G.J. E.D.T.,Vol.3(4):28-40 (July-August, 2014) ISSN:

Transcription

1 THE EFFECTS OF THERMAL RADIATION, HEAT GENERATION, VISCOUS DISSIPATION AND CHEMICAL REACTION ON MHD MICROPOLAR FLUID PAST A STRETCHING SURFACE IN A NON-DARCIAN POROUS MEDIUM B. Lavana 1 & A. Leela Ratnam 1 Department of Mathematics, Priadarshini College of Engineering & Technolog, Nellore. Department of Applied Mathematics, Sri Padmavathi Mahila Visva Vidalaam, Tirupathi. Abstract The main concern of present stud is to investigate the effects of thermal radiation and chemical reaction on a stead to-dimensional laminar flo of a viscous incompressible electricall conducting micropolar fluid past a vertical isothermal stretching surface embedded in a non-darcian porous medium in the presence of viscous dissipation and heat generation. The governing equations of momentum, angular momentum, energ, and species equations are solved numericall using Runge-Kutta fourth order method ith the shooting technique. The effects of various parameters on the velocit, microrotation, temperature and concentration field as ell as skin friction coefficient, Nusselt number and Sherood number are shon graphicall and tabulated. It is observed that the micropolar fluid helps in the reduction of drag forces and also acts as a cooling agent. An excellent agreement is observed beteen some of the obtained results of the current stud and those of previousl published studies. Introduction The free convection processes involving the combined mechanism of heat and mass transfer are encountered in man natural processes, in man industrial applications and in man chemical processing sstems. The stud of free convective mass transfer flo has become the object of extensive research as the effects of heat transfer along ith mass transfer effects are dominant features in man engineering applications such as rocket nozzles, cooling of nuclear reactors, high sinks in turbine blades, high speed aircrafts and their atmospheric re-entr, chemical devices and process equipments. Ostrach [1], the initiator of the stud of convection flo, made a technical note on the similarit solution of transient free convection flo past a semi infinite vertical plate b an integral method. Sakiadis [] analzed the boundar laer flo over a solid surface moving ith a constant velocit. This boundar laer flo situation is quite different from the classical Blasius problem of boundar flo over a semi-infinite flat plate due to en- trainment of ambient fluid. Erickson et al. [3] extended the ork of Sakiadis for suction or injection of a smooth surface. The flo over a stretching surface is an important problem in man engineering processes ith applications in industries such as extrusion, melt-spinning, the hot rolling, ire draing, glass fiber production, manufacture of plastic and rubber sheets, cooling of a large metallic plate in a bath, hich ma be an electrolte, etc. In industr, polmer sheets and filaments are manufactured b continuous extrusion of the polmer from a die to a indup roller, hich is located at a finite distance aa. In man environmental and industrial flos the classical theor of Netonian fluids is unable to explain the microfluid mechanical characteristics observed. Micropolar fluids are fluids ith microstructure belonging to a class of complex fluids ith nonsmmetrical stress tensor referred to as micromorphic fluids. Phsicall the represent man industriall important liquids consisting of randoml oriented particles suspended in a viscous medium. The classical theories of continuum mechanics are inadequate to explicate the microscopic manifestations of such complex hdrodnamic behaviour. Eringen [4] presented the earliest formulation of a general theor of fluid microcontinua taking into account the inertial characteristics of the substructure particles, hich are alloed to sustain rotation and couple stresses. Later Eringen [5] generalized the theor to incorporate thermal effects in the so-called thermo micropolar fluid. The theor of micropolar fluids and its extension, the thermo micropolar fluid constitute suitable non-netonian hdrodnamic and thermo-hdrodnamic models hich can simulate the flo dnamics of colloidal fluids, liquid crstals, polmeric suspensions, haemotological fluids etc. Man numerical studies of micropolar heat and mass transfer have been communicated in the literature. Hassanien and Gorla [6] investigated the heat transfer to a micropolar fluid from a non-isothermal stretching sheet ith suction and bloing. Flo over a porous stretching sheet ith strong suction or injection as examined b Kelson and Farell [7]. Transport of momentum and thermal energ in fluid saturated porous media ith lo porosities, such as rocks, soil, sand, etc., is commonl described b using Darc s model for conservation of momentum and b using an energ equation based on the velocit field found from this model [8 ]. In contrast to rocks, soil, sand and other media that do fall in this categor, certain porous materials, such as foam metals and fibrous media, usuall have high porosit. Raptis [9] studied the boundar laer flo of a micropolar fluid through a non-darcian porous medium. Magnetoconvection plas an important role in agriculture, petroleum industries, geophsics and in astrophsics. Important applications are found in the stud of geological formations, in exploration and thermal recover of oil and in the assessment of aquifers, geothermal reservoirs and underground nuclear aste storage sites. MHD flo has applications in metrolog, solar phsics and in motion of the earth s core. Also, it has applications in the field of stellar and planetar magnetospheres, aeronautics, chemical engineering and electronics. The effects of a transversel applied 8

