MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and zero mass flux conditions

Transcription

1 ULLETN OF THE POLSH ACADEMY OF SCENCES TECHNCAL SCENCES Vol. 65 No. 07 DO: 0.55/bpasts MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and ero mass flux conditions A. RAUF S.A. SHEHZAD * T. HAYAT M.A. MERAJ and A. ALSAED Department of Mathematics COMSATS nstitute of nformation Technolog Sahiwal 5700 Pakistan Department of Mathematics Quaid--Aam Universit slamabad 00 Pakistan Department of Mathematics King Abdulai Universit Jeddah 589 Saudi Arabia Abstract. n this article the stagnation point flow of electricall conducting micro nanofluid towards a shrinking sheet considering a chemical reaction of first order is investigated. nvolvement of magnetic field occurs in the momentum equation whereas the energ and concentrations equations incorporated the influence of thermophoresis and rownian motion. Convective boundar condition on temperature and ero mass flux condition on concentration are implemented. Partial differential equations are converted into the ordinar ones using suitable variables. The numerical technique is utilied to discuss the results for velocit microrotation temperature and concentration fields. Ke words: micro nanofluid shrinking sheet MHD chemical reaction.. ntroduction The analsis of fluid flow over stretching/shrinking surfaces has been of immense interest for the researchers in the fields of engineering and chemical process [ ]. Especiall the fluid flow generated b the surface is important in the extrusion and manufacturing process of material polers cooling sstems plastic industr petrochemical industr paper production geophsical sstems power stations chemical plants air conditioning refrigeration etc. Out of all the above-mentioned sstems most attention was given to enhancing the energ generation and transfer of heat. Several methods have been proposed in this regard but the techniques the emplo are not suitable due to lesser thermal conductivit of heat transfer fluid. Therefore energ materials are introduced. These energ materials known as nanomaterials contain tin particles of the same sie as the de roglie wave [5]. Therefore nanoparticles have attracted researchers because of the abundance of applications thereof in technological and engineering processes. Nanoparticles in the base fluid known as nanofluids displa thermophoresis and rownian motion properties which enhance the thermal performance and thermal conductivit of base fluids [6 7]. Ultra-high heat transfer rates and extreme stabilit are the two main features of nanofluids [8] and do not cause problems such as erosion sedimentation and pressure drop. Magnetohdrodnamics (MHD) is significant for chemistr mathematics phsics and engineering and is applied in biological transportation pumps mixing of samples cooling of strips drug deliver MHD generators etc. External applied magnetic field is ver helpful when controlling heat transfer and flow. As nanoparticles increase the thermal and electrical conductivit of nanofluids making them liable to influence the magnetic * ali_qau70@ahoo.com field. Magnetic nanofluids are useful in nanocrosurger aerodnamics sensors blood flow analsis nuclear plants artificial kidnes and smartfluids for vibration damping. A number of researchers have discussed different models of hdromagnetic nanofluid flow over stretching surfaces [9 6]. Man biological fluids as well as the fluids that are used in industrial applications such as printer inks animal blood detergents paint food stuff poler liquids etc. change their flow characteristics when subjected to applied shear stress and thus differ from Newtonian fluids. These materials are called non-newtonian fluids. Researchers have discussed several non-newtonian fluid flow models such as Maxwell fluid power law fluid second or third grade fluid etc. Eringen [7 8] introduced the theor of micropolar fluids for the first time. This theor deals with the intrinsic motion and local microstructure of the fluid particles and can be valuable when investigating the impact of poler suspensions colloidal solutions biological and mudd fluids etc. Furthermore the impact of mass and heat transfer combined with the impact of chemical reaction has in