American Journal of Applied Mathematics 18; 6(): 34-41 http://.sciencepublishinggroup.com/j/ajam doi: 1.11648/j.ajam.186.1 ISSN: 33-43 (Print); ISSN: 33-6X (Online) Numerical Analysis of Non-Uniform Heat Source/Sink in a Radiative Micropolar Variable Electric Conductivity Fluid ith Dissipation Joule Heating Rasaq Adekunle Kareem 1, Sulyman Olakunle Salau, Jacob Abiodun Gbadeyan 3 1 Department of Mathematics, Lagos State Polytechnic, Ikorodu, Nigeria Department of Mathematics, Landmark University, Omu-aran, Nigeria 3 Department of Mathematics, University of Ilorin, Ilorin, Nigeria Email address: To cite this article: Rasaq Adekunle Kareem, Sulyman Olakunle Salau, Jacob Abiodun Gbadeyan. Numerical Analysis of Non-Uniform Heat Source/Sink in a Radiative Micropolar Variable Electric Conductivity Fluid ith Dissipation Joule Heating. American Journal of Applied Mathematics. Vol. 6, No., 18, pp. 34-41. doi: 1.11648/j.ajam.186.1 Received: February 1, 18; Accepted: March 6, 18; Published: March 6, 18 Abstract: Computational analysis of radiative heat transfer of micropolar variable electric conductivity fluid ith a noneven heat source/sink and dissipative joule heating have been carried out in this article. The flo past an inclined plate ith an unvarying heat flux is considered. The transformed equations of the flo model are solved by the Runge-Kutta scheme coupled ith shooting method to depict the dimensionless temperature, microrotation and velocity at the boundary layer. The results sho that the coefficient of the skin friction and the temperature gradient at the all increases for regular electric conductivity and non-uniform heat sink/source. Keyords: Radiation, Dissipation, Hydromagnetic, Joule Heating, Micropolar Fluid 1. Introduction A micropolar fluid comprises of gyratory micro-segments that cause fluid to sho non-netonian activities. The inventions of micropolar fluid flo as discovered useful in exploratory liquid crystals, colloidal suspensions, body fluids, lubricants, turbulent flo of shear, fluid polymeric, preservative suspensions, flos in vessels and microchannels. The theory of micropolar fluids pioneered by [5] has been an energetic research area for long time till present days. The phenomenons of the model and the utilizations of micropolar fluid as examined by [], hile [9] did tentatively investigation to demonstrate that the present of minute stabilizers polymeric in fluids can diminish the flo impact at the all up to 5 to 3 percent. The decrease that as depicted by the theory of micropolar substances as announced in [14], the cerebrum fluid that is a case of body fluids can be adequately detailed as micropolar fluids. Convective flo fluids containing microstructure have many applications, for example, eaken polymer fluids substances, various types of suspensions and fluid gems. Convective free flo of micropolar fluids past a bended or level surfaces has entranced the psyche of researchers from the time hen the flo model as conceived. Many studies have accounted for and examined results on micropolar fluids. Investigated to the impact of radiation on MHD convective heat and mass transfer flo as analyzed by [3, 7]. Also, [1] considered micropolar boundary layer fluid flo along semiinfinite surface utilizing similarity solution to change the models to ordinary different equations. Moreover, [1] reported on the viscous dissipation impacts on MHD micropolar flo ith ohmic heating, heat generation and chemical reaction. Numerous convective flo are brought on by heat absorption or generation hich might be as a result of the fluid chemical reaction. The occurrence of heat source or sink can influence the fluid heat profile that modifies the rate of deposition of particle in the structures for example, semiconductor afers, electronic chips, atomic reactors and so on. Heat absorption or generation has been thought to be temperature dependent heat generation and surface dependent heat generation. The analyzed the effect of non-homogenous
