IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 78-1684, -ISSN : 3 334X PP 7-14 www.iosrjournals.org Thermal Radiation and Heat Transfer Effects on MHD Micro olar Fluid Flow Past a Vertical Plate with Chemical Reaction K.BhagyaLakshmi 1, P.V.Satya Narayana and N.V.R.V.Prasad 3 1 (Deartment of Mathematics, CMR Technical Camus, Medchel, Hyderabad,Telangana,India) (Deartment of Mathematics, SAS, VIT University, Vellore 63 14, T.N, India) 3 (Deartment of Mathematics, S.V.G.S. Junior College, Nellore,A.P, India) Abstract: We examine the instantaneous effects of chemical reaction and radiation absortion on unsteady MHD free convective heat and mass transfer flow for a micro olar fluid bounded by a semi infinite vertical late in the resence of heat generation and thermal radiation. The late is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts erendicular to the orous surface in which absorbs micro olar fluid with a suction velocity varying with time. The dimensionless governing equations of the flow, heat and mass transfer are solved analytically using regular erturbation method. The effects of various ertinent arameters on flow, heat and mass transfer roerties are discussed analytically and exlained grahically. Also the velocity rofiles of micro olar fluid is comared with the corresonding flow roblem for a Newtonian fluid and found that the olar fluid velocity is decreasing. Keywords: Chemical reaction, heat transfer, micro olar fluid, MHD, thermal radiation I. Introduction The theory of microolar fluids was first introduced and formulated by Eringen [1]. This theory dislays the effects of local rotary inertia and coule stress. The theory is exected to a mathematical model for the non-newtonian fluid behavior observed in certain fluid such as exotic lubricants, colloidal fluids, liquid crystals etc., which is more realistic and imortant from a technological oint of view. The theory of thermo microolar fluids was develoed by Eringen [] by extending his theory of microolar fluid. Sharma and Guta [3] studied the effects of medium ermeability on thermal convection in microolar fluids. Pratha Kumar et al. [4] have studied the roblem of fully develoed free convective flow of microolar and viscous fluids in a vertical channel. Muthu et.al. [5] studied eristaltic motion of microolar fluid in circular cylindrical tubes. Srinivasacharya et.al. [6] analyzed the unsteady stokes flow of microolar fluid between two arallel orous lates. Muthuraj and Srinivas [7] investigated, fully develoed MHD flow of a microolar and viscous fluids in a vertical orous sace using HAM. Kim [8] investigated the effects of heat and mass transfer in the MHD microolar fluid flow ast a vertical moving late. The role of thermal radiation on the flow and heat transfer rocess is major imortance in the design of many advanced energy conversion systems oerating at higher temeratures. Thermal radiation within the system is the result of emission by hot walls and the working fluid. Effects of chemical reaction and thermal radiation on heat and mass transfer flow of MHD microolar fluid in rotation frame of reference is investigated by Das [9]. A comlete analytic solution to heat transfer of a microolar fluid through a orous medium was analyzed by Rashidi et al. [1].MHD flow of a microolar fluid towards a vertical ermeable late with rescribed surface heat flux is investigated by Nor Azizah et al. [11]. Ezzat et al. [1] steadied the combined heat and mass transfer for unsteady MHD flow of erfect conducting microolar fluid with thermal relaxation. Peristaltic motion of a magneto hydrodynamic micro olar fluid in tube is analyzed by Yongqi Wang et al. [13]. Patil, Kulkarni [14] studied the effects of chemical reaction on free convective flow of a olar fluid through a orous medium in the resence of internal heat generation. Kesavaiah et.al, [15] investigated, effects of the chemical reaction and radiation absortion on an unsteady MHD convective heat and mass transfer flow ast a semi-infinite vertical ermeable moving late. Effects of chemical reaction on unsteady MHD heat and mass transfer flow ast a semi infinite vertical orous moving late in the resence of viscous disation is analyzed by Bhagya Lakshmi[16]. Many researchers [17 ] have studied the roblem of non-newtonian fluid flows analytically and numerically over various flow geometries. In this aer we examine the instantaneous effects of chemical reaction and radiation absortion on unsteady MHD free convective heat and mass transfer flow for a microolar fluid bounded by a semi infinite vertical late in the resence of heat generation and thermal radiation. The late is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts erendicular to the orous surface in which absorbs microolar fluid with a suction velocity varying with time. The dimensionless 7 Page [i- CAMK16] DOI: 1.979/1684-16533714
Thermal radiation and heat transfer effects on MHD microolar fluid flow ast a vertical late with.. governing equations of the flow, heat and mass transfer are solved analytically using regular erturbation method. The effects of various ertinent arameters on flow, heat and mass transfer roerties are discussed analytically and exlained grahically. Also the velocity rofiles of microolar fluid is comared with the corresonding flow roblem for a Newtonian fluid and found that the olar fluid velocity is decreasing. II. Formulation An unsteady two-dimensional hydromagnetic laminar free convection with mass transfer flow of an incomressible, electrically conducting microolar fluid ast a semi-infinite vertical ermeable moving late embedded in a saturated orous medium in the resence of thermal radiation and chemical reaction is considered. The x -axis measured along the orous late, in the uward direction and a magnetic field of uniform strength B is alied in the y -direction which is normal to the flow direction. Since, the motion is two dimensional and length of the late is very large so all the hysical variables are indeendent of x and they are functions of normal distance y and t only. It is assumed that there is no alied voltage which imlies the absence of an electrical field. The transversely alied magnetic field and magnetic Reynolds number are assumed to be very small so that the induced magnetic field is negligible. It is assumed that the hole size of the orous late is significantly larger than a characteristic microscoic length scale of the orous medium. The suction velocity is assumed to be (1 t V ) Ae where A is a real ositive constants and and A are small values less than unity, and V is scale of suction velocity which is non-zero ositive constant. The negative sign indicates that the suction is towards the late. The fluid roerties are assumed to be constant excet that the influence of density variation with temerature and concentration has been considered in the body-force term. The concentration of diffusing secies is very small in comarison to other chemical secies, the concentration of secies far from the wall C, is infinitesimally small and hence the Soret and Dufour effects are neglected. The chemical reactions are taking lace in the flow and all thermo hysical roerties are assumed to be constant. Under these conditions, the governing equations can be written in a Cartesian frame of reference as: v y u u 1 P u ( ) ( ) ( ) r g f T T gc C C t y x y u K Bu j ( ) t y y r y T T K T Q 1 q t y c y c c y r ( T T ) Q ( ) l C C c c c ( D R C C ) t y y The aroriate boundary conditions for the velocity, angular velocity, temerature and concentration fields are u u T T T T e (6) t, w ( w ), (5) (4) (1) () (3) u, ( ) t n C C w CwC e at y (7) y u U U (1 e ), T T, C C, at y (8) t in which t is the time, δ is a scalar constant,u is a scale of free stream velocity. Outside the boundary layer, equation () gives [i- CAMK16] DOI: 1.979/1684-16533714 8 Page
Thermal radiation and heat transfer effects on MHD microolar fluid flow ast a vertical late with.. 1 P du U B U x dt K The radiative heat flux term by using the Rosseland aroximations given by 4 4 T qr 3K y Where 1 T 4T 3T (11) 4 3 4 We introduce the dimensionless variable as follows: u =U u, V,, y, UV u UU y U U U,, V V j s c j t v K, t, T T ( T ) w T, c c c( cw c),, K, V V V Q R B gb f gbc, H, M, Gr ( T ) w T, Gc ( c w c ), D c V V UV UV c Pr, R K K c 4T KK 1, (1) (9) (1) Furthermore, the sin-gradient viscosity γ which, gives same relationshi between the coefficients of viscosity and micro inertia is defined as. ( ) j j (1 ) (13) where, In view of equations (9) (13), the governing equations () (5) reduce to the following dimensionless form: U u t du u u u (1 Ae ) (1 ) Gr GcC M U u t y dt y K y t y y t (1 Ae ) t 1 4R (1 Ae ) (1 ) H Qlc t y Pr 3 y c t c 1 c (1 Ae ) c t y Sc y Where ε = /(+β) = µj / γ (18) The boundary conditions (6) and (8) are then given by the following dimensionless form: t u t u UP, 1 e, n, c 1 e at y (19) y t u U (1 e ),,, c at y () (14) (15) (16) (17) [i- CAMK16] DOI: 1.979/1684-16533714 9 Page
Thermal radiation and heat transfer effects on MHD microolar fluid flow ast a vertical late with.. III. Solution of The Problem In order to reduce the above system of artial differential equations in dimensionless form, we may reresent the linear and angular velocities, temerature and concentration as: t u u ( y) e u ( y) o( ) (1) 1 t ( y) e 1( y) o( ) t ( y) e 1( y) o( ) t ( ) 1( ) ( ) () (3) c c y e c y o (4) By substituting the above equations (1) (4) into equations (14) (17), equating the harmonic and nonharmonic terms and neglecting the higher-order terms of O(ε ), we obtain the following equations. Zeroth- order: (1 ) u u N u N Gr Gcc (5) (6) (3 4 R) 3 Pr( H ) 3 PrQ l c (7) c Scc Scc (8) First-order: (1 ) u u ( N ) u u Gr Gc c A u (9) 1 1 1 1 1 1 A 1 1 1 (3 4 R) 3Pr 3 P ( H) 3 APr 3 Pr Q c (31) 1 1 r 1 l 1 c Scc Scc AScc 1 1 1 Where N = ( M + 1/K) the rimes denote differentiation with resect to y. The corresonding boundary conditions can be written as u U, u, nu, nu, 1, 1, c 1, c 1 at y (33) P 1 1 1 1 1 u 1, u 1,,,,, c, c at y (34) 1 1 1 1 The solutions of equations (5) (3) with satisfying boundary conditions (33) and (34) are given by R1y by 3 b1y y u ( y) 1 a e a e a e a e (35) 1 6 5 t, y y c y t c e c (36) 1 ( y) c11 e y (37) hy Ac 6 11 y 1 ( y) c13 e e (38) by 3 ( y) (1 z ) e by 1 z e (39) 3 3 ( y) c e z e z e z e (4) h4 y by 3 b y b1 y 1 9 4 6 8 ( ) hy 1 c y e (41) c ( y) (1 z ) e z e (4) by b1y 1 3 The Skin friction, Coule stress coefficient,nusselt number and Sherwood number are given by w c f [1 (1 n) ] u () VU c m M w (1 ) () ju (3) (3) 1 4R NuxRe x (1 ) 3 y vx Re x Where y= Sh c y= y 1 xre x ( ) [i- CAMK16] DOI: 1.979/1684-16533714 1 Page
Thermal radiation and heat transfer effects on MHD microolar fluid flow ast a vertical late with.. IV. Results And Discussions For different values of the magnetic field arameter M, the translational velocity and microrotation rofiles are lotted in Fig. 1 and. It is obvious that the effect of increasing values of the arameter M results in a decreasing velocity distribution across the boundary layer. The effect of magnetic field is more rominent at the oint of eak value i.e. the eak value drastically decreases with increases in the value of magnetic field, because the resence of magnetic field in an electrically conducting fluid introduce a force called the Lorentz force, which acts against the flow if the magnetic field is alied in the normal direction, as in the resent roblem. This tye of resisting force slows down the fluid velocity as shown in this figure. The results also show that the magnitude of the microrotation on the orous late increases as M increases. Figure 3 and 4 illustrates the variation of velocity and microrotation distribution across the boundary layer for several values of late moving velocityu in the direction of the fluid flow, resectively. Although we have different initial late moving velocities, the velocity decrease to the constant values for given material arameters. The results also show that the magnitude of microrotation on orous late increases as U increases. For various values of Q, the rofiles of translational velocity and the microrotation across the l boundary layer are shown in Fig.5 and 6. It is clear from the grah that the translational velocity increases with an increase of Q l, which is due to the fact that when heat is absorbed the buoyancy force which accelerated the flow rate. The results also show that the magnitude of microrotation on orous late decreases as Ql increases. The effects of heat generation H on the translational velocity and microrotation rofiles across the boundary layer are resented in Fig.7 and 8. It is shown that the translational velocity across the boundary layer increases with an increasing of H. The results also show that the magnitude of microrotation decreases with an increasing of H. Fig.9 and 1 reresents the temerature rofiles for various values of thermal radiation arameter R and randtl number Pr in the boundary layer. This figure indicate that the effect of thermal radiation is to enhance heat transfer because of the fact that thermal boundary layer thickness increases with increase in the thermal radiation. Thus it is oint out that the radiation should be minimized to have the cooling rocess at a faster rate. But the temerature decreases with increasing the values of Prandtl number Pr in the boundary layer. From this lot it is evident that temerature in the boundary layer falls very quickly for large value of the Prandtl number, because of the fact that thickness of the boundary layer decreases with increase in the value of the Prandtl number. Tyical variations of the temerature rofiles along the sanwise coordinate y are shown in Fig.11 for varies values of Q l.the results show that as an increasing Q l, the temerature rofiles decreases.the temerature rofiles with resect to san wise coordinate y for various values of ζ is shown in Fig.1.The results show that as an increasing of the chemical reaction arameter ζ, the thermal boundary layer increases. Figure 13 and 14 resents tyical rofiles of the velocity and angular velocity for various values of Schmidt number Sc. It is noted from figure that an increase in the value of Sc, leads to a decrease in the velocity of fluid. Further it is observed that the velocity of the olar fluid is found to decrease in comarison with Newtonian fluid (α= and β=). This is due to the fact that as the Schmidt number increases the concentration decreases which leads to decrease in concentration buoyancy effect, hence a decrease in fluid velocity. The oosite behavior is found in the case of angular velocity. The negative values of the angular velocity indicate that the micro rotation of sub structures in the olar fluid is clock-wise. Fig.1. Velocity rofiles for various values of M Fig.. Micro-rotation rofiles for various values of M [i- CAMK16] DOI: 1.979/1684-16533714 11 Page
Thermal radiation and heat transfer effects on MHD microolar fluid flow ast a vertical late with.. Fig.3. Velocity rofiles for various values of U Fig.4. Micro-rotation rofiles for various values of U Fig.5. Velocity rofiles for various values of Q l Fig.6. Micro-rotation rofiles for various values of Q l Fig.7. Velocity rofiles for various values of H Fig.8. Micro-rotation rofiles for various values of H Fig.9. Temerature rofiles for various values of R Fig.1. Temerature rofiles for various values of Pr 1 Page [i- CAMK16] DOI: 1.979/1684-16533714
Thermal radiation and heat transfer effects on MHD microolar fluid flow ast a vertical late with.. Fig.11. Temerature rofiles for various values of Q l, Fig.1. Temerature rofiles for various values of δ Fig. 14. Effects of Schimidt number Sc on Fig. 13. Effects of Schimidt number Sc on angular velocity rofiles velocity rofiles. References [1]. Eringen A.C. Theory of microolar fluids, J.Math.Mech.Vol 16 (1966).1-18. []. Eringen A.C., J.Math.Anal.Al.Vol 38(197) 48-496. [3]. R.C.Sharma, U.Guta, Thermal Convection in microolar fluids in orous medium, Int.J.Eng.Sci.33 (1995) 1887-189. [4]. J. Prata kumar, J.C.Umavathi, J.Chamkha Ali, I.Po, Fully develoed free convective flow of microolar and viscous fluids in a vertical channel, Alied Mathematical Modeling, 34 (1) 1175-1186. [5]. P.Muthu, B.V.Rathish Kumar, Peeyush Chandra, Peristalic motion of microolar fluid in circular cylindrical tubes: Effect of wall roerties, Alied Mathematical Modelling, 3(8) 19-33. [6]. D.Srinivacharya, J.V.Ramana murthy, D.Venugoalam, Unsteady Stokes flow of microolar fluid between two arallel orous lates, Int.J.Engg,Sci.39(1) 1557-1563. [7]. R. Muthuraj, S.Srinivas, Fully develoed MHD flow of a microolar and viscous fluids in a vertical orous sace using HAM, Int.J.of Al. Math and Mech. 6(1) 55-78. [8]. Y.J.Kim, Heat and mass transfer in MHD microolar flow over a vertical moving late in a orous medium, J.Trans Porous Media, 56(4) 17-37. [9]. K. Das, Effects of chemical reaction and thermal reaction on heat and mass transfer flow of MHD microolar fluid in a rotating frame of reference, International Journal of heat and mass mass transfer, 54(11) 355-3513. [1]. M.M. Rashid, S.A. Mohimanian Pour, S.Abbasbandy, Analytic aroximate solutions for heat transfer of a microolar fluid through a orous medium with radiation, Communications in Non-linear science and numeric al simulation, 16(11) 1874-1889. [11]. Nor Azizah, Anular Ishak, MHD flow of a microolar fluid towards a vertical ermeable late with rescribed surface heat flux, Chemical Engineering Research and Design, 89(11) 91-97. [1]. M.Ezzat, A.A. Ei-Bary, S. Ezzat, Combined heat and mass transfer for unsteady MHD flow of erfect conducting microolar fluid with thermal relaxation, Energy conversion and management, 5(11) 934-945. [13]. Yongqi Wang, Nasir Ali, Tasawaar Hayat, Martin Oberlack, Peristalic motion of a magnetohydrodynamic microolar fluid in a tube, Alied Mathematical modeling, 35(11) 3737-375. [14]. P.M. Patil, P.S. Kulkarni, Effects of chemical reaction on free convective flow of a olar fluid through a orous medium in the resence of internal heat generation, International journal of thermal sciences, 47(8) 143-154. [i- CAMK16] DOI: 1.979/1684-16533714 13 Page
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