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Ct33572581

  1. 1. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581572 | P a g eEffects Of Radiation On Mhd Mixed Convection Flow Of AMicropolar Fluid Over A Heated Stretching Surface With HeatGeneration/AbsorptionP. Bala Anki Reddy*and D. Vijaya Sekhar***Department of Mathematics, Fluid dynamics division, VIT University, Vellore-632014, Tamil Nadu.**Department of Mathematics, NBKRIST, Vidyanagar, Nellore, A.P., India.AbstractA theoretical analysis is performed tostudy the flow and heat transfer characteristicsof magnetohydrodynamic mixed convection flowof a micropolar fluid past a stretching surfacewith radiation and heat generation/absorption.The transformed equations solved numericallyusing the shooting method. The effects of thevelocity, the angular velocity, and thetemperature for various values of differentparameters are illustrated graphically. Also, theeffects of various parameters on the local skin-friction coefficient and the local Nusselt numberare given in tabular form and discussed. Theresults show that the mixed convectionparameter has the effect of enhancing both thevelocity and the local Nusselt number andsuppressing both the local skin-frictioncoefficient and the temperature. It is found thatlocal skin-friction coefficient increases while thelocal Nusselt number decreases as the magneticparameter increases. The results show also thatincreasing the heat generation parameter leads toa rise in both the velocity and the temperatureand a fall in the local skin-friction coefficient andthe local Nusselt number. Furthermore, it isshown that the local skin-friction coefficient andthe local Nusselt number decrease when the slipparameter increases.Keywords: Radiation, stretching surface,micropolar fluid, heat transfer, heat source.1.IntroductionThe fluid dynamics due to a stretchingsurface and by thermal buoyancy is of considerableinterest in several applications such as aerodynamicextrusion of plastic sheets, the boundary layer alonga liquid film in condensation processes, paperproduction, glass blowing, metal spinning anddrawing plastic films. Free and mixed convectionsof a micropolar fluid over a moving surface havebeen studied by many authors [1–9] under differentsituations. Desseaux and Kelson [10] studied theflow of a micropolar fluid bounded by a linearlystretching sheet while Bhargava et. al. [11] studiedthe same flow of a micropolar flow over a non-linear stretching sheet. The theory of micropolarfluid was first introduced and formulated by Eringen[12]. LaterEringen [13] generalized the theory to incorporatethermal effects in the so-called thermo micropolarfluid. Crane [14] first obtained an elegant analyticalsolution to the boundary layer equations for theproblem of steady two dimensional flow due to astretching surface in a quiescent incompressiblefluid. Gupta and Gupta [15] extended the problemposed by Crane [14] to a permeable sheet andobtained closed-form solution, while Grubka andBobba [16] studied the thermal field and presentedthe solutions in terms of Kummer’s functions. The3-dimensional case has been considered by Wang[17]. Chen [18] studied the case when buoyancyforce is taken into consideration, and Magyari andKeller [19] considered exponentially stretchingsurface. The heat transfer over a stretching surfacewith variable surface heat flux has been consideredby Char and Chen [20], Lin and Cheng [21],Elbashbeshy [22] and very recently by Ishak et al.[23], Mostaffa Mahmoud and Shimaa Waheed [24].All of the above mentioned studies dealt withstretching sheet where the flows were assumed to besteady.The aim of this work is to investigate theeffect of radiation on the flow and heat transfer of amicropolar fluid over a vertical stretching surface inthe presence of heat generation (absorption) andmagnetic field, where numerical solutions areobtained using shooting method.2. Formulation of the ProblemConsider a steady, two-dimensionalhydromagnetic laminar convective flow of anincompressible, viscous, micropolar fluid with aheat generation (absorption) on a stretching verticalsurface with a velocity ( )wv x . The flow is assumedto be in the x-direction, which is taken along thevertical surface in upward direction and y-axisnormal to it. A uniform magnetic field of strength B0is imposed along y-axis. The magnetic Reynoldsnumber of the flow is taken to be small enough sothat the induced magnetic field is assumed to benegligible. The gravitational acceleration g acts inthe downward direction.