35 Rasaq Adekunle Kareem et al.: Numerical Analysis of Non-Uniform Heat Source/Sink in a Radiative Micropolar Variable Electric Conductivity Fluid ith Dissipation Joule Heating heat absorption or generation and variable electric conductivity on micropolar fluid as carried out by [15]. It as seen that the plate dependent heat generation is loer contrasted to temperature dependent heat generation. Also, [11, 1] carried out study on the influence of thermaldiffusion and non-even heat source/sink on radiative micropolar MHD fluid past a porous medium. The effect of dissipation on hydromagnetic fluid and heat transfer processes has turned out to be noteorthy in the industry. A lot of engineering practices happen at high temperature ith viscous dissipation heat transfer. Such flos have been explored by [17] ho examined the influence of viscous dissipation on heat and mass transfer of magnetodyrodynamic micropolar fluid ith chemical reaction hile [16] considered hydromagnetic micropolar fluid of heat and mass transport in a porous medium ith expanding plate, chemical reaction, heat flux and variable micro inertia. In [18], investigation into the MHD Micropolar flo fluid ith joule heating, viscous dissipation, constant mass and heat fluxes as carried out. It as noticed from the study that the flo profile rises first ithin η 1 as the microrotation parameter rises. Afterard, the flo gradually decreases for η > 1 as the microrotation parameter rises. Also, microrotation moves from negative to positive in the boundary layer. Considering the referred literature, the aim of the present ork is to study the viscous dissipation of variable electrically conducting micropolar fluid behavior past an inclined plate in permeable media ith thermal radiation, heat fluxes and joule heating for high speed fluid in nonhomogenous heat generation/absorption hich have not been considered by many researchers. The study is necessary because of the industrial application of micropolar fluid. Therefore, it is important to study the flo velocity, temperature and microrotation boundary layer at the surface.. The Flo Mathematical Formulation Convective flo of to-dimensional viscous, micropolar, laminar fluid through a semi-finite plate that is inclined at an angle α to the vertical is considered. The magnetic field varies in strength as a function of x that is assumed to be in y-direction and defined as B = (, B( x )). The Reynolds number is minuet hile the outer electric field is assumed as zero. Accordingly, the applied external magnetic field is high contrasted to the stimulated magnetic field. The density ( ρ ) of the fluid is inert ( U = ) ith the buoyancy forces causing the convective motion. The fluid viscosity µ is considered to be unvarying hile the body forces and the pressure gradient are ignored. Rosseland diffusion approximation for radiation is adopted for the flo Considering the assumptions above, the convective micropolar fluid taking after the Boussinesq approximation may be described by the geometry and subsequent equations. Figure 1. The flo geometry. u v + = x u v µ + r u r σ ( B( x)) ν u + v = + + g β ( T T ) cosα u u x ρ ρ ρ K (1) ()
American Journal of Applied Mathematics 18; 6(): 34-41 36 b r u u + v = + x ρ j ρ j r ' µ σ ( ( )) p p p p T T k T q + r u B x q u + v = + + u + x ρc ρc ρc ρc (3) (4) ith the boundary conditions u T q u =, v =, = a, = m at y = k u = U =, =, T = T as y here u and v are the fluid velocity in x and y coordinates respectively, K is the permeability of the porous medium, ν = µ is the kinematic viscosity, µ is the dynamic viscosity, ρ ρ is the fluid density, is the microrotaion in x and y r components, b = ( µ + ) j is the micropolar viscosity, j is the micro-inertial per unit mass hich is assumed to be constant, r is the microrotation coefficient, k is the thermal conductivity, T is the fluid temperature, c p is the specific heat at constant pressure, g is the acceleration due to gravity and β is the thermal expansion coefficient. u A linear correlation involving the surface shear and microrotation function is picked for studying the influence of varying surface circumstances for microroation. Note that the microroation term a = implies =, that is the microelement at the all are not siveling but hile a =.5 implies varnishing of the anti-symmetric module of the stress tensor that stand for feeble concentration. This confirms that for a fine particle suspension at the all, the particle sivel is the same as the fluid velocity but a = 1 represents the turbulent boundary layer flos. Using Rosseland diffusion