Thermal Instability of Rivlin-Ericksen Elastico-Viscous Rotating Fluid in Porous Medium in Hydromagnetics

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1 vailabl at ppl. ppl. Math. ISSN: Vol. 7, Issu (Jun ), pp pplications and pplid Mathmatics: n Intrnational Journal (M) Thrmal Instability of Rivlin-Ericksn Elastico-Viscous Rotating Fluid in orous Mdium in Hydromagntics S. K. Kango Dpartmnt of Mathmatics Govrnmnt Collg, Haripur (Manali) Himachal radsh 7536 (INDI) drsanjaykanungo@in.com Vikram Singh Dpartmnt of Mathmatics Jwalaji Dgr Collg, Jwalamukhi Himachal radsh 763 (INDI) bstract Rcivd: Fbruary, ; ccptd: January, Th thrmal instability of a layr of Rivlin-Ericksn lastico-viscous rotating fluid in a porous mdium in hydromagntics is considrd. For stationary convction, th Rivlin-Ericksn lastico-viscous fluid bhavs lik an ordinary (Nwtonian) fluid. Th magntic fild is found to hav a stabilizing ffct on th thrmal instability of a layr of Rivlin-Ericksn fluid in th absnc of rotation whras th mdium prmability has a dstabilizing ffct on thrmal instability of Rivlin-Ericksn fluid in th absnc of rotation. Rotation always has a stabilizing ffct. Th magntic fild, mdium prmability and rotation introduc oscillatory mods in th systm, which wr non-xistnt in thir absnc. Th cas of ovr stability is also considrd and th sufficint conditions for th non-xistnc of ovr stability ar obtaind in th procss. Th study finds applications in gophysics, chmical tchnology and nginring. mong th applications in nginring disciplins on can find th food procss industry, chmical procss industry, solidification and cntrifugal casting of mtals and rotating machinry. Kywords: Thrmal Instability; Rivlin-Ericksn Fluid; Viscolasticity MSC : 76, 76D5, 76E5, 76S5 48

2 M: Intrn. J., Vol. 7, Issu (Jun ) 49. Introduction Thrmal convction in a Nwtonian fluid layr in th prsnc of magntic fild and rotation was discussd in dtail by Chandraskhar (98). Bhatia and Stinr (97) studid th problm of thrmal instability of a Maxwll fluid in th prsnc of rotation and found that rotation has a dstabilizing influnc in contrast to th stabilizing ffct on an ordinary (Nwtonian) fluid. Bhatia and Stinr (973) also studid th thrmal instability of a Maxwll fluid in th prsnc of a magntic fild whil th thrmal convction in an Oldroyd fluid in hydromagntics was studid by Sharma (975). In th physical world, th invstigation of th flow of th Rivlin-Ericksn fluid through a porous mdium has bcom an important topic du to th rcovry of crud oil from th pors of rsrvoir rocks. Flows in porous rgions ar a crping flow. Whn a fluid prmats a porous matrial, th actual path of th individual particls cannot b followd analytically. Whn th dnsity of a stratifid layr of a singl-componnt fluid dcrass upwards, th configuration is stabl. This is not ncssarily th cas for a fluid consisting of two or mor componnts which can diffus rlativ to ach othr. Th rason lis in th fact that th diffusivity of hat is usually much gratr than th diffusivity of a solut. displacd particl of fluid thus loss xcss hat, if any, mor rapidly than th xcss solut. Th rsulting buoyancy forc may tnd to incras th displacmnt of th particl from its original position and thus caus instability. Thr ar many lastico-viscous fluids that cannot b charactrizd by Maxwll s constitutiv rlations or Oldroyd s constitutiv rlations. Two such classs of lastico-viscous fluids ar th Rivlin-Ericksn fluid (955) and th Waltrs B' fluid (96). Waltrs proposd th constitutiv quations of such lastico-viscous fluids. Waltrs (96) rportd that th mixtur of polymthyl mthacrylat and pyridin at 5 C containing 3.5g of polymr pr litr bhavs vry narly as th Waltrs B' lastico-viscous fluid. Rivlin-Ericksn (955) proposd a thortical modl for yt anothr lastico-viscous fluid. Such typs of polymrs ar usd in agricultur, communication appliancs and in bio-mdical applications. Spcific xampls of ths includ th filtration procss, packd bd ractor, insulation systm, cramic procssing, nhancd oil rcovry, chromatography tc. Ths polymrs ar also usd in th manufactur of parts of spac-crafts, aro plan parts, tirs, blt convyrs, rops, cushions, sat foams, plastics, nginring quipmnts, adhsivs and contact lns. Sharma and Kumar (996) studid th ffct of rotation on thrmal instability in th Rivlin-Ericksn lastico-viscous fluid whras th thrmal convction in lctrically conducting Rivlin-Ericksn fluid in th prsnc of magntic fild was studid by Sharma and Kumar (997).Thrmal convction in th Rivlin-Ericksn lastico-viscous fluid in a porous mdium in hydromagntics was studid by Sharma and Kango (999). Thrmal convction in a rotating layr of a porous mdium saturatd by a homognous fluid is a subjct of practical intrst for its applications in nginring. mong th applications in nginring disciplins on can find th food procss industry, chmical procss industry, solidification and cntrifugal casting of mtals and rotating machinry. n application of th rsult of flow through a porous mdium in th prsnc of a magntic fild is in th gothrmal rgions. Th rotation of th Earth distorts th boundaris of a hxagonal convction clls in fluid

