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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204 Radiation and Mass Transfer Effects on MHD Flow of a Micropolar Fluid towards a Stagnation Point on a Vertical Stretching Sheet E. Manjoolatha, N. Bhaskar Reddy and T.Poornima Department of Mathematics, Sri Venkateswara University, Tirupati - 517502, A.P.ABSTRACT This paper focuses on the study of magnetic field on the micropolar fluid problem aremagnetohydrodynamic mixed convection flow very important.of a micropolar fluid near a stagnation point on The problem of stretching sheet under thea vertical stretching sheet in the presence of influence of magnetic field is also an interestingradiation and mass transfer. Using the problem of research. Such investigations of MHDsimilarity transformations, the governing flows are industrially important and haveequations have been transformed into a system applications in different areas of research such asof ordinary differential equations. The resultant petroleum production, geophysical flows, coolingdimensionless governing equations are solved of underground electric cables etc. It has beenemploying fourth order Runge-Kutta method found that the application of a magnetic fieldalong with shooting technique. The effects of reduces the heat transfer at the stagnation point andvarious governing parameters, namely, material increases the skin friction. These features are usefulparameter, radiation parameter, magnetic in the design of heat shield for protecting theparameter and velocity ratio parameter on the spacecraft entering or re-entering the atmosphere.velocity, microrotation, temperature and Abo-Eldahaband Ghonaim [5] investigatedconcentration, as well as the skin friction, the convective heat transfer in an electricallyrate of heat transfer and the rate of mass conducting micropolar fluid at a stretching surfacetransfer have been computed and shown with uniform free stream. Mohammadein and Gorlagraphically. It is observed that the micropolar [6] investigated the transverse magnetic field onfluid helps in the reduction of drag forces and mixed convection in a micropolar fluid flowing onalso act as a cooling agent. a horizontal plate with vectored mass transfer. They analyzed the effects of a magnetic field withKeywords - Micropolar fluid, Mixed convection, vectored surface mass transfer and inducedStagnation point, Stretching sheet, Radiation, Mass buoyancy stream wise pressure gradients on heattransfer, MHD . transfer. Bhargava et al. [7] studied the numerical solution of the problem of free convection1. INTRODUCTION micropolar fluid flow between two parallel porous In recent years, the dynamics of vertical plates in the presence of magnetic field.micropolar fluids, originating from the theory of Srinivasacharya and Shiferaw [8] presentedEringen [1], has been a popular area of research. numerical solution of the steady conductingThis theory takes into account the effect of local micropolar fluid through porous annulus under therotary inertia and couple stresses arising from influence of an applied uniform magnetic field.practical microrotation action. This theory is Rawat et el. [9] investigated the steady, laminarapplied to suspensions, liquid crystals, polymeric free convection flow of an electrically-conductingfluids and turbulence. This behavior is familiar in fluid between two vertical plates embedded in amany engineering and physical applications. Also, non-Darcy porous medium in the presence ofthe study of boundary layer flows of micropolar uniform magnetic field with heatfluids over a stretching surface has received much generation/absorption and variable thermalattention because of their extensive applications in conductivity effects.the field of metallurgy and chemical engineering Stagnation flow, fluid motion near thefor example, in the extrusion of polymer sheet from stagnation region, exists on all solid bodies movinga die or in the drawing of plastic films. Na and Pop in a fluid. Problems such as the extrusion of[2] investigated the boundary layer flow of a polymers in melt-spinning processes, glassmicropolar fluid past a stretching wall. Desseaux blowing, the continuous casting of metals, and theand Kelson [3] studied the flow of a micropolar spinning of fibers all involve some