  2. 2. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581573 | P a g eThe temperature of the micropolar fluid far awayfrom the plate is T∞, whereas the surfacetemperature of the plate is maintained at Tw, where( )wT x T ax  , a > 0 is constant, and Tw > T∞.The temperature difference between the bodysurface and the surrounding micropolar fluidgenerates a buoyancy force, which results in anupward convective flow. Under usual boundarylayer and Boussinesq approximations, the flow andheat transfer in the presence of heat generation(absorption) are governed by the followingequations:0u vx y   (1) 2202Bu u k u k Nu v g T T ux y y y                  (2)2022N N N k uu v Nx y j y j y            (3) 2021 rp p pQqT T k Tu v T Tx y c y c y c            (4)Subject to the boundary conditions:( )wu u x cx  , 00,uv N my , ( ) 0wT T x at y 0, 0,u N T T as y    (5)where u and v are the velocity components in the xand y directions respectively, T is the fluidtemperature, N is the component of themicrorotation vector normal to the x-y plane, ρis thedensity, j is the microinertia density, μ is thedynamic viscosity, k is the gyro-viscosity (or vortexviscosity), β is the thermal expansion coefficient, σis the electrical conductivity, pc is the specific heatat constant pressure, κ is the thermal conductivity, cis a positive constant of proportionality, x measuresthe distance from the leading edge along the surfaceof the plate, and 0 is the spin-gradient viscosity.Thermal radiation is simulated using the Rosselanddiffusion approximation and in accordance with this,the radiative heat flux rq is given by:* 4*143rTqk y  Where* is the Stefan–Boltzman constant and*1kis the Rosseland mean absorption coefficient.We follow the recent work of the authors [25, 26] byassuming that 0 is given by0 12 2k Kj j              (6)This equation gives a relation between thecoefficient of viscosity and micro inertia, where K =k/μ (> 0) is the material parameter, j = ν/c, j isthe reference length, and m0 (0 ≤ m0 ≤ 1) is theboundary parameter. When the boundary parameterm0 = 0, we obtain N = 0 which is the no-spincondition, that is, the microelements in aconcentrated particle flow close to the wall are notable to rotate (as stipulated by Jena and Mathur(27)). The case m0 = 1/2 represents the weakconcentration of microelements. The casecorresponding to m0 = 1 is used for the modelling ofturbulent boundary layer flow (see PeddiesonandMcnitt (28)).We introduce the following dimensionlessvariables: 1 1 22 2 102, , , , , ,wBT Tc cy N cx g u cxf v c f MT T c                  * *01 1 12 * 3 * 33 3, ,Pr , , ,16 16ppc Qk k k kg a kR K Rc T k cc T          (7)Through (7), the continuity (1) is automaticallysatisfied and (2) – (4) will give then 2 1 0K f ff f Kg Mf       (8)  1 2 02Kg fg f g K g f        (9) 1 11 0Prf fR          (10)The corresponding boundary conditions are01, 0, , 1 at 00, 0, 0 asf f g m ff g            (11)where the primes denote the differentiation withrespect to , M is the magnetic parameter, λ is thebuoyancy parameter, γ is the heat generation (> 0)or absorption (<0) and R is the radiation parameter.The physical quantities of interest are the local skin-friction coefficient Cfx and the local Nusselt numberNux , which are respectively proportional to    0 , 0f   are worked out and theirnumerical values presented in a tabular form.
