LETTERE AL IWUOVO CI~]~NTO

VOL. 36, N. 1

The Generalized Burgers' E q u a t i o n in Radiative

1 Gennaio 1983

Magnetogasdynamies

(*).

~r BARTUCCE]~T~I Dipartimento di Matematica, Universith della Calabria . Cosenza, Italia

(rieevuto il 21 Giugno 1982; manoseritto revisionato ricevuto il 6 0 t t o b r e 1982) PACS. 52.35. - Waves, oscillations and instabilities in plasma.

The propagation of plane waves in r a d i a t i v e magnetogasdynamics was recently investigated b y GnEco and GIAMB6 (1) and S~DOV and NARIBOLI (2) b y using the reduction-perturbation methods developed b y TANIUTI and WEI (a,4). They obtain that, when the dissipative effects are considered, the equation ruling the propagation is a Burgers' equation t h a t can be integrated b y using the Cole-Hopf transformation (~). I n this letter we investigate the two-dimensional propagation b y using a more general perturbation-reduction method (6), obtaining a Burgers' equation with nonhomogeneities. The plane where the propagation occurs is supposed to be orthogonal to the magnetic field. The field equations governing a viscous, heat-conducting and electrically conducting gas at a sutlleiently high temperature (T > l0 ~ K) with the termal radiation are (7)

(1)

30 - - + V-(ev) = o, ~t

(2)

~ ~+v.V

v=--V(P+P~)+#(VxH)xH+V.~,

(3) j2

= (P + P~)(V.v) + (~.V)v + -ff

(*) (1) (2) (3) (~) (~) (~) (~)

+ V(zVT) +

V(D~E~)

This work was supported by CNP~-GNFM. S. GIAlvIBb a n d A. GRECO: t~iv. Mat. Univ. Parma, 5, 471 (1979). SEDOV-NARIBOLI: J. Phys. Soc. Jpn., 44, 1380 (1978). To TANIUTI a n d C. C. W E I : J. Phys. Soc. Jpn., 24, 941 (1968). To TANI~YTI: Suppl. Prog. Theor. Phys., 55, 1 (1974). G. B. ~VITHAM: Linear and N o n Linear Waves ( N e w Y o r k , N . Y . , 1974). S. GIAMBb, A. GRECO, P. PANTANO: C. R. Acad. Sci: Paris Set. ~4, 289, 553 (1979). S. I. I~ ~lagnetogasdynamics and Plasma Dynamics (Berlin, 1962).

21

22

~ . BARTUCCELLI

with the Maxwell equations

(4)

~ H = v x(vxH)--V x(~VxH), ~t

where ~ and v are the material density and the gas velocity, P and PR the gas dynamical and radiation pressure, respectively, # the magnetic permeability, H the magnetic field and T the viscous stress tensor _ [ ~v r

~vs~

~v k

\i~xs

~X r]

~X ~

=

r,s = 1,2,3,

'

where 5r~ is the Kronecker symbol, with ~ and 0 being the viscosity coefficients; v being the magnetic diffusibility, C r is the specific heat at constant volume, T the absolute temperature, J = V • H, ~ is the electric conductivity, E R the radiation energy density, Z the heat conductivity and D~ the diffusion coefficient of radiation. We also confine our research to the so-called thick gas approximation, by imposing P~ = E g / 3 and E ~ = a R T ~, where a n is the Stefan-Boltzmann constant. Finally we have the following state equation:

(5)

P = RqT.

We now suppose that the flow occurs in the plane O X Y orthogonal to the magnetic field H, and all the functions depend on X, Y and t only. Then, following (6), we look for a solution Q, v ~ ( u , v , O ) , H ~ (0,0, H) and T, of the system (1)-(4) which describes a perturbation of a given constant s61ution ~0, Vo = 0, H0(0, 0, H0), To expressed as an asymptotic series in terms of a small parameter e

q~ ~

(6)

sqz 1 +

s2u2 +

...,

v =sv 1 +s2v~+...,

~=~o

+~H1 + ~ g 2 + ....

