MHD STEADY FLOW IN A CHANNEL WITH SLIP AT THE PERMEABLE BOUNDARIES

Transcription

1 GENERAL PHYSICS MHD STEADY FLOW IN A CHANNEL WITH SLIP AT THE PERMEABLE BOUNDARIES O.D. MAKINDE, E. OSALUSI Appled Mathematcs Department, Unversty of Lmpopo, Prvate Bag X116, Sovenga 77, South Afrca Receved June 4, 5 The hydromagnetc stea flow of a vscous conductng flud n a channel wth slp at the permeable boundares s nvestgated. Analytcal solutons are constructed for the governng nonlnear boundary-value problem usng perturbaton method together wth Padé approxmaton technque based on computer extended seres soluton and the mportant propertes of the overall flow structure are dscussed. Key words: Channel flow, permeable walls, magnetc feld, slp coeffcent, Padé approxmants. 1. INTRODUCTION The stu of flow of an electrcally conductng flud through a channel permeable walls not only possesses a theoretcal appeal but also model many bologcal and engneerng problems such as magnetohydronamcs (MHD) generators, plasma studes, nuclear reactors, geothermal energy extracton, the boundary layer control n the feld of aeronamcs, blood flow problems, etc. A survey of MHD studes n the technologcal felds can be found n Moreau ([15]). Furthermore, an extensve theoretcal work has carred out on the hydromagnetc flud flow n a channel under varous stuatons e.g., Hartmann ([17]), Borkakat and Pop (1984), Maknde (3), etc. Meanwhle, Beavers and Joseph ([5]) n ther expermental work on boundary condtons at a naturally permeable wall confrmed the exstence of slp at the nterface separatng the flow n the channel and the permeable boundares. The mportance of slp velocty on ultra-fltraton performance has been well llustrated by Sngh and Lawrence ([17]). Pal et al. ([16]) nvestgated the effect on slp on longtudnal dsperson of tracer partcles n a channel bounded by porous meda. The problem of lamnar flow n channels of slowly varyng wdth permeable boundares was nvestgated n Maknde ([1]). Rom. Journ. Phys., Volume 51, Nos. 3 4, p , Bucharest, 6

2 3 O.D. Maknde, E. Osalus The man objectve of the present paper s to stu the combned effect of magnetc feld and permeable walls slp velocty on the stea flow of an electrcally conductng flud n a channel of unform wdth. Usng Berman ([1]) smlarty approach, the governng Naver-Stokes equatons s reduced to a nonlnear boundary-value fourth order parameter dependent ordnary dfferental equaton and sem-analytcal and sem-numercal technques of computer extended seres soluton coupled wth Padé approxmants s suggested and utlsed for ts soluton, Maknde ([11]), Maknde ([14]). In the followng sectons, the problem s formulated, analysed and dscussed.. MATHEMATICAL FORMULATION Consder the stea lamnar flow of an ncompressble vscous conductng flud n a channel wth slp at the permeable boundares under the nfluence of an externally appled homogeneous magnetc feld. It s assumed that the flud has small electrcal conductvty and the electro-magnetc force produced s also very small. We choose a Cartesan coordnate system (x,y) where x les along the center of the channel, y s the dstance measured n the normal secton such that y=a s the channel s half wdth. Let u and v be the velocty components n the drectons of x and y ncreasng respectvely. Then, the contnuty, Naver-Stokes equatons governng the flow are: v u + =, x (1) e u u 1 P σ B u u + v = + v u, x ρ x ρ () v v 1 P u + v = + v v, x ρ (3) were = +, p the flud pressure, ρ the flud densty, v the kenematc x vscosty coeffcent, σ e the flud conductvty, B = ( µ eh) the electromagnetc nducton, µ e the magnetc permeablty and H the ntensty of magnetc feld.

3 3 MHD stea flow 31 Bo (Magnetc feld ntensty) v y=a (permeable wall) u Bo y=a (Permeable (permeable wall) Fg. 1. Schematc dagram of the problem. In order to complete the formulaton of the problem, the boundary condtons have to be specfed. These can be wrtten as follows: and u =, v =, on y =. e. symmetry ( ) u µ = β u, v= V, on y= a. (4) (5) The boundary condton (5) s the well known Beavers and Joseph (1967) slp condton, µ the namc vscosty coeffcent, β the coeffcent of sldng frcton and V characterstc wall sucton velocty. The followng dmensonless varables are ntroduced nto Eqs. (1 5); x y ap u v x=, y =, P=, u =, v=, a a ρvv V V Va aσ B µ Re =, H =, k =, v ρv aβ (6) and we obtan (neglectng the prme for clarty): u v + =, x (7)