  3. 3. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581574 | P a g e3. Solution of the problemThe governing boundary layer equations(8) - (10) subject to boundary conditions (11) aresolved numerically by using shooting method. Firstof all higher order non-linear differential equations(8) - (10) are converted into simultaneous lineardifferential equations of first order and they arefurther transformed into initial value problem byapplying the shooting technique. From the processof numerical computation, the skin-frictioncoefficient, the Nusselt number and Sherwoodnumber which are respectively proportional to 0f and  0 are also sorted out and theirnumerical values are presented in a tabular form.4. Results and DiscussionTo study the behavior of the velocity, theangular velocity, and the temperature profiles,curves are drawn in Figures 1–18. The effect ofvarious parameters, namely, the magnetic parameterM, the material parameter K, the radiation parameterR, the buoyancy parameter λ, the heat generation(absorption) parameter γ, and the Prandtl number Prhave been studied over these profiles.Figures 1–3 illustrate the variation of thevelocity f , the angular velocity g, and thetemperature θ profiles with the magnetic parameterM. Figure 1 depict the variation of f with M. It isobserved that f decreases with the increase in Malong the surface. This indicates that the fluidvelocity is reduced by increasing the magnetic fieldand confines the fact that application of a magneticfield to an electrically conducting fluid produces adrag like force which causes reduction in the fluidvelocity. The profile of the angular velocity g withthe variation of M is shown in Figure 2. It is clearfrom this figure that g increases with an increase inM near the surface and the reverse is true away fromthe surface. Figure 3 shows the resultingtemperature profile θ for various values of M. It isnoted that an increase of M leads to an increase of θ.Figure 4 illustrates the effects of thematerial parameter K on f . It can be seen from thisfigure that the velocity decreases as the materialparameter K rises near the surface and the oppositeis true away from it. Also, it is noticed that thematerial parameter has no effect on the boundarylayer thickness. The effect of K on g is shown inFigure 5. It is observed that initially g decreases byincreasing K near the surface and the reverse is trueaway from the surface. Figure 6 demonstrates thevariation of θ with K. From this figure it is clear thatθ decreases with an increase in K.It was observed from Figure 7 that thevelocity increases for large values of λ while theboundary layer thickness is the same for all valuesof λ. Figure 8 depicts the effects of λ on g. Theangular velocity g is a decreasing function of λ nearthe surface and the reverse is true at larger distancefrom the surface. Figure 9 shows the variations of λon θ. It is found that θ decreases with an increase inλ.Figure 10 shows the effect of the heatgeneration parameter (γ > 0) or the heat absorptionparameter (γ < 0) on f . It is observed that fincreases as the heat generation parameter (γ > 0)increases, but the effect of the absolute value of heatabsorption parameter (γ < 0) is the opposite. Theeffect of the heat generation parameter (γ > 0) or theheat absorption parameter (γ < 0) on g within theboundary layer region is observed in Figure 11. It isapparent from this figure that g increases as the heatgeneration parameter (γ > 0) decreases, while gincreases as the absolute value of heat absorptionparameter (γ < 0) increases near the surface and thereverse is true away from the surface. Figure 12displays the effect of the heat generation parameter(γ > 0) or the heat absorption parameter (γ < 0) on θ.It is shown that as the heat generation parameter (γ> 0) increases, the thermal boundary layer thicknessincreases. For the case of the absolute value of theheat absorption parameter (γ < 0), one sees that thethermal boundary layer thickness decreases as γincreases.The effect of the Prandtl number Pr on thevelocity, the angular velocity, and the temperatureprofiles is illustrated in Figures 13, 14, and 15. Fromthese figures, it can be seen that f decrease withincreasing Pr, while g increases as the Prandtlnumber Pr increases near the surface and the reverseis true away from the surface. The temperature θ ofthe fluid decreases with an increase of the Prandtlnumber Pr as shown in Figure 15. This is inagreement with the fact that the thermal boundarylayer thickness decreases with increasing Pr. Figures16, 17, and 18 depict the effect of the radiationparameter on f , g, and θ, respectively. It is seenthat f and θ decrease as α increases, while gincreases as α increases.The local skin-friction coefficient in termsof − (0)f and the local Nusselt number in termsof − (0) for various values of M, λ, K, γ, R andPr are tabulated in Table 1. It is obvious from thistable that local skin-friction coefficient and the localNusselt number increases with the increase of themagnetic parameter M. It is noticed that as K or λincreases, the local skin-friction coefficient as wellas the local Nusselt number decreases. It is foundthat the local Nusselt number decreases with theincrease of the heat generation parameter (γ > 0),while it increased with the increase of the absolutevalue of the heat absorption parameter (γ < 0). It isfound that both the local skin friction coefficient andthe local Nusselt number increase with increasing Pror R
  4. 4. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581575 | P a g e0 1 2 3 4 50.00.20.40.60.81.0fM = 0.5M = 1.0M = 1.5Fig. 1. Velocity profiles for variousvalues of M.0 1 2 3 4 5-0.050.000.050.100.150.200.250.300.350.400.45gM = 0.5M = 1.0M = 1.5Fig. 2. Angular Velocity profiles forvarious values of M.0 1 2 3 4 50.00.20.40.60.81.0M = 0.5M = 1.0M = 1.5Fig. 3. Temperature profiles for various values of M.0 1 2 3 4 50.00.20.40.60.81.0fK = 0.5K = 1.2K = 2.5Fig. 4. Velocity profiles for various values of K.
