Received 2 August 2014; revised 2 September 2014; accepted 10 September 2014

Transcription

1 Ameicn Jounl of Computtionl Mthemtics, 4, 4, Published Online eptembe 4 in cires. Effect of Vible Viscosity, Dufou, oet nd Theml Conductivity on Fee Convective Het nd Mss Tnsfe of Non-Dcin Flow pst Poous Flt ufce Isc L. Animsun, Anselm O. Oyem Deptment of Mthemticl ciences, Fedel Univesity of Technology, Akue, Nigei Deptment of Mthemtics, Fedel Univesity Lokoj, Lokoj, Nigei Emil: Onyekchukwu.oyem@fulokoj.edu.ng, ellityo@yhoo.co.uk Received August 4; evised eptembe 4; ccepted eptembe 4 Copyight 4 by uthos nd cientific Resech Publishing Inc. This wok is licensed unde the Cetive Commons Attibution Intentionl License (CC BY). Abstct The motion of incompessible fluid of vible fluid viscosity nd vible theml conductivity with theml dition, Dufou, oet with het nd mss tnsfe ove linely moving poous veticl semi-infinite plte with suction is investigted. The govening equtions e tnsfomed into system of coupled nonline odiny diffeentil equtions using simility tnsfomtions with dimensionless vibles nd solved numeiclly using shooting method with Runge- Kutt fouth-ode method nd Newton-Rphson s intepoltion scheme implemented in MATLAB. The esult showed tht with incese in Dufou nd oet pmete, fluid velocity inceses nd tempetue inceses with incese in vition of Dufou while, tempetue deceses with incese in oet. The effects of vible fluid viscosity, vible theml conductivity, theml dition, oet, Dufou, Pndtl nd chmidt pmetes on the dimensionless velocity, tempetue nd concenttion pofiles e shown gphiclly. Keywods Vible Viscosity, Vible Theml Conductivity, Non-Dcin Flow, Fee Convection, Dufou, oet, Poous Flt ufce. Intoduction Fluid flow poblems of fee convective, het nd mss tnsfe though poous medium hd been given tten- How to cite this ppe: Animsun, I.L. nd Oyem, A.O. (4) Effect of Vible Viscosity, Dufou, oet nd Theml Conductivity on Fee Convective Het nd Mss Tnsfe of Non-Dcin Flow pst Poous Flt ufce. Ameicn Jounl of Computtionl Mthemtics, 4,

