Compressibility and collisional effects on thermal instability of a partially ionized medium

Transcription

1 Pram~na, Vl. I0, N. 3, March 978, pp , printed in India. Cmpressibility and cllisinal effects n thermal instability f a partially inized medium R C SHARMA and K C SHARMA Department f Mathematics, Himachal Pradesh University, Simla MS received 23 June 977 Abstract. The thermal instability f a finitely cnducting hydrmagnetic cmpsite and cmpressible medium is studied t include the frictinal effects with neutrals. The effect f cmpressibility is fund t be stabilizing. In cntrast t the nnscillatry mdes fr (Cp/g)~> in the absence f a magnetic field; C~,/3 and g being specific heat at cnstant pressure, unifrm adverse temperature gradient and acceleratin due t gravity respectively, the presence f magnetic field intrduces scillatry mdes in the system. The verstable case is als discussed. The magnetic field is fund t have a stabilizing effect n the system fr (Cp/g)~>. Keywrds. Cmpressibility; cllisins; thermal instability.. Intrductin The prblem f thermal instability f fluids, bth in hydrdynamics and hydrmagnetics, has been treated in detail by Chandrasekhar (96). The Bussinesq apprximatin has been used thrughut. The equatins gverning the system becme quite cmplicated when the fluids are cmpressible. Spiegel and Vernis (960) made the fllwing assumptins: (i) the fluctuatins in density, pressure and temperature intrduced due t mtin, d nt exceed their ttal static variatins and (ii) the depth f fluid layer is much less than the scale height as defined by them. Using the abve assumptins, Spiegel and Vernis (960) have fund the flw equatins t be the same as fr incmpressible fluids except that the static temperature gradient is replaced by its excess ver the adiabatic. Fllwing Hans (968) the medium has been idealized as a cmpsite mixture f a hydrmagnetic (inized) cmpnent and a neutral cmpnent, the tw interacting thrugh mutual cuisinal (frictinal) effects. As it is quite frequent that the medium is nt fully inized and may be permeated with neutral atms, the abve idealizatin f the medium as a cmpsite mixture was called fr. Hans (968) als fund that these cllisins have stabilizing effect n the Rayleigh-Taylr instability. Sharma (976) studied the thermal hydrmagnetic instability f a partially inized medium. The bject f the present paper is t study the cmpressibility and cllisinal effects n thermal instability f a cmpsite medium. 267

2 268 R C Sharma and K C Sharma 2. Frmulatin f the prblem and dispersin relatin Cnsider an infinite hrizntal cmpressible and cmpsite layer f thickness d cnsisting f a finitely cnducting hydrmagnetic fluid f density p, permeated with neutrals f density Pa, acted n by a unifrm vertical magnetic field H(0, 0, H) and gravity frce g(0, 0, --g). This layer is heated frm belw such that a steady adverse temperature gradient/3(-----[dt/dz[) is maintained. Bth the inized fluid and neutral atms are assumed t behave like cntinuum fluids. Fllwing Spiegel and Vernis (960), Sharma (976), the basic linearized equatins gverning the mtin f cmpressible and cmpsite medium are: _ pcgq=_ V 8P + g Sp + pv V~q q-~rr (ST h) H -t- pav (qa--q), () 8t ~gq d = _ vc (qa -- q), (2) cgt ~gh --V (q H) q- ~ ~7~h, (3) ~gt V.q:O, V.h=O, (4) ~O_[[36._~_]wt ~ +K A~O, (5) /gt ~ C t, I where q(u, v, w), h(h~, h, hz), 0, 8p and 8p dente respectively the perturbatins in velcity, magnetic field H, temperature T, density p and pressure p; g[ C, t~, v (-=--/~/pm), K',, (:, '/pmcp), vc, q~ and ~ stand fr the adiabatic gradient, the viscsity, the kinematic viscsity, the thermal cnductivity, the thermal diffusivity, the cllisinal frequency between the tw cmpnents f the cmpsite medium, the velcity f the neutral cmpnent and the resistivity respectively, am(= a, say) is the cefficient f thermal expansin and pm (= p, say) is the density f the inized medium. In writing eq. (2), it is assumed that the effects n the neutral cmpnent resulting frm the fields f gravity and pressure are neglected and that the neutral particles are nncnducting. Analyzing in terms f nrmal mdes, we seek slutins f the abve equatins, whse dependence n space crdinates x, y, z and time t is f the frm [w, 0, hz] = [W(z), O(z), K(z)] exp (ik~xf ikyyj-nt), (6) where k,, k, are hrizntal wave numbers f the harmnic disturbance, k~=kx2+k, ~ and n is the frequency. Eliminating qd between eqs (I) and (2) and using expressin (6), eqs (I)-(5) give,c dg/, i W --a)dk- (DS--a 2) ( D ~ -- a ~ -- + a"~v~-~vl W + 4rrpv gadz a 20 = 0, (7) V