2 magnetic field on the flo of an electricall conducting fluid past an impulsivel started infinite isothermal vertical plate ere studied b Soundalgekar et al. [10]. MHD effects on impulsivel started vertical infinite plate ith variable temperature in the presence of a transverse magnetic field ere studied b Soundalgekar et al. [11]. The dimensionless governing equations ere solved using the Laplace transform technique. Radiative heat and mass transfer pla an important role in manufacturing industries for the design of fins, steel rolling, nuclear poer plants, and gas turbines. Various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering applications. If the temperature of the surrounding fluid is rather high, radiation effects pla an important role and this situation exists in space technolog. In such cases, one has to take into account the effect of thermal radiation and mass diffusion. England and Emer [1] studied thermal radiation effects of an opticall thin gra gas bounded b a stationar vertical plate. Radiation effects on mixed convection along an isothermal vertical plate ere studied b Hossain and Takhar [13]. Raptis and Perdikis [14] studied the effects of thermal radiation and free convection flo past a moving vertical plate, the governing equations ere solved analticall. Das et al. [15] analzed radiation effects on flo past an impulsivel started infinite isothermal vertical plate. Haat and Qusim [16 ] proposed the effects of thermal radiation on MHD flo of a micropolar fluid ith mass transfer. The radiation effect on stead free convection flo near isothermal stretching sheet in the presence of a magnetic field is studied b Ahmed [17]. The stud of heat and mass transfer ith chemical reaction is of great practical importance in man branches of science and engineering. Das et al. [18]studied the effects of mass transfer flo past an impulsivel started infinite vertical plate ith constant heat flux and chemical reaction. Anjalidevi and Kandasam [19] studied effects of chemical reaction, heat and mass transfer on laminar flo along a semi-infinite horizontal plate. Pal and Chatterjee [0] studied heat and mass transfer in MHD non-darcian flo of a micropolar fluid over a stretching sheet embedded in a porous media ith nonuniform heat source and thermal radiation. Chamkha et al. [ 1] studied the coupled heat and mass transfer b MHD natural convection of micropolar fluid about a truncated cone in the presence of radiation and chemical reaction. Intensive studies have been carried out to investigate effects of chemical reaction on different flo tpes [ -5 ]. Vajravelu and Rollins [6] studied the heat transfer characteristics in an electricall conducting fluid over a stretching sheet ith variable all temperature and internal heat generation or absorption. Mostafa and Shimaa [7] studied the MHD flo and heat transfer of a micropolar fluid over a stretching surface ith heat generation and slip velocit. Jat et al. [8] studied the MHD flo and heat transfer near the stagnation point of a micropolar fluid over a stretching surface ith heat generation/absorption. Abo-Eldahb and El Aziz [9] found the heat transfer in a micropolar fluid past a stretching surface embedded in a non-darcian porous medium ith heat generation. Mohammed Ibrahim [ 30] proposed the radiation and mass transfer effects on MHD free convection flo of a micropolar fluid past a stretching surface placed in a non-darcian porous medium in presence of heat generation. In all the above papers viscous dissipation is neglected. But hen the motion is under strong gravitational field, or flo field is of extreme size, the viscous dissipative heat cannot be neglected. Rahman [31] analzed the stead laminar free-forced convective flo and heat transfer of micropolar fluids past a vertical radiate isothermal permeable surface in the presence of viscous dissipation and Ohmic heating. Abel et al. [3] studied the MHD flo, and heat transfer ith effects of buoanc, viscous and Joules dissipation over a nonlinear vertical stretching porous sheet ith partial slip. Hsiao and Lee [33] analzed the conjugate heat and mass transfer for MHD mixed convection ith viscous dissipation and radiation effect for viscoelastic fluid past a stretching sheet. Gebhart [34] has shon that the viscous dissipation effect plas an important role in natural convection in various devices processes on large scales (or large planets). Also, he pointed out that hen the temperature is small, or hen the gravitational field is of high intensit, viscous dissipations is more predominant in vigorous natural convection processes. Govardhan et al. [35] studied the radiation effect on MHD stead free convection flo of a gas at a stretching surface ith a uniform free stream ith viscous dissipation. Salem [36 ] investigated the effects of viscous dissipation and chemical reaction on MHD micropolar fluid along a permeable stretching sheet in non-darcian porous medium ith variable viscosit. Mahmoud [37] found that the effects of viscous dissipation and heat generation on MHD flo of a micropolar fluid over a moving permeable surface embedded in a non-darcian porous medium. Hoever the interaction of chemical reaction ith thermal radiation of an electricall conducting micropolar fluid past a stretching surface has received little attention. Hence an attempt is made to investigate the thermal radiation effects on a stead free convection flo near an isothermal vertical stretching sheet in the presence of a magnetic field, a non- Darcian porous medium, viscous dissipation and heat generation. The governing equations are transformed b using similarit transformation and the resultant dimensionless equations are solved numericall using the Runge-Kutta fourth order method ith the shooting technique. The effects of various governing parameters on the velocit, temperature, concentration, skin-friction coefficient, the Nusselt number, and Sherood number are shon in the figures and tables and analzed in detail. Mathematical Formulation Let us consider a stead, to-dimensional laminar, free convection boundar laer flo of an electricall conducting dissipative and heat generating micropolar fluid through a porous medium bounded b a vertical isothermal stretching sheet coinciding ith the plane = 0, here the flo confined to > 0. To equal and opposite forces are introduced along the x - axis so that the sheet is linearl stretched keeping the origin fixed ( See Figure A). A uniforml distributed transverse magnetic field of strength B0 is imposed along the - axis. The magnetic Renolds number of the flo is taken to be small enough so that the induced distortion of the applied magnetic field can be neglected. It is also assumed that microscopic inertia term involving J ( here J is the square of the characteristic length of microstructure) can be neglected for stead to-dimensional boundar laer flo in a micropolar fluid ithout introducing an 9