2 I. L. Animsun, A. O. Oyem tion due to its pplictions in mny engineeing poblems such s nucle ecto design, geotheml systems, petoleum engineeing pplictions, evpotion t the sufce of wte body, contol of pollutnts in gound wte, enegy tnsfe in wet cooling towe, food pocessing coole [] [], nd the poblem of het nd mss tnsfe flow of lmin boundy lye ove stetching sheet in stuted poous medium hs n impotnt ppliction in the metllugy nd chemicl engineeing fields. Lyek et l. [3] consideed the effects of theml dition nd vible fluid viscosity on fee convective flow nd het tnsfe pst poous stetching sufce while, Alhbi et l. [4] investigted the het nd mss tnsfe of MHD viscoelstic fluid flow though poous medium ove stetching sheet with chemicl ections, due to the impotnce of oet (Theml-diffusion) nd Dufou (Diffusion-themo) effects on the fluids with vey light molecul weight s well s medium molecul weight. Dcy s empiicl flow model epesents simple line eltionship between flow te nd pessue dop in poous medi; ny devition fom the Dcy flow scenio is temed non-dcy flow. Fochheime [5] cied out esech on flowing of gs though col beds nd his epot showed tht the eltionship between flow te nd potentil gdient ws non-line t sufficiently high velocity nd this non-lineity incesed with flow te. He dded second ode velocity tem to epesent the micoscopic inetil effect, nd modified the Dcy eqution into the Fochheime eqution, which ws widely consideed to descibe the inetil effects due to dditionl fiction obseved fo high velocity flow, Dcy [6]. Alm, et l. [7], studied the effects of Dufou nd oet on two-dimensionl stedy MHee convective nd mss tnsfe flow pst semi-infinite veticl poous plte in poous medium numeiclly. Alm, et l. [8] futheed the esech using Locl simility solution given by ctte et l. [9] nd Hshimoto [] to tnsfom the nonline ptil equtions govening the poblem to lowe ode of coupled odiny diffeentil eqution. In ll of the bove mentioned studies, fluid viscosity nd fluid theml conductivity wee ssumed to be constnt thoughout the boundy lye. Howeve, it is known tht the physicl popeties of the fluid my chnge significntly with tempetue. Fo lubicting fluids, het geneted by the intenl fiction nd the coesponding ise in tempetue ffects the viscosity of the fluid nd so the fluid viscosity cn no longe be ssumed constnt. The incese of tempetue leds to locl incese in the tnspot phenomen by educing the viscosity coss the momentum boundy lye nd so the het tnsfe te t the wll is lso ffected getly. Extenl heting such s the mbient tempetue nd high she tes cn led to high tempetue being geneted within the fluid. This my hve significnt effect on the fluid popeties. It is well known fct in fluid dynmics, tht the popety which is most sensitive to tempetue ise is viscosity nd theml conductivity, Anykoh, []. Fo instnce, the viscosity of wte deceses bout 4% when the tempetue inceses fom C to 5 C. The viscosity of i is t K,.389 t K,.86 t 4 K nd 3.65 t 8 K s epoted by Cebeci, et l. []. Mukhopdhyy [3] dopted Btchelo s model of tempetue dependent fluid viscosity when he studied the effect of dition nd vible fluid viscosity on flow nd het tnsfe long symmetic wedge ssuming constnt theml conductivity. lem, et l. [4] investigted the effects of vible popeties on MHD het nd mss tnsfe flow ne stgntion point towds stetching sheet in poous medium with theml dition nd dopted the model of Psd, et l. [5] fo tempetue dependent viscosity nd theml conductivity nd lso incopoted the stgntion point velocity into thei momentum eqution. In this ppe, we wnt to study the effects of vible fluid viscosity, vible theml conductivity, Dufou nd oet on fee convective het nd mss tnsfe of non-dcin flow pst poous flt sufce tking into ccount dition effects using Rosselnd ppoximtion in modeling the ditive het tnsfe.. Mthemticl Fomultion A stedy two-dimensionl lmin fee convection flow of viscous incompessible vible viscosity nd vible theml conductivity fluid long poous veticl sufce in the pesence of suction nd het genetion nd bsoption is consideed. The govening mss, momentum, enegy nd concenttion equtions tkes the fom of [6] u v + = () x ( T ) u u u µ b u + v = µ ( T) + gβ( T T) + gβ ( C C) u u x ρ ρ K K () 358