T =T o +eT x +e2T2 + ....

where (7)

~ = ~n(~ 0r ~) ,

Un = U n ( I f x, ~) ,

- ~ n = H n ( ~ cr ~) ,

Tq~ = T n ( ~ % ~) ,

~'b = 1, 2 . . . .

with

(8)

~ = e~(X~),

~ = 0, 1, 2, X o = t .

By inserting (6) in (1)-(4), and by using (7) and (8), we equate to zero the coefficients of

THE GENERALIZED BURGERS~ EQUATION IN RADIATIVE I~AGNETOGASDYNAMICS

23

the obtained power series in e. Then, at first order in e we obtain C~Vl

--2-~ ~ §

nl + ~ o ~ n ~ =

0,

(9) eH 1

eV1

-- 2 ~

+ NoMo

?lI + N o M o

~ = O,

where, as usually, we have p u t (s)

(]o)

,~

~0t

and

-

n --

V
57+1

with M--

N = (Cv + 4a~T*) 1,

2ER 3

S = R - } - - - - T4 a R 3 @

P,

3

W e h a v e n o n t r i v i a l s o l u t i o n s for t h e first d e r i v a t i v e of ~ , u~, v 1, H~, T1, w i t h r e s p e c t t o ~, if w e s u p p o s e t h a t tlie p h a s e ~ s a t i s f i e s t h e e i k o n a l e q u a t i o n

where A = -L V Z - +

NoMo% + #H~ @o

I n this ease we obtain t h a t (11)

~1 = ~0//,

Ul :

"~H~'tl,

Vl :

~I/~2,

H1 : I t o l I ,

T1 : N o M o I I

with II=H(v ~, ~) being a function of its arguments, to be determined in the next stage. The second order gives

54@1

{~1

~Vl~

[

[~gl

~A

(s)

G. BOILLAT: La ~)ropagation des ondes (Paris, 1965).

~V2 \ ]

avA l=

~

lVL BARTUCCELLI

8T~ ]

[

8u~

RT~ 8~ ----§

~ o + ~-o ~-~ + ~ o ~ + - ~o ~o-- ~

~ ~ J ~+ ~o ~

~+

+lV~l L-

8~ + \-~-o ~ + ~ ~

+

02) rv av~

+ lV~l ~

~.,~,

~,~

e~,~-o~-~,,~--~v~§

~1 - l V v l ~

~v~

~

+-

. ~ - I V ~ l ~#

= o

r ,~_u-~ 8u~ ~,~ ] § ~ +_,u,,-~-~,:~§

+ lVvl L-

with ~ = 2~7 + O, ~ = g + 4aRD~ TS. B y (12) we can derive, as usually (6), an algebraic compatibility equation t h a t can be written as r-3H c3 8H ~H - ~ + / / = - [ l o g O ] + ~ / / - = . = fi 8 ~ '

(13) where

,

1

]

:~---- RTo +-~(?oeo + ,V § MoNogo) + t~H~ +-~So(Wo + MoNo) ]Vg[, l [~o33 + /zvtt~~ + ~oNoMoSo] lVgl~ with

y = R/q(MN-- T),

(2 = 4aRT2/~(N M ) , - T/3),

W = ~(MN)~ + MN(MN/o .

Here a is a contracted time along the rays, solution of

(14)

d~ ~ da

~YJo 8~

d~a da

~Wo 8~

THE G:ElqERALIZED BURGERS ~ EQUATION IN RADIATIVE MAGI~ETOGASDYlqAMICS

25

where

E q u a t i o n (13) g o v e r n s t h e t r a s p o r t of t h e first o r d e r of t h e p e r t u r b a t i o n a l o n g t h e r a y s . I n t h i s case t h e e q u a t i o n e x p l a i n s t h e b a l a n c e b e t w e e n t h e n o n l i n e a r t e r m w o r k i n g for t h e s h o c k f o r m a t i o n , t h e d i s s i p a t i v e t e I n l a n d t h e g e o m e t r i c a l t e r m , l i n e a r i n H. W e h a v e a g r o w i n g or d a m p i n g for H a c c o r d i n g to d O / d a > 0 or d O / d a < O. B e c a u s e t h e p r o p a g a t i o n occurs i n a c o n s t a n t s t a t e , t h e r a y e q u a t i o n (14) c a n b e integrated, obtaining ~7~ = ~

+ X~l~,

a = 1, 2 .

