The strength of the shock in a thermally radiative gas has been calculated for the steady flow of gas under the condition that the gradient of total pressure vanishes on the flow side of the shock

Ojha, S.N.

Singh, S.N.

English

Defence Science Journal

SHOCK STRENGTI! IN A THERMALLY RADIATIVE GASS. N. OJ'HA & S. N. SINGHs. a. College, Ballia (U. P.)(Received 27 November 1975)The 8trength of the 8hock in-& thermally radiative gas has been calculated for the steady flow of ga8 underthe condition that the gradient of total pressure vanishes on the flow side of the shock., ~e.rma and, ~ishralhave derived jump,relations for therma~ly,radiative shocks in ~lasma. Usingcom-patIbIhty COndItIons of Thomas2, propagatIon of thermally radiatIve shocks has been iliscussed by Ojha &Verma3 under certain bo~ndary .conditions. The str~n~h ~f the shock h~s .bee~ derived by many authorse.g., Kanwal4.; Pant & MIshrao In the presence of dISSIpatIve and nOn-dISSIpatIve mechanisms and in ab-sence of thermal radiation. Our aim in this pa per is to derive the density shock strength in a thermally radia-tive gas. For simplicity we have taken steady flow of gas an~ stationary normal shocks in course of deriva-tion. We have assumed that the gradient of total pressure vanishes on the flow side of the sh(jck so that-:JL = O and Thomas2 compatibilit y conditions have been used.a"SHOCK CONDITIONS AND STRENGTH OF THE SHOCKEquations of con.servation of ~ass and momentum for steady optically thick radiative gas flow (Pai6)using Thomas2 equatIons (7 & 9) yIeldp~~ + Un 9 + A = 0 (I)(2)where(3)(4)~ = [PI i] ni; '\i = ['liilj] nj; 11. = (p*, i] niA = U~ P,~ + B«,8 Xil; 'Ii.,~and(5)a .8 .B .- pu . 9 u ."'. - pua u ., -3 "a ..'3'.8 -.,~The ~hock condition~ across the shock sufraceXi = Xi (ya) (i= 1, 2, 3 ; IX = I, 2)&r~(6)[p Un) = 01~5DEF. SOl. J" VaL. 27. JULY 1977{7)Pi U1n [Ut] = ~4aTc3p(8)pUn ( ! U2 + h -1- = lK T, ii] + [F.. n..]wherch= 'YPo p(y--l) (9)and Xi are cartesian co-ordinates, ya are curvilinear co-ordinates and a comma (,) indicates partial differen-tiation with respect to curvilinear co-ordinates. Assume that the shock strength tp is defined by[p]Pi4>=Then we hav64> (11)[Ut] = ~.~ Uln nt(12)[ aT4 ] cf> ,p + ~ = ~Pl Ul"2From equation of state, we have[p, i] = R {T [p, i] + p [T, i]}Thuspua 'Ui,a niUnwhereaT'R1' = -sp--Also we can write (8) in the form[ P'U~(! U2 +h + , -f- ) ]= K*[T,i]n.126aT4]3n;OJHA & SINGIl : Shock Strength in a Thermally Radiative GaswhereK*=(K+ ~)and is called the ecffective condutivity due to radiation. Eliminating[T,iJni between (15) and (16) we getpuau'n'Ui,a ni4aT43p;:;::pUn (1 + 481;)which is a first order differential equation in p. and y" that can be solved in principle. For simplicityconsider the stationary norm~l shock for which, u(J = 0~Ul"),ani (19)-f/JUi,a = [U£],a = ~) Ul" ~,a -[ P ] 1 ~ -~ PIp ={l+.~).-.~UI2n- ~ ~andUl~'11,,= U2n =(1+ <f»Also, we have-cP(2 + cP)-2 (1 + cP)2-U21n[! U2] = ! {['/1t] ['/1t] + 2Ult ['/1t]} =Now making use of (18) to (23), we gett/J=whereO 2 -NPl1 -p!J~7D.I'. S(j!. J., VOL. 27. JULY 1977REFERENCESI. VEBMA, B. G. & MlsHBA, S. N., Rev. de. Mat. y. Fiaica, (Accepted).2. TROMAS, T. Y., J. Math. Mech, 6 (1957), 311.8. OJHA, S. N. & VJ:RMA, B. G., Pure J: Applied GeophV.ic., 102 (1973), 148.4. KANWAL, R. P., Jour. Math. Mach., 9 (1960),681.5. PANT, J. C. & MISHRA, R. S. Rendiconti del Cire., Mat di. Palermo., 12 (1963), 1.23.6. PAl, S. I., 'Radiation Gas Dynamics', (Springer Verla.g), 1966, p. 41."l~

Shock Strength in a Thermally Radiative Gas

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Shock Strength in a Thermally Radiative Gas

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