3 I. L. Animsun, A. O. Oyem subject to the boundy conditions: q DK u v T T T T T T q t C + = κ ( ) + ( ) + x y ρcp y y ρcp y ρcp CC P y C C C DK T u + v = D + x T t y m y m u = Bx v = V T = C C = C t y = (5) w W W u T T C C s y (6) whee x nd y epesents coodinte xes long the continuous sufce in the diection of motion nd pependicul to it nd u nd v e the Dcin velocity components long x- nd y-xis espectively, ρ is the density, g is the foce of gvity, b is the empiicl constnt, µ is the viscosity, ϑ is the kinemtic viscosity, B T is the coefficients of theml expnsion, β is the coefficient of volumetic expnsion, K is the pemebility of the poous medium, κ is the theml conductivity, C p is the specific het t constnt pessue, q is the dimensionl het bsoption coefficient, C is the concenttion, C is the mbient concenttion, C p is the specific het t constnt pessue, K t is the theml diffusion tio, T m is the men fluid tempetue nd V w is the suction velocity coss the stetching sheet. In this esech, we conside vible viscosity, whee the fluid viscosity µ ( T ) is ssumed to vy s line function of tempetue [6] nd κ is the constnt vlue of the coefficient of theml conductivity f fom the sheet nd δ is constnts, µ is the constnt vlue of the coefficient of viscosity f b >, Psd et l. [7]. Thus, fom the sheet nd b constnt ( ) µ ( T) = µ + bt ( T) κ( T) = κ + δ ( T T) Using Rosselnd ppoximtion fo dition w. (7) 4 4σ T q = 3k whee σ is the tefn-boltzmnn constnt nd k is known s the bsoption coefficient. Assuming tht the tempetue diffeence within the flow is such tht T 4 my be expnded in Tylo seies nd expnding T 4 bout T nd neglecting highe odes, Equtions (3) becomes ( T ) T T T T κ T 6σ T T q DK C u + v = T T T +. (9) x ρc ρc T ρc CC Intoducing stem functions, Equtions (), (4), (5), (6) nd (9) becomes 3 t κ ( ) ( ) P P 3k ρcp P P ψ ψ u =, v = x. 3 ψ ψ ψ ψ θ ψ ψ 3 = υ ξ + υ + θ ξ + βθ w x x ( ) g ( T T ) ( ) υ + θ ξ ψ b ψ + gβφ( Cw C ), K K ( Cw C ) ( ) 3 6σ T q DKt ε [ θε ] θ P P 3k ρcp P P w ψ θ ψ θ κ θ κ θ θ φ = () x x ρc ρc ρc CC T T subject to ( Tw T ) ( ) ψ φ ψ φ φ DK θ = D + x x T C C t y m w (3) (4) (8) () () 359

4 I. L. Animsun, A. O. Oyem ψ m ψ = Bx ; = Vw; θ ; φ t y = (3) x ψ ; θ ; φ s y. (4) Intoducing the following dimensionless vibles, B T T C C = y ; ψ = υ xb f ( ) ; θ = ; φ =. (5) υ x T T C C Equtions ()-(4) educe to system of coupled nonline odiny diffeentil equtions [ ] w w [ + ξ θξ ] 3 d f dθ d f d f df Fs df + ξ θξ ξ f J 3 Gξθ JGTξφ d d d d DR e d D d = (6) 4 d θ dθ dθ d φ + θε + + P Ω f + ε + Pθ + PD f = 3N d d d d (7) subject to d φ d d c f φ θ c + + = d d d (8) df ; f fw ; θ ; φ t y d = = = = = (9) df ; θ ; φ s y. () d 3. Numeicl olution The set of coupled odiny diffeentil Equtions (6)-(8), subject to (9) nd () ws solved numeiclly using Runge-Kutt fouth ode technique long with shooting method ws implemented using MATLAB. The esultnt initil vlue poblem is solved by employing Runge-Kutt fouth ode technique. The step size =.5 is used to obtin the numeicl solution with fifteen deciml plce ccucy s the citeion of convegence. Fom the numeicl computtion, kin-fiction coefficient, Nusselt Numbe nd the hewood Numbe, which e espectively popotionl to f ( ), θ ( ) nd φ ( ) e lso soted out nd thei vlues pesented in tbul fom. 4. Results nd Discussion Figue shows the velocity pofiles fo sevel vlues of ξ nd ε using N =.7, P, Ω =. in the pesence of suction ( f w =.3). Velocity inceses with n incese in tempetue dependent viscosity ξ nd theml conductivity ε within 6.88, while the velocity pofiles to deceses with n incese in the vlue of ξ nd ε, mking the fluid move fste s shown in Tble. Fom Figue, we cn obseve tht incese in tempetue-dependent fluid viscosity nd theml conductivity pmetes esults in decese in theml boundy lye thickness nd tempetue pofile. The effect of vible viscosity nd theml conductivity on mss tnsfe fo diffeent vlues of ξ nd ε nd fixed vlue of c shows tht mss tnsfe occus t highe te coss the fluid flow of high viscosity nd theml conductivity thn the lowe theml conductivity in Figue 3. Fom Figue 4-6, Locl Fochheime inceses s velocity deceses with incese in tempetue nd concenttion, see Tble. While fom Figues 7-9, we obseve tht s velocity inceses, velocity of the poous plte tends to decese nd D inceses with deceses in tempetue nd concenttion see Tble Conclusion In this ppe, the effect of vible viscosity, theml conductivity Dufou nd oet of non-dcin flow on 36

