MHD Flow of a Third-Grade Fluid Iduced by No-Coaxial Rotatios of a Porous Disk Executig No-Torsioal Oscillatios ad a Fluid at Ifiity TAHIRA HAROON COMSATS Istitute of Iformatio Techology, Abbottabad, NWFP, Pakista. T. HAYAT 1, S. ASGHAR 1, ad A. M. SIDDIQUI 2 1 Departmet of Mathematics, Quaid-I-Azam Uiversity, Islamabad, Pakista. 2 Departmet of Mathematics, Pesylvaia State Uiversity, York Campus, York, Pesylvaia 17403, U. S. A. Abstract :- The problem of magetohydrodyamics (MHD) flow of a coductig, icompressible fluid due to o-coaxial rotatios of a porous disk, executig oscillatios i its ow plae, ad a fluid at ifiity is cosidered i the presece of a uiform trasverse magetic field. The porous character of disk ad the o-liearity of the fluid icrease the order of the differetial equatio. The solutios for the cases, whe the agular velocity is greater, smaller or equal to the frequecy of oscillatio are examied. The structure of the velocity distributios ad the associated boudary layers are ivestigated icludig the case of blowig ad resoat oscillatios. It is foud that ulike the hydrodyamic situatio for the case of blowig ad resoace, the hydromagetic steady state solutio satisfies the boudary coditio at ifiity. The iheret difficulty ivolved i the purely hydrodyamic problem associated with the case of blowig ad the resoat frequecy has bee resolved i this paper by the additio of the magetic field. Key-Words :- Icompressible MHD flow, No-coaxial rotatio, Oscillatio, No-Newtoia fluid, Porous disk. 1 Itroductio Exact solutios for the flow due to a sigle disk i a variety of situatios have bee obtaied by a umber of workers. Berker [1] has cosidered the viscous flow due to o-coaxial roatatios of a disk ad a fluid at ifiity. Thorley [2] has studied the flow due to otorsioal oscillatios of a sigle disk i semiifiite expase of fluid i a rotatig frame of referece. The MHD effect o the Ekma layer over a statioary ifiite horizotal plate i a electrically coductig liquid, rotatig with uiform agular velocity about a vertical axes has bee studied by Gupta [3]. The flow due to rotatios of a porous disk ad a fluid at ifiity, about differet axes has bee studied by Erdoga [4]. Murthy ad Ram [5] have cosidered the MHD flow ad heat trasfer due to eccetric rotatios of a porous disk
ad a fluid at ifiity. Rajagopal i [6] ad [7] has cosidered the flow of a simple fluid i a orthogoal rheometer ad the flows of Newtoia ad o-newtoia fluids betwee parallel disks rotatig about a commo axis. Kasiviswaatha ad Rao [8] discussed the flow due to o-coaxial rotatios of a disk, executig o-torsioal oscillatios i its ow plae ad a fluid at ifiity. The usteady flow due to o-coaxial rotatios of a disk ad a fluid at ifiity which are impulsively started was ivestigated by Pop [9]. Later, Erdoga [4] poited out by that if the disk ad the fluid at ifiity are iitially at rest the problem becomes three dimesioal ad the solutio caot be obtaied easily ad suggested a chage i iitial coditio ad proposed that the disk ad the fluid are iitially rotatig about z -axis ad suddely sets i motio; the disk rotatig about z-axis ad fluid about z -axis. He showed that ow the problem is solvable ad presets a aalytic solutio for the velocity field. I aother paper, Erdoga [10] foud a exact solutio of the time-depedet Navier-Stokes equatios for the flow due to o-coaxial rotatios of a oscillatig disk ad a fluid at ifiity. I this paper, umerical solutio of the timedepedet equatios is give for the magetohydrodyamic icompressible flow due to o-coaxial rotatios of a porous disk ad a third grade fluid at ifiity. Additioally, the disk is executig oscillatios i its ow plae. The porous disk ad o-liear fluid behavior cosidered i this study results i the icrease of the order of the o-liear differetial equatio with costat complex coefficiets obtaied by isertig the velocity field ito the equatios of motio. It is apparet from physical cosideratios that suctio ad blowig have opposite effects o the boudary layer flows. Ideed, the suctio prevets the imposed otorsioal oscillatios from spreadig far away from the disk by viscous diffusio for all values of the frequecy parameter. O the cotrary, the blowig promotes the spreadig of the oscillatios far away from the disk. I the case of blowig ad resoace, the oscillatory boudary layer flows are o loger possible. Thus, it remais to aswer the questio of fidig a meaigful solutio for the case of blowig ad the resoat frequecy. A attempt is made to aswer this questio by posig a hydromagetic boudary o iitial-boudary value problem. It is show that ulike the hydrodyamic situatio for the case of blowig ad resoace, the hydromagetic steady state solutio satisfies the boudary coditio at ifiity. 