approximation for radiation [8, 13]. q r 4 4σ T = 3 δ here σ and δ are the Stefan-Boltzmann and the mean absorption coefficient respectively, taken the temperature 4 difference in the flo to be sufficiently small such that T may be consider as a linear function of temperature, introducing 4 Taylor series to expand T around the free stream T and ignore higher order terms, this gives the approximation (5) (6) 4 3 4 4 3 T T T T (7) Using equation (7), equation (6) can be express as. r 3 q 16σ T T = 3δ The non-uniform heat absorption or generation is represented as [11] ku q T T T T e ν x ' η = λ( ) λ ( ) + here λ and λ denote the heat sink/source and space coefficients temperature dependent respectively hile T is the free stream temperature. Here, λ > and (8) (9) λ > stand for heat source but λ < and λ < depict heat sink. In this study, it is assumed that the introduced magnetic field strength B( x ) is capricious and it is represented as ( ) = B B x x, here B is constant. Also, the electrical conductivity σ depends on the fluid velocity and is defined as σ = σ u, here σ is constant see Cortell (7) Using the folloing dimensionless variables; 3 T T U ψ = ν U Ux f ( η ), η = y, ( ) =, = h( ) ν x θ η T T ν x η (1) here ψ is the stream function, U is the reference velocity q T and T = k ν x U hile u = ψ and v = ψ then, x νu u = U f ( η), v = ( f ( η) η f ( η)) (11) x Using equations (8) and (9) along ith the electric conductivity dependent fluid velocity, variable magnetic field and equation (1) in equations (1)-(5) to obtain, (1 + δ ) f + ff + δh + G θcosα Mf φ f = (1) r εh δ ( h + f )) + ε( f h + g h) = (13) 4 3 1 + R θ + (1 + δ ) Pr Ec f + MPr Ec f + Pr ( f θ θ f ) + ( λθ + λ e η ) = 3 (14)
37 Rasaq Adekunle Kareem et al.: Numerical Analysis of Non-Uniform Heat Source/Sink in a Radiative Micropolar Variable Electric Conductivity Fluid ith Dissipation Joule Heating The boundary conditions become f = f =, h = af, θ = 1 at η = f =, h =, θ = as η here δ = r µ is the vortex viscosity term, σ B M = ρ the magnetic field term, (15) = x υ φ is the permeability U K parameter, ε = ju ν x is the micro-inertia density term, 4σ T R = is the radiation parameter, U E = is the c δ k ( T T ) c Eckert number, gxβ ( T T ) Gr = is the thermal Grashof U µ cp number and Pr = is the Prandtl number. k The essential engineering quantities of interest for this flo are the local skin friction C and Nusselt number N given as: f p is u τ q and Therefore, are respectively taken as τ = µ u T, q = k y= y= 1 1 1 1 1 = ( ) f x [ 1 (1 ) δ ] (), x = ( x ) (17) C Re + a f Nu Re (18) θ () The computational values for from equations (18) Table 1. Comparison of values of a. f ' () C f and Nu x are obtained for δ = ε = Gr = α = R = Ec =, various a [4] [6] [15] Present results..67547.67555.67498.63534..766758.766837.76766.766831.5.889477.889544.89366.891784.75.953786.953975.956365.95453 1. 1. 1. 1.15 1.1986 C f τ xq =, = Nu ρu ( ) k T T (16) Table. Values of f ' () and θ ' () for different values of M, E c, λ, λ, φ, and δ on PP-Physical Parameters. PP values ' f () ' θ () PP values ' f () M 1 1.11795 6.33597 R.1 1.1758 5.3674 3 1.15638 7.747.5 1.8877 5.55485 5 1.1996 8.9674.7 1.1531 5.8463 7 1.4187 1.11891 1. 1.395 6.393 E c. 1.1111 5.94768 φ.7 1.114 5.6761.5 1.17319 6.35698.1 1.1117 5.8543.7 1.199 6.6656.3 1.1116 6.893 1. 1.9671 7.19343.5 1.1133 6.378 λ..586.79851 λ. 1.1758 5.3674.1.6853 3.1943.5 1.1111 5.94768.3.89 4.974 1. 1.19538 6.48375.5 1.1111 5.94768 1.5 1.799 6.98646 ' () θ 3. Results and Discussion The computational outcomes for the coupled differential equations are gotten for the dimensionless microrotation, temperature and velocity profiles. In the analysis, the default parameters value are taken as: a =.5, P =.73, G =.5, β =., ε =, R =.1, E =., r r c M =.5, α = 3, δ = 1, λ =.5 and λ =.5. The values depend on the decision of existing articles in vie of inaccessibility of investigation figures for vortex viscosity and micro-inertia density parameters, appropriate values are chosen to confirm the polar impact on flo properties. Table 1 depicts the computational outcomes that demonstrate the action of microrotation parameter a on the fluid flo parts of the present result contrasted ith the existing results. The comparison is observed to be in a superb agreement as appeared in the table. Table represents the computational results, that sho the influence of some fluid properties on the flo and heat part of the investigation. It is noticed from the table that an increase in the values of parameters M, E c, λ, λ and β enhances the skin friction and the temperature gradient at the plate due to an increase in the flo and temperature boundary layer thickness. Additionally, an increase in values of δ causes shrinking in the flo at the plate but enhances the