5 I. L. Animsun, A. O. Oyem Tble. Effects of vition of ξ nd ε. F s, D., =,, J GT, P ; D =.3, Ω =., =., =.4, f =.3. c W f ( ) θ ( ) φ ( ) ξ = ε = ξ = ε = ξ = ε ξ = ε ξ = ε f Tble. Effects of vition of F. ξ = ε., F = Vies, D, =,, J GT, P ; D =.3, Ω =., =., =.4, f =.3. f c W s f ( ) θ ( ) φ ( ) F = F = F = F F Tble 3. Effects of vition of. ξ = ε., F, D = Vies, =,, J GT, P ; D =.3, Ω =., =., =.4, f =.3. f c W s f ( ) θ ( ) φ ( ) D = D = D = D D ξ = ε =.6 ξ = ε =.9 ξ = ε. ξ = ε.5 ξ = ε.8 f ' ( ).5 ξ nd ε ξ = ε = Vies =. = =.3, =.4 Ω =. =. 5 5 Figue. Velocity pofiles fo sevel vlues of ξ nd ε. 36

6 I. L. Animsun, A. O. Oyem θ ( ) ξ nd ε ξ = ε =.6 ξ = ε =.9 ξ = ε. ξ = ε.5 ξ = ε.8 ξ = ε = Vies =. = =.3, =.4 Ω =. = Figue. Tempetue pofiles fo sevel vlues of ξ nd ε. φ ( ) ξ nd ε ξ = ε =.6 ξ = ε =.9 ξ = ε. ξ = ε.5 ξ = ε.8 ξ = ε = Vies =. = =.3, =.4 Ω =. = Figue 3. Concenttion pofiles fo sevel vlues of ξ nd ε. f ' ( ) =.5 =.5 = ξ = ε. = Vies. = =.3, =.4 Ω =. =. 5 5 Figue 4. Velocity pofiles fo sevel vlues of. 36

7 I. L. Animsun, A. O. Oyem θ ( ) =.5 =.5 = ξ = ε. = Vies. = =.3, =.4 Ω =. = Figue 5. Tempetue pofiles fo sevel vlues of. φ ( ) =.5 =.5 = ξ = ε. = Vies. = =.3, =.4 Ω =. =. 5 5 Figue 6. Concenttion pofiles fo sevel vlues of..4 =.5. =.5 = f ' ( ) ξ = ε.. = Vies = =.3, =.4 Ω =. =. 5 5 Figue 7. Velocity pofiles fo sevel vlues of locl Dcy. 363

8 I. L. Animsun, A. O. Oyem θ ( ) =.5 =.5 = ξ = ε.. = Vies = =.3, =.4 Ω =. = Figue 8. Tempetue pofiles fo sevel vlues of locl Dcy. φ ( ) =.5 =.5 = ξ = ε.. = Vies = =.3, =.4 Ω =. =.. Figue 9. Concenttion pofiles fo sevel vlues of locl Dcy. fee convective het nd mss tnsfe pst poous flt plte ws investigted using Runge-Kutt method with shooting method nd Newton-Rphson technique. The pocedue evels ccutely thei effects nd vious solution bnches. It is obseved tht with n incese in tempetue-dependent fluid viscosity nd theml conductivity pmete, velocity inceses, tempetue deceses nd concenttion deceses coss the flow egion. Incesing Pndtl numbe s well s dition pmete on the velocity boundy lye, educes the velocity field, nd deceses tempetue with incesing vlue of theml dition, theeby cusing concenttion nd Dufou to incese togethe with decese in oet pmete nd slightly inceses velocity. Ou esults nd pocedues cn be used s n effective tool to investigte nonline boundy-poblems in science nd engineeing. Acknowledgements Authos e thnkful to the efeee(s) fo the vluble comments on the elie dft of the ppe. Refeences 5 5 [] Inghm, D.B. nd Pop, I. (5) Tnspot Phenomen in Poous Medium III. Elsevie, Oxfod. [] Vfi, K. (5) Hndbook of Poous Medi. nd Edition, Tylo nd Fncis, New Yok. 364

