Fully developed free convective flow of a Williamson fluid in a vertical channel under the effect of a magnetic field

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1 Available online at Advances in Applied Science Reseach,, 3 (4): ISSN: CODEN (USA): AASRFC Fll developed fee convective flow of a Williamson flid in a vetical channel nde the effect of a magnetic field B. Ana Kmai a, K. Ramakishna Pasad b and K. Kavitha c a Depatment of Mathematics, Chadalawada Ramanamma Engineeing College, Renignta Road, ipati-57 56, A.P, INDIA b Depatment of Mathematics, S V Univesit, ipati-57 5, A.P, INDIA _ INRODUCION he analsis of fee convection in vetical channels occs in man indstial pocesses and natal phenomena. Most of the inteest in this sbject is de to its applications, fo instance, in the design of cooling sstems fo electonic devices and in the field of sola eneg collection. Some of the elated papes on this topic, sch as Ang and Wok (986), Cheng et al. (99), Baletta (998,999), El-Din (3), Bolama and Galanis (4), Baletta et al. (5) deal with the evalation of the tempeate and velocit pofiles fo the vetical paallel-flow fll developed egime. In all the above stdies of fee and mixed convection flow in vetical channels ae based on the hpothesis that the flids ae Newtonian. Howeve, becase of thei fndamental and technological impotance, theoetical stdies of fee, foced and mixed convection flow of non-newtonian flids in channels and tbes ae ve impotant in seveal indstial pocesses. Szei and and Rajagopal (985) have stdied the flow of a thid gade flid between heated paallel plates cased b extenal pesse gadient and obtained similait soltions of the eneg eqation, nmeicall. Akıldız () have stdied the flow of thid gade flid between heated paallel plates. Chamka et al. () have stdied the fll developed fee connective flow of - micopla flid between two vetical paallel plates analticall. Recentl, Siddiqi et al. () have investigated the flow of a thid gade non- Newtonian flid between two paallel plates sepaated b a finite gap b sing the Adomian decomposition method. Williamson flid is chaacteized as a non-newtonian flid with shea thinning popet, i.e., viscosit deceases with inceasing ate of shea stess (Dapa and Scapi, 7). he se of electicall condcting flids nde the inflence of magnetic fields in vaios indsties has led to a enewed inteest in investigating hdomagnetic flow and heat tansfe in diffeent geoemeties. Fo example, Spaow and Cess (96) consideed the effect of a magnetic field on the fee convection heat tansfe fom a sface. Gaandet et al. (99) have stdied boanc diven convection in a ectangla enclose with a tansvese magnetic field. Chamkha (999) have investigated fee convection effects on thee-dimensional flow ove a vetical stetching sface in the pesence of a magnetic field. Bhagava et al. (3) have stdied the effect of magnetic field on the fee convection flow of micopola flid between two paallel poos vetical plates. Haat et al. (4) have stdied the Hall effects on the nstead hdomagnetic oscillato flow of a second gade flid. Hazeem attia (5) have investigated the nstead flow of a dst condcting flid between paallel poos plates. Sanal and Adhikai (6) have stdied the effects of adiation on MHD flid flow in vetical channel. In view of these, we stdied the fll developed fee convection flow of a Williamson flid in a vetical channel nde the effect of magnetic field. he govening non-linea eqations ae solved fo the velocit field and tempeate field sing the petbation techniqe. he effects of vaios emeging paametes on the velocit field and tempeate field ae stdied thogh gaphs in detail.. Mathematical fomlation he eqations govening the flow of an incompessible Williamson flid ae given b 49

2 . V (.) dv ρ ρ f +. τ (.) dt whee ρ denotes the constant flid densit, V is the velocit vecto and f epesents the bod foce pe nit mass. he opeato d / dt denotes the mateial time deivative and τ is the stess tenso. Fig. he phsical model he constittive eqation fo a Williamson flid is given b ( )( ) τ η + η + η Γ& γ & γ (.3) Whee τ is the exta stess tenso, η is the infinite shea ate, viscosit η o is the zeo shea ate viscosit, Γ is the time constant and γ& is defined as & γ & γ & ijγ ji π (.4) i j whee π is the second invaiant stess tenso. We conside in the constittive Eq. (.3) the case fo which η and Γ & γ < so we can wite. τ η + Γ& γ & γ (.5) ( ) he above model edces to Newtonian fo Γ We conside the lamina fee convection flow of a Williamson flid between two plates at distance h a apat, as shown in Fig..We choose co-odinates sstem, with X - axis paallel to the flow while Y - axis is nomal to the flow. A nifom magnetic field B is applied in the tansvese diection to the flow. he flow assme stead and fll developed, i.e., the tansvese velocit is zeo. It is also assmed that the walls ae heated nifoml bt thei tempeates ma be diffeent eslting in asmmetic heating sitation nde these assmptions the eqations that descibe the phsical sitation ae 493

3 µ σ ρ β d d d d d d + Γ B + g ( ) whee σ is the electical condctivit. (.6) (.7) Sbject to the bonda conditions ( ), ( ), ( h), ( h) Intodcing the following non-dimensional vaiables x UΓ,, x, We, θ, U h h η h (.8) into Eqs. (.6) and (.7), we get (afte dopping the bas) d d G + We M θ + d d d Re (.) d θ d (.) whee M Bh σ gβ ( ) 3 h is the Hatmann nmbe, G is the Gashof nmbe η ν and Re Uh ν is the Renolds nmbe. he coesponding dimensionless bonda conditions ( ) θ ( ), ( ) θ ( ), (.9), (.) 3. Petbation Soltion Eq. (.) is non-linea and it is difficlt to get a closed fom soltion. Howeve fo vanishing We, the bonda vale poblem is ageeable to an eas analtical soltion. In this case the eqation becomes linea and can be solved. Nevetheless, small Γ sggests the se of petbation techniqe to solve the non-linea poblem. Accodingl, we wite + We (3.) and θ θ + Weθ (3.) Sbstitting eqations (.) and (.) into Eqs. (.8) and (.9) and bonda conditions (.) and then eqating the like powes of We, we obtain 3. Zeoth-ode sstem ( We ) d d G M θ (3.3) Re d θ (3.4) d ogethe with bonda conditions 494