3 Thermal instability f cmpressible and cmpsite medium 269 (D2--a~--V~a)K=--(H-~-) DW, (8) =- -- fl-- g IV, (9) K where a==kd, ~,~-nd~/v, px=v/k, pz~v]~l, %=pa]p, D=d/dz and x, y, z stand fr the crdinates in the new unit f length d. Use has als been made f the Bussinesq equatin f state 8p= -- apo. Eliminating O and K between eqs (7)-(9), we get ( D~--a~)( D2--a~--pz~) [(~-a'-,, ~,)(D'-a'-,, l + +vd,l~! a vcda/v ~ -- QD'] W = -- R(~.~)(D'--a'--p,g)a'W, (0) where R--~gafld4/vtc is the Rayleigh number, Q:Had~/4,rpv77 is the Chandrasekhar number and G:Cdt]g. Cnsider the case in which bth the bundaries are free and the medium adjining the fluid is nncnducting. The bundary cnditins apprpriate fr the prblem are (Chandrasekhar 96): W=D~W=O, 0=0 X=0 and h is cntinuus at z---o and. () In the absence f any surface current, the tangential cmpnents f magnetic field are cntinuus. Hence the bundary cnditins in additin t () are DK=O, n the bundaries. Using the abve bundary cnditins, it can be shwn that all the even rder derivatives f W must vanish fr z=o and and hence the prper slutin f W characterizing the lwest mde is (2) W= Wshl*r z, (3) where W 0 is a cnstant. Substituting (3) in eq. (0), we btain the dispersin relatin Rx =(~)I( + b)( + b-l-p~-~) b( +b+p.~)

4 270 R C Sharma and K C Sharma where a 2 _ Q = R and b = The statinary cnvectin When instability sets in as rdinary cnvectin, the marginal state will be characterized by =0 and eq. (4) reduces t (5) Fr the fixed value f Q, let the nndimensinal number G accunting fr the cmpressibility effects be als kept as fixed, then we find that where Rc and Rc dente respectively the critical Rayleigh number in the absence and presence f cmpressibility. The effect f cmpressibility is, thus, t pstpne the nset f thermal instability. Hence we btain a stabilizing effect f cmpressibility. The cases G< and G= crrespnd t negative and infinite values f critical Rayleigh numbers in the presence f cmpressibility which are nt relevant fr the present prblem. 4. Stability f the system and nn-scillatry mdes Multiplying eq. (7) by W*, the cmplex cnjugate f W, and using eqs (8) and (9), we get s ~ a~/~ ~ '7 (8 + v, ~*,) i~ + + g-~..~, ~/ i~ + ff-~p~ + Cp a Ka z (i 6 + Px * 6) = 0, (7),,( -- G) where I~= f (ID~WI~+2a~IDWI~+a4WI~)dz, ~=f (I D~KI ~ + 2a~l DKI ~ + a~ IKI ~) dz,

