Asymptotic Behaviour of the Solutions. of the Falkner-Skan Equations. Governing the Swirling Flow

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1 Adv. Thor. Appl. Mch., Vol. 3,, no. 4, 5-58 Aymptotic Bhaviour o th Solution o th Falknr-Skan Equation Govrning th Sirling Flo J. Singh Dpartmnt o Civil Enginring, ntitut o Tchnology, Banara Hindu Univrity, Varanai 5 U.P., ndia B. B. Singh and. M. Chandarki Dpartmnt o Mathmatic, Dr. Babaahb Ambdkar Tchnological Univrity Lonr 43, Dit. Raigad M.S., ndia brijbhaningh@yahoo.com, imran4@yahoo.co.in Abtract Hr th aymptotic natur, a th indpndnt imilarity variabl tn to ininity, o th olution o th quation govrning th irling lo in a laminar compribl boundary layr ovr an axi-ymmtric urac ith variabl cro-ction ha bn tudid; th rult bing bad on th aymptotic intgration o cond ordr linar dirntial quation. Kyor: Aymptotic bhaviour, principal and linarly indpndnt Solution, injction rat, prur gradint, irling lo, longitudinal acclration, nthalpy dirnc ratio ntroduction Ma addition i an intrting phnomnon o boundary layr blo o hr larg amount o luid ar addd into th boundary layr illing th rgion nar th all and cauing igniicant altration in th proil o variabl. On account o thi act, or larg rat o injction th boundary

2 5 J. Singh, B. B. Singh and. M. Chandarki layr ar charactrizd by i an innr layr clo to th urac hr vicou orc ar ngligibl and ii a rlativly thin outr vicou layr in hich tranition rom th innr to th invicid xtrnal lo tak plac. Du to mallr prur gradint and gratr il o intgration, th uual hooting mtho o handling th problm ail or larg bloing paramtr. Thi i du to th poor convrgnc and gratr intability o th mtho ud. Sinc th prdiction o maiv bloing on lndr arodynamic bodi i o tchnological igniicanc, hnc problm o uch typ hav bn tudid by a numbr o invtigator [4-] and [5-7]. Th orkr hav ud th mtho o matchd aymptotic xpanion, invr mtho and numrical mtho obtaind by combining th init dirnc tchniqu ith quai-linarization. Th objctiv o th prnt papr li in th tudy o th aymptotic bhaviour, a th indpndnt imilarity variabl tn to ininity, o th olution o th Falknr Skan quation hich govrn th abov problm or larg rat o injction. Th rult ar bad on a igniicant mthod, namly th aymptotic intgration o cond ordr linar dirntial quation, o inding olution o cond ordr linar dirntial quation or vry larg approaching ininity valu o th indpndnt variabl. n th prnt papr, hav conidrd th acclration paramtr lying in < th rang β and th ca β ha bn dipod o. t i du to th act that th lattr ca i not o practical igniicanc a rgar th prnt problm hr vry high vlociti o lo ar rquird to b attaind ith th hlp o larg injction rat. Though th ca β i o mathmatical igniicanc, but th tudy o xitnc and uniqun o th olution o th Falknr-Skan quation govrning th prnt problm or thi ca i ull o complication and variou aumption ar rquird to b impod on α and β. Thr ar vn crtain aumption hich ar l likly to occur in actual practic. On account o thi act, it i by ar th bt to conidr th ca β >. < Analyi Th imilarity quation govrning th lo-pd irling laminar compribl boundary layr lo o a prct ga ith dnity ρ, contant pciic hat C p, vicoity μ proportional to tmpratur T and Prandtl numbr unity caud by r vortx on th longitudinal lo ovr an axi-ymmtric urac o radiu r ith larg injction at th urac ar [7] " " β [ G g g ] α[ G g g G ] = G " G =

3 Aymptotic bhaviour o olution 53 undr th boundary condition =, =, = 3 =, G = G 4 Hr i a dimnionl tram unction dind in uch a ay that = u u hr u and u ar th longitudinal vlocity componnt in ε-dirction inid and outid th boundary layr rpctivly. G tan or both th normalizd irl vlocity componnt and nthalpy dirnc ratio and i givn by g g g G = v v = 5 hr v i th irl vlocity in th η - dirction, g = H H, H = c p T u v 6 and uix and dnot valu at th all and th dg o th boundary layr rpctivly. Alo α, β, g and ar th irl, longitudinal acclration, all tmpratur and ma tranr paramtr rpctivly and ar givn by X dr v H X du α = r dx u, h = H u dx h / H ρ W X g =, H rρ μu β, = 7 ξ X = ρ μur dξ, h = c pt Hr i th vlocity componnt normal to th urac in ξ -dirction. Prim dnot dirntiation.r.t. th indpndnt imilarity variabl dind by = ξ rρ u / X ρ ρ dξ 8 Th quation, 3 can a ll b rittn a " = " β α G 9 undr th boundary condition = =, = n th abov quation, hav takn g = to avoid complication in th dicuion o th problm. Morovr, th abov orm o th quation hall hlp u to tudy th ct o larg injction rat on th lo ild. b th olution o th quation 9, lt u put h = in it to obtain h" h β α β G h =

4 54 J. Singh, B. B. Singh and. M. Chandarki To rmov th middl trm in, lt u ubtitut = k h xp 3 in it to obtain " = k Q k 4 hr = 4 G Q α β Sinc Q i a continuou complx valud unction or atiying < d Q l l or om l on th rang l, hnc olloing [4], p. 384 th quation 4 hall hav a olution atiying Q k xp, o k k = 6 and a olution atiying Q k xp and k k, a 7 Uing 5 in 6 and 7, hav in vi o 3 that thr xit a olution atiying h xp, 8a a o h xp 8b and a olution atiying h xp, 9a h xp, a 9b hr = 4 G β α β Sinc a, hnc t Con tan,or that a.subtituting thi into 8a and 8b, hav 4 xp J a