2 Basic Equatios We itroduce a Cartesia coordiate system with the z-axis ormal to the porous disk which lies i the plae z = 0. The regio z > 0 is occupied by a icompressible thirdgrade fluid. The axis of rotatio, of both the disk ad the fluid, are assumed to be i the plae x = 0, with the distace betwee the axes beig l. The disk ad the fluid at ifiity are iitially rotatig about the z -axis with the same agular velocity, ad at time t = 0, the disk starts to oscillate suddely alog the x-axis ad to rotate impulsively about the z- axis with the same agular velocity ad the fluid at ifiity cotiues to rotate about the z -axis with the same agular velocity. The fluid is electrically coductig ad assumed to be permeated by a magetic field B havig o compoets i the x ad y directios. The velocity field is chose as follows: u = y + f(z, t), v = x + g(z, t), w = w, (1) where u, v, w are the compoets of the velocity vector V, i the directios x, y, z respectively. Obviously w > 0 is the suctio velocity ad w < 0 is the blowig velocity. The velocity field satisfies. V = 0, which is othig else tha the icompressibility coditio. For the problem uder cosideratio, the boudary ad the iitial coditios ca be
writte i the followig form u = y + U cos t or y + U si t, v = x, w = w, at z = 0, t > 0, u = (y l), v = x, w = w, as z, for all t, u = (y l), v = x, w = w, at t = 0, for z > 0, (2) where is the frequecy of the o-torsioal oscillatios ad U the velocity. The hydromagetic equatios of motio for a electrically coductig, icompressible fluid are ρ DV Dt =. T + J B, (3). B = 0, (4) B = µ m J, (5) E = B t, (6) ad Ohm s law for a movig coductor J = σ(e + V B). (7) D I above equatios, ρ is the desity, the Dt material time derivative, J the electric curret desity, µ m the magetic permeability, E the electric field, B the total magetic field so that B = B + b, b the iduced magetic field, σ the electric coductivity of the fluid. I equatios (4-7) the magetic Reyolds umber R m [11] is assumed to be small as is the case with the most of the coductig fluids ad hece the iduced magetic field is small i compariso with the applied magetic field ad is therefore ot take ito accout. The magetic body forces J B ow becomes σ(v B) B. The costitutive equatio of third-grade fluid is T = pi + µa 1 + α 1 A 2 + α 2 A 2 1 + β 1 A 3 + β 2 (A 1 A 2 + A 2 A 1 ) + β 3 (tra 2 1)A 1, (8) where T is the stress tesor, I is the idetity, A 1, A 2 ad A 3 the Rivli-Erickse tesors of the first, secod ad third orders, respectively, p the static fluid pressure (p = p(x, y, z)), µ the dyamic viscosity coefficiet,α 1, α 2, β 1, β 2 ad β 3 are material costats. The first, secod- ad third-order Rivli-Erickse tesors are respectively: A 1 = (grad V) + (grad V) T, A i = DA i 1 + A i 1 (grad V) Dt + (grad V) T A i 1, i > 1. The thermodyamics of fluid modeled by equatio (8) has bee object of a detailed study by Fosdick ad Rajagopal i [12] ad Du ad Rajagopal [13]. They show that the equatio (8) to be compatible with thermodyamics ad the free eergy to be miimum whe the fluid is at rest, the material costats should satisfy the relatios µ 0, α 1 0, β 1 = β 2 = 0, β 3 0, 24µβ 3 α 1 + α 2 24µβ 3 (9) ad specific Helmholtz free eergy Ψ has the form Ψ = ˆΨ(θ, L) = ˆΨ(θ, 0) + α 1 4ρ L + LT 2. (10) I above expressios L = grad V. (11) I our aalysis we assume that the fluid is thermodyamically compatible; hece the stress costitutive relatio (8) reduces to T = pi + µa 1 + α 1 A 2 + α 2 A 2 1 + β 3 (tra 2 1)A 1. (12) Substitutig equatio (12) ad J B = σ B 2 V ito equatio (3) ad the elimiatig the modified pressure oe obtais [ F + i F F ] w t z = ν 2 F z σ 2 ρ B2 (F l) + i 2 l + α [ 1 3 F ρ t z w 3 F 2 z i F ] 3 2 z 2 ( ) +2β 3 F 2 F, (13) z z z where F = f + i g, (14)
