 .

• ggvg

[vgt

+ vgvgx

]= vg

[ g

pgx

+( +

Cs2

)(vs vg)

].

ss

[ t

(v2s2

)+ vs

x

(v2s2

)]= vs

[ s

psx

+ (pg ps)sx

( +

Cs2

)(vs vg)

](20)

gg

[ t

(v2g2

)+ vg

x

(v2g2

)]= vg

[ g

pgx

+( +

Cs2

)(vs vg)

]. (21)

. (5), (6) (20), (21) .

ss

[ t

(es +

v2s2

)+ vs

x

(es +

v2s2

)]= vss

psx

+ vs

[(pg ps)

sx

( +

Cs2

)(vs vg)

] sps

vsx

+

x

(ks

Tsx

) h(Ts Tg) + (ps s)F

gg

[ t

(eg +

v2g2

)+ vg

x

(eg +

v2g2

)]= vgg

pgx

+

+vg

( +

Cs2

)(vs vg) gpg

vgx

+

x

(kg

Tgx

)+ h(Ts Tg) +

+(ps s)F (vs vg)pgsx

+ (vs vg)2 (es eg)Cs .

ss

[Est

+ vsEsx

]=

x(svsps) + vs

[pg

sx

( +

Cs2

)(vs vg)

]+

+

x

(ks

Tsx

) h(Ts Tg) (ps s)F (22)

gg

[Egt

+ vgEgx

]=

x(gvgpg) + vg

( +

Cs2

)(vs vg) +

+

x

(kg

Tgx

)+ h(Ts Tg) (ps s)F vspg

sx

+ (vs vg)2 (es eg)Cs (23)

Es = es +v2s2; Eg = eg +

v2g2.

- (20) (21) , , , . , .

(22) (23) , .

2014, 7, 2 33

• .. , ..

. (22) (23) .

ss

[Est

+ vsEsx

]+ gg

[Egt

+ vgEgx

]=

x(svsps + gvgpg) + (24)

+

x

(ks

Tsx

)+

x

(kg

Tgx

) Cs

[(es eg) +

1

2(vs vg)2

].

, ( ) (24) - , (24): - .

, , .. [10]. , . (24) - , [10],

[ssEst

+ssvsEs

x

]+

[gsEgt

+ggvgEg

x

]=

x(svsps + gvgpg). (25)

(24) .

[ssEst

+ssvsEs

x

]+

[gsEgt

+ggvgEg

x

]=

x(svsps + gvgpg) +

+

x

(ks

Tsx

)+

x

(kg

Tgx

) Cs (v2g vsvg).

[10], , ,

[ssEst

+ssvsEs

x

]+

[gsEgt

+ggvgEg

x

]= (26)

= x

(svsps + gvgpg) Cs (v2g vsvg).

- [10] , (26) Cs (v2g vsvg).

34 .

• 1. (1) (7) - , (1) (2), (3) (4), - (5) (6), (7) .

2. , .

3. , - [9, 10], - , , .

.. -

.

13 01 00072.

1. , .. / .. , .. // . 1989. . 308, 5. . 10741078.

2. , .. / .. , .. // . 1989. . 25, 6. . 7279.

3. , .. / . -, . // . : . 1997. . 3. . 3943.

4. , .. /.. . .: ,1978. 336 .

5. , .. / .. //- . 2011. . 84, 1. . 7492.

6. .. / .. // . 1956. . 20, . 27. . 184195.

7. , .. - / .. , .. // . : -. . . 2012. . 6, 11 (270). . 47.

8. , .. - / .. , .. - // . : . 2012. 27 (286), . 12. . 6973.

9. Baer, M. F Two-Phase Mixture Theory for the Deagration-to-Detonation Transition (DDT)in Reactive Granular Materials / M. F. Baer, J. Nunziato// Int. J. Multiphase Flow. 1986. V. 12. P. 861889.

2014, 7, 2 35

• .. , ..

10. / .. ,.. , .. , .. // , . 1981. . 3. . 3943.

, - , , - , - - (. , ), yum_kov@mail.ru.

, , , - , (. , ), ea_kov@mail.ru.

