Science World Journal Vol 12(No 4) 2017 www.scienceworldjournal.org ISSN 1597-6343

EFFECT OF BRINKMAN NUMBER AND MAGNETIC FIELD ON LAMINAR IN A VERTICAL PLATE CHANNEL

1* 2 3 Joseph Kpop Moses ; Peter Ayuba and Abubakar S. Magaji Full Length Research Article

1, 2, 3 Department of Mathematical Sciences, Kaduna State University – Nigeria.

*Corresponding Author’s e-mail address: [email protected]

ABSTRACT Gav et al (1992) showed experimentally for mixed convection that The effect of Brinkman number and magnetic field on laminar the increasing buoyancy parameter will make the reversal flow convection in a vertical plate channel with uniform and wider and deeper taking into consideration the effects of viscous asymmetric temperatures has been studied. The dimensionless dissipation. Barletta (1998) demonstrated that in the case of form of momentum and energy balanced equations has been upward flow reversal decreases with viscous dissipation. solved using one term perturbation series solution. The solution of However, in the case of downward flow, the flow reversal the dimensionless velocity u and temperature θ were obtained. increases with viscous dissipation. He also calculated criteria for The numerical values of the skin friction and the the onset of flow reversal for both directions of flow, but without were obtained and tabulated. It was found that the effect of taking viscous dissipation into account. magnetic parameter enhances the effect of flow velocity by Barletta (1999) investigated the fully developed and laminar suppressing the turbulence while Brinkman number has convection in a parallel – plate channel by taking into account accelerating effect on the temperature profile. both viscous dissipation and buoyancy. Uniform and asymmetric temperatures are prescribed at the channel walls. The velocity Keywords: MHD, Laminar, Convection, vertical channel, field is considered as parallel. A perturbation method is employed Brinkman number to solve the momentum balance equation and energy balance equation. A comparison between the velocity and temperature NOMENCLATURES profiles in the case of laminar forced convection with viscous ν - Kinematic viscosity dissipation is performed in order to point out the effect of 훼 - Thermal conductivity buoyancy. The case of convective boundary conditions is also 휇- Dynamic viscosity discussed. 푇푏 - Bulk temperature Daniel et.al. (2010) investigated the fully developed and laminar 푈푚 − Mean velocity convection in a vertical parallel plate channel by taking into ∆푇 - Temperature difference account both viscous dissipation and buoyancy. Uniform and 푀 표푟 퐻푎 − magnetic parameter asymmetric temperatures are prescribed at the channel walls. 퐺푟 - The velocity field is considered as parallel. A perturbation method 푅푒 - is employed to solve the momentum balance equation and the 𝑔 - Acceleration due to gravity, energy balance equation. A comparison with the velocity and 퐵푟 - Brinkman number temperature profiles in the case of laminar forced convection with various dissipation is performed in order to point out the effect the INTRODUCTION effect of buoyancy. The case of convective boundary condition is The study of magnetohydrodynamic (MHD) flows with viscous also discussed. dissipation and buoyancy effects has many importance in Joseph et.al. (2013) studied the MHD forced convection in a industrial, technological and geothermal applications such as horizontal double – passage with uniform wall heat flux by taking temperature plasmas, cooling of nuclear reactors, liquid metals into account the effect of magnetic parameter. They assumed the etc. A comprehensive review of the study of MHD flows in flow of the fluid to be laminar, two – dimensional, steady and fully relations to the applications to the above areas has been made by developed. The fluid is incompressible and the physical properties several authors. are constants. Dileep and Rashin (2012) investigated the fully Mixed convection in a vertical parallel – plate channel with developed magntohydrodynamic mixed convection in a vertical uniform wall temperatures has been studied by Tao (1960). channel, partially filled with clear fluid and partially, filled with a Javerci (1975) dealt with the effects of viscous dissipation and fluid – saturated porous medium is by taking into account viscous Joule heating on the fully developed MHD flow with heat transfer and ohmic dissipations. Alim, et al (2007) investigated the in a channel. The exact solution of the energy equation was pressure work and viscous dissipation effects on MHD natural derived for constant heat flux with small magnetic Reynolds convection along a sphere. The laminar natural convection flow number. Sparrow et al (1984) observed experimentally flow from a sphere immersed in a viscous incompressible fluid in the reversal in pure convection in a one – sided heated vertical presence of magnetic field has been considered in this channel. The mixed convection in a vertical parallel – plate investigation. channel with asymmetric heating, where one plate is heated and Barik (2013) analyzed the free convection heat and mass transfer the other is adiabatic has been studied by Habchi and Acharya MHD flow in a vertical channel in the presence of chemical (1986). reaction. Ranuka et al (2009) studied the MHD effects of unsteady heat convective mass transfer flow past an infinite vertical porous plate with variable suction, where the plate’s Effect of Brinkman Number and Magnetic Field an Laminar Convection in a 58 Vertical Plate Channel