I t follows t h a t t h e r a y s are, a t e v e r y i n s t a n t , o r t h o g o n a l to s p a t i a l surfaces, i . e . t h e w a v e s are <~p a r a l l e l )). I n t h i s case, w e c a n e x p r e s s t h e e x p a n s i o n p a r a m e t e r 0 as

(15)

0 = ~/]•

(~.o/ro)~,

w h e r e r o is t h e i n i t i a l r a d i u s . O b v i o u s l y r o is n o t c o n s t a n t for e v e r y rays. E q u a t i o n (13), for p l a n e w a v e s r e d u c e s t o a B u r g e r s ' e q u a t i o n s i m i l a r t o t h a t o b t a i n e d i n (1). B y u s i n g t h e f o r m (15), cq. (13) c a n b c w r i t t e n as

(16)

5~' ~- 2~'-- + ~ ~ 7 + ~ - ~ = 0 ,

where ~(2 - - ( ~ / f i ) H ,

d r ' -- (-- ro/~0) d a

and

~' = - - 2.

E q u a t i o n (16) is t h e so-called (, c y l i n d r i c a l )) B u r g e r s ' e q u a t i o n , t h a t is s i l n i l a r to t h a t o b t a i n e d i n (2). O b v i o u s l y t h i s e q u a t i o n h a s t h e s a m e f o r m a l e x p r e s s i o n for e v e r y ray, h u t r 0 is n o t c o n s t a n t for e v e r y ray. r 0 is c o n s t a n t o n l y w h e n we c o n s i d e r w a v e fron~s w i t h c i l y n d r i c a l s y m m e t r y . A n e q u a t i o n s i m i l a r t o (16) arises i n a large class of p h y s i c a l s y s t e m s (9) w h e n a b a l a n c e b e t w e e n d i s s i p a t i o n , g e o m e t r y a n d n o n l i n e a r i t y exists. F o r t h i s e q u a t i o n , u p to now, a n a n a l y t i c a l s o l u t i o n is n o t k n o w n , a n d t h e i n f o r m a t i o n is o b t a i n e d b y i l n p o s i n g (lo)

(17)

{

.Q = ( 2 r ~,~ = ( 2 ~ ' ) - ~ .

(9) S. GIAMB0, A. GRECO and P. ~&NT:kNO: C. R. A c a d . Sci. P a r i s , 288, 85 (1979); D. G. CRIGttTON: ,Am. Rev. F l u i d M e c h . , 11, 11 ( 1979) ; S. GIAMBb, &. PALUMBOand P. PXNTANO: -4nn. M a t . P u r a A p p l . , 129, 143 (1981); D. FUSCO: Proc. R . Soe. E d i m b u r g S e t . -4, 82, 102 (1979); D. G. CRIOHTON and ft. F. SCOTT: P h i l . T r a n s . R . Soc. L o n d o n Ser. ,A, 29, 13 (1979). (i0) T. B~UGA~IXO, P. CARBONARO and P. PAIgTANO: A t t i - 4 I ~ I E T A (Genova, 1982).

26

1VI. B A R T U C C E L L I

B y substituting (17) in (16), we have o)"= oY((5--w). This equation can be integrated, obtaining for ~9 9 ~ (2v')-~{(2~')-+~'-- [2 log2'((2v')- 89189 where a is a constant of iategration and

~:{f~,lo~ ~+o,~,~} -~

~- a]}+,