aspect of flowfluid bounded by a stretching sheet. Hady [4] over a stretching sheet or cylindrical fiber bystudied the solution of heat transfer to micropolar Paullet and weidman [10]. Ishak et al. [11]fluid from a non-isothermal stretching sheet with theoretically studied the similarity solutions for theinjection. However, of late, the effects of a steady magnetohydrodynamic flow towards a 195 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204stagnation point on a vertical surface immersed in studied the oscillating plate-temperature flow of aan incompressible micropolar fluid. Lokendra polar fluid past a vertical porous plate in theKumar [12] has analyzed the effect of MHD flow presence of couple stresses and radiation. Rahmanof micropolar fluid towards a stagnation point on a and Sattar [19] studied transient convective heatvertical stretching sheet. Ishak et al. [13] transfer flow of a micropolar fluid past ainvestigated the steady two-dimensional stagnation continuously moving vertical porous plate withpoint flow of an incompressible micropolar fluid time dependent suction in the presence of radiation.towards a vertical stretching sheet. Abd-El Aziz and Cairohave [20] analyzed the Due to recent advances in space thermal radiation effects on magnetohydrodynamictechnology and nuclear energy the study of natural mixed convection flow of a micropolar fluid past aconvection still continues to be a major area of continuously moving semi-infinite plate for highinterest. The natural convection in fluids with temperature differences. Rahman and Sultana [21]microstructure has been an important area of discussed the radiative heat transfer flow ofresearch. Flows in which buoyancy forces are micropolar fluid with variable heat flux in a porousdominant are called natural and the respective heat medium.transfer being known as natural convection. This Many transport processes exist in natureoccurs at very small velocity of motion in the and in industrial applications in which thepresence of large temperature differences. These simultaneous heat and mass transfer occur as atemperature differences cause density gradients in result of combined buoyancy effects of thermalthe fluid medium and in the presence of diffusion and diffusion of chemical species. A fewgravitational body force, free convection effects representative fields of interest in which combinedbecome important. In forced convection case, the heat and mass transfer plays an important role arenatural convection effects are also present because designing of chemical processing equipment,of the presence of gravitational body force. A formation and dispersion of fog, distribution ofsituation where both the natural and forced temperature and moisture over agricultural fieldsconvection effects are of comparable order is and groves of fruit trees, crop damage due toknown as mixed or combined convection. freezing, and environmental pollution. In this The study of mixed convection is very context, Callahan and Marner [22] considered theimportant in view of several physical problems. In transient free convection flow past a semi-infiniteseveral practical applications, there exist significant vertical plate with mass transfer. Unsteady freetemperature differences between the surface of the convective flow on taking into account the massbody and the free stream. In such flows the flow transfer phenomenon past an infinite vertical plateand thermal fields are no longer symmetric with was studied by Soundalgekar and Wavre[23].respect to stagnation line. The friction factor and Kumar [24] analyzed the effect of heat and masslocal heat transfer rate can be quite different under transfer in the hydromagnetic micropolar fluid flowthese conditions related to the forced convection along a stretching sheet. Bhargava et al. [25]case. Ishak et al [14] studied mixed convection studied coupled fluid flow, heat and mass transferstagnation point flow of a micropolar fluid towards phenomena of micropolar fluids over a stretchinga stretching surface. Hassanien and Gorla [15] have sheet with non-linear velocity.considered the mixed convection in stagnation The aim of the present study is to analyzeflows of micropolar fluids with an arbitrary the effects of magnetohydrodynamic mixedvariation of surface temperature. The