4 ( ) ( ), θ ( ), ( ) 3. Fist-ode sstem( We ) θ (3.5) d d d G M θ d d d Re (3.6) d θ d (3.7) ogethe with bonda conditions ( ) ( ), θ ( ), ( ) 3.3 Zeoth-ode soltion Solving Eqs. (3.3) and (3.4) sing the bonda conditions (3.8), we get ( ) θ (3.8) θ + (3.9) G A sinh M cosh M + ( ) + Re M (3.) cosh M hee A. sinh M 3.4 Fist-ode soltion Solving Eq. (3.7) sbject to the bonda conditions in Eq. (3.8), we get θ (3.) Sbstitting the Eqs. (3.) and (3.) into the Eq. (3.6) and then solving the eslting eqation with the coesponding conditions, we get A3 A cosh M + A 7 sinh M sinh M G 3M 3M Re M A3 A4 A5 + cosh M + sinh M cosh M 3M M M 3 3 A M + A, A A M, A4 ( ) M, A5 A ( ) M, 4 whee ( ) 3 (3.) A sinh M A3 cosh M A4 sinh M A5 cosh M A6 +, 3M 3M M M A3 cosh M A7 A6 + sinh M. 3M Finall, the petbation soltions p to fist ode fo θ and ae given b θ θ + Γ θ θ + (3.3) and ( ) + Γ (3.4) RESULS AND DISCUSSION Fig. shows the effect of Weissenbeg nmbe We on fo M,.5, G obseved that, velocit fist deceases and then inceases with inceasing We. and Re. It is he effect of Hatman nmbe M on fo We.,.5, G and Re 3. It is fond that, the velocit deceases with an incease in Hatmann nmbe M. is epesented in Fig. 495

5 .9 We.3,.,., Fig.. Effect of Weissenbeg nmbe We on fo G, M,.5 and Re...8 M M M.6 M Fig. 3. Effect of Hatmann nmbe M on fo We.,.5, G and Re. 496

6 ..5 G. G G.5.5 G Fig. 4. Effect of Gashof nmbe G on fo M,.5, We.and Re..9 Re.6 Re.3 Re 3 Re Fig. 5. Effect of Renolds nmbe Re on fo M,.5, We. and G. 497

7 Fig. 6. Effect of wall tempeate paamete on fo M, G, We.and Re θ Fig. 7. Effect of wall tempeate paamete on θ. 498

8 Fig. 4 depicts the effect of Gashof nmbe G on fo M,.5, We. obseved that, the velocit inceases with inceasing Gashof nmbeg. and Re. It is he effect of Renolds nmbe Re on fo M,.5, G and We. is shown in Fig. 5. It is noted that, the velocit deceases with an incease in Renolds nmbe Re. Fig. 6 illstates the effect of wall tempeate paamete on fo M, We., G and Re. It is fond that, the velocit inceases with inceasing. Fig. 7 shows the effect of wall tempeate paamete with an incease in. on θ. It is obseved that, the tempeateθ inceases REFERENCES [] Akıldız, F.. Int. J. Non-Linea Mech. 36(), [] Ang, W. and Wok, G. J. Heat ansfe, 8 (986), [3] Bhagava, R., Kma, L. and akha, H. S., Int. J. Engng. Sci., 4(3), [4] Baletta, A., Int. J. Heat Mass ansfe, 4 (998), [5] Baletta, A. ASME J. Heat ansfe, (999), [6] Baletta, A. and Zanchini, E. Int. J. Heat Mass ansfe, 4 (999), ] [7] Baletta, A., Magai, E. and Kelle, B. Int. J. Heat Mass ansfe, 48 (5), [8] Bolama, K. and Galanis, N. J. Heat ansfe, 6 (4), [9] Chamkha, A.J. Int. J. Heat Flid Flow, (999), [] Chamkha, A.J., Gosan,.. and Pop, I. Int. Commn. Heat Mass ansfe, 9 (), -96. [] Cheng, C. H., Ko, H. S. and Hang, W.H., J. hemophsics, 3(99), [] Dapa, I. and Scapi, G., Intenational Jonal of Rock Mechanics and Mining Sciences, 44()(7), [3] El-Din, M.M. S., Int. J. hemal Sci. 4 (3), [4] Haat,., Wang, Y. and Htte, K., Intenat. J. Non-LineaMech. 39 (4) [5] Hazeem Attia, k. J. Phs., 9(5), [6] Gaandet, J.P., Albossiee,. and Moea, R. Int. J. Heat Mass ansfe 35 (99) [7] Sanal, D.C. and Adhikai A (6). Blletin of Calctta Mathematical Societ, 98(5), pp [8] Siddiqi, A.M., Hameed, M., Siddiqi, B.M. and Ghoi, Q.K., Commn Nonlinea Sci. Nme. Simlat. 5 (), [9] Spaow, E.M. and Cess, R.D., Int. J. Heat Mass ansfe, 3(96), [] Szei, A. Z. and Rajagopal, K. R. Int. J. Non-Linea Mech. (985),