Science World Journal Vol 12(No 4) 2017 www.scienceworldjournal.org ISSN 1597-6343

2 2 2 2 temperature oscillates with the same frequency as that of variable 푑 푇 ν 푑푈 𝜎푒퐵0 푈 훼 2 + ( ) + = 0 (2) suction velocity with soret effects. The governing equations are 푑푌 푐푝 푑푌 𝜌0푐푝 solved numerically by using implicit finite difference method. Where the acceleration due to gravity is g, ν is the kinematic 휇 Saleh and Hashim (2009) analysed flow reversal phenomena of viscosity which is defined by ν= , 푃 = 푝 + 휌푏𝑔푋 is the the fully – developed laminar combined free and forced MHD 𝜌푏 difference between the pressure p and the hydrostatic convection in a vertical plate – channel. The effect of viscous pressure P. 훼 is thermal conductivity and 휇 = 휌 휈 is the dissipation is taken into account. Tamia and Samad (2010) 푏 dynamic viscosity studied and analyzed the radiation and viscous dissipation effects Equations (2) and the boundary condition on푇, shows that 푃 on a steady two – dimensional magneto hydrodynamics free 휕푃 depends on 푋; 푇 depends only on 푌; 푇 is a constant; is convection flow along a stretching sheet with heat generation. 푏 휕푋 The non – linear partial differential equations governing the flow a constant. field under consideration have been transformed by a similarity The bulk temperature 푇푏 and the mean velocity 푈푚 (Barletta, transformation into a system of non – linear ordinary differential 1999) are given by. equations and then solved numerically by applying Nachtsheim – 퐿 퐿 Swigert shooting iteration technique together with sixth order 1 and 1 (3) 푇푏 = ∫0 푈푇푑푌 푈푚 = ∫0 푈푑푌 Runge – Kutta integration scheme. Resulting in non – 퐿푈푚 퐿 dimensional velocity, temperature and concentration profiles are The boundary conditions are: then presented graphically for different values of the parameters 푈(0) = 0 , 푇(0) = 푇0, of physical engineering interest. 푈(퐿) = 0 , 푇(퐿) = 푇0 (4) Idowu et.al (2013) investigated viscous dissipation and buoyancy Due to the symmetry of the problem, 푈 푎푛푑 푇 are symmetric effects on laminar convection with transpiration. Uniform and functions of 푌, so that they must fulfill the condition asymmetric temperatures are prescribed at the channel walls. 푈′(0) = 0, 푇′(0) = 0 (5) The velocity field is considered as parallel. A perturbation method Introducing the dimensionless quantities below 푈 푇−푇 푌 −퐿2 푑푃 is employed to solve the momentum balance equation and the 푢 = , 휃 = 푏 , 푦 = , 휆 = , 퐺푟 = 푈푚 ∆푇 퐿 휇푈푚 푑푋 energy balance equation. A comparison with the velocity and 3 64퐿 𝑔훽∆푇 , temperature profiles in the case of laminar forced convection with 휈2 various dissipation is performed in order to point out the effect of 4퐿푈 퐺푟 휇푈2 푅푒 = 푚 , 퐸 = , 퐵푟 = 푚 buoyancy. The case of convective boundary condition is also 휈 푅푒 훼∆푇 discussed. Where the temperature difference ∆푇 is given by 2 Das et.al. (2015) studied the fully developed mixed convection 휇푈푚 ∆푇 = flow in a vertical channel filled with nanofluid in the presences of 푘 uniform transverse magnetic field. Anjali Devi and Ganga (2010) While the Grashof number 퐺푟 is always positive, the Reynolds studied the Viscous and Joule dissipation effects on MHD non – number 푅푒 and the parameter 퐸 can be either positive or linear flow and heat transfer past a stretching porous surface negative. In particular, in the case of upward flow (푈푚 > embedded in a porous medium under a transverse magnetic field. 0), both 푅푒 and 퐸 are positive, while, for downward flow(푈푚 < Patrulescu, et al (2010) investigated the steady mixed convection 0), these dimensionless parameters are negative. flow in a vertical channel for laminar and fully developed flow regime. In the modeling of heat transfer the viscous dissipation Therefore, as a consequence of the dimensional quantities term was also considered. Temperature on the right wall is together with the momentum balance equation (1), the energy assumed constant while a mixed boundary condition (Robin balance equation (2) and equations (3) – (5) yields; boundary condition) is considered on the left wall. 푑2푢 퐸휃 The mixed convection of Newtonian fluid between vertical parallel − 푀푢 = − + 휆 plates channel with MHD effect and variation in Brinkman number 푑푦2 16 2 2 has been studied by Alizadeh Rasul et.al. (2014). The effect of 푑 휃 2 푑푢 2 + 퐵푟푀푢 = −퐵푟 ( ) (6) MHD and Brinkman number on laminar mixed convection of 푑푦 푑푦 𝜎 훽2퐿2 Newtonian fluid between vertical parallel plates channel was Where 푀 = 0 0 which is the magnetic parameter and 𝜌0휈 investigated by Salehi et.al (2013). The boundaries are 1 퐵푟 = considered to be isothermal with equal temperatures. 𝜌푏푐푝 This work presents a solution of the problem of laminar MHD 푢(1) = 0, 푢′(0) = 0, 휃(1) = −휑, 휃′(1) = 1 (7) 1 1 convection in a vertical channel with uniform and symmetric ∫ 푢휃푑푦 = 0, ∫ 푢푑푦 = 1 (8) temperatures. The solution is found by using one term 0 0