5 Thermal instabibty f cmpressible and cmpsite medium 27 ~=f (I D Kl~ a'l Kl')dz, z~= f (ID~l' a'l~l')dz, ~= f (I[~)az, (is) which are all psitive definite and ~* is the cmplex cnjugate f ~. Putting ~ -----,, + i - t and then equating the real and imaginary parts f eq. (7), we btain 6 + (~'+" d~/~) {~"+~"+~" (" d~/") (I +%)} Z~+ ~ (I~+p~, Z~) (~, + v~d2/v) 2+ et ~ 4~pv Cp ~,ca ~ t r + p~ c,,/6) = 0, +,,(I_G) vs (9) and [~ ' + a, (red'/v) (I + %) + (v d'/v)" (I + %) + ~,' z~ (~, + v~ d~/v) ~ + ~fl P2/4 q- Cp a Ka 9 ] 4~pv 7(6:~ p~ z, :. (20) It fllws frm eq. (20) that if G>I and if the magnetic field is absent, cry=0, which means that the scillatry mdes are nt allwed fr G>I and in the absence f magnetic field. The presence f magnetic field, in cntrast t nnscillatry mdes fr G > in the absence f magnetic field, intrduces scillatry mdes in the system. 5. The verstable ease In this sectin we cnsider the pssibility f whether instability may ccur as an verstability. Put G,/~-=igl, it being remembered that, may be cmplex. Since fr verstability, we wish t determine the critical Rayleigh number fr the nset f instability via a state f pure scillatins, it suffices t find cnditins fr which (4) will admit f slutin with, real. Equatin (4) becmes I G \ [- I ( II b( + b + ipagl) (2)

6 272 R C Sharma and K C Sharma Equating real and imaginary parts f eq. (2) and eliminating R~ between them, we btain (t,~2,:~/a ~) [p:0 ~, + 0 +b)( +pb (,:~/d'~)] ~: + (l+b) [p~ ( +p+%) +Pl (l+b)(:rid ~) %vc + ( +b)" (l+pb (,::/a') + Q~ (p~-p,) (,::/a,)]~' + (l+b) [( +b)uvc 9 (I+p+%) + Q: (p:-pz)vc ~] = 0. (22) Equatin (22) is quadratic in ~,2 and des nt allw any f its rts t be such that Re(c,a) is psitive s that vl is imaginary when Pl >~P2. Hence Pl >~P2 i.e. ~ ~<~, (23) is a sufficient cnditin fr the nnexistence f verstability. The cnditin (23) is the same as in the absence f cmpressibility as well as frictinal (cllisinal) effects with neutrals n thermal instability (Chandrasekhar 96). Thus K ~<~7 is a sufficient cnditin fr the nnexistence f verstability which hlds bth in the presence r absence (Chandrasekhar 96) f cmpressibility and cllisinal effects with neutrals n thermal instability f a cmpsite and cmpressible medium. T study the effect f magnetic field n thermal instability f a cmpressible and cmpsite medium, we examine the nature f dr/dq. It fllws frm eq. (4) that (24) The imaginary part f eq. (24) equated t zer gives pl =,02. (25) Equating real parts f eq. (24) and substituting (25) in it, we btain dr _ (26) It fllws frm eq. (26) that dr:/dq is psitive if G>I. Hence fr G>I, the Rayleigh number increases as the magnetic field increases, shwing the stabilizing effect f magnetic field. The cases G< and G:-I are nt allwed here as thse wuld mean negative and infinite Rayleigh numbers respectively which are nt relevant in the present discussin. References Chandrasekhar S 96 Hydrdynamic and ttydrmagnetic Stability (Oxfrd: Clarendn Press) Chap. 4 Hans H K 968 Nucl. Fusin 8 89 Sharma R C 976 Physica C8 99 Spiegel E A and Vernis G 960 Astrphys. J