American Journal of Applied Mathematics 18; 6(): 34-41 38 thermal gradient at the surface. Figures and 3 portray the fluid flo and microrotation profiles for various values of the magnetic field term M. It is seen from Figure that the flo diminishes as the values of M increases because of the influence of Lorentz forces that retard the convective fluid motion. Figure 3, affirmed that as the values of the magnetic field parameter increases the microrotation near the plate ithin the range η 1. increases but reduces as it distance aay from the surface ithin the range 1. η 5. The outcomes demonstrate that the magnetic field might be use to maintain the flo and heat transfer properties. Figures 4 and 5 sho the effect of the vortex viscosity δ on the fluid velocity and microrotation field. It is found that a rise in the values of δ initial abatement the fluid velocity at η 1.6 but increases at range 1.6 η 5 hile microrotation profiles near to the surface at η decreases and later increases as it moves aay from the surface. The reason is because the viscous friction ithin the fluid tends to organize the flo into a collection of irrotational vortices and a moving vortex carries ith it some angular and linear momentum and energy. Figure 4. Velocity fields for different values of δ. Figure. Velocity fields for different values of M. Figure 5. Microrotation fields for different values of δ. Figure 3. Microrotation fields for different values of M. Figures 6-9 represent the microrotation and temperature fields for different values of temperature dependent and
39 Rasaq Adekunle Kareem et al.: Numerical Analysis of Non-Uniform Heat Source/Sink in a Radiative Micropolar Variable Electric Conductivity Fluid ith Dissipation Joule Heating surface temperature heat absorption or generation terms λ and λ. Figure 6 and 8 demonstrated that a rise in the values of λ and λ decreases the microrotation near the surface at η 1.5 and it increases as it far aay from the surface. Also, in Figures 7 and 9 the temperature distribution increases as the parameters λ and λ increases respectively. This is because the transfer of thermal energy across a ell-defined boundary decreases that it in turn increases the amount of heat ithin the system and thereby causes rise the profiles. Figure 8. Microrotation fields for different values of λ. Figure 6. Microrotation fields for different values of λ. Figure 9. Temperature fields for different values of λ. Figure 7. Temperature fields for different values of λ. Figure 1. Velocity fields for different values of φ.
American Journal of Applied Mathematics 18; 6(): 34-41 4 Figure 1 depicts the effect of porosity term φ on the dimensionless velocity. It is evidenced that the flo field decreases as the porosity parameter rises. This is as a result of the all of the plate that gives an additional opposition to the flo mechanism by influencing the fluid to move at a decelerated rate. particles through space or through a material medium out of the system. Figure 1 represent the consequent of variation in the Eckert number E c on the temperature distribution. Eckert number expresses the relationship beteen kinetic energy and enthalpy difference, and is used to characterize heat dissipation. It is noticed that the temperature field increases ith a rise in the values of E c. This is due to increase in the temperature boundary layer thickness that reduces the amount of energy dissipating out of the system and thereby causes a rise in the heat distribution 4. Conclusion Numerical simulation as carried out for dimensionless boundary layer equations of convective heat transfer of a hydromagnetic and micropolar fluid past inclined surface ith non-uniform heat sink/source. It is observed from the study that, the magnetic field decreases the flo rate and angular