  5. 5. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581576 | P a g e0 1 2 3 4 50.000.050.100.150.200.250.300.350.400.45gK = 0.5K = 1.2K = 2.5Fig. 5. Angular velocity profiles for various valuesof K.0 1 2 3 4 50.00.20.40.60.81.0K = 0.5K = 1.2K = 2.5Fig. 6. Temperature profiles for various values of K.0 1 2 3 4 50.00.20.40.60.81.0f = 0.1 = 0.5 = 1.0Fig. 7. Velocity profiles for various values of  .0 1 2 3 4 50.000.050.100.150.200.250.300.350.400.45g = 0.1 = 0.5 = 1.0Fig. 8. Angular velocity profiles for various valuesof  .
  6. 6. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581577 | P a g e0 1 2 3 4 50.00.20.40.60.81.0 = 0.1 = 0.5 = 1.0Fig. 9. Temperature profiles for various values of .0 1 2 3 4 50.00.20.40.60.81.0fFig. 10. Velocity profiles for various values of  .0 1 2 3 4 50.000.050.100.150.200.250.300.350.400.45g = - 0.3 = 0.3 = 0.6Fig. 11. Angular velocity profiles for various valuesof  .0 1 2 3 4 50.00.20.40.60.81.0 = - 0.3 = 0.3 = - 0.6Fig. 12. Temperature profiles for various values of .
  7. 7. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581578 | P a g e0 1 2 3 4 50.00.20.40.60.81.0fPr = 0.72Pr = 1.0Pr = 2.0Fig. 13. Velocity profiles for various values of Pr.0 1 2 3 4 50.000.050.100.150.200.250.300.350.400.45gPr = 0.72Pr = 1.0Pr = 2.0Fig. 14. Angular velocity profiles for various valuesof Pr.0 1 2 3 4 50.00.20.40.60.81.0Pr = 0.72Pr = 1.0Pr = 2.0Fig. 15. Temperature profiles for various values ofPr.0 1 2 3 4 50.00.20.40.60.81.0fR = 0.5R = 1.0R = 1.5Fig. 16. Velocity profiles for various values of R.