problem of convection flow of a micropolar fluid near aunsteady mixed convection in two dimensional stagnation point on a vertical stretching sheet bystagnation flows of micropolar fluids on a vertical taking the radiation and mass transfer into account.flat plate when the plate temperature varies linearly Using the similarity transformations, the governingwith the distance along the plate has been equations have been transformed into a set ofconsidered by Lok et al. [16]. Bhargava et al. [17] ordinary differential equations, and the resultantanalyzed the effect of surface conditions on the equations are solved using Runge-Kutta methodmixed convection micropolar flow driven by a along with shooting technique. The behavior of theporous stretching sheet. velocity, microrotation, temperature and In the context of space technology and in concentration, as well as the skin friction, the ratethe processes involving high temperatures, the of heat transfer and the rate of mass transfer foreffects of radiation are of vital importance. Recent variations in the thermo-physical parameters,developments in hypersonic flights, missile re- namely, material parameter, magnetic parameter,entry, rocket combustion chambers, power plants radiation parameter and velocity ratio parameterfor inter planetary flight and gas cooled nuclear have been computed and discussed in detail.reactors, have focused attention on thermalradiation as a mode of energy transfer, and 2. MATHEMATICAL FORMULATIONemphasized the need for improved understanding of A steady, two-dimensional flow of anradiative transfer in these processes. Ogulu [18] incompressible, electrically conducting micropolar 196 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204fluid near a stagnation point on a vertical heated boundary layer and C - the fluid concentration instretching sheet is considered. The x-axis is taken the boundary layer, -the kinematic viscosity, Talong the sheet and y-axis normal to it. The fluidoccupies the half plane (y>0). It is assumed that the - the thermal expansion coefficient, C - thevelocity of the flow external to the boundary layer coefficient of expansion with concentration, B0 -ue ( x ) , velocity of the stretching sheet u w ( x ) , the the magnetic field of constant strength, q r - thetemperature of the sheet Tw ( x ) and concentration radiative heat flux, D - the coefficient of massof sheet are proportional to the distance x from the diffusivity. Further, μ, κ, ρ, j, N, γ, α and k arestagnation point, i.e. ue ( x) ax, uw ( x ) cx, respectively the dynamic viscosity, vortex viscosityTw ( x) T bx Cw ( x) C dx , where (or the microrotation viscosity), fluid density, and micro-inertia density, microrotation vector (ora, b, c and d are constants, Tw ( x ) T with T angular velocity), spin gradient viscosity, thermalbeing the uniform temperature of the fluid and diffusivity and thermal conductivity. Following the work of many authors, we assume Cw ( x) C with C being the uniform that γ = (μ +κ/ 2) j = μ (1+ K/ 2) j, where K =κ /μconcentration of the fluid. A uniform magnetic field is the material parameter. This assumption isof strength B0 is assumed to be applied in the invoked to allow the field of equations predicts thepositive y-direction normal to the plate. The correct behavior in the limiting case when themagnetic Reynolds number of the flow is taken to microstructure effects become negligible and thebe small enough so that the induced magnetic field total spin N reduces to the angular velocity. The lastis negligible. The level of concentration of foreign terms on the right-hand side of (2.2) represents themass is assumed to be low, so that the Soret and influence of the thermal and solutal buoyancyDufour effects are negligible. Under the usual forces on the flow field and ‘ ’ indicates theboundary layer approximation, the governing buoyancy assisting and the opposing the flowequations are regions, respectively.u v By using the Rosseland approximation (Brewster 0 (2.1) [26]), the radiative heat flux qr is given byx y 4 * T 4 qr (2.7) u u du 2u N 3K yu v ue e 2 x y dx y y where * is the Stefan-Boltzmann constant and B 2 K - the mean absorption coefficient. It should be (u u ) g T (T T ) g C (C C ) 0 noted that by using the Rosseland approximation, e the present analysis is limited