the last few ears been investigated with regard to possible applications in hdrometallurgical and chemical plants including fruit-processing methods freee damage of crops temperature distribution and growth of trees and heat and mass transfer in cooling towers. Heat transfer due to surface convection and ero mass flux at a stretching/shrinking surface has gained significance in material ding hot wiring nuclear plants transpiration process production of glass fiber heat exchangers prevention of energ etc. The numerical solution of the problem of MHD stagnation point flow of micropolar fluid towards a moving sheet was presented b Ashraf and ashir [9]. The extension of the above problem was performed b Rashidi et al. [0]. The added the term of mixed convection to the problem and solved it analticall. Rauf et al. [] numericall analed the MHD flow of micropolar fluid over a stretchable disk. The effects of a po- ull. Pol. Ac.: Tech. 65() 07 55

2 A. Rauf S.A. Shehad T. Haat M.A. Meraj and A. Alsaedi rous medium along with heat and mass transfer were studied. Ashraf and Wehgal [] solved the problem of MHD flow of micropolar fluid confined between two fixed porous disks with heat transfer characteristics. Shehad et al. [] applied an analtical technique based on HAM to the problem of unstead micropolar fluid and heat transfer influenced b a stretching sheet. Jalilpour et al. [] applied HPM to the problem of MHD nanofluid flow over stretching sheet immersed in a porous medium. Pal et al. [5] analed stagnation point radiative flow of nanofluid over a surface with porous medium. The above-mentioned problem was extended to mixed convective nanofluid flow with chemical reaction b Pal and Mandal [6]. Haat et al. [7] considered the problem of mixed convective flow over a stretching surface with chemical reaction. The emploed the convective-tpe boundar condition at the surface of a sheet. Kunetsov and Nield [8] considered the problem of boundar laer flow of nanofluid past a vertical plate. The implemented boundar conditions which impl that the flux of nanoparticles is ero at the boundar. The problem of Maxwell nanofluid flow over a stretching surface was solved b Haat et al. [9]. Here the Kunetsov and Nield condition [8] of ero mass flux at the surface of a sheet was emploed. The different flow problems under convective surface conditions have been modeled and addressed b Haat et al. [0 ] and mtia et al. []. Our main objective here is to find numerical solutions for MHD stagnation point flow of an incompressible electricall conducting flow of micro nanofluid over a heated shrinking sheet. The sheet obes the convective condition on temperature and the ero-mass condition. The variations of individual parameters of interest are examined.. Problem formulation Here two-dimensional laminar incompressible stagnation point flow of an electricall conducting micro nanofluid impinging in normal direction over a heated shrinking sheet is considered. We investigate the impact of magnetic field of strength 0. Magnetic field is utilied in transverse direction of the flow field. The governing equations are: u v () x C C u v D x C D T T T k C C (5) where: u and v are the velocit components in the x and directions respectivel p is the pressure υ is the microrotation ρ is the densit μ is the viscosit k is the vortex viscosit j is the microinertia densit γ is the spin gradient viscosit σ e is the electrical conductivit of fluid α is the thermal diffusivit of fluid τ = (ρc) p (ρc) f is the ratio of nanoparticle heat capacit and the base fluid heat capacit D is the rownian diffusion coefficient D T is the thermophoretic diffusion coefficient and k is the reaction coefficient. The boundar conditions are: u x0 bx v x0 x0 T k h f T f T C D T D T ux U a x x T x T Cx C where b < 0 corresponds to the sheet shrinking rate and h f is the heat transfer coefficient. Considering the similarit transformations: u a x f v a f u a a a x g T T C C T f T C equations ( 5) become f R g M f (6) (7) R f f f 0 (8) f g f g fg 0 C (9) g R A u u du u v U x dx k U u e 0 u k () Pr f Pr N N Sc f N Pr b t Nt t (0) Sc 0. () u j u v k () x T T T u v C T DT T D () x D σ Here M = e 0 ρα denotes the magnetic parameter R = μ k the vortex viscosit parameter C = γ μj the spin gradient viscosit A = μ ρja microinertia densit parameter Pr = μc p k 0 the Prandtl number N b = (ρc) pd C (ρc) f the rownian motion parameter N t = (ρc) pd T (T f T) υ (ρc) f the thermophoresis parameter Sc = υ υt D the Schmidt number and γ = k a the chemical reaction parameter. 56 ull. Pol. Ac.: Tech. 65() 07

