In this article, irreversibility analysis of thermal

radiation with slip condition on MHD Poiseuille flow through

porous medium is investigated. The upper and lower walls are

kept constant with the same temperature. The radiative heat

flux in the energy equation is assumed to follow Roseland

approximation. Semi-analytical solutions of the non-linear

boundary value problems obtained from the governing

equations is constructed using Adomian decomposition method,

and the effects of some fluid parameters on fluid motion,

temperature, entropy generation and Bejan number are

presented

Agboola, O.O.

Okagbue, H. I.

Opanuga, A. A.

Covenant University Repository

  Abstract—In this article, irreversibility analysis of thermal radiation with slip condition on MHD Poiseuille flow through porous medium is investigated. The upper and lower walls are kept constant with the same temperature. The radiative heat flux in the energy equation is assumed to follow Roseland approximation. Semi-analytical solutions of the non-linear boundary value problems obtained from the governing equations is constructed using Adomian decomposition method, and the effects of some fluid parameters on fluid motion, temperature, entropy generation and Bejan number are presented.  Index Terms— Irreversibility, radiation, MHD, Poiseuille flow, slip condition  I. INTRODUCTION ecent researches reveal that more attention has been devoted to the preservation of scarce resources. This has led to the investigation of the causes of irreversibility in various flow systems; some of these are found in Refs. [1-5].  In addition, Arikoglu [6] submitted that, all energy producing, converting and consuming systems must be re-examined carefully and possible available-work destruction mechanisms be removed.   Available research works show that the effect of velocity slip on entropy generation of plane Poiseuille flow has not been fully addressed. Few investigations on this subject are [7-9]. Motivated by [8, 9], this article examines the entropy generation due to thermal radiation and velocity slip on MHD Poiseuille flow through porous medium.  Numerous semi-analytical methods for solving boundary value problems are found in literature, most of these techniques have difficulties in relation to the size of computational work and convergence. However the technique of Adomian Decomposition Method (ADM) [10-, 12] applied in this article is easy to apply with high accuracy and rapid convergence.  Manuscript received February, 13, 2017; revised March 10, 2017. This work was supported by the Centre for Research and Innovation, Covenant University, Ota, Nigeria.  A. A. Opanuga, H.I. Okagbue, and O.O. Agboola are with the Department of Mathematics,Covenant University, Ota, Nigeria.(e-mail: abiodun.opanuga@covenantuniversity.edu.ng, hilary.okagbue@covenantuniversity.edu.ng, ola.agboola@covenantuniversity.edu.ng). II.  MATHEMATICAL FORMULATION The assumptions made include: The flow is steady, electrically conducting and incompressible; the fluid is viscous and flow through parallel porous medium; both plates are fixed and maintained at uniform temperature; uniform transverse magnetic field 0B  is applied neglecting the induced magnetic field and the Hall effect; Navier slip boundary condition is assumed at the fluid-solid interface; the fluid is optically thick following Roseland approximation. The governing equations are given as [8, 9]                                2202B ud u bu dpd K d                                               (1)                                                   2 2 22 2020rB ud T du bukd d Kdqd                              (2)                                     2 2 2020 0 020GB uk dT duET d T d TuT K                              (3)                                               1 2(0) (0)(0) , ( )du duu u hd d     ;0(0) , ( ) hT T T h T                                                        (4) The Roseland approximation term for optimally thick fluid is written as  4443cr cdTqk d                                                                (5)                              The temperature term 4( )T   in equation (5) can be expressed in term of its linearity function as given by Raptis et al. [13], then the expansion in Taylor series about 0T  gives        24 4 3 20 0 0 03 40 0 04 64T T T T T T T TT T T T T                             (6) Irreversibility Analysis of a Radiative MHD Poiseuille Flow through Porous Medium with Slip Condition A. A. Opanuga* Member, IAENG, H.I. Okagbue, O.O. Agboola  R Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.ISBN: 978-988-14047-4-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2017 Using equations (5) and (6) in equation (2) and neglecting higher order terms, we obtain                                      2 2 22 2023 202163ccB ud T du ukd d KT d Tk d                                (7)          The dimensionless expressions for the present problem are: 2002 2020 02 2 30 2 0 0021 21 2, , , ,Pr , , ,( ) ( )4, , ,, ,fp Gf fcfcT Tu h dpu Ah U T T dxv c T h EUBr Nsk k T T k T TkT T B h TH RT kkhK h h                         (8)         Applying the above dimensionless variables in equations   (1, 3, 4, 7) yields 22 220d uH u u Ad                                               (9)                                                                                                          222 222 24130d duR BrH ud dBr u                                (10)                                                                              2222 2 2 2413dud BrNs R ddH u u                            (12)                                                              1 2(0) (0)(0) , (1) ;(0) 0, (1) 1du duu ud d                               (13)                                      Solving equations (9-10) by ADM yields the solution of the boundary value problems. III. ENTROPY GENERATION The dimensionless entropy generation expression in equation (11) provides four sources of irreversibility, that is equation (11) is of the form; 22413dHTI TRI Rd      heat transfer and thermal radiation irreversibility; 2Br duVDId     viscous dissipation irreversibility; 2 2BrH uMFI  magnetic field irreversibility and  2 2Br uPI porosity irreversibility. The Bejan number assumes values between 0 and 1. 0Be  for  VDI , 1Be  for  HTI  and 0.5Be   is when both VDI  and HTI  contribute equally to entropy generation. Then setting 1 211,1N NBeNs N                                          (14)                                                                            where  21 222 2 2 2241 ,     3dN RdBr duN H u ud                              (15)  IV. RESULTS AND DISCUSION  In this article, the effect of Navier slip and thermal radiation are investigated on the entropy generation of MHD Poiseuille flow through porous medium. The effects of some parameters on fluid velocity, temperature, entropy generation and Bejan number are presented in this section.  Figs. 1 and 2 depict the effect of slip parameters on fluid velocity. It is observed from Fig. 1 that fluid velocity increases with increase in lower wall slip parameter while the situation is reversed with upper wall slip parameter in Fig. 2. In Fig. 3, we present the effect of radiation parameter on the temperature. It is obvious that fluid temperature is lowered with increased values of radiation parameter. This is caused by the absorption of heat emitted by the absorptivity parameter. Figs. 4 and 5 reveal that fluid temperature is enhanced by increase in slip parameter.   Furthermore, Figs. 6 and 7 depict that entropy generation is retarded at the lower wall while it is enhanced at the upper wall. Also, Fig. 8 is the plot of thermal radiation effect on entropy generation. The Figure shows that entropy generation is significantly increased with increase in radiation parameter (R).    Finally, Figs. 9 and 10 show similar results. In the plots the Bejan number increases at the lower wall while there a reduction in the middle and upper walls of the channel. This is an indication that heat transfer irreversibility dominates entropy generation at the lower wall while viscous dissipation irreversibility is the major contributor to irreversibility at the upper wall. In Fig. 11 a rise in thermal radiation parameter leads to an increase in Bejan number across the channel. This shows that heat transfer irreversibility is the dominant contributor to entropy generation.  ACKNOWLEDGMENT The authors sincerely appreciate Covenant University for the financial support, and the constructive comments of the reviewers.  Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.ISBN: 978-988-14047-4-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2017  Fig 1: Velocity versus lower wall slip parameter              Fig 2: Velocity versus upper wall slip parameter  Fig 3: Temperature versus thermal radiation parameter        Fig 4: Temperature versus lower wall slip parameter     Fig 5: Temperature versus upper wall slip parameter  Fig 6: Entropy generation versus lower wall slip parameter  Fig 7: Entropy generation versus upper wall  slip  parameter   Fig 8: Entropy generation versus Thermal radiation     parameter   Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.ISBN: 978-988-14047-4-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2017  Fig 9: Bejan number versus lower wall parameter  Fig 10: Bejan number versus upper wall parameter   Fig 11: Bejan number versus thermal radiation parameter      REFERENCES [1] S.O. Adesanya and O. D. Makinde, Effects of couple stress on entropy generation rate in a porous channel with convective heating, Comp. Appl. Math.,34(2015):293–307 [2] A.O. Ajibade, B.K. Jha and A. Omame, Entropy generation under the effect of suction/injection, Applied Mathematical Modelling, 35(2011):4630-4646 [3] A.A. Opanuga, J.A. Gbadeyan, S.A. Iyase and H.I. Okagbue, Effect of Thermal Radiation on the Entropy Generation of Hydromagnetic Flow Through Porous Channel, The Pacific Journal of Science and Technology, 17( 2)(2016):59-68  [4] S. O. Adesanya, S. O. Kareem, J. A. Falade and S. A. Arekete, Entropy generation analysis for a reactive couple stress fluid flow through a channel saturated with porous material, Energy, 93(2015): 1239-1245 [5] S. O. Adesanya, Second law analysis for third-grade fluid with variable properties, Journal of Thermodynamics Volume 2014 (2014), 8pages http://dx.doi.org/10.1155/2014/452168 [6] A. Arikoglu, I. Ozkol and G. Komurgoz, effect of slip on entropy generation in a single rotating disk in MHD flow, Applied Energy 85 (2008): 1225–1236 [7] T. Hayat, S. Hina and N. Ali, Simultaneous effects of slip and heat transfer on the peristaltic flow, Commun Nonlinear Sci Numer Simulat 15 (2010):1526–1537 [8] S.O. Adesanya and O. D. Makinde, Entropy generation in couple stress fluid flow through porous channel with fluid slippage, International Journal of Exergy, 15(3)(2014):344 – 362. [9] A.S. Eegunjobi and O.D. Makinde , Effects of Navier slip on entropy generation in a porous channel with suction/injection, Journal of Thermal Science and Technology, 7(4)(2012): 522-535 [10] A.A. Opanuga, O.O. Agboola and H.I. Okagbue, “Approximate solution of multipoint boundary value problems”, Journal of Engineering and Applied Sciences, vol. 10, no 4, pp. 85-89, 2015. [11] A.A. Opanuga, O.O. Agboola , H.I. Okagbue and J.G. Oghonyon (2015a), “Solution of differential equations by three semi-analytical techniques”, International Journal of Applied Engineering Research, vol. 10, no 18, pp. 39168-39174, 2015. [12] A.A. Opanuga, H.I. Okagbue, E.A. Owoloko, and O.O. Agboola,“Modified Adomian decomposition method for thirteenth order boundary value problems”, Gazi University Journal of Science, (in press) [13] A. Raptis, Perdikis, H.S. Takhar, “Effect of thermal radiation on MHD flow”, Applied Mathematics and Computation, vol. 153: pp. 645–649      u      axial velocity       dynamic viscosity p     fluid pressure h     channel width      fluid density T      fluid temperature 0T     initial fluid temperature fT    final fluid temperature k     thermal conductivity of the fluid pc    specific heat at constant pressure     electrical conductivity of the fluid Navier slip       coefficients 0B     uniform transverse magnetic field rq      radiative heat flux u    dimensionless velocity     dimensionless temperature Pr    Prandtl number Br    Brinkman number       parameter that measures the temperature          difference between the two heat reservoirs b       empirical constant in the second order (porous        inertia resistance) Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.ISBN: 978-988-14047-4-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2017 H    magnetic field parameter Be    Bejan number A    axial pressure gradient R    thermal radiation parameter K    porous media permeability 1,2    Navier slip parameters respectively      porous media shape parameter GE    local volumetric entropy generation rate Ns    dimensionless entropy generation rate      is the kinematic viscosity c    Stefan-Boltzman constant ck     mean absorption coefficient for thermal radiation.                     Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K.ISBN: 978-988-14047-4-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2017

Irreversibility Analysis of a Radiative MHD

Poiseuille Flow through Porous Medium with

Slip Condition

Irreversibility Analysis of a Radiative MHD Poiseuille Flow through Porous Medium with Slip Condition

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Irreversibility Analysis of a Radiative MHD Poiseuille Flow through Porous Medium with Slip Condition

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