ν = µ ρ, F is the complex cojugate of F. From equatios (1), (14) ad coditios (2) we have U cos t F (0, t) = or, (15) U si t F (, t) = l, F (z, 0) = l. O itroducig o-dimesioal parameters F ξ = z, τ = t, F (ξ, τ) = 2ν l 1, F F (ξ, τ) = l 1, U = U β = 3 l 2 β 3, l ρν 2 α = α 1 ρν, ɛ = w, N = σ 2ν ρ B2 (16) equatio (13) ad coditio (15) becomes α 3 F τ ξ F 2 αɛ 3 ξ 3 +2ɛ F ξ 2 F τ +β ( F ξ ξ F (0, τ) = + (1 iα) 2 F ξ 2 2(i + N)F ) 2 F ξ U cos t 1 or U si t 1 = 0, (17) F (, τ) = 0, F (ξ, 0) = 0., (18) We ote that the equatio (17) is a third order ad o-liear partial differetial equatio. As a result, it seems to be impossible to obtai the geeral solutio i closed form for arbitrary values of all parameters arisig i this o-liear equatio. Further, equatio (17) is parabolic with respect to time which allows a time marchig solutio to the equatio. The o-liearity of above equatio must be supressed i applyig the Vo Neuma stability aalysis by takig solutio-depedet coefficiets multiplyig derivatives, tempoarily froze. The modified equatio approach to aalyzig o-liear computatioal algorithm is applicable [14] but the appearace of products of higher-order derivatives makes the costructio of more accurate schemes less precise tha the case of liear equatios. As this problem is time depedet ad has mixed derivative with respect to time ad space coordiates so we are forced to use a implicit scheme. Applyig implicit scheme to oliear equatio (17) a umber of choices are available. Numerical methods for a parabolic partial differetial equatio iclude both 1) a boudary value problem ad 2) a iitial value problem. Combiig these two problems may result i very complicated or, at least, iefficiet methods e. g. higher order Ruge-Kutta methods or predictor-corrector methods [15]. This limitatio leads us to cosideratio of the simplest group of umerical methods for iitial value problems. A modified Crak-Nicolso implicit formulatio with forward time ad cetral fiite differece space approximatio is used, so that equatio (17) is trasformed ito algebraic equatio of the form a j Fj 1 +1 + b j Fj +1 + c j Fj+1 +1 = d j, (19) where ( ) α a (1 iα) τ j = +, ( ) α b (1 iα) τ j = 2 + + 1, ( ) α c (1 iα) τ j = +, ( α d j = (1 iα) τ + (Fj+1 2Fj + Fj 1) ɛα τ 2h 3 ) (F j+2 2F j+1 + 2F j 1 Fj 2) + ɛ τ h (F j+1 Fj 1) + 2(1 τ(i + N))Fj + β τ 4h ((F 4 j+1 Fj 1) 2 ( F j+1 2 F j + F j 1) + 2(F j+1 F j 1)(F j+1 2F j + F j 1)( F j+1 F j 1)). (20) Here ξ = [ξ j ] j=m j=1 is take as strictly icreasig sequece of discrete poits such that 0 = ξ 1 <
ξ 2 < ξ 3 < < ξ M ad h = ξ i ξ i 1 = ξ M ξ 1, M 1 where M is the umber of grid poits i space coordiates ad τ = τ +1 τ is time iterval. The right had side of equatio (19) is cosidered i some fashio as kow, say from the previous time step ad left had side as the depedet variable i a umerical solutio of a usteady flow problem. The steady state approached asymptotically at large times. The implicit methods are ucoditioally stable uless o-liear effects cause istability, which is cotrolled by suitable choice of τ ad h. The equatio (19) must be writte at all iterior grid poits resultig i a system of algebraic equatios of order M from which the ukows Fj +1 for all j ca be solved simultaeously, usig a implicit algorithm [16]. Let us cosider system of equatios of the form A F = B. (21) F is a vector of ukow odal values, A cotais the algebraic coefficiets arisig from discretizatio ad B is made up of algebraic coefficiets associated with discretisatio ad kow values of F o previous time step ad give by the boudary coditios. Cosiderig every ode, equatio (21) yields b 1 c 1 a 2 b 2 c 2 a j b j c j a M 1 b M 1 c M 1 a M b M F 1 d 1 F 2 d 2 F j = d j, F M 1 F M d M 1 d M (22) where a j, b j, c j ad d j are give by equatio (20), o-zero values of d j are associated with source terms or for d 1 ad d M with boudary coditios. All terms i A other tha those show are zero. To prevet illcoditioig it is ecessary that b j > a j + c j. 3 Numerical Discussio Equatio( 19) has bee solved by usig a modified Crak Nicolso implicit formulatio with forward time ad cetral differece space approximatio usig 100 grid poits (ξ = 10) for sufficiet