3 MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and ero mass flux conditions The dimensionless boundar conditions are: f Nt f 0 N f 0 g0 0 i 0 Nb0 g () where i = h f υ k a is the heat transfer iot number and N = b a is the shrinking parameter.. Numerical solution The Runge-Kutta-Fehlberg (RKF5) method is ver helpful when solving d dx = f(x ) (x l) = l. The procedure involves suitable step sie which guarantees the accurateness in solution of the initial value problem. Ever proper step contains two different tpes of approximations to the solution which are computed and compared. When answers match closel the approximation is valid. f answers are not accurate enough then the step sie is decreased. f the answers meet more than the significant digits the step sie is incremented. n each step the following six steps are required: 5 6 h f x m m h f xm h m xm h 8 h f 9 m 9 xm h m 97 h f xm h 6 h f xm h 8 5 h f The approximation of order to the solution is: m xm () () 0 5 A better approximation of order 5 to the solution is given b: 6 m m (5) Finall dh stands for optimal step sie and is obtained b multipling h with a scalar d Here d is determined b: 0.8 tol h d (6) m where tol stands for the error tolerance. A finite difference technique based on RKF5 method with a shooting technique [ 6] is implemented to obtain the numerical solution of the sstems (8 ) with the corresponding boundar conditions (). The following are set: f g 5 f g f (7) where prime stands for the derivative with respect to. Using (7) into (8 ) we obtain the reduced first-order sstem of differential equations: R 0 (8) R M 5 C (9) 0 5 R A 5 7 Pr 7 Pr Nb 7 9 Pr Nt (0) Nt 7 Sc 8 0. () N 9 Sc 9 The boundar conditions () become 7 Nt 0 0 N a 0 i Nb a a () The Newton-Raphson algorithm and the shooting method are used to guess the conditions a a and a in (). Finall the problem is integrated to obtain the boundar conditions at = 0. The convergence criteria are set to at least Results and discussion This section is devoted to illustrating our findings in graphical as well as tabular forms. The dimensionless parameters ull. Pol. Ac.: Tech. 65() 07 57

4 A. Rauf S.A. Shehad T. Haat M.A. Meraj and A. Alsaedi N = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = Nb = 0. N = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = Nb = 0. M = f() M = f () g() M = N = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = Nb = 0. Fig.. f() for different values of M Fig.. f () for different values of M Fig.. g() for different values of M of our interests are the micropolar parameters R C and A magnetic parameter M shrinking parameter N Prandtl number Pr heat transfer iot number i Schmidt number Sc thermophoretic parameter Nt rownian motion parameter Nb and chemical reaction parameter γ. To have the best knowledge of the phsics of our model we chose to describe shear and couple stresses and heat and mass transfer rates at the sheet considering different values of the phsical parameters. We fix R = C = 0. A = 0. M = 0.5 N = 0.5 Pr = 0. Sc = 0. Nt = 0. Nb = 0. i = 0. and γ = 0. into our computation procedure altering one parameter at a time as discussed through graphs and tables. We adjusted = 7 5 in order to have asptotic behavior of velocit microrotation temperature and concentration profiles. Figures are drawn to explore the behavior of magnetic parameter M on velocit and microrotation. Here M = 0 shows the hdrodnamic flow and M(>0) represents hdromagnetic flow. ncreasing M results in an enhancement in normal velocit profiles f() and streamwise velocit profiles f (). A reverse flow region can be seen near the surface because of the shrinking of the sheet [9]. Fig. shows that large M can be helpful to stop the reverse flow phenomenon. The imposed magnetic field produces a frictional force called the Lorent force which offers a resistance in a flow field and due to its velocit the boundar laer pushes towards the wall of