3 5 S. K. Kango and Vikram Singh through a porous mdium and th distortion plays an important rol in th xtraction of nrgy in th gothrmal rgions. Kping in mind th rlvanc and growing importanc of non-nwtonian fluids in gophysical fluid dynamics, chmical tchnology and industry; th prsnt papr attmpts to study th thrmal instability of th Rivlin-Ericksn lastico-viscous rotating fluid in a porous mdium in hydromagntics.. Formulation of th roblm and rturbation Equations Considr an infinit, horizontal, incomprssibl layr of an lctrically conducting Rivlin- Ericksn lastico-viscous fluid of dpth d in a porous mdium which is actd on by a uniform horizontal magntic fild H ( H,,), gravity forc g (,, g) and uniform rotation (,, ). This layr is hatd from blow such that a stady advrs tmpratur gradint β (= dt/dz ) is maintaind. Lt p, ρ, T,, g,,,, k and q ( uvw,, ) dnot, rspctivly, th fluid prssur, dnsity, tmpratur, thrmal cofficint of xpansion, gravitational acclration, mdium porosity, lctrical rsistivity, magntic prmability, mdium prmability and fluid vlocity. Th hydromagntic quations [Chandraskhar (98), Josph (976), Rivlin-Ericksn (955)] ar q g (. ) p ( ) q q t H H 4, k t q q (). q, () T E ( q. ) T T, t (3). H, (4) t H H q H. (5) (. ) Hr, E = ε + (- ε) scs /( C) is a constant ; is th thrmal diffusivity ; s, C s and, C stand for th dnsity and hat capacity for solid and fluid, rspctivly. Th quation of stat is T T, (6)

4 M: Intrn. J., Vol. 7, Issu (Jun ) 5 whr th suffix zro rfrs to valus at th rfrnc lvl z = and in writing qn.(), us has bn mad of th Boussinsq approximation. Th stady stat solution is q,,, T T z whr, z, (7) T T / d is th magnitud of th uniform tmpratur gradint and is positiv as tmpratur dcrass upwards. Hr, w us th linarizd stability thory and normal mod analysis mthod. Considr a small prturbation on th stady stat solution and lt q ( uvw,, ), p,, θ and h ( hx, hy,, hz) dnot th prturbations in vlocity (,,), prssur p, dnsity ρ, tmpratur T and magntic fild H ( H,,) rspctivly. Th chang in th dnsity, causd by th prturbation θ in tmpratur, is givn by. (8) Thn th linarizd hydromagntic prturbation qns. bcom q p g hh q q, t 4 k t (9). q, () E w t, (). h, () h H. q h. (3) t Writing Equations (9)-(3) in scalar form, using (8) and liminating u,v, hh x, y, p btwn thm, w obtain H w g h, z w t x y 4 z k t z (4) H w, (5) t k t 4 z z