  8. 8. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581579 | P a g e0 1 2 3 4 50.000.050.100.150.200.250.300.350.400.45gR = 0.5R = 1.0R = 1.5Fig. 17. Angular velocity profiles for various valuesof R.0 1 2 3 4 50.00.20.40.60.81.0R = 0.5R = 1.0R = 1.5Fig. 18. Temperature profiles for various values ofR.Table 1.M  K  R Pr (0)f (0)0.50.51.20.30.50.720.7484020.2976231.00.51.20.30.50.720.90297 0.3779530.51.01.20.30.50.720.5566240.2005480.50.52.50.30.50.720.6489540.2151110.50.51.2-0.30.50.720.7682840.3040270.50.51.20.60.50.720.7453970.2966260.50.51.20.31.00.720.7634540.3023690.50.51.20.30.51.0 0.7603630.3013845. ConclusionsIn the present work, the effects of heatgeneration (absorption) and a transverse magneticfield on the flow and heat transfer of a micropolarfluid over a vertical stretching surface with radiationhave been studied. The governing fundamentalequations are transformed to a system of nonlinearordinary differential equations which are solved byusing shooting technique. The velocity, the angularvelocity, and the temperature fields as well as thelocal skin-friction coefficient and the local Nusseltnumber are presented for various values of theparameters governing the problem. From the results,we can observe that, the velocity decreases withincreasing the magnetic parameter, and the absolutevalue of the heat absorption parameter, while itincreases with increasing the buoyancy parameter,the heat generation parameter, and the Prandtlnumber. Also, it is found that near the surface thevelocity decreases as the slip parameter and thematerial parameter increase, while the reversehappens as one moves away from the surface. Theangular velocity decreases with increasing thematerial parameter, the slip parameter, the buoyancyparameter, and the heat generation parameter, whileit increases with increasing the magnetic parameter,the absolute value of the heat absorption parameter,and the Prandtl number near the surface and thereverse is true away from the surface. In addition thetemperature distribution increases with increasingthe slip parameter, the heat generation parameter,and the magnetic parameter, but it decreases withincreasing the Prandtl number, the buoyancyparameter, the material parameter, and the absolutevalue of the heat absorption parameter. Moreover,the local skin-friction coefficient increases withincreasing the magnetic parameter and the absolutevalue of the heat absorption parameter, while thelocal skin-friction decreases with increasing the
  9. 9. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581580 | P a g ebuoyancy parameter, the slip parameter, and the heatgeneration parameter. Finally, the local Nusseltnumber increases with increasing the buoyancyparameter, and the absolute value of the heatabsorption parameter, and decreases with increasingthe magnetic parameter, the slip parameter, and theheat generation parameter.References1. Y. J. Kim and A. G. Fedorov, “Transientmixed radiative convection flow of amicropolar fluid past a moving, semi-infinite vertical porous plate,” InternationalJournal of Heat and Mass Transfer, vol.46, no. 10, pp. 1751–1758, 2003.2. R. Bhargava, L. Kumar, and H. S. Takhar,“Mixed convection from a continuoussurface in a parallel moving stream of amicropolar fluid,” Heat and Mass Transfer,vol. 39, no. 5-6, pp. 407–413, 2003.3. M. M. Rahman and M. A. Sattar,“Transient convective flow of micropolarfluid past a continuously moving verticalporous plate in the presence of radiation,”International Journal of AppliedMechanics and Engineering, vol. 12, pp.497–513, 2007.4. M. M. Rahman and M. A. Sattar,“Magnetohydrodynamic convective flow ofa micropolar fluid past a continuouslymoving vertical porous plate in thepresence of heat generation/absorption,”Journal of Heat Transfer, vol. 128, no. 2,pp. 142–152, 2006.5. Ishak, R. Nazar, and I. Pop, “Heat transferover a stretching surface with variable heatflux in micropolar fluids,” Physics LettersA, vol. 372, no. 5, pp. 559–561, 2008.6. Ishak, R. Nazar, and I. Pop, “Mixedconvection stagnation point flow ofamicropolar fluid towards a stretchingsheet,” Meccanica, vol. 43, no. 4, pp. 411–418, 2008.7. Ishak, R. Nazar, and I. Pop,“Magnetohydrodynamic _MHD_ flow of amicropolar fluid towards a stagnation pointon a vertical surface,” Computers andMathematics with Applications, vol. 56, no.12, pp. 3188–3194, 2008.8. Ishak, Y. Y. Lok, and I. Pop, “Stagnation-point flow over a shrinking sheet in amicropolarfluid,” Chemical EngineeringCommunications, vol. 197, pp. 1417–1427,2010.9. T. Hayat, Z. Abbas, and T. Javed, “Mixedconvection flow of a micropolar fluid overa non-linearly stretching sheet,” PhysicsLetters A, vol. 372, no. 5, pp. 637–647,2008.10. A. Desseaux, N. A. Kelson, Flow of amicropolar fluid bounded by a stretchingsheet, ANZIAM J., Vol. 42, pp. C536-C560, 2000.11. R. Bhargava, S. Sharma, H.S. Takhar, O.A. Beg, P. Bhargava, Nonlinear Analysis:Modelling and Control, Vol. 12, pp. 45-63,2007.12. A.C. Eringen, “Theory of micropolarfluids,” Journal of Mathematics andMechanics, vol. 16, pp. 1–18, 1966.13. A. C. Eringen, “Theory of thermomicropolar fluids,” Journal of AppliedMathematics, vol. 38, pp. 480– 495, 1972.14. L.J. Crane, “Flow past a stretching plate,”J. Appl. Math. Phys. (ZAMP), vol. 21, pp.645 – 647, 1970.15. P. S. Gupta and A. S. Gupta, “ Heat andmass transfer on a stretching sheet withsuction or blowing, “ Can. J. Chem. Eng,vol. 55, pp. 744-746, 1977.16. L. J. Grubka and K. M. Bobba, “Heattransfer characteristics of a continuousstretching surface with variabletemperature,” ASME J.Heat Transfer, vol.107, pp. 248-250, 1985.17. C. Y. Wang, “ The three-dimensional flowdue to a stretching surface,” Phys. Fluids,vol. 27, pp. 1915-1917, 1984.18. C. H. Chen, “Laminar mixed convectionadjacent to vertical, continuously stretchingsheets,” Heat and Mass Transfer, vol. 33,pp.471-476, 1998.19. E. Magyari and B. Keller, “ Heat and masstransfer in the boundary layers on anexponentially stretching continuoussurface,” J. Phys. D: Appl. Phys, vol. 32,pp. 577-585, 1999.20. M. I. Char and C. K. Chen, “Temperaturefield in non-Newtonian flow over astretching plate with variable heat flux,”Int. J. Heat Mass Transfer, vol. 31, pp.917-921, 1988.21. C. R. Lin and C. K. Chen, “Exact solutionof heat transfer from stretching surfacewith variable heat flux,” Heat MassTransfer, vol. 33, pp. 477-480, 1998.22. E. M. A. Elbashbeshy, “Heat transfer overa stretching surface with variable surfaceheat flux,” J. Phys. D: Appl. Phys. Vol. 31,pp. 1951-1954, 1998.23. Ishak, R. Nazar and I. Pop, “ Heat transferover a stretching surface with variable heatflux in micropolar fluids,” Phys. Lett. A,vol. 372, pp. 559-561, 2008.24. Mostaffa Mahmoud and Shimaa Waheed,Effects of slip and heatgeneration/absorption on MHD mixedconvection flow of a micropolar fluid overa heated stretching surface, Hindawi
  10. 10. P. Bala Anki Reddy, D. Vijaya Sekhar / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.572-581581 | P a g ePublishing Corporation MathematicalProblems in Engineering, Vol. 2010,Article ID 579162, 20 pages,doi:10.1155/2010/579162.25. G. Ahmadi, “Self-similar solution ofincompressible micropolar boundary layerflow over a semi infinite plate,”International Journal of EngineeringScience, vol. 14, pp. 639–646, 1976.26. K. A. Kline, “A spin-vorticity relation forunidirectional plane flows of micropolarfluids,” International Journal ofEngineering Science, vol. 15, no. 2, pp.131–134, 1977.27. S. K. Jena and M. N. Mathur, “Similaritysolutions for laminar free convection flowof a thermomicropolar fluid past a non-isothermal vertical flat plate,” InternationalJournal of Engineering Science, vol. 19,no. 11, pp. 1431–1439, 1981.28. J. Peddieson and R. P. Mcnitt, “Boundarylayer theory for a micropolar fluid,” RecentAdvances in Engineering Science, vol. 5,pp. 405–426, 1970.

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