accuracy. For all computatios we have take h =.1 ad τ =.001. For the case α = 0, β = 0 we get solutio for the Newtoia fluid, ad results matched with the aalytical solutio [10]. Numerical solutios cofirm that for large times the startig solutios ted to the steady-state solutios. For some times after the iitiatio of motio, the velocity field cotai trasiets the these trasiets disappear ad steady state is achieved. The time required to attai steady flow for the cosie ad sie oscillatios is obtaied for >, = ad <. The value of this time for cosie oscillatio is shorter tha that for sie oscillatio ad it depeds o the ratio of the frequecy of oscillatio to the agular velocity of the disk ad the ratio U. l I the case of >, the oscillatio of the disk domiates, the the time required to attai steady flow both for the cosie ad sie oscillatios becomes very short. However, i the case of < the time required to attai steady flow for the sie oscillatio become large. Numerically, we have computed the magitudes of f ad g with the distace from the l l disk for cosie ad sie oscillatios keepig amplitude costat U = 4 at =.5, 1, 2, 5 l with varyig time correspodig to three types of flows (Newtoia, secod-grade ad third-grade), with ad without suctio. The full lies deote startig velocities ad dotted lies show steady-state velocities. Results described below report solutios up to ξ = 2, where free stream velocities have ot yet bee reached i most cases. For Newtoia fluids, we observed that without oscillatios steady state is achieved after τ = 10. Whe cosie or sie oscillatios
are itroduced, time to reach steady state is reduced. With the cosie oscillatios, whe suctio (ɛ = 2) is itroduced, its value is reduced further ad boudary layer thickess is also f reduced due to suctio. Magitudes of l g also reduced but the magitudes of ear l the disk are icreased due to suctio. Whe oly disk is rotatig the time to reach its steady state is τ = 10. Itroducig cosie oscillatio =.5 < 1 this time is reduced to τ = 5, ad for sie oscillatio its values is 7. For = 2 > 1 due to cosie oscillatio the value of τ whe we get steady-state is 4.5 while for the sie oscillatio its value is 5.5. For = 5 > 1 system with cosie oscillatio approaches to its steady-state at τ = 2 ad for system with sie oscillatio the value is τ = 3 (By itroducig oscillatios time to reach steady-state is reduced. This time is shorter for cosie oscillatios as compared to sie oscillatios). No oscillatory behavior ca be observed for the Newtoia fluids with ad without suctio. For the secod-grade fluid (α 0) time to reach steady-state is icreased ad also oscillatory behavior become visible i fluid velocities ( f ad g ). Near the disk the l l magitudes of the velocities ( f ad g ) is l l also icreased. Boudary layer thickess is also icreased due to α. By itroducig suctio (ɛ = 2) i the secodgrade fluid, time to reach steady-state is decreased but oscillatory behavior becomes very promiet, boudary layer thickess is icreased. That is to say, that the oscillatory behavior is due to both o-newtoia fluids ad suctio (become visible for o- Newtoia fluid ad ehaced by suctio, i our case). By icreasig the value of α time to reach steady state is also icreased. Boudary layer thickess is also icreased due to o-newtoia ature of the fluid ad oscillatory behavior. For the third-grade fluid (α 0, β 0) time to reach steady state is delayed further but oscillatory behavior dimiish whe β is itroduced. Iclusio of suctio icreases oscillatory behavior. By icreasig oscillatio of the disk steady state is achieved much earlier. Icreasig oscillatios reduces boudary layer thickess. For the third-grade fluid (α 0, β 0) time to reach steady state is icreased further but due to oscillatios its value is reduced. Whe =.5 steady-state is achieved at τ = 3, for = 2 we get τ >.5 ad whe = 5 its value becomes =.3. The time to reach steady state for the third-grade fluid i the presece of cosie oscillatios is reduced. I the presece of suctio (ɛ = 2) time to reach steady state is icreased i cosiderable amout. Whe =.5 its value is τ = 5, for = 2, τ becomes 4 ad for = 5, τ = 1. i. e. time to reach steady state is further delayed i the third grade fluid (whe