9 I. L. Animsun, A. O. Oyem [3] Lyek, G.C. nd Mukhopdhyy,. (8) Effects of Theml Rdition nd Vible Fluid Viscosity on Fee Convective Flow nd Het Tnsfe pst Poous tetching ufce. Intentionl Jounl of Het nd Mss Tnsfe, 5, [4] Alhbi-leh, M., Mohmed, A.A. nd Mhmoud,.E. () Het nd Mss Tnsfe in MHD Visco-Elstic Fluid Flow though Poous Medium ove tetching heet with Chemicl Rection. cientific Resech, Applied Mthemtics,, [5] Fochheime, P.Z. (9) Wssebewegungduch Boden. Zeitschift des Veeins Deutsche Ingenieue, 45, [6] Dcy, H. (856) Les fontinepubligues de l ville de Dijoin. Dlmont. [7] Alm, M.. nd Rhmn, M.M. (5) Dufou nd oets on MHD Fee Convective Het nd Mss Tnsfe Flow pst Veticl Flt Plte Embedded in Poous Medium. Jounl of Nvl Achitectue nd Mine Engineeing,, [8] Alm, M.., Rhmn, M.M. nd md, M.A. (6) Dufou nd oet Effects on Unstedy MHD Fee Convective nd Mss Tnsfe Flow pst Veticl Poous Plte in Poous Medium. Nonline Anlysis:Modeling nd Contol,, 7-6. [9] ctte, M.A. nd Hossin, M.M. (99) Unstedy Hydomgnetic Fee Convection Flow with Hll Cuent nd Mss Tnsfe long n Acceleted Poous Plte with Time-Dependent Tempetue nd Concenttion. Cndin Jounl of Physics, 7, [] Hshimoto, H. (957) Boundy Lye Gowth on Flt Plte with uction o Injection. Jounl of the Physicl ociety of Jpn,, [] Anykoh, M.W. () New chool Physics. 3d Edition, Aficn Fist Publishe, Enugu, [] Cebeci, T. nd Bdshw, P. (984) Physicl nd Computtionl Aspects of Convective Het Tnsfe. pinge, New Yok. [3] Mukhopdhyy,. (9) Effects of Rdition nd Vible Fluid Viscosity on Flow nd Het Tnsfe long ymmetic Wedge. Jounl of Applied Fluid Mechnics,, [4] lem, A.M. nd Fthy, R. () Effects of Vible Popeties on MHD Het nd Mss Tnsfe Flow ne tgntion Point towds tetching heet in Poous Medium with Theml Rdition. Chinese Physics B,, Aticle ID: 547. [5] Psd, K.V., Vjvelu, K. nd Dtti, P.. () The Effects of Vible Fluid Popeties on the Hydomgnetic Flow nd Het Tnsfe ove Non-Linely tetching heet. Intentionl Jounl of Theml ciences, 49, [6] Btchelo, G.K. (987) An Intoduction to Fluid Dynmics. Cmbidge Univesity Pess, London. [7] Lognthn, P. nd Asu, P.P. () Lie Goup Anlysis fo the Effects of Vible Fluid Viscosity nd Theml Rdition on Fee Convective Het nd Mss Tnsfe with Vible tem Condition. cientific Resech Jounl,,

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