perturbation series solution. From equations (6) to (8), the functions 푢(푦) 푎푛푑 휃(푦) can be

determined as well as the constants 흀 and 푅, if the value of MATERIALS AND METHODS dimensionless parameter E is given. In particular, it is easily The problem of laminar and fully – developed flow of a Newtonian verified that the choice 퐸 = 0 correspond to the absence of fluid in a vertical parallel – plate with effect of magnetic field and buoyancy forces, i.e., to forced convection. viscous dissipation and buoyancy effects were considered.

The momentum and energy balance equations (Saleh and Perturbation Series Solution Hashim (2009)) are; 2 2 The solutions of equations (6) to (8) are obtained by the 1 휕푃 푑 푈 𝜎푒퐵0 푈 𝑔훽(푇 − 푇푏) − + ν 2 − = 0 (1) perturbation method. 𝜌푏 휕푋 푑푌 𝜌푏 Effect of Brinkman Number and Magnetic Field an Laminar Convection in a 59 Vertical Plate Channel

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Let us expand the functions ( ) and the constants γ and √푀 푢 푦 , 휃(푦) 푢 (푦) = [푐표푠ℎ√푀푦 + 푐표푠ℎ√푀], 휆 = 휑 as a power series in the parameter 퐸 , namely 0 푠𝑖푛ℎ√푀−푀푐표푠ℎ√푀 0 2 −푀√푀푐표푠ℎ√푀 푢(푦) = 푢0(푦) + 푢1(푦)퐸 + 푢2(푦)퐸 + ⋯ = (17) 푠𝑖푛ℎ√푀−푀푐표푠ℎ√푀  n ( y) (9) un E By substituting equation (17) in equation (14), one obtains the n0 function 휃0(푦) and the constant 휑0, which are given by