suctio is applied, perhaps this is due to oscillatory behavior). Whe we cosider the sie oscillatios the we ote that the time to reach steady-state is agai reduced ad with the icrease i oscillatios the time to reach steady-state is decreased further. By itroducig suctio (ɛ = 2) time to reach steady-state is reduced i a cosiderable amout. Boudary layer thickess is also reduced. The magitude of f is reduced by suctio while the magitude l g of are icreased ear the disk. l For the secod-grade fluid the time to reach steady-state is icreased. Oscillatory behavior ca be see ear the disk. The magitude f of ad g ear the disk is icreased. l l Boudary layer thickess is also icreased for the secod grade fluid. Whe suctio is applied to the secod grade fluid oscillatory behavior becomes promiet ad ca be see up to cosiderable distace from the disk. Boudary layer thickess is icreased due to this oscillatory behavior. Time to reach steady-state is reduced. For > boudary layer thickess is also reduced but oscillatory behavior is promiet ad with the icrease i frequecy the value of boudary layer thickess is reduced further. For = 1 (resoat case) boudary coditios at ifiity for the steady case are ot met whe fluid is Newtoia or o-
Newtoia. Whe suctio is applied the coditio at ifiity is fulfilled but for blowig, the coditio at ifiity is ot satisfied. If we cosider electrically coductig fluid the boudary coditio at ifiity is also satisfied for blowig ad resoace. It is likely that the magetic field provides some mechaism to cotrol the growth of the boudary layer thickess at the resoat frequecy. 4 Cocludig Remarks The most distictive feature is that ulike the hydrodyamic situatio for the case of the resoat oscillatios, the solutio satisfies the boudary coditio at ifiity for all values of the frequecy parameter, ad the associated boudary layers remai bouded for all values of the frequecy icludig = 2. The physical implicatio of this coclusio is that for the case of resoace ad blowig, the ubouded spreadig of the oscillatios away from the disk is cotrolled by the exteral magetic field. Cosequetly, the hydromagetic oscillatios are cofied to the ultimate boudary layers. Refereces [1] R. Berker. Hadbook of Fluid Dyamics, volume VIII/3. Spriger-Verlag, Berli, 1963. [2] Claire Thorley. O Stokes ad Rayleigh Layers i a Rotatig System. Quart. Jr. Mech. ad Applied Math., XXI:451 460, 1968. [3] A. S. Gupta. Ekma layer o a porous plate. Phys. Fluids, 155:903 931, (1972). [4] M. E. Erdoga. Usteady Flow of a Viscous Fluid due to No-Coaxial Rotatios of a Disk ad a Fluid at Ifiity. It. J. No-Liear Mechaics, 32(2):285 290, (1997). [5] Murthy S. N. Ram R. K. P. MHD Flow ad Heat Trasfer due to Eeccetric Rotatios of a Porous Disk ad a Fluid at Ifiity. It. J. Egg. Sci., 16:943 949, (1978). [6] Rajagopal K. R. O the Flow of a Simple Fluid i a Orthogoal Rheometer. Arch. Rat. Mech. Aal, 79:39 47, (1982). [7] Rajagopal K. R. Flow of viscoelastic fluids betwee rotatig disks. Theor. Comput. Fluid Dyamics, 3:185 206, (1992). [8] Kasiviswaatha S. R. Rao A. R. A Usteady Flow due to Eccetrically Rotatig Porous Disk ad a Fluid at Ifiity. It. J. Egg. Sci., 25:1419 1425, (1987). [9] Pop I. Usteady Flow due to No- Coaxially Rotatig a Disk ad a Fluid at Ifiity. Bull. Tech. Ui. Ist., 32:14 18, (1979). [10] M. E. Erdoga. A Exact Solutio of the Time-Depedet Navier-Stokes Equatios for the Flow due to No- Coaxial Rotatios of a Oscillatig Disk ad Fluid at Ifiity. It. J. Egg Sci., 38(2000)175. [11] Shercliff J. A. A textbook of Magetohydrodyamics. Pergamo, (1965). [12] R. L. Fosdick, K. R. Rajagopal. Thermodyamics ad Stability of Fluids of Third-Grade. Proc. Roy. Soc. Lod. Ser., A,339(1980)351. [13] J. E. Du, K. R. Rajagopal. Fluids of Differetial Type. It. J. Egg. Sci., 21(1983)487. [14] G. H. Klopfer, D. S. McRae. No-liear Trucatio Error Aalysis of FDF for the Euler Equatios. AIAA J., 21(1983)487. [15] R. D. Richtmyer ad K. W. Morto. Differece Methods for Iitial-Value Problems. Number 4 i Itersciece Tracts i Pure ad Applied Mathematics. Itersciece Publishers, secod editio editio, 1967. [16] C. A. J. Fletcher. Computatioal Techiques for Fluid Dyamics, volume I. Spriger-Verlag Berli Heidelberg, 1988.