N igerian Journal of M athematics and Applications V olume 24, (2015),

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1 N igeian Jounal of M athematics an Applications V olume 24, 205), c N ig. J. M ath. Appl. http : // Flow of an Incompessible MHD Thi Gae Flui Though a Cylinical Pipe with Isothemal Wall an Joule Heating Aiyesimi, Y. M. 2 Okeayo, G. T. 3 Obi, B. I. Abstact In this pape, we consiee the flow of an incompessible MHD thi gae flui though a cylinical pipe with isothemal wall an Joule heating. Axial pessue-gaient was assume to have inuce the motion. The govening equations of the flow fiel ae solve using petubation metho. Results ae pesente.. Intouction With the emegence of polyme inusty, petoleum inusties an othe types of pulp inusties, in ecent yea, the non-newtonian fluis have become vey much impotant. Also with its complexity, it becomes ifficult to suggest a single moel which will exhibit all the popeties of non-newtonian fluis, as such vaious empiical an semi-empeical moel have been put fowa. Non- Newtonian fluis can be classifie mainly into two goups such as iffeential type fluis an ate types flui. Within these two goups, the iffeential type of fluis have eceive much attention fom eseaches. Example of such fluis in this goup is the secon gae flui which is most acceptable because of its mathematical simplicity when compae with the thi gae an the fouth fluis. Nagis Khan an Tahi Mahmoo 202) stuie the influence of slip conition on the thin film flow of a thi oe flui. Receive Septembe 3, 205. Coesponing autho. - appliebon@yahoo.com 200 Mathematics Subject Classification. 76Dxx & 46T2. Key wos an phases. Incompessible MHD, isothemal wall, Joule heating. Depatment of Mathematics, Feeal Univesity of Technology, Minna, Nigeia 2 Depatment of Mathematical Sciences, Ono State Univesity of Science an Technology, Okitipukpa, Nigeia 3 Depatment of Mathematics, Imo State Univesity, Owei, Nigeia 228

2 Flow of an Incompessible MHD Thi Gae Flui Though a Cylinical To solve the couple equations of the thi gae flui equies excessive computational effot an analytical solution which ae almost impossible to obtain except stong simplification ae mae. Okoya consiee the isappeaance of citicality fo eactive thi gae flui with moel viscosity in a flat channel [5]. Tahi Mahmoo an Nagis Khan 202) on thin film flow of a thi gae flui though poous meium ove an incline plane, employe the petubation an homotopy petubation methos an the esults obtaine fom the two methos ae in close ageement. Masoui an Chistie [0] analyze the effects of tempeatue-epenent viscosity fo the thee sepaate cases teate by Szei an Rajagopal [2]. Accoing to Aiyesimi et al [3] on the analysis of unsteay MHD thin film flow of a thi gae flui with heat tansfe own an incline plane. They iscovee that the vaiation of the velocity an tempeatue pofile with the magnetic fiel paamete an the gavitational paamete epene on time. Aiyesimi et al [4] ealt with the effects of magnetic fiel on the MHD flow of a thi gae flui though incline channel with Ohmic heating. They analyze the couette flow, poiseuille an Couette-poiseuille flow an solve the esulting non- linea iffeential equation by employing egula petubation technique. Aiyesimi et al [] stuie MHD flow of a thi gae flui with heat tansfe an slip bounay conition own an incline plane. It was iscovee that the esult obtaine by the taitional petubation metho was in close ageement with that obtaine by homotopy petubation technique. Aiyesimi et at [2] examine the viscous issipation effect on the MHD flow of a thi gae flui own an incline plane with mathematical fomulation of the poblem is establishe an solve in the next two sections espectively. 2.0 Mathematical Fomulation Following Makine [9], we consie the steay flow of an incompessible MHD thi gae flui though a cylinical pipe with isothemal wall. Axial pessuegaient was assume to have inuce the flui motion. Fo magnetohyoynamically evelope flow, both the velocity an the tempeatue fiels epen on only. Consieing [5, 0, 4] an neglecting the eacting viscous flui assumption, the govening equations fo the momentum an enegy balance can be epesente as: ) 2) k µ u ) + β T ) + ) u 2 µ + β 3 ) ) ) u 3 σb 2 0u = p ) u 2 + γb 0 u 2 = 0

3 230 Aiyesimi, Y. M., Okeayo, G. T. an Obi, B. I. Subject to the axial-symmetic an bounay conitions: 3) u T 0) = 0) = 0, ua) = 0, T a) = 0 whee T is the tempeatue of the cyline, u is the flui velocity, T 0 is the plate tempeatue, µ is the coefficient of ynamic viscosity of the flui, U is the chaacteistic flui velocity, B 0 is the magnetic flui effect, P is the pessue of the system, B 3 is the mateial coefficient an γ is the heat eaction. Intoucing the following non-imensional vaiables into equations ) - 3) fo non-imensionalizationn. 4) θ = T T 0, = a, u = u u 0 an we obtain the following imensionless momentum an enegy equations: 5) 6) u ) + γ θ ) u + B u ) 3 Mu = ) 2 ) u 4 + 2B + Mu 2 = 0 with 7) u θ 0) = 0) = 0, u) = 0, θ) = 0 whee γ an M ae the imensionless non-newtonian an viscous heating paametes espectively 2.0 Metho of Solution In oe to solve equation 5), we intouce petubation seies as follows: u = u 0 + = γu + 0γ) 2 an θ = θ 0 + γθ + 0γ) 2 8) with M = γm an obtain the oe γ 0 an oe γ poblems espectively as: 9) u ) 0 =