휃(푦) = 휃 (푦) + 휃 (푦)퐸 + 휃 (푦)퐸2 + ⋯ = 2 0 1 2 휃0(푦) = −퐵푟퐴 {4푐표푠ℎ2√푀푦 + 16푐표푠ℎ(푦 + 1)√푀 +  2 2 n 16 cosh(푦 − 1) √푀 + 4푀푦 푐표푠ℎ2√푀 + 4푀푦 } + 퐵 + (y) (10) 3 3  n E [1 + 퐵푟퐴 퐶 + 퐵퐷]퐸 , 휑0 = [−1 − 퐵푟퐴 퐶 + 퐵퐷]퐸 (18) n0 The Skin Friction  The wall shear stress i.e. the skin friction at the wall is given by 2 n γ = 훾 + 훾 퐸 + 훾 퐸 + ⋯ = 푑푢 √푀푠𝑖푛ℎ√푀 0 1 2  n E 휏 = ( ) = (19) n0 푑푦 푦=0 푠𝑖푛ℎ√푀−푀푐표푠ℎ√푀 (11)  Nusselt Number n 휑 = 휑 + 휑 퐸 + 휑 퐸2 + ⋯ = The rate of heat Transfer i.e. the heat flux at the wall in terms of 0 1 2 Rn E n0 Nusselt number is given by 푑휃 (12) 푁푢 = ( ) = −32퐵푟퐴2 푠𝑖푛ℎ√푀 (20) 푑푦 푦=0 These power series expansions are substituted in equations (6) to (8). Thus, the original boundary value problem is mapped into a Where; √푀 sequence of boundary value problems which can be solved in 퐴 = ; 퐵 = 퐵푟퐴2[20푐표푠ℎ2√푀 + 2푠𝑖푛ℎ 푀−푀푐표푠ℎ 푀 succession, in order to obtain the coefficients of the power series √ √ 4푀푐표푠ℎ2√푀 + 4푀 + 16]; expansion in equations (9) to (12). A detailed analysis on the 106 38 퐶 = 푠𝑖푛ℎ3√푀 + 푠𝑖푛ℎ√푀 + 8푐표푠ℎ√푀 − application of perturbation methods in heat transfer problems can 3√푀 √푀 4푀 4푀 be found 8푐표푠ℎ3√푀 + 푠𝑖푛ℎ√푀 + 푐표푠ℎ3√푀 + 4푐표푠ℎ√푀 + √푀 3 2푀 8 8 푒푥푝(3√푀) − 4푀푒푥푝(−3√푀) + 푒푥푝(−√푀) − ; The equation which corresponds to 푛 = 0 in the series is the √푀 √푀 √푀 following: 2 1 2 퐷 = 푠𝑖푛ℎ√푀 + 2푐표푠ℎ√푀 ; 퐸 = 2 푑 푢0 ′ √푀 푠𝑖푛ℎ√푀+2푐표푠ℎ√푀 − 푀푢0(푦) = −휆0 , 푢0(0) = 0 , (13) √푀 푑푦2 2 2 푑 휃0 2 푑푢0 DISCUSSION OF RESULTS 2 + 퐵푟푀[푢0(푦)] = −퐵푟 ( ) , (14) 푑푦 푑푦 Here, 1 – term perturbation series is employed to evaluate the Subject to 1 dimensionless velocity profile 푢 and temperature profile 휃 since ( ) 푢0 1 = 0 , ∫0 푢0푑푦 = 1 the other terms resulted in a trivial solutions. The resulted 1 ′ ( ) ( ) equations were solved with MATLAB and the results were plotted 휃0 0 = 0 , 휃0 1 = −휑0 , ∫0 푢0휃0푑푦 = 0 graphically. The governing equation that corresponds to 푛 = 1 is given The effect of Brinkman number and magnetic field in a vertical as: channel, the velocity profile and the temperature profile are depicted graphically against 푦 for different values of the magnetic 푑2푢 parameter 푀 and Brinkman number 퐵푟 respectively. The skin 1 − 푀푢 (푦) = −휆 , (15) 푑푦2 1 1 friction 휏 and the Nusselt number 푁푢 were also determined numerically for different values of magnetic and Brinkman number 푑2휃 푑푢 푑푢 1 + 2퐵푟푀[푢 (푦)푢 (푦)] = −퐵푟 0 1 , (16) respectively. 푑푦2 0 1 푑푦 푑푦 Figure 1 demonstrates the variation of velocity 푢 for different 1 values of the magnetic parameter 푀. It is observed from figure 1 ′ 푢1(0) = 0 , 푢1(1) = 0 , ∫ 푢1푑푦 = 0 that the velocity decreases with increase in the magnetic 0 parameter. This means that the magnetic parameter suppressed 1 the turbulence of the flow. ′ 휃1(0) = 0 , 휃1(1) = −휑1 , ∫(푢0휃1 + 푢1휃0) = 0 Figure 2 depicts the effect magnetic parameter 푀 on 0 temperature profile 휃. It is clear that the temperature rises with Equations (13) and (14) are easily solved, because the function increase in the Brinkman number 퐵푟. The Brinkman number is a 휃0(푦) does not affect the function 푢0(푦). The latter, together dimensionless number that show the ratio of heat produced by with the constant휆0, is determined by solving equation (13), viscous dissipation and heat transported by moleculer conduction. namely The higher its value, the slower the conduction of heat produced and hence the larger the temperature rise. This is applicable to

Effect of Brinkman Number and Magnetic Field an Laminar Convection in a 60 Vertical Plate Channel