5 5 S. K. Kango and Vikram Singh t E w, t H, z w hz H, t z whr. x y z (6) (7) (8) v u hy and x y x rspctivly. hx y stands for th z-componnts of th vorticity and currnt dnsity, 3. Disprsion Rlation nalyzing th disturbancs into normal mods, w assum that th prturbation quantitis ar of th form [ w,, h,, ] [ W z, z, K z, Z z, X z ]xp ik xik y nt, (9) z x y whr k x, k y ar th wav numbrs along th x and y dirctions, rspctivly; k = is th rsultant wav numbr and n is, in gnral, a complx constant. Using xprssion (9), Equations (4) (8), in non-dimnsional form, bcom k k x y 4 ga d ikxhd i kxd F D a W D a K, 4 l () ikxhd i kxd F X W, 4 l () d D a Ep W, () x D a p, ik Hd (3)

6 M: Intrn. J., Vol. 7, Issu (Jun ) 53 ik Hd x D a p K W, (4) whr w hav put a kd, nd /, x* x/ d, y* y/ d, z* z/ d and D d / dz*. p / is randtl numbr, p / is magntic randtl numbr, / l k d is th dimnsionlss mdium prmability and F / d is th dimnsionlss kinmatic viscolasticity. Elimination Θ, K, Z and X btwn Equations ()-(4), w obtain l Ra D a p k Qd D a D a Ep D a D a EpD a p F x D a p Fkx Qd W l T k d D a Ep D a p W (5) x, whr Q = / 4 is th Chandraskhar numbr, R = gd 4 / H d 4 numbr and T 4 d / is th Taylor numbr. is th Rayligh Considr th cas whr both boundaris ar fr as wll as maintaind at constant tmpraturs whil th adjoining mdium is prfctly conducting. Th cas of two fr boundaris is a littl artificial but it nabls us to find analytical solutions and to mak som qualitativ conclusions. Th appropriat boundary conditions, with rspct to which Equations ()-(4) must b solvd, ar (Chandraskhar (98)):,,, at z and z W D W DZ and DX, K on th prfctly conducting boundaris. (6) Th cas of two fr boundaris, though littl artificial, is th most appropriat for stllar atmosphrs (Spigl (965)).Using th abov boundary conditions, it can b shown that all th vn ordr drivativs of W must vanish for z and z and hnc th propr solution of (5) charactrizing th lowst mod is W W sin z, (7)

7 54 S. K. Kango and Vikram Singh whr W is a constant. Substituting (7) in (5) and ltting a x, R 4 R/, T T /, Q Q/, k kcos, i /,, w obtain th disprsion rlation 4 x l F x x Epi cos x R xepi i Qcos x x ip T xepi x ip F xip i Qxcos (8) Equation (8) is th rquird disprsion rlation studying th ffcts of magntic fild, kinmatic viscolasticity, mdium prmability and rotation on thrmal instability of Rivlin-Ericksn fluid. 4. Th Stationary Convction For th cas of stationary convction, = and Equation (8) rducs to R Q x x T cos x cos x x Qx cos, (9) which xprsss th modifid Rayligh numbr R as a function of th dimnsionlss wav numbr x and th paramtrs Q, and T. For stationary convction th paramtr F accounting for th kinmatic viscolasticity ffct vanishs and thus th Rivlin-Ericksn lasticoviscous fluid bhavs lik an ordinary Nwtonian fluid. To invstigat th ffcts of magntic fild, mdium prmability and rotation, w xamin th dr bhavior of dq, dr d and dr analytically. Equation (9) yilds dt dr dq 4 T x x cos x cos x Qx cos, (3)

8 M: Intrn. J., Vol. 7, Issu (Jun ) 55 x dr In th absnc of rotation, cos, which is always positiv. Th magntic fild, dq thrfor, has a stabilizing ffct on thrmal instability of Rivlin-Ericksn fluid in th absnc of rotation. This stabilizing ffct of magntic fild is in good agrmnt with arlir work of Sharma and Kango (999). In th prsnc of rotation, th systm is stabl if 4 T x x cos x x Qx cos cos. Similarly, it can b shown from qn. (9) that dr x T x d x x 3 cos Qx cos x. (3) dr In th absnc of rotation,, which is always ngativ. Thr is an analogous d x rlation for thrmal convction in Rivlin-Ericksn lastico-viscous fluid in porous mdium in hydromagntics as drivd by Sharma and Kango (999). Th mdium prmability, thrfor, has a dstabilizing ffct on thrmal instability of Rivlin-Ericksn fluid in th absnc of rotation. In th prsnc of rotation, th systm is stabl if x T x x x 3 cos Qx cos. Similarly, it can b shown from qn. (9) that dr dt x x cos Qx cos, (3) which is always positiv. Th rotation, thrfor, always has a stabilizing ffct on thrmal instability of Rivlin-Ericksn fluid. Th kinmatic viscolasticity has no ffct for stationary convction.