velocity hile the temperature dependent and surface dependent temperature heat sink or source parameters as ell as viscous dissipation parameter increases angular velocity and temperature distributions. The vortex viscosity parameter decreases the flo and microrotation profiles near the all. Figure 11. Temperature fields for different values of R. References [1] Ahmadi, G. (13). Self-similar solution of incompressible micropolar boundary layer flo over a semi-infinite plate, Int. J. Eng. Sci., 14, 639-646. [] Ariman, T., Turk, M. A., Sylvester, N. D. (1974). Microcontinuum fluid mechanics, a revie. Int. J. Eng. Sci. 1, 73. [3] Bhuvanesari, M. Sivasankaran, S., Kim, Y. J. (1). Exact analysis of radiation convective flo heat and mass transfer over an inclined plate in porous medium, World Appl. Sci. J., 7, 774-778. [4] Cortell, R. (7). Viscous flo and heat transfer over a nonlinearly stretching sheet. Appl. Math. Comput., 184, 864-873. [5] Eringen, A. C. (1966). Theory of micropolar fluids, J. Math. Mech., 16, 1-18. [6] Hayat, T., Abbas, Z., Javed, T. (8). Mixed convection flo of micropolar fluid over a nonlinearly stretching sheet. Phys. Lett. A, 37, 637-647. Figure 1. Temperature fields for different values of E c. The influence of radiation on the temperature profiles is given in figure 11. It is noticed that as the values of R increases, there is corresponding enhancement in the temperature fields hich results in an increase in the thermal boundary layer thickness and thereby decreases the amount of emission or transmission of energy in the form of aves or [7] Hayat, T., Qasim, M. (1), Influence of thermal radiation and Joule heating MHD flo of a Maxell fluid in the presence of thermophoresis, Int. J. Heat Mass Transfer, 53, 478-4788. [8] Helmy, K. A. (1995). MHD boundary layer equations for poer la fluids ith variable electric conductivity. Meccanica, 3, 187-. [9] Hoyt, J. W., Fabula, A. G. (1964). The Effect of additives on fluid friction, US Naval Ordnance Test Station Report.
41 Rasaq Adekunle Kareem et al.: Numerical Analysis of Non-Uniform Heat Source/Sink in a Radiative Micropolar Variable Electric Conductivity Fluid ith Dissipation Joule Heating [1] Khilap, S., Manoj, K. (15). Effect of viscous dissipation on double stratified MHD free convection in micropolar fluid flo in porous media ith chemical reaction, heat generation and ohmic Heating, Chemical and Process Engineering Research, 31, 75-8. [11] Mabood, F., Ibrahim, S. M. (16). Effects of soret and nonuniform heat source on MHD non-darcian convective flo over a stretching sheet in a dissipative micropolar fluid ith radiation, Journal of Applied Fluid Mechanics, 9, 53-513. [1] Mabood, F., Ibrahim, S. M., Rashidi, M. M., Shadloo, M. S., Giulio L. (16). Non-uniform heat source/sink and Soret effects on MHD non-darcian convective flo past a stretching sheet in a micropolar fluid ith radiation, International Journal of Heat and Mass Transfer, 93, 674-68. [13] Mebarek-Oudina, F., Bessaih, R. (14). Numerical modeling of MHD stability in a cylindrical configuration. Journal of the Franklin Institute, 351, 667-681. [14] Poer, H. (1998). Micropolar fluid model for the brain fluid dynamics, in: Intl. Conf. on bio-fluid mechanics, U.K. [15] Rahman, M. M., Uddin, M. J., Aziz, A. (9). Effects of variable electric conductivity and non-uniform heat source (or sink) on convective micropolar fluid flo along an inclined flat plate ith surface heat flux, Int. Journal of Thermal Sciences, 48, 331-34. [16] Raat, S., Kapoor, S., Bhargava, R. (16). MHD flo heat and mass transfer of micropolar fluid over a nonlinear stretching sheet ith variable micro inertia density, heat flux and chemical reaction in a non-darcy porous medium, Journal of Applied Fluid Mechanics, 9, 31-331. [17] Siva, R. S., Shamshuddin, M. D. (15). Heat and mass transfer on the MHD flo of micropolar fluid in the presence of viscous dissipation and chemical reaction, Int. Conference on computational Heat and Mass transfer, 17, 885-89. [18] Ziaul Haquea, Md., Mahmud Alam, Md., Ferdos, M., Postelnicu, A. (1). Micropolar fluid behaviors on steady MHD free convection and mass transfer flo ith constant heat and mass fluxes, joule heating and viscous dissipation, Journal of King Saud University Engineering Sciences, 4, 71-84.