3 appreciable error in the solution. Under the above assumptions and upon treating the fluid saturated porous medium as continuum, including the non-darcian inertia effects, and assuming that the Boussinesq approximation is valid, the boundar laer form of the governing equations can be ritten as ( Willson [38], Nield and Bejan [39]). Continuit equation Figure A. Sketch of the phsical model u v 0 x (1) Momentum Equation B u v g T T g C C k u u C u x K u u u * () Angular Momentum equation u G1 0 (3) Energ equation T T ke T 1 qr Q0 u u v T T x cp cp cp cp (4) Species equation C C C u v D K r C C x (5) Subject to the boundar conditions u bx, v 0, T T, C C, 0 at = 0, u u, T T, C C, 0 as (6) here x and are the coordinates along and normal to the sheet. u and v are the components of the velocit in the x and - directions, respectivel.,k1 and G 1 are the microrotation component, coupling constant, and microrotation constant, respectivel. k,, e C1 K are the effective thermal conductivit, permeabilit of the porous medium, transport propert related to the inertia effect. T is fluid temperature, C is fluid concentration. T is the surface temperature, C is the surface concentration, T be the ambient temperature of fluid, C is the ambient * concentration of fluid,,,u and g are the coefficient of thermal expansion, coefficient of concentration expansion, free steam velocit, and acceleration due to gravit, respectivel. 0 densit, be the kinematic viscosit, be the dnamic viscosit, cp be the specific heat at constant pressure of the 30 be the electrical conductivit, be the fluid

4 fluid, Q0 be the volumetric rate of heat generation, b be the constant, D be the diffusion coefficient, and Kr is the chemical reaction parameter. B using the Rosseland approximation ( Brester [40] ), the radiative heat flux in direction is given b q r 4 * 3k T * 4 (7) * here is the Stefan-Boltzmann constant and equation (4) becomes * k is the mean absorption coefficient. B using equation (7), the energ * 4 T T ke T 4 T Q0 u u v * T T x cp 3k cp cp cp (8) It is convenient to make the governing equations and conditions dimensionless b using bx b u v u x, R, u, v R, R, u u u u c TT, T T C C C C M Pr B b 0 0, c p k g T T Gr bu * 3 * kk F, 4 T T T r T,, Gc * g C C p bu u Ec Q c ( T T ), Q bc Da Kb Cu b (9) p Sc D Kr Kr b here R is the Renolds number. In vie of the equation (8), the equations (1), (), (3), (8) and (5) reduce to the folloing non-dimensional form. u v 0 x (10) u u u 1 x Da u v Gr Gc N M u u (11) u G 0 (1) u v r r r Q Ec u x Pr 3F Pr (13) 1 u v Kr x Sc (14) The corresponding boundar conditions are u = x, v = 0, 0, 1, 1 at = 0, u = 1, 0, 0, 0 as (15) Proceeding ith the analsis, e define a stream function ( x, ) such that 31

5 u, v (16) x No, let us consider the steam function as if ( x, ) f ( ) xg( ) (17) xh( ) (18) In vie of equation (16) (18), the continuit equation (10) is identicall satisfied and the momentum equation (11), angular momentum equation (1), energ equation (13) and species equation (14) becomes 3 1 Gr Gc M 3 x x Da (19) u G 0 (0) r 3r 1 r Q Ec x x Pr 3F