the sheet as shown in Fig.. An increase in the magnetic parameter causes a reduction in microrotation profiles as described in Fig.. Figures 6 are presented to investigate the impact of the shrinking parameter on f() f () and g(). t is noted from Figs. and 5 that f() and f () decrease b enhancing N. The reverse flow in the vicinit of surface of the sheet is also observed for increased values in magnitude of N. Fig. 6 illustrates the influence of N on microrotation profiles g(). The profiles are increased due to the enhanced values of magnitude of N. Fig. 7 explores the behavior of R C and A in microrotation. nfluence of different values of micropolar parameters causes an enhancement in profiles g(). Fig. 8 illustrates the effect of Prandtl number on temperature profiles. Phsicall the Prandtl number is inversel proportional to the thermal diffusivit. Hence larger values of Pr produce weaker thermal diffusivit. This corresponds to a reduction in both temperature and the M = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = Nb = 0. M = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = Nb = 0. N = f() N = f () g() N = M = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = Nb = 0. Fig.. f() for different values of N Fig. 5. f () for different values of N Fig. 6. g() for different values of N 58 ull. Pol. Ac.: Tech. 65() 07

5 MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and ero mass flux conditions M = 0.5 N = 0.5 Sc = γ = 0. i = 0. Pr = Nt = Nb = 0. M = 0.5 N = 0.5 R = C = Sc = γ = 0. A = i = 0. Nt = Nb = 0. M = 0.5 N = 0.5 R = C = Sc = γ = 0. A = 0. Pr = Nt = Nb = 0. g() Case : R = C = 0. A = 0. Case : R =.5 C = 0.5 A = 0. Case : R = C = 0. A = 0. Case : R = C = 0. A = 0.6 θ() Pr = θ() i = Fig. 7. g() for different four cases Fig. 8. θ() for different values of Pr Fig. 9. θ() for different values of i Sc = Nt = Nb = ϕ() ϕ() ϕ() M = 0.5 N = 0.5 R = C = γ = 0. A = i = 0. Pr = Nt = Nb = 0. M = 0.5 N = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nb = 0. M = 0.5 N = 0.5 R = C = Sc = γ = 0. A = i = 0. Pr = Nt = 0. Fig. 0. ϕ() for different values of Sc Fig.. ϕ() for different values of Nt Fig.. ϕ() for different values of Nb associated boundar laer thickness. Fig. 9 is plotted to explore the impact of i on θ(). Phsicall the iot number is the ratio of internal thermal resistance at the surface of the bod to the boundar laer thermal resistance. Therefore enhancing the values of i shows an increase in temperature profiles and its related boundar laer thickness. Fig. 0 is sketched for a better understanding of the impact of Sc on concentration profiles ϕ(). Phsicall Schmidt number is inversel proportional to the mass diffusion therefore an increase in Sc causes a reduction in nanoparticle concentration profiles as well as in related boundar laer thickness. Figures are designed to depict the effect of thermophoretic and rownian motion parameters on ϕ(). n thermophoresis the small particles are pushed awa from the hot surface and are driven towards a cold surface therefore increasing the values of Nt which causes an increase in nanoparticle concentration profiles (Fig. ). The rownian motion comes into pla due to the ig-ag movement of nanoparticles. Such motion then results in an increase of kinetic energ of particles and hence the collision between the particles increases. Therefore the rownian motion is affected b the increasing values of Nb which then reduces ϕ() with the relevant boundar laer thickness as shown in Fig.. Figure shows that the profiles ϕ() and the appropriate boundar laer thickness decrease with de- ϕ() γ = M = 0.5 N = 0.5 R = C = Sc = 0. A = i = 0. Pr = Nt = Nb = 0. Fig.. ϕ() for different values of γ ull. Pol. Ac.: Tech. 65() 07 59