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screw extruder, where the energy supplied to the polymer melt written in a dimensionless form. The solution of the dimensionless comes from the viscous heat generated by shear between velocity 푢 and the dimensionless temperature 휃 has been elements of the flowing liquid moving at different velocities and determined. direct heat conduction from the wall of the extruder. A perturbation method has been employed to evaluate the Tables 1 and 2 show the variation of magnetic parameter and dimensionless velocity 푢 and temperature 휃. The Skin Friction Brinkman number on skin friction and Nusselt number and the Nusselt number were also determined. respectively. It is observed from table 1 that the skin friction It can be concluded that been the effect of magnetic parameter decreases with increase in magnetic material. While table 2 enhances the effect of flow velocity by suppressing the turbulence shows that increase in Brinkman number bring rise to Nusselt while Brinkman number has accelerating effect on the number. temperature profile. The results of this work is in agreement to Daniel et.al. (2010) and This study has potential application in screw extruder. Idowu et.al. (2013). The difference is the buoyancy and transpiration term. REFERENCES Alim, M. A, Alam, Md. M and Chowedburry, Md. M. K., (2007): Pressure Work and Viscous Dissipation Effects on MHD Natural Convection Flow along a Sphere, Proceedings of the International Conference of Mechanical Engineering (ICME). Anjali Devi, S. P and Ganga. B., (2010): Dissipation Effects on MHD Non – Linear Flow and Heat Transfer Past a Porous Surface with Prescribed Heat Flux, J. Appl. Fluid Mechanics, 3, (1), 1 – 6. Barik R.N., (2013): Free Convection Heat and Mass Transfer MHD Flow in a Vertical Channel in the Presence of Chemical Reaction. Int. J. Analysis and Application, 2 (2), Figure 1: Effect of 푀표푟 퐻푎 on Velocity profile 151 – 181. Barletta A., (1999): Laminar Convection in a Vertcal Channel with Viscous Dissipation and Buoyancy Effects,Int. Comm. Heat Mass Transfer, 26, (2), 153 – 164. Barletta A., (1998): Effect of Viscous Dissipation in Laminar Forced Convection within Ducts Int. J. Heat Mass Transfer, 41, 3501. Das S., Jana R.N. and Makinde O.D. (2015): Mixed Convective Magnetohydrodynamic Flow in a Vertical Channel filled with Nanofluids.Int. j. Eng. Science and Tech. 18 (2), 244 – 255. Dileep Singh Chauchan and Rashim Agrawal (2012): Magnetohydrodynamic Convection Effects with Viscous and Ohmic Dissipation in a Vertical Channel Partially Figure 2. Effect of 퐵푟 on temperature profile filled by a Porous Medium. Journ. App. Science and Eng. 15 (1), 1 – 10. Table 1: Variation of Skin Friction 휏 with different values of Ebrahim Salehi, Rasul Alizadeh and Alireza Darvish (2013); The Magnetic parameter 푀 Effect of MHD and Brinkman number on Laminar mixed Magnetic 푀 = 1 푀 = 3 푀 = 5 푀 = 7 convection of Newtonian Fluid between Vertical Parallel Parameter Plates Channel, Proceedings of the 2013 International Skin −3.1945 −0.7895 −0.5433 −0.4358 Conference on Mechanics, Fluids, Heat, Elasticity and Friction 휏 Electromagnetic Fields. Gau C, Yih K.A and Aung W., (1992): Flow Reversal in Pure Convection in a One – Sided Heated Vertical Channel Table 2: Variation of Nusselt number 푁푢 with different values of with Increasing Buoyancy Parameter. ASME J. Heat Brinkman number 퐵푟 Transfer, 14, 928. Brinkman 퐵푟 = 1 퐵푟 = 3 퐵푟 = 5 퐵푟 = 7 Habchi S. and Acharya S., (1986): Mixed Convection in a Vertical number Parallel – Plate Channel with Asymmetric Heating. Numer. Heat Transfer, 9, 485. Nusselt Number 0.2224 0.3335 0.4447 0.5559 Idowu A.S, Joseph K.M, Onwubuoya C and Joseph W.D (2013): 푁푢 Viscous Dissipation and Buoyancy Effects on Laminar Convection in a Vertical Channel with Transpiration. Int. Summary and Conclusion J. Applied Math. Research, @ (4), 455 – 463. The effect of Brinkman number and Magnetic field on laminar Javeri, V. B., (1975): Combined Influence of Hall Effect, Ion Slip, convection in a vertical plates channel with uniform and Viscous Dissipation and Joule Heating on MHD Heat asymmetric temperatures has been studied. The governing Transfer in a Channel. Warme and Stoffubertraging.8, equations (momentum and energy balanced equations) has been 193 – 202. Effect of Brinkman Number and Magnetic Field an Laminar Convection in a 61 Vertical Plate Channel

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Effect of Brinkman Number and Magnetic Field an Laminar Convection in a 62 Vertical Plate Channel