5 Aymptotic bhaviour o olution 55 " O xp J 4 b and into 9a and 9b, hav xp J 4 a " xp J 4 a b 4 3 hr J = 6 6 β 4 8 W no ubtitut τ = G in to obtain τ" τ = 3 undr th boundary condition τ =, τ = 4 Th aymptotic bhaviour o 3 and 4 hav arlir bn dicud by th author [5], hnc thy do not nd any thorough invtigation hr. Th rult obtaind in th abov papr ar only rquird hr to tudy th aymptotic natur o th olution in th prnt t-up. Th author [] invtigatd that, 4 hav principal olution atiying, a, xp, G c c G G 5 and linarly indpndnt olution atiying, a, G c, G hr c > and c ar th contant. 3 Rult and Dicuion n calculu, th natur o th olution o th dirntial or dirntial dirnc quation i tudid a th indpndnt variabl tn to ininity. Thi proprty i compltly ulilld by th tudy o th aymptotic natur o th olution. A olution hich tn to zro or to an ininitimal limit a th indpndnt variabl tn to ininity i aid to xhibit aymptotic natur, or i aid to b aymptotically tabl. t i alo o much mor practical igniicanc in boundary valu problm and hnc ha bn tudid by a numbr o orkr [-3] and [3-4]. On th contrary, tho olution hich do not bhav in th abov ahion ar aid to b aymptotically untabl.

6 56 J. Singh, B. B. Singh and. M. Chandarki Th critria ovr hich our ntir dicuion i bad ar lim =, lim " =, lim G = and lim G =. Th critria rquir a littl clariication. Th cond critrion can b drivd rom th olloing thorm: > Thorm: Givn that α, < β thr xit a uniqu olution o 9, uch that "> on, and lim " =. Th proo o th abov thorm ollo rom [3] and rom [4],p.5. Only tylitic chang ar rquird to b carrid out. Th third critrion i obviou, or lim =. Th lat critrion ollo rom th act that Sinc G = G a, hnc G π / a = π / or larg. Thror, G a. W impo th critria on th LHS o th olution and obrv hthr RHS o th olution ar alo tnding to th am limit or not. th anr i airmativ, thy ill ho aymptotic natur a, and on th contrary not. Sinc larg injction rat hav ovrriting inlunc in aronautical nginring hr vry high vlociti o th light ar attaind in modrn tim, hnc our major concrn i to tudy th ct o on vlocity proil and nthalpy dirnc ratio. Hr that th LHS o th aymptotic rlation a and b tnd to zro a and th RHS ar alo approaching th am limit i.. zro. Hnc th rlation hall hold tru a. Contrary to it, th rlation a and b do not xit a, or thir RHS do not tnd to th limit zro to hich th LHS tnd. Hr on noticabl act i that th irl paramtr α and longitudinal acclration β do not inlunc th aymptotic natur o th olution, or thy ar abnt in thm. Scond important thing i that vn i tak = or > uction, th aymptotic natur o th olution i th am a in th ca larg injction. Finally, that 5 hall xit a, hra 6 hall not. Rrnc [] B. B. Singh and A.K. Singh, Aymptotic bhaviour o MHD Falknr Skan and corrponding hat tranr quation, ndian J. Pur Appl. Math , 64. dz

7 Aymptotic bhaviour o olution 57 [] B. B. Singh, Aymptotic bhaviour o imilar proil, ndian J. Pur Appl. Math , 67. [3] B. Gabuti, On th quation o imilar proil, J. Appl. Math. Phy. AMP35 984, 65. [4] D. P. Wam and P. G. Wam, An aymptotic init dormation analyzing or an iotropic compribl hypr-latic hal pac ubjctd to a tnil point load, SAM Journal o Applid Mathmatic, 73. [5] D. R.Kaoy, On laminar boundary layr blo o, J. Fluid Mch , 9. [6] E. J. Waton, Th quation o imilar proil in boundary layr thory ith trong bloing, Proc. Roy. Soc. A 94966, 8. [7] H. L. Back, Flo and hat tranr in boundary layr ith irl, AAA J. 7969, 98. [8]. C. Walton, Boundary layr lo at a thr dimnional tagnation point ith trong bloing, Quart. J. Mch. Appl. Math. 6973, 43. [9] J. A. Morion and K. G. Ramakrihnan, Aymptotic olution to an invr problm or a hard unburd rourc, SAM Journal o Applid Mathmatic, 63. [] J. Aroty and J. D. Col, Boundary layr lo ith larg injction rat, R and Corp., Rp. RM [] J. Prtch, Analytic olution o th boundary layr ith axi-ymmtric uction and injction, J. Ang. Math. Mch., 4 944, 64. [] M. Mci, Aymptotic bhaviour o olution or a dgnrat hyprbolic ytm o vicou conrvation la ith boundary ct, J. Appl. Math. Phy. 5999, 67. [3] P. Hartman, On th aymptotic bhaviour o th olution o a dirntial quation in boundary layr thory,. Ang. Math. Mch , 3. [4] P. Hartman, Ordinary dirntial quation, Wily, N York, 964. [5] P. R. Nachthim and M. J. Grn, Numrical olution o boundary layr lo ith maiv bloing, AAA J. 9977, 533. [6] T. Kubota and F. L.Frmandz, Boundary layr lo ith larg injction and hat tranr, AAA J. 6968,.

8 58 J. Singh, B. B. Singh and. M. Chandarki [7] T. M. Liu and P. R. Nachthim, Shooting mthod or olution o boundary layr lo ith maiv bloing, AAA J. 973, 584. Rcivd: July, 9

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