6 A. Rauf S.A. Shehad T. Haat M.A. Meraj and A. Alsaedi structive chemical reaction (γ > 0) whereas an opposite trend is seen for generative chemical reaction (γ < 0). Table is formed to describe the arbitrar values of micropolar parameters [9]. Table is drawn to present the impact of the magnetic parameter shrinking parameter and micropolar parameters on shear and couple stresses. t is noted that shear stresses intensif with an increase in M while a reverse trend is observed for R C and A. An increase in the shear stresses is seen for 0.5 < N < 0.75 whereas the opposite behavior can be noted for 0.75 < N <. The couple stresses are increasing for increasing values of micropolar parameters magnetic parameter and shrinking parameter as shown in Table. Table displas the effects of heat transfer rate for various values of Pr and i. Rising values of the Prandtl number and heat transfer iot number lead to an increase in heat transfer rate at the sheet. Table presents the impact of the Schmidt number thermophoretic parameter rownian motion parameter and destructive/ generative chemical reaction parameter on ϕ (). t is seen that the mass transfer rate is a decreasing function of Sc Nb and γ. However the opposite trend is noted for larger values of Nt. Table 5 was made to present the validit of the numerical results. t is apparent that the results are compared well with the previousl published literature work. Table Different values of R C and A for the four cases discussed Case No. R C A Table Shear and couple stresses at sheet for different values of M N and four Cases of R C and A M N Case No. ( + R )f (0) g (0) Table Heat transfer rate at the sheet for different values of Pr and i Pr i θ (0) Table Mass transfer rate at the sheet for different values of Sc Nt Nb and γ Sc Nt Nb γ ϕ (0) Table 5 Comparison of numerical values of shear and couple stresses at sheet for various values of M [9] M ( + R )f (0) g (0) results from [9] present stud results from [9] present stud ull. Pol. Ac.: Tech. 65() 07

7 MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and ero mass flux conditions 5. Conclusion The following conclusions can be drawn from the presented analsis: ) Couple stresses are enhanced b increasing M N R C and D. ) Pr and i increase θ (). ) Mass transfer rate enhances with the increase of Nt whereas a reverse trend is noted in case of increasing Sc Nb and γ. ) Temperature profiles and thermal boundar laer thicknesses are decreasing functions of Pr while the opposite behavior is seen in case of enhancing the values of i. 5) Concentration profile decreases for increasing values of the Schmidt number and rownian motion parameter. On the other hand the profiles increase for larger values of the thermophoretic parameter. References [] S.A. Shehad Z. Abdullah A. Alsaedi F.M. Abbasi and T. Haat Thermall radiative three-dimensional flow of Jeffre nanofluid with internal heat generation and magnetic field J. Magnet. Magnet. Mater (06). [] L. Zheng J. Niu X. Zheng and L. Ma Dual solutions for flow and Radiative heat transfer of micropolar fluid over stretching/ shrinking sheet nt. J. Heat Mass Transf. 55 (5 6) (0). [].J. Gireesha A.J. Chamkha S. Manjunatha and C.S. agewadi Mixed convective flow of a dust fluid over a vertical stretching sheet with non-uniform heat source/sink and radiation nt. J. Numer. Meth. Heat Fluid Flow () (0). [].J. Gireesha G.K. Ramesh M.S. Abel and C.S. agewadi oundar laer flow and heat transfer of a dust fluid flow over a stretching sheet with non-uniform heat source/sink nt. J. Multiphase Flow 7 (8) (0). [5] A. Sharma V.V. Tagi C.R. Chen and D. uddhi Review on thermal energ storage with phase change materials and application Renew. Sustain. Energ Rev. () 8 5 (009). [6] S.U.S. Choi and J.A. Eastman Enhancing thermal conductivit of fluids with nanoparticles ASME nternational Engineering Congress and Exposition San Francisco (995). [7] D.K. Devendiran and V.A. Amirtham A review on preparation characteriation properties and applications on nanofluids Renew. Sustain. Energ Rev. 6 0 (06). [8] M.M. Rahman W.A. Al-Maroui F.S. Al-Hatmi M.A. Al-Lawatia and.a. Eltaeb The role of a convective surface in models of the radiative heat transfer in nanofluids Nuclear Eng. Design (0). [9] S.A. Shehad T. Haat and A. Alsaedi nfluence of convective heat and mass conditions in MHD flow of nanofluid ull. Pol. Ac.: Tech. 6 () 65 7 (05). [0] S.A. Shehad Z. Abdullah F.M. Abbasi T. Haat and A. Alsaedi Magnetic field effect in three-dimensional flow of an Oldrod- nanofluid over a radiative surface J. Magnet. Magnet. Mater (06). [] T. Haat T. Muhammad. Ahmad and S.A. Shehad mpact of magnetic field in three-dimensional flow of Sisko nanofluid with convective condition J. Magnet. Magnet. Mater. 8 (06). [] M.M. Rashidi N.V. Ganesh A.K.A. Hakeem and. Ganga uoanc effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation J. Mol. Liq (0). [] T. Haat T. Muhammad S.A. Shehad G.Q. Chen and.a. Abbas nteraction of magnetic field in flow of Maxwell nanofluid with convective effect J. Magnet. Magnet. Mater (05). [] T. Haat M. mtia and A. Alsaedi Unstead flow of nanofluid with double stratification and magnetohdrodnamics nt. J. Heat Mass Transf (06). [5] T. Haat M. Waqas M.. Khan and A. Alsaedi Analsis of thixotropic nanomaterial in a doubl stratified medium considering magnetic field effects nt. J. Heat Mass Transf. 9 (06). [6] T. Haat M.. Khan M. Waqas T. Yasmeen and A. Alsaedi Viscous dissipation effect in flow of magnetonanofluid with variable properties J. Mol. Liq. 7 5 (06). [7] A.C. Eringen Simple micropolar fluids nt. J. Eng. Sci (96). [8] A.C. Eringen Theor of micropolar fluids J. Appl. Math. Mech. 6 8 (966). [9] M. Ashraf and S. ashir Numerical simulation of MHD stagnation point flow and heat transfer of a micropolar fluid towards a heated shrinking sheet nt. J. Numer. Methods Fluids 69 () 8 98 (0). [0] M.M. Rashidi M. Ashraf. Rostami M.T. Rastegari and S. ashir Mixed convection boundar-laer flow of a micropolar fluid towards a heated shrinking sheet b homotop analsis method Thermal Sci. 0 () (06). [] A. Rauf M. Ashraf K. atool T. Hussain and M.A. Meraj MHD flow of a micropolar fluid over a stretchable disk in a porous medium with heat and mass transfer AP Adv (05). [] M. Ashraf and A.R. Wehgal MHD flow and heat transfer of a micropolar fluid between two porous disks Appl. Math. Mech. () 5 6 (0). [] S.A. Shehad M. Waqas A. Alsaedi and T. Haat Flow and heat transfer over an unstead stretching sheet in a micropolar fluid with convective boundar conditions J. Appl. Fluid Mech. 9 () 7 5 (06). []. Jalilpour S. Jafarmadar D.D. Ganji A.. Shotorban and H. Taghavifar Heat generation/absorption on MHD stagnation flow of nanofluid towards a porous stretching sheet with prescribed surface heat flux J. Mol. Liq (0). [5] D. Pal G. Mandal and K. Vajravelu Flow and heat transfer of nanofluids at a stagnation point flow over a stretching/shrinking surface in a porous medium with thermal radiation Appl. Math. Comput (0). [6] D. Pal and G. Mandal nfluence of thermal radiation of mixed convection heat and mass transfer stagnation-point flow in nanofluids over stretching/shrinking sheet in a porous medium with chemical reaction Nuclear Eng. Design (0). [7] T. Haat M.. Ashraf S.A. Shehad and A. Alsaedi Mixed convection flow of Casson nanofluid over a stretching sheet with convectivel heated chemical reaction and heat source/sink J. Appl. Fluid Mech. 8 () 80 8 (05). ull. Pol. Ac.: Tech. 65() 07 6

8 A. Rauf S.A. Shehad T. Haat M.A. Meraj and A. Alsaedi [8] A.V. Kunetsov and D.A. Nield Natural convective boundar-laer flow of a nanofluid pas a vertical plate: A revised-model nt. J. Thermal Sci (0). [9] T. Haat T. Muhammad S.A. Shehad and A. Alsaedi Three-dimensional boundar laer flow of Maxwell nanofluid: mathematical model Appl. Math. Mech. 6 (6) (05). [0] T. Haat S. Farooq A. Alsaedi and. Ahmad Hall and radial magnetic field effects on radiative peristaltic flow of Carreau-Yasuda fluid in a channel with convective heat and mass transfer J. Magnet. Magnet. Mater (06). [] T. Haat M. mtia and A. Alsaedi MHD D flow of nanofluid in presence of convective conditions J. Mol. Liq (05). [] T. Haat Y. Saeed S. Asad and A. Alsaedi Convective heat and mass transfer in flow b an inclined stretching clinder J. Mol. Liq (06). [] M. mtia T. Haat A. Alsaedi and. Ahmad Convective flow of carbon nanotubes between rotating stretchable disks with thermal radiation effects nt. J. Heat Mass Transf (06). [] W.M.K.A.D. Zaimi. idin N.A.A. akar and R.A. Hamid Applications of Runge-Kutta-Fehlberg method and shooting technique for solving classical lasius equation World Appl. Sci. J (0). [5] A.K. Jhankal and M. Kumar Magnetohdrodnamic (MHD) plane poiseuille flow with variable viscosit and unequal wall temperature ran. J. Chem. Eng. () 6 68 (0). [6] O.D. Makinde S. Khamis M.S. Tshehla and O. Franks Analsis of heat transfer in erman flow of Nanofluids with Navier slip viscous dissipation and convective cooling Adv. Math. Phs. 0 () (0). 6 ull. Pol. Ac.: Tech. 65() 07