to optically thick ( 2.2) fluids. If temperature differences within the flow are sufficiently small, then equation (2.7) can be N N 2 N u 4 linearized by expanding T into the Taylor series ju v 2N x y y y 2 about T , which after neglecting higher order terms (2.3) takes the form T T T qr 2 T 4 4T T 3T 3 4u v (2.4) (2.8) x y y 2 k y In view of equations (2.7) and (2.8), equation (2.4) reduces to C C 2C 16 * T 2Tu v D (2.5) T T 3 x y y 2 u v 1 The boundary conditions for the velocity, x y 3K k y 2 (2.9)temperature and concentration fields are The continuity equation (2.1) is satisfied by the 1 u Cauchy Riemann equationsu uw ( x), v 0, N , T Tw ( x), C Cw ( x) at 2 y u v y0 y and x (2.10)u ue ( x), N 0, T T , C C as where ( x, y ) is the stream function. y In order to transform equations (2.2), (2.6) (2.3), (2.9) and (2.5) into a set of ordinarywhere u and v are the velocity components along differential equations, the following similaritythe x and y axes, respectively, g - the acceleration transformations and dimensionless variables aredue to gravity, T - the fluid temperature in the introduced. 197 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204 c Nu and Sherwood number Sh ,which are defined f ( ) y c x , 1/ 2 , as w xqw T T Cf Nu h( ) N , j , ( ) , u / 22 w , k (Tw T ) , c(c ) x 1/ 2 c Tw T xM n Sh C C B02 Gr D (Cw C ) ( ) , M , 2 , Cw C c Re x where k is the thermal conductivity and, the wall shear stress w , the heat flux q w and mass flux g T (Tw T ) x 3Gr , M n are given by 2 g C (Cw C ) x3 Gc u Gc , , Pr , w ( ) N , 2 2 Re x y y 0 K k ue x T C Sc .R , Re x qw k M n D 4 * T 3 D y y 0 y y 0(2.11) Using the similarity variables (2.11), we obtainwhere f ( ) is the dimensionless stream function, 1 Kθ is the dimensionless temperature, is the C f Re1/ 2 1 f (0), x 2 2dimensionless concentration, η is the similarityvariable, j is the characteristic length, M is the Nu / Re x (0) , Sh / Re1/ 2 (0) 1/ 2 xmagnetic parameter, Gr is the local thermal Our main aim is to investigate how the values of fGrashof number, Gc is the local solutal Grashof ′′(0), (0) and (0) vary in terms of thenumber, Pr is the Prandtl number, is the thermal radiation and buoyancy parameters.buoyancy parameter, is the solutal buoyancy 3 SOLUTION OF THE PROBLEMparameter, Sc is the Schmidt number and R is the The set of coupled non-linear governingradiation parameter. boundary layer equations (2.12) - (2.15) togetherIn view of equations (2.10) and (2.11), the with the boundary conditions (2.16) are solvedequations (2.2), (2.3), (2.9) and (2.5) transform into numerically using Runge-Kutta method along with shooting technique. First of all, higher order non- 2 a a linear differential equations (2.12) - (2.15) are(1 K ) f ff f 2 Kh M ( f ) 0 converted into simultaneous linear differential c c (2.12) equations of first order and they are further transformed into initial value problem by applying(1 K / 2)h fh f h K (2h f ) 0 the shooting technique (Jain et al.[27]). The resultant initial value problem is solved by (2.13) employing Runge-Kutta fourth order method. The 1 R " f f 0 (2.14) step size =0.05 is used to obtain the numerical Pr R solution with five decimal place accuracy as the criterion of convergence. From the process of " Scf Scf 0 (2.15) numerical computation, the skin-friction coefficient, the Nusselt number and the SherwoodThe corresponding boundary conditions are number, which are respectively proportional to f 0, f 1, h 1 f , 1, 1 at 0 f (0), (0) and (0) , are also sorted out 2 and their numerical values are presented in graphs. a f , h 0. as (2.16) 4 RESULTS AND DISCUSSION c In order to get a physical insight of thewhere the primes denote differentiation with problem, a representative set of numerical results isrespect to shown graphically in Figures 1-16, to illustrate theThe physical quantities of interest are the skin influence of physical parameters viz., materialfriction coefficient C f , the local Nusselt number parameter, magnetic parameter, radiation parameter and velocity ratio parameter on the velocity, microrotation, temperature and concentration as 198 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204well as the skin-friction coefficient, Nusselt number decreases for opposing flow. The concentrationand Sherwood number for both the cases of