25 2013 .

Bulletin of the South Ural State University.Series "Mathematical Modelling, Programming & Computer Software",

2014, vol. 7, no. 2, pp. 2937.

MSC 76N15 DOI: 10.14529/mmp140202

A Mathematical Study of the Conservation Equationfor Two-Phase Mixtures

Yu.M. Kovalev, South Ural State University, Chelyabinsk, Russian Federation,yum_kov@mail.ru,E.A. Kovaleva, Chelyabinsk State University, Chelyabinsk, Russian Federation,ea_kov@mail.ru

We study the invariance under the Galilean transformations of the Baer Nunziato

equations for interpenetrating interacting ows which describe the transition from

combustion to explosion in two-phase mixtures. We show that the original Baer Nunziato

model is invariant. In addition, we establish the invariance of the kinetic and total energy

equations for the components and mixture. But the conservation equations for the total

energy of the mixture in the Baer Nunziato model and in the model of Nigmatulin's group

have dierent behavior. Thus, additional study is required to choose the model describing

more adequately the transition from combustion to explosion in two-phase mixtures.

Keywords: mathematical model; invariance; multicomponent mixture.

References

1. Grishin .., Kovalev Yu.. [Experimental Study on the Impact of the Explosion ofCondensed Explosives on the Front Crown Forest Fire]. Doklady Akademii Nauk, 1989,vol. 308, no. 5, pp. 10741078. (in Russian)

2. Grishin .., Kovalev Yu.. [Experimental and Theoretical Study of the Interaction of theExplosion on the Front Crown Forest Fire]. Combustion, Explosion, and Shock Waves, 1989,vol. 25, issue 6, pp. 724730. DOI: 10.1007/BF00758739

3. Kovalev Yu.., Cheremokhov A.Yu. [The Weakening of the System of Air Shock WaveGratings]. Voprosy atomnoy nauki i tekhniki. Seriya: Matematicheskoe modelirovaniezicheskikh protzessov [Problems of Atomic Science and Technology. Series: MathematicalModelling of Physical Processes], 1997, no. 3, pp. 3943. (in Russian)

36 .

• 4. Nigmatulin R.I. Osnovy mekhaniki geterogennykh sred [Fundamentals of Mechanics ofHeterogeneous Media]. Moscow, Nauka, 1978. 336 p.

5. Kuropatenko V.F. [New Models of Continuum Mechanics]. [Novie modeli mechanicisploshnikhsred]. Journal Engineering Physics and Thermophysics , 2011, vol. 84, no. 1,pp. 7492. (in Russian) DOI: 10.1007/s10891-011-0457-0

6. Rakhmatulin K.A. [Fundamentals of gas dynamics of interpenetrating motions ofcompressible media]. Prikladnaya matematika i mekhanika [Journal of Applied Mathematicsand Mechanics], 1956, vol. 20, no. 27, pp. 184195. (in Russian)

7. Kovalev Yu.., Kuropatenko V.F. Analysis of the Invariance of Some Mathematical Modelsof Multi-Media. Bulletin of the South Ural State University. Series "Mathematics, Mechanics,Physics", 2012, no. 11 (270), pp. 47. (in Russian)

8. Kovalev Yu.., Kuropatenko V.F. Analysis of the Invariance Under the GalileanTransformation of Some Mathematical Models of Multi-Media. Bulletin of the South UralState University. Series "Mathematical Modelling, Programming & Computer Software",2012, no. 27 (286), issue 12, pp. 6973. (in Russian)

9. Baer M.F, Nunziato J. Two-Phase Mixture Theory for the Deagration-to-DetonationTransition (DDT) in Reactive Granular Materials. Int. J. Multiphase Flow, 1986, vol. 12,pp. 861889. DOI: 10.1016/0301-9322(86)90033-9

10. Vaynshteyn P.B., Nigmatulin R.I., Popov V.V., Rakhmatulin H.A. Nonstationary Problemsof the Combustion of Aerosuspensions in Fuel that Contains the Oxidant. Fluid Dynamics,1981, vol. 16, no. 1, pp. 1419. DOI: 10.1007/BF01094807