The differential effects of the shock wave in conducting fluid have been discussed purely a vector technique. While discussing the vorticity and current density generated behind shock wave, it has been observed that the vorticity depends on the thermodynamical behaviour of the fluid

Pandey, R. K.

Upadhyaya, K. S.

English

Defence Science Journal

ON DIFPERENTIAL EFFECTS OF THE SHOCK- WAVE IN CONDUCTING FLUID --  - -'- -* K. S- U~ADHTAYA & R. E. PANDEY - University of &lahabaa, Allahbad " - -- - - ,. - (&&ed 28 Nay 1913) The differential effeots of the sho& wave in conducting fluid have been discussed purely e * - rsdm tsahnique. While diaaovlng the rorticity and c q e n t  dpaity -generatad behind tWwk wave, it has been observeJ that expression -for the vorticity depends on the the'pmdynamioal behaviour of the fluid. -, m - Some authorsl-6 have discussed &h& differsntial bffkts of shock'waves using tensorial technique in which vector quantities-vorticity and current density are obtained by the contraction of indiaes of tensor quantities of order two. In the present paper: vector technique method proposed by Hayes6 and developed by the authors7'8 has been used. The motive behind it is that the vector quantities oap be obtained dirw- tly from the set of vector equations and thedetermination of the tensor quanti; ies of higher order are not necessary. This very technique . i ~  used for the determination. of ' the derivatives pf the flow and field parameters vorticity, current density and the curvatures of stream line and magnetia line. % - D I S C U S S I O N  O N  T H E  F L O W  \ The equations governing the flow of an ideal conducting fluid are9 wherep, p, i, 2 and q stand for the vorticity pressure, density, vebcity vector, nragna0io die& veotor' and entropy respectively of the fluid under consideration. From the thermodynamical relation of the fluid we know that ' - - Iu view of the relation (5)  the equation of state i.e., (4) reduces to where C2 stands for A A If the velocity of the shock be Gn (41 is unit normal vector to shook surfme), t henbe  maferial de*iv* tive of any flow or field parameter along the normal trajectory to the shock surface is given bylo where f is any flow or field parameter including the components of the velocity and magnetic field vectors. Resolving the del operator aiong normal to the shock surfaoe and in the plane of the shock surfa(3o we havd F Dm. S ~ I .  J., VOL %6, JANUABY 1976 where V1 denotes the resolved part of V in the tangent plane to the shock s h e .  In view of the - relations (7) and (8), the equations (1) to (4) reduce to A 6j * + p n . -  + a = o ,  Vn S; an (4 a5 " a p  - 1  r a -  PVm =+. - aB an .(a. z) -H. - ) + L O ,  an (10) (11) ap ( P V ,  -+ 8===0, Vn -- ma 89r (12) where, Vn and &an& for the normal components of vehoity and-magnetic field with respect to shook. SP 9 SP a  =---+G.VI)P+P(VI.~), st B = - - @ - + ( ~ . v ~ ) ~ - @ ( ~ . v ~ ) P ,  6t St 1 The resolved components of (10) and (11) along the normal to the shook surf- - - are A as' 1 A 5 2  A pVn n.  - +2 +.-i;; ( g . 3  - H n n .  -)-+f . n = O ,  an an Brt P (14) A aE Eliminating ra . - from (14) and (16) we get an A * p Vn:n.-  an (16) The resolved components of (10) and (11) along the magnetic line are : and Eliminating a . * from (17) and (18) we get, an aB A & -  -(an* + E . Z  + H . B = o .  Vn B. -+ H 2 a .  - H n  an a + -p V. a@ - -  (19) Solving (16) and (19), we have - A i. an { B - ~ w p ~ s ~ }  + ~ { X  - 4 a v n  + 8 . 8 -  n n s . n +  s. P v n  Hn - - - A + -2Z.A-4a Vn A-sn=a. P v n  (@) 10 But in view of the equation (9) , the eqKation (12) g i q  v, A -  -6- -- " -1 + p n . -  + a + @  B = O ,  C2 dn an (21) 1 C 2 ~ + P  + 4 n p c 2 { ~  ( ~ - l ) - ( p ~ - l ) }  t .  $-r( ) A Hn A + B ) - ~ ~ v ~  n .  (a+-  B ) = o ,  (22) where Vna Ha2 a -and ba = R2 Ca 4 ~p Van DETE$%iKINATION 0 8  T B E  DIBBEREWTIAL E F F E C T S  OF TfilE S H O C K  W A V E  It is well-known that the appearmice of the  hock wave is associated with tlre change in the vorticity, current density, derivatives of the flow and iield parameters and the shape of the path of the fluid particles. Por the differential effects of the shock wave it is sufficient to determine the jumps in the vorticity . and current density, the normal derivatives of the flow and field parameters the curvatures and torsions of the stream and magnetic lines. In this connection we have the fohwing theorems : Thewem 1 The normal derivatives of the flow and field parameters are given by 3% (23) an (24) A -= " ti +Dn, i m (25) aij - A - = E + R n ,  . an (26) where - 1 C = -  Vn (1  - ban) an a n  P 1 A a 3  md n - is given by the equation (22). arb Proof:-Equations (23) and (24) can be obtained easily with the help of 6) and (12). Substituting the - aa a5 aB value of 3 from (24) and H -- from (19) in (10) and (11) respectively and solving them for - snd -ava m arb m - - we get the relations (26) and (26). Theorern 2 The expressions for the vorticity asd omrent density generated behind the shock wave are givw by Proof :-We know from the defmitions of the vortioity and current density veotors that@ In view of relation (a), the equations (29) and (30) yield - A *  W = % x - + V , X U ,  av, biib. W. J., V&. 2b, Q&u&Y 1978 A d hubstituting the values from -(25) and (26) in (31) and (32) and remembering that n X n ql), . + - . - . - .  we get the required equathns. Thewern 3 The jumps in vorticity and current density across a magaetogasdynamie shock wave depend on the thermodynemioal behaviour of the fluid. (This theorem is in contrast to the simibr theorem obtained in non-conducting fluid by Trumdell" and the aubhorss.) ?"- ,. - .. ai Proof :-The expressions for the vortieiby and current density c~loulacsd-above in&de n . - . But for - an A s, the determination of n. - we have t~ izse energy equation. s Theorern 4 The curvatures o; the stream line and magnetic line behind the magnietogasdynami: shock wave are given by ,- - ~8 = 02 [{ 7% E + (6. r ~ i )  6 l2 + v*% PI + 2 V. Vn Em+ (C b ~ d  }I T33) - where v = l G l :  a q d - R - - l j j 1  Proof :-From the definition of the curvature of a curve y e  hav,e , - - - - -  - av K2 = * . - , (35); ,I aH K =- . - .  as as wFs' as' where, s and s' denote the arc length along stream line and mqpetic l h e  respectively. But in view of re1atio.n (8) the derivatives along stream line and magnetic line are .- - w - 1 - ;aH as v - . . gu ai? Substituting the values of -and -- from relations (25) and (26) in above equations we get 3n a% In view of equations (39) and (48) we get the required expressions for K and K' as given in -eqtiatioris (33j and (34). - A C K N O W L E D G E M E N T S  .. !Vhe authors are thankful to Dr. U.S. Upadhyaya, Department of Mathematics, Banaras Hindu Univer- sify, Varanasi for his valuable suggestions in the preparation of this paper. R E F E R E - N O E S  - 1. TXOMAS, T. Y., J ,  XMh. Phye., 26 (1947), p. 62-68. 2. MWBRA, R. S., India?, J .  Math., 2 (1959)s 13. 19-28. r 3, UPADHYAYA, K. S., A n n a b  de la societe' Sciemtifiole de Bruxellea, Tome 81, 111 (1967), p. 292-297, 4. UPADHYAYA, K S. & UPADEYAYA, U.  S., TENSYIR, N. S., 22 (1971). 5.  K..u?wa~, R. P., Proo. Roy. floe., 257 (1960), p. 268-268. 6. IUYIOS, W. D., J .  Fluid $ ' ~ h . ,  2 (1957)s p. 595-600. 7 .  UPADHYAYA, K. S. & PANDEY, R .  K., Anmles de la Societe' scienti~ic& de ~ruz&es, T. 86,111 (1972), p. 231-298. 8. UPADHYAYA, K. S., & P A ~ D E Y ,  R .  K., "On the Differential. Effect o f  the  Shock Wave" (Communicated). 9. LAXDAU, L.D. bc L I B ~ I T Z ,  E. M., LLE1ectPodynami~ o f Continuum Media", (Pergarno% Prees, New York), 1967. 10. Q ~ w u ,  R. P., J.  &the dfd., 9 (1960), p. 881-695. . , - 11. T s o s s n m ,  0.. J. Aero. hi., I@ (1962), p. 826.828, " 22 ' 

On Differential Effects of the Shock Wave in Conducting Fluid

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On Differential Effects of the Shock Wave in Conducting Fluid

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