4 3 O.D. Maknde, E. Osalus 4 u u P u u Re u + v = + + H u, x y x x Re v v u v P v v + = + +, x y y x u =, v =, on y =, u u = k, v= 1, y = 1, (8) (9) (1) (11) where Re s the flow Reynolds number (wth Re > ndcatng sucton and Re < s for njecton), H the Hartmann number and k s the slp parameter. We elmnate pressure p from the dmensonless governng Eqs. (8 9) and ntroduce streamfuncton Ψ and vortcty ω n the followng manner: u = ψ, v= ψ, ω= ψ ψ. x x (1) The governng equaton (7) (11) then become (, ) ( xy, ) ω ψ ψ ω= Re H, ω= ψ, (13) ψ k ψ ψ =, = 1, 1, y y x on y = ψ ψ =, =, y =, x (14) (15) Followng Berman (1953), we ntroduce smlarty varable: ( ), xg( y) ψ= x F y ω= (16) The dmensonless governng equatons (13) together wth the boundary condtons (14) (15) n terms of smlarty varables F, G can be wrtten as G df dg d F = G F H G G + = Re,, (17)

5 5 MHD stea flow 33 df d F = k, F = 1, on y = 1, (18) d F =, F =, on y =. (19) Equatons (17) (19) above can be easly combned to form a nonlnear boundary-value fourth order parameter dependent ordnary dfferental equaton. 3. METHOD OF SOLUTION The non-lnear nature of the equatons (17) (19) peclude ts soluton exactly, hence, we seek the soluton n the form of power seres n Re.e., () = = F = Re F, G = Re G We substtute the above expresson () nto (17) (19) and collect the coeffcents of lke powers of Re, the resultng equatons are: Zeroth Order df d F G d G d F =, =, (1) = k, F = 1, on y = 1, () Hgher Order (n 1) d F F =, =, on y =, (3) G n dfn 1 dgn 1 d Fn = Re G F + H Gn 1, Gn =, (4) = dfn Fn d = k, Fn =, on y = 1, (5) d Fn =, F =, on y =, (6) n

6 34 O.D. Maknde, E. Osalus 6 We have wrtten a MAPLE program that calculates successvely the coeffcents of the soluton seres. In outlne, t conssts of the followng segments: 1. Declaraton of arrays for the soluton seres coeffcents e.g. F=array(..), G=array(..).. Input the leadng order term and ther dervatves.e. F, G. 3.Usng a MAPLE loop procedure, terate to solve equatons (4) (6) for the hgher order terms.e F n, G n, n = 1,,3, Compute the skn frcton, axal pressure gradent and centerlne axal velocty coeffcents. Some of the soluton for stream-functon and vortcty are then gven as follows: 31 ( ) ( k ) ( ) ( ( k ) ) ( ) 3 y k y Re y y 1 F = + y + y k + H y k + H y ykH + y + 3yk 18k 7H 7Hk 147kH + ORe, (7) 3y Rey 4 G = 7H y + 1y 4H 9 336H k k ( ( 3 k ) ) ( ) yk+ 4H yk 63k + 63ykH 63kH + ORe. (8) The non-dmensonal form of the wall skn frcton t w n terms of stream-functon can be wrtten as µ xv d F tw =, on y = 1, (9) a where µ s the namc vscosty. From the results n (7), we obtan the expresson for t w explctly as ( H k H ) µ xv 3 Re Re ( tω = H 3 5 a 1+ 3k 14( 1+ 3k) 334( 1+ 3k ) k 1848H k H k 13356k H (3) H k 1663H k 1751H k 9568H k + O( Re ). From equaton (8), we can determne the flud pressure dstrbuton. Let

7 7 MHD stea flow 35 P = xa, x usng equatons (1) and (16), we obtan 3 d F df d F df A= Re F H 3 + and explcty as 3 3Re 3 3 A= 315H k + 315k + 15k H 1k 1+ 3k ( 3 ( k ) ) ( 5 ( + k ) Re kH 1k 14H H k + 693H k k H k 541k 3793k H 56k + 13H k ) ( ) H k 693H + 77H 34 + O Re. (31) (3) (33) 4. SERIES SUMMATION AND IMPROVEMENT We extend the soluton seres usng a computer symbolc algebra package (MAPLE) n order to examne the effect of nertal forces, the Hartmann number and slp velocty on the flow structure. The frst 15 coeffcents for the above seres representng the flow characterstcs were obtaned. We recast the seres nto several dagonal Padé approxmants [M/M] n order to mprove ts usefulness. For nstance, the seres for the wall shear stress s transformed as follows, N A= f Re = = M = M a c = Re Re (34) where N = M + M s the order of the seres requred for each approxmant. The dea s to match the Taylor seres expanson as far as possble, Maknde (1996). Ths method often evaluates accurately functons beyond the radus of convergence of the correspondng nfnte seres. It fals when we are evaluatng near the zeros of the denomnator of the fracton.