4 Flow of an Incompessible MHD Thi Gae Flui Though a Cylinical ) u ) + u ) 3 0 Mu 0 = 0 Then solving equations9) an 0) using equation 7), we obtain: ) u 0 = 4 4 ) ) 2) u = M M Heat Tansfe Analysis Fom equation 6), we obtain: 3) θ ) [ + γ λ u ) 2 ) ] u 4 2B + mu 2 = 0 Using the seies 8) in 3) an simplifying, we obtain 4) γ 0 : θ ) ) 2 0 u0 + B = 0 5) γ : θ ) u 0 u + 2B + EcMu2 0 = 0 Using equation ) in equation 4), we obtain: [ ] 6) θ 0 = B Using equations ) an 2) in equation 5), we obtain: 7) θ = B B M 4 B M 4 EcM2 + EcM EcM6 576 B 52 B M 28 + B M 576 EcM 64 EcM 28 + EcM 576 The solutions of the momentum an enegy equations ae espectively: 8) u) = { ) ) + γ 48 + M M } 64

5 232 Aiyesimi, Y. M., Okeayo, G. T. an Obi, B. I. 9) ) { θ) = B 64 4 B 8 + γ B M 4 B M 4 EcM2 + EcM EcM6 576 B 52 B M 28 + B M 576 EcM 64 EcM 28 + EcM Results an Discussion The tempeatue θ) an momentum u) pofiles wee examine with vaious values of the physical paamtes. Figues an 2 shows that velocity inceases with incease in magnetic paamete an γ espectively. While in figues 3,4,5 an 6 show that incease in γ, Binkman numbe, M an the Ecke numbe also inceases the tempeatue pofile.

6 Flow of an Incompessible MHD Thi Gae Flui Though a Cylinical

7 234 Aiyesimi, Y. M., Okeayo, G. T. an Obi, B. I.

8 Flow of an Incompessible MHD Thi Gae Flui Though a Cylinical Conclusion In this pape, we applie petubation technique to examine the flow of an incompessible MHD thi gae flui though a cylinical pipe with isothemal wall an Joule heating. The technique shows excellent ageement with Makines esult on themal stability of a eactive thi gae flui with Hemite-Pae appoximation technique. Refeences [] Aiyesimi Y. M., Okeayo G. T. an Lawal O. W., MHD flow of a thi gae flui with heat tansfe an slip bounay conition own an incline plane, J. of Mathematical Theoy an Moelling, 9 202). [2] Aiyesimi Y. M., Okeayo G. T. an Lawal O. W., Viscous issipation effect on the MHD flow of a thi gae flui own an incline plane with Ohmic heating, J. of Mathematical Theoy an Moelling, 2 9), 20. [3] Aiyesimi Y. M., Okeayo G. T. an Lawal O. W., Analysis of unsteay MHD thin film flow of a thi gae flui with heat tansfe own an incline plane, J. of Applie an computational Mathematics, 204). [4] Ayub M., Rashee A. an Hayat T., Exact flow of a thi gae flui past a poous plate using homotopy analysis metho, Int. J. Eng. Sci., 4 209) 2003). [5] Khan N. an Mahmoo T., The influence of slip conition on the thin film flow of a thi oe flui, Int. J. of Nonlinea Science, 202). [6] Mahmoo T. an Khan N., Thin film flow of a thi gae flui though poous meium ove an incline plate, Int. J. of Nonlinea Science, 202). [7] Makine O. D., Stong exothemic explosions in a cylinical pipe: A case stuy of seies summation technique, Mech. Res. Common., ), [8] Makine O. D., Themal stability of a eactive thi gae flui in a cylinical: An exploitation of Hemite-Pae appoximation technique, Elsevie Science Diect. Applie Mathematics an Computation, ),

9 236 Aiyesimi, Y. M., Okeayo, G. T. an Obi, B. I. [9] Massoui M. an Chistie I., Effects of vaiable viscosity an viscous issipation on the flow of a thi gae flui in a pipe, Int.J. of Nonlinea Mech., 30 5) 995), [0] Okoya S. S., Disappeaance of citicality fo eactive thi-gae flui with Ranols moel viscosity in a flat channel Nonl., Int. J. of Nonlinea Mech., 20). [] Szei A. Z. an Rajagopal K. R., Flow of a non- Newtonian flui between heate paallel plates, Int. J. of Nonlinea Mech., 20 2) 985), 9-0. [2] Yuusoy M., Bayakceken H., Kapuen M. an Aksoy F., Entopy analysis fo thigae flui flow with Vogel moel viscosity in an annula pipe, Int. J. of Nonlinea Mech., ), [3] Yuusoy M. an Pakemili M., Appoximate analytical solution fo the flow of a thi-gae flui in a pipe, Int. J. of Nonlinea Mech., ),