9 56 S. K. Kango and Vikram Singh 5. Stability of th Systm and Oscillatory Mods Hr w xamin th possibility of oscillatory mods, if any, in th stability problm du to th prsnc of magntic fild, kinmatic viscolasticity and rotation. Multiplying Equation () by W*, th complx conjugat of W, intgrating ovr th rang of z and making us of Equations () - (4) togthr with th boundary conditions (6), w obtain whr ga l 4 F I I Ep * I I p * I d * * l 4 d F I I p I, (33), I DW a W dz I D a dz, I 3 dz, 4 4, 5 I DK a DK a K dz I DK a K dz, 6 I DX a X dz, I7 X dz, I8 Z dz. (34) Th intgrals I,, I 8 ar all positiv dfinit. utting r ii and quating th ral and imaginary parts of Equation (33), w obtain F ga d F r I EpI 3 pi5 pi7 d I8 l 4 4 l ga d d I I I4 I6 I8, (35) l 4 4 l i F d d I8 pi7 l 4 F ga I EpI3 pi5 l 4. (36)

10 M: Intrn. J., Vol. 7, Issu (Jun ) 57 It follows from Equation (35) that r may b positiv or ngativ which mans that th systm may b stabl or unstabl. It is clar from (36) that i may b zro or non-zro, maning that th mods may b non-oscillatory or oscillatory. Th oscillatory mods ar introducd du to th prsnc of kinmatic viscolasticity, magntic fild and rotation which wr non-xistnt in thir absnc. 6. Th Cas of Ovr Stability Hr w discuss th possibility of whthr instability may occur as ovr stability. Sinc w wish to dtrmin th Rayligh numbr for th onst of instability via a stat of pur oscillations, it suffics to find conditions for which (8) will admit of solutions with ral. If w quat ral and imaginary parts of (8) and liminat R btwn thm, w obtain c c c, (37) 3 3 whr w hav put c, b x and F Epp F 3 b p b, (38) F Ep Q cos F b b 3 b b T 3 4 EpQ cos Ep T Qcos b b Ep p 4 4 b b Q cos F Ep Q cos Q cos b b Ep p T Q Ep p cos 3 cos b b Q Ep p. (39) 3 6 s roots of (37) is and this has to b positiv. is ral for ovr stability, th thr valus of c 3 must b positiv. Th product of th

11 58 S. K. Kango and Vikram Singh It is clar from Equations (38) and (39) that and 3 ar always positiv if Ep p and Q cos T, which implis that C C s s (4) and H cos 8 k. (4) Equations (4) and (4) ar, thrfor, th sufficint conditions for non-xistnc of ovr stability, th violation of which dos not ncssarily imply th occurrnc of ovr stability. 7. Nomnclatur p = prssur = fluid dnsity p = th prturbation in prssur = th prturbation in dnsity = kinmatic viscosity = kinmatic viscolasticity = mdium porosity k = mdium prmability g = acclration du to gravity H = magntic fild = lctrical rsistivity = magntic prmability = rotation vctor = thrmal diffusivity q u, v, w = prturbation in fluid vlocity q (,, ) k x, k y = wav numbrs in th x and y dirctions, rspctivly / x y k k k = wav numbr of th disturbanc 8. Conclusion Th study of viscolastic fluids finds applications in gophysics and chmical tchnology. Thr ar many lastico-viscous fluids that cannot b charactrizd by Maxwll s constitutiv rlations or Oldroyd s constitutiv rlations. Rivlin-Ericksn is on such class of lastico-viscous fluids. layr of lctrically conducting Rivlin-Ericksn lastico-viscous fluid hatd from blow has bn considrd in th prsnc of a uniform horizontal magntic fild and uniform rotation in a