8 36 O.D. Maknde, E. Osalus 8 5. GRAPHICAL RESULTS AND DISCUSSION Snce the flud s ncompressble and vscous, the above mathematcal analyss s very sutable for lqud. It s very mportant to note the an ncrease n the postve value of flow Reynolds number (Re) represents an ncrease n the flud sucton whle an ncrease n the negatve value of Re represents an ncrease n the flud njecton. In order to mprove our results at moderately large sucton and njecton Reynolds number for varous flow characterstcs shown n the fgures below, we have compared the numercal results obtaned from several dagonal Padé approxmants [M/M]. Fgs. and 3 show the flud velocty profles. A parabolc axal velocty profle s observed wth maxmum value at the channel centerlne and mnmum value at the walls. However, a general decrease n the magntude of both axal and normal velocty profles are notced wth an ncrease n both wall slp (k) and magnetc feld ntensty (H). The occurrence of negatve axal velocty near the channel walls due to slp ndcates the possblty of flow reversal near the walls. The wall skn frcton wth respect to flow Reynolds number are shown n the Fgs. 4 and 5. The magntude of the wall skn frcton ncreases wth sucton and decreases wth njecton. Meanwhle, a general decrease n wall skn frcton s observed wth an ncrease n wall slp and a decrease n magnetc feld ntensty. u 1. k= k=.1 k=. v 1. k= k= k= Fg.. Velocty profles for dfferent values of k; Re = u H= H= H=4 1. v H= H=.5.5 H= Fg. 3. Velocty profles for dfferent values of H; Re = 1..

9 9 MHD stea flow 37 t w 3. k= k=. k= Re 1. Fg. 4. Wall Skn Frcton for dfferent values of k, H =.5. t w.4 H=.4 H=. H= Re 1. Fg. 5. Wall Skn Frcton for dfferent values of H, k=.1. CONCLUSION We nvestgated the combned effects of wall slp and magnetc feld on the stea flow of conductng vscous ncompressble flud n a channel wth permeable boundares. Our results revealed that the flud velocty s reduced by both magnetc feld and wall slp. We also notced the presence of flow reversal near the wall due to wall slp. Generally, wall skn frcton ncreases wth sucton and decreases wth njecton, however, both wall slp and magnetc feld also have great nfluence on wall skn frcton. REFERENCES 1. W. H. H. Banks, M. B. Zaturska, On flow through a porous annular ppe, Phys. Fluds A, 4, 1131(199).. G. A. Baker, Jr., Essentals of Padé Approxmants. Academc Press, New York (1975). 3. S. Berman, Lamnar flow n channels wth porous walls, J. Appl. Phys, 4, 13(1953). 4. G. S. Beavers, D. D. Joseph, Boundary condtons at a naturally permeable wall, J. Flud Mech. 3, 197(1967). 5. A.K. Borkakat, I. Pop, MHD heat transfer n the flow between two coaxal cylnders, Acta Mechanca, 97(1984). 6. A.J. Guttamann, Asymptotc analyss of power-seres expansons. Phase Transtons and Crtcal Phenomena, C. Domb and J. K. Lebowtz, eds. Academc Press, New York, pp.1 (1999).

10 38 O.D. Maknde, E. Osalus 1 7. J. Hartmann, Hg-Dynamcs-I. Math-Fys. Medd., 15, No. 6(1937). 8. D. L. Hunter, G. A. Baker, Methods of seres analyss III: Integral approxmant methods, Phys. Rev. B, 19, 388 (1979). 9. Y. J. Km, Unstea MHD convectve heat transfer past a sem-nfnte vertcal porous movng plate wth varable sucton, Int. J. Eng. Sc., 38, 833(). 1. O. D. Maknde, Lamnar flow n a channel of varyng wdth wth permeable boundares, Rom. Jour. Phys., 4, Nos. 4-5, 43(1995). 11. O. D. Maknde, Computer extenson and bfurcaton stu by analytc contnuaton of porous tube flow problem, Journal of Math. Phy. Sc., 3, 1(1996). 1. O. D. Maknde, MHD stea flow and heat transfer on the sldng plate, A. M. S. E., Modellng, Measurement & Control, 7, No. 1, 61(1). 13. O. D. Maknde, Magneto-Hydromagnetc Stablty of plane-poseulle flow usng Mult-Deck asymptotc technque, Mathematcal & Computer Modellng, 37, No. 3 4, 51 (3). 14. O. D. Maknde, Strongly exothermc explosons n a cylndrcal ppe: a case stu of seres summaton technque, Mechancs Research Communcatons, 3, 195(5). 15. R. Moreau, Magnetohydronamcs. Kluwer Academc Publshers, Dordrecht (199). 16. D. Pal, R. Veerabhadraah, P. N. Shvakumar, N. Rudraah, Longtudnal dsperson of tracer partcles n a channel bounded by porous meda usng slp condton, Int. J. Math. Math. Sc., 7, 755(1984). 17. R. Sngh, R. L. Lawrence, Influence of slp velocty at a membrane surface on ultra-fltraton performance-ii (Tube flow system), Int. J. Mass Transfer, 1, 731(1979). 18. R. M. Terrll, P. W. Thomas, Lamnar flow through a unformly porous ppe, Appl. Sc. Res., 1, 37(1969).