distribution with the variation of material parameterassisting and opposing flows. Throughout the is shown in Fig.8. It has been observed that forcomputations, the Prandtl number and Schmidt assisting flow the concentration increases with thenumber are taken to be 0.71 and 0.22(unless increase in the material parameter while forotherwise stated). opposing flow it decreases. Thus the temperature Fig.1 illustrates the velocity profiles for and concentration can be controlled by the materialdifferent values of the magnetic parameter M. It is parameter. The desired temperature or theobserved that the velocity decreases with an concentration can be generated by controlling theincrease in the magnetic parameter for assisting buoyancy parameters or the material parameter.flow, whereas it increases for opposing flow. The distribution for the velocity with theMaximum for assisting flow and minimum for velocity ratio parameter a/c has been displayed inopposing flow shift away from the wall with an Fig.9. The value of a/c>1 shows that the stretchingincrease in the magnetic parameter. Thus the fluid velocity is less than the free stream velocity whileflow can be effectively controlled by introducing value a/c<1 shows that the stretching velocity isthe magnetic field to the Newtonian fluid. From greater than the free stream velocity. It is observedFig.2, it is noticed that as the magnetic parameter that the velocity increases with an increase in theincreases, the microrotation increases for assisting velocity ratio parameter. Thus the fluid flow can beflow while it decreases for opposing flow. It is also effectively controlled by controlling a/c. Near theclear that for opposing flow the microrotation boundary it is observed that the velocity is higherbecomes negative away from the boundary which for assisting flow as compared to the opposingshows the reverse rotation. For assisting flow the flow. The microrotation profiles for the variation ofmicrorotation is negative near the boundary. velocity ratio parameter a/c are shown in Fig.10. Fig.3 represents the temperature Clearly the angular rotation decreases with andistribution with the variation of magnetic increase in a/c for both assisting and opposingparameter M. For assisting flow it increases with flows. It is obvious from the figure that foran increase in magnetic parameter, while a reverse opposing flow microrotation is higher than thosepattern is observed for opposing flow. Fig.4 depicts for assisting flow. For large values of a/c,the concentration distribution with the variation of microrotation becomes negative which shows themagnetic parameter. For assisting flow it increases reverse rotation.with an increase in magnetic parameter, while a The temperature profiles for the variationreverse pattern is observed for opposing flow. Thus of velocity ratio parameter a/c are presented inthe high temperature or the high concentration can Fig.11. Temperature decreases with an increase inbe controlled by magnetic parameter or the a/c for both assisting and opposing flows. It is clearbuoyancy parameters, which is required in many from the figure that the temperature is higher forengineering applications. opposing flow when compared with the Fig.5 depicts the velocity profiles for temperature for assisting flow. The concentrationdifferent values of the material parameter (K), for profiles for the variation of velocity ratio parameterboth assisting and opposing flows. It is observed a/c are presented in Fig.12. Concentrationthat the velocity decreases with an increase in the decreases with an increase in a/c for both assistingmaterial parameter (K) for assisting flow whereas it and opposing flows. It is clear from the figure thatincreases for opposing flow. Maximum for the concentration is higher for opposing flow whenassisting flow and minimum for opposing flow shift compared with the concentration for assisting flow.away from the wall with increasing material Thus the temperature or the concentration can beparameter. Thus the fluid flow can be effectively effectively controlled by increasing or decreasingcontrolled by introducing micro-constituents to the the stretching or the free stream velocities.Newtonian fluid. Fig.13 depicts the velocity profiles for Fig.6 shows the microrotation distribution different values of the radiation parameter R, forwith the variation of material parameter. For both assisting and opposing flows. It is observedassisting flow the microrotation decreases with an that the velocity decreases with an increase in theincrease in the material parameter while it increases radiation parameter for assisting flow whereas itfor opposing flow. It is clear from the figure that increases for opposing flow. Maximum forfor opposing flow the microrotation becomes assisting flow and minimum for opposing flow shiftnegative near the boundary. For assisting flow the away from the wall with increasing radiationmicrorotation is negative away from the boundary parameter. Fig.14 shows the microrotationwhich shows the reverse rotation. The temperature distribution with the variation of radiationdistribution with the variation of material parameter parameter. For assisting flow the microrotationis shown in Fig.7. It has been observed that for increases with an increase in the radiationassisting flow the temperature increases with the parameter, while it decreases for opposing flow. Itincrease in the material parameter while it is clear from the figure that for opposing flow the 199 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204microrotation becomes negative away from theboundary which shows the reverse rotation. For 1.3assisting flow the microrotation is negative near the M=1,2,3,4boundary. 1.2 Assisting flow Opposing flow - - - The temperature profiles with the variationof radiation parameter are presented in Fig.15. 1.1Temperature decreases with an increase in the 1.0radiation parameter for both assisting and opposing fflows. It is clear from the figure that the 0.9temperature is higher for opposing flow whencompared with the temperature for assisting flow. 0.8Thus the temperature can be effectively controlledby the radiation parameter.The concentration 0.7 M=1,2,3,4profiles for the variation of the radiation parameter 0.6 Pr=0.71,K=2, Sc=0.22, R=a/c=1are presented in Fig.16. Concentration increases 0.0 0.5 1.0 1.5 2.0 2.5 3.0with an increase in the radiation parameter for bothassisting and opposing flows. It is clear from the figure that the concentration is higher for opposingflow when compared with the concentration for Fig.1 Velocity profiles for different Massisting flow. The skin friction, rate of heat transfer andrate of mass transfer with the buoyancy parameter λ 0.5for different values of the radiation parameter R, Assisting flow 0.4are shown graphically in Figs. 17-19. From Fig.17, Opposing flow - - - M=1,2,3,4it is observed that for opposing flow the skin 0.3friction increases with the increase of the radiation 0.2parameter while reverse pattern is observed for 0.1assisting flow. From Fig.18, it is noticed that the h 0.0Nusselt number increases with the increase of the -0.1radiation parameter for both assisting and opposingflows. From Fig.19, it is observed that for opposing -0.2flow the Sherwood number decreases with the -0.3 M=1,2,3,4increase of the radiation parameter while reverse -0.4pattern is observed for assisting flow. K=2, Pr=0.71,, Sc=0.22, R=a/c=1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.05 CONCLUSIONS An attempt is made to study the effects ofradiation and mass transfer on MHD mixed Fig.2 Microrotation profiles for differentconvection flow of a micropolar fluid near a Mstagnation point on a vertical stretching sheet. Thegoverning equations are approximated to a systemof non-linear ordinary differential equations bysimilarity transformation. The behavior of the 1.0velocity, microrotation, temperature and 0.9 Assisting flowconcentration as well as the skin friction, Nusselt 0.8 Opposing flow - - -number and Sherwood number are shown 0.7graphically for various values of the governingparameters. In the buoyancy-assisting flow region, 0.6the skin friction decreases with increasing values of 0.5 M=1,2,3,4the radiation parameter, whereas, the opposite 0.4behavior is observed in the buoyancy-opposing 0.3flow region. For both assisting and opposing flowsthe rate of heat transfer increases with an increase 0.2 M=1,2,3,4in the radiation parameter. In the buoyancy- 0.1assisting flow region, the Sherwood number 0.0 K=2, Pr=0.71,, Sc=0.22, R=a/c=1increases with increasing values of the radiation 0.0 0.5 1.0 1.5 2.0 2.5 3.0parameter, whereas, the opposite behavior is observed in the buoyancy-opposing flow region. Fig.3 Temperature profiles for different M 200 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204 1.0 1.0 0.9 Assisting flow Opposing flow - - - Assisting flow 0.8 0.8 Opposing flow ---- 0.7 0.6 M=1,2,3,4 0.6 K=2,5 0.5 0.4 0.3 0.4 M=1,2,3,4 K=2,5 0.2 0.1 0.2 K=2, Pr=0.71, Sc=0.22, R=a/c=1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 M=0.1, Pr=0.71, Sc=0.22, R=a/c=1 0.0 0 1 2 3 4 Fig.4 Concentration profiles for different M Fig.7 Temperature profiles for different K 1.16 1.14 K=2,3,4,5 1.0 1.12 Assisting flow Opposing flow ---- 1.10 Assisting flow 1.08 0.8 Opposing flow ---- 1.06 1.04 1.02 0.6 f 1.00 0.98 0.96 0.4 K=2,5 0.94 0.92 0.90 0.2 K=2,5 0.88 K=2,3,4,5 0.86 M=0.1, Pr=0.71, Sc=0.22, R=a/c=1 M=0.1, Pr=0.71, Sc=0.22, R=a/c=1 0 1 2 3 4 0.0 0 1 2 3 4 Fig.5 Velocity profiles for different K Fig.8 Concentration profiles for different K 1.6 0.16 a/c=0.2,0.5,1.0,1.5 Assisting flow 1.4 0.12 K=2,3,4,5 Opposing flow ---- 1.2 0.08 0.04 1.0 h f 0.8 0.00 -0.04 0.6 -0.08 0.4 -0.12 K=2,3,4,5 0.2 M=0.1, Pr=0.71, Sc=0.22, R=a/c=1 M=0.1, Pr=0.71,K=2, Sc=0.22, R=1 -0.16 0 1 2 3 4 0 1 2 3 4 Fig.9 Velocity profiles for different a/c Fig.6 Microrotation profiles for differentK 201 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204 1.6 0.4 R=0,0.5,1.0,1.5 Assisting flow a/c=0.2,0.5,1.0,1.5 1.4 0.3 Opposing flow ---- 1.2 0.2 1.0 0.1 f 0.8 h 0.0 -0.1 0.6 -0.2 0.4 R=0,0.5,1.0,1.5 -0.3 0.2 M=0.1, Pr=0.71,K=2, Sc=0.22, R=1 M=0.1,Pr=0.71,K=2,Sc=0.22,a/c=1 -0.4 0 1 2 3 4 0 1 2 3 4 Fig.10 Microrotation profiles for different a/c Fig.13 Velocity profiles for different R 1.0 M=0.1, Pr=0.71,K=2, Sc=0.22, R=1 0.8 0.8 M=0.1, Pr=0.71,K=2, Sc=0.22, a/c=1 a/c=0.2,0.5,1.0,1.5 0.6 Assisting flow Opposing flow ---- R=0,0.5,1.0,1.5 0.6 0.4 0.2 0.4 h 0.0 0.2 -0.2 -0.4 R=0,0.5,1.0,1.5 0.0 0 1 2 3 4 -0.6 0 1 2 3 4 Fig.11 Temperature profiles for different a/c Fig.14 Microrotation profiles for different R 1.0 M=0.1, Pr=0.71,K=2, Sc=0.22, R=1 1.0 M=0.1, Pr=0.71,K=2, Sc=0.22, a/c=1 0.8 Assisting flow 0.8 Opposing flow - - - R=0.5,1.0,1.5 0.6 a/c=0.2,0.5,1.0,1.5 0.6 0.4 0.4 0.2 0.2 0.0 0 1 2 3 4 0.0 0 1 2 3 4 Fig.12 Concentration profiles for different a/c Fig.15 Temperature profiles for different R 202 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204 1.0 M=0.1, Pr=0.71,K=2, Sc=0.22, a/c=1 1 .0 R = 0 ,0 .5 ,1 .0 ,1 .5 Assisting flow 0.8 Opposing flow - - - A s s is tin g flo w r e g io n 0 .5 0.6 - ( 0 ) 0 .0 0.4 O p p o s in g flo w r e g io n -0 .5 0.2 R=0,0.5,1.0,1.5 R = 0 ,0 .5 ,1 .0 ,1 .5 -1 .0 0.0 0 1 2 3 4 -3 -2 -1 0 1 2 3 Fig.19 Sherwood number for different R (M=0.1, K=2, λ=δ a/c=1) Fig.16 Concentration profiles for different R 2 .0 REFERENCES 1 .5 R = 0 ,0 .5 ,1 .0 ,1 .5 [1] Eringen, A. C., (1966), Theory of O p p o s in g flo w r e g io n micropolar fluids, Journal of Mathematics 1 .0 and Mechanics, Vol.16, pp. 1-18. 0 .5 [2] Na, T. Y. and Pop, I., (1997), Boundary-f ( 0 ) 0 . 0 layer flow of micropolar fluid due to a stretching wall, Archives of Applied - 0 .5 Mechanics, Vol.67, No.4, pp. 229-236. A s s is tin g flo w r e g io n [3] Desseaux, A. and Kelson, N. A., (2000), - 1 .0 Flow of a micropolar fluid bounded by a - 1 .5 R = 0 ,0 .5 ,1 .0 ,1 .5 stretching sheet, Anziam J., Vol.42, - 2 .0 pp.536-560. [4] Hady, F. M., (1966), on the solution of - 2 .5 -3 -2 -1 0 1 2 3 heat transfer to micropolar fluid from a non-isothermal stretching sheet with injection, International Journal of Fig.17 skin Numerical Methods for Heat and Fluid friction for different R Flow, Vol.6, No.6, pp. 99-104. (M=0.1, K=2, λ=δ a/c=1) [5] Abo-Eldahab, E. M., and Ghonaim, A. F., (2003), Convective heat transfer in an electrically conducting micropolar fluid at a stretching surface with uniform free 1 .2 stream, Appl. Math. Comput., Vol.137, pp. O p p o s in g flo w re g io n A s s is tin g flo w re g io n 323-336. 1 .0 R = 1 .5 [6] Mohammadein, A. A., and Gorla, R. S. R., R = 1 .0 0 .8 (1998), Effects of transverse magnetic R = 0 .5 field on mixed convection in a micropolar - ( 0 ) 0 .6 fluid on a horizontal plate with vectored mass transfer, Acta Mechanica, Vol.118, 0 .4 pp.1-12. R = 0 .0 [7] Bhargava, R., Kumar, L., and Takhar, 0 .2 H.S., (2003b), Numerical solution of free convection MHD micropolar fluid flow 0 .0 between two parallel porous vertical -3 -2 -1 0 1 2 3 4 plates, Int. J. Engng Sci., Vol.41, pp. 123- 136. [8] Srinivasacharya, D., and Shiferaw, M., (2008), Numerical solution to the MHD Fig.18 Nusselt number for different R flow of micropolar fluid between two 203 | P a g e
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E. Manjoolatha, N. Bhaskar Reddy, T.Poornima / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.195-204 concentric porous cylinders, Int. J. of porous plate in the presence of radiation, Appl. Math and Mech., Vol.4, No.2, pp. Int. J. App. Mech. Engi., Vol.12, No.2, 77-86. pp.497–513.[9] Rawat, S., Bhargava, R. and Anwar Bég, [20] Abd-El Aziz, M. and Cairo, (2006), O., (2010), Hydromagnetic micropolar Thermal radiation effects on free convection heat and mass transfer in a magnetohydrodynamic mixed convection darcy-forchheimer porous medium with flow of a micropolar fluid past a thermophysical effects: finite element continuously moving semi-infinite plate solutions, Int. J. of Appl. Math and Mech., for high temperature differences, Acta Vol.6, No.13, pp. 72-93. Mechanica, Vol.187, pp.113–127.[10] Paullet, J. and Weidman, P., (2007), [21] Rahman, M.M. and Sultana, T., (2008), Analysis of stagnation point flow toward a Radiative Heat Transfer Flow of stretching sheet, International Journal of Micropolar Fluid with Variable Heat Flux Non-Linear Mechanics, Vol. 42, pp.1084– in a Porous Medium, Nonlinear Analysis: 1091. Modelling and Control, Vol. 13, No. 1,[11] Ishak, A., Nazar, R. and Pop, I., (2008b), pp.71–87 Magnetohydrodynamic (MHD) flow of a [22] Callahan G.D. and Marner, W.J., (1976), micropolar fluid towards a stagnation Transient free convection with mass point on a vertical surface”. Comp. Math. transfer on an isothermal vertical flat Appls., Vol.56, pp.3188–3194. plate, Int. J. Heat Mass Transfer, Vol.19,[12] Lokendra Kumar, Singh B., Lokesh pp.165-174. Kumar and Bhargava, R., (2011), Finite [23] Soundalgekar, V.M. and Wavre, P. D., Element Solution of MHD Flow of (1977), Unstedy free convection flow past Micropolar Fluid Towards a Stagnation an infinite vertical plate with constant Point on a vertical stretching sheet, Int. J. suction and mass transfer, Int. J. Heat of Appl. Math and Mech., Vol.7, No.3, Mass Trasfer, Vol.20, pp.1363-1373. pp.14-30. [24] Kumar, L., (2009), Finite element analysis[13] Ishak, A., Nazar, R. and Pop, I., (2006), of combined heat and mass transfer in Mixed convection boundary layers in the hydromagnetic micropolar flow along a stagnation-point flow toward a stretching stretching sheet. Comp. Mater. Sci., 46, vertical sheet. Meccanica, Vol.41, pp. pp. 841-848. 509–518. [25] Bhargava, R., Sharma S, Takhar, H.S.,[14] Ishak, A., Nazar, R. and Pop, I., (2008a), Bég, O. A. and Bhargava, P., (2007), Mixed convection stagnation point flow of Numerical solutions for micropolar a micropolar fluid towards a stretching transport phenomena over a nonlinear sheet, Meccanica, Vol.43, pp.411-418. stretching sheet, Nonlinear Analyis:[15] Hassanien, I.A. and Gorla, R.S.R., (1990), Modelling and Control, Vol.12, pp. 45–63. combined forced and free convection in [26] Brewster, M.Q., (1992), Thermal radiative stagnation flows of micropolar fluids over transfer and properties, John Wiley & vertical non-isothermal surfaces, Int. J. sons. Inc., New York. Engng Sci., Vol.28, pp. 783-792. [27] Jain, M.K., Iyengar, S.R.K. and Jain, R.K.,[16] Lok, Y.Y., Amin, N. and Pop, I., (2006), (1985), Numerical Methods for Scientific Unsteady mixed convection flow of a and Engineering Computation, Wiley micropolar fluid near the stagnation point Eastern Ltd., New Delhi, India. on a vertical surface, Int. J. Therm.Sci., Vol.45, pp.1149-1157.[17] Bhargava, R., Kumar, L. and Takhar, H.S., (2003a), Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet, Int. J. Engng Sci., Vol.41, pp. 2161-2178.[18] Ogulu, A., (2005), On the oscillating plate-temperature flow of a polar fluid past a vertical porous plate in the presence of couple stresses and radiation, Int. comm. Heat Mass Transfer, Vol.32, pp. 1231– 1243.[19] Rahman, M.M. and Sattar, M.A., (2007), Transient convective flow of micropolar fluid past a continuously moving vertical 204 | P a g e
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