12 M: Intrn. J., Vol. 7, Issu (Jun ) 59 porous mdium. For stationary convction, th Rivlin-Ericksn lastico-viscous fluid bhavs lik an ordinary (Nwtonian) fluid. Th magntic fild is found to hav a stabilizing ffct on th thrmal instability of th layr of Rivlin-Ericksn fluid in th absnc of rotation whras th mdium prmability has a dstabilizing ffct on thrmal instability of Rivlin-Ericksn fluid in th absnc of rotation. Rotation always has a stabilizing ffct. Th magntic fild, mdium prmability and rotation gnrat oscillatory mods in th systm that wr non-xistnt in thir absnc. Th cas of ovr stability is also considrd and th sufficint conditions for th nonxistnc of ovr stability ar dtrmind. Th sufficint conditions for th non-xistnc of ovr stability for thrmal instability in Rivlin- Ericksn lastico-viscous fluid in th prsnc of magntic fild and rotation in porous mdium ar, rspctivly, C C s s C C s s cos and H 8 k. cos and H 8 k ar, thrfor, th sufficint conditions for nonxistnc of ovr stability, th violation of which dos not ncssarily imply th occurrnc of ovr stability. cknowldgmnt Th author is gratful to th rviwrs for thir usful commnts and valuabl suggstions. REFERENCES Bhatia,. K. and Stinr, J. M (97). Convctiv instability in a rotating viscolastic fluid layr, Z. ngw. Math. Mch. Vol. 5 pp Bhatia,. K. and Stinr, J. M 973). Thrmal instability in a viscolastic fluid layr in hydromagntics, J. Math. nal. ppl. Vol. 4 pp Chandraskhar, S. (98). Hydrodynamic and Hydromagntic Stability, Dovr ublication, Nw York. Josph, D.D. (976). Stability of Fluid Motions II, Springr-Vrlag, Nw York. Rivlin, R. S. and Ericksn, J. L. (955). Strss-dformation rlaxations for isotropic matrials, J. Rational Mch. nal. Vol. 4 pp Sharma, R. C. (975). Thrmal instability in a viscolastic fluid in hydromagntics, cta hysica Hungarica Vol. 38 pp. 93. Sharma, R. C. and Kumar,. (996). Effct of rotation on thrmal instability in Rivlin-Ericksn lastico-viscous fluid, Z. Naturforsch. Vol. 5a pp. 8. Sharma, R. C. and Kumar,. (997). Thrmal instability in Rivlin-Ericksn lastico-viscous fluids in hydromagntics, Z. Naturforsch. Vol. 5a pp. 369.

13 6 S. K. Kango and Vikram Singh Sharma, R. C. and Kango, S.K. (999). Thrmal convction in Rivlin-Ericksn lastico-viscous fluid in porous mdium in hydromagntics, Czch.J.hys. Vol. 49 No. pp Spigl, E.. (965). Convctiv instability in a comprssibl atmosphr, strophys. J. Vol. 4 pp. 68. Waltrs, K. (96).Th motion of an lastico-viscous liquid containd btwn coaxial Cylindrs, Quart. J. Mch. ppl. Math., Vol.3 pp Waltrs, K. J. (96). Mcaniqu Vol. pp utobiography of th author (S. K. Kango) I, Dr. S.K. Kango, am 4 yars old and working as ssistant rofssor in th subjct Mathmatics in Govrnmnt Collg Haripur (Manali) of Himachal radsh in India. I hav don h.d. in Mathmatics in th faculty of hysical Scincs of H Univrsity, Shimla (India) in 999 undr th suprvision of Lat rofssor R.C. Sharma. I hav attndd a 4-wk orintation programm, 3-wk rfrshr cours in Mathmatics, prsntd rsarch paprs in an intrnational confrnc and national sminar on Mathmatics. Biography of th scond author (Vikram Singh) H is 38 yars old and working as Lcturr in Mathmatics in Jwalaji Dgr Collg Jwlamukhi of Himachal radsh in India. H passd his M.hil Dgr in th subjct of Mathmatics from Madurai Kamraj Univrsity, Madurai (India) in 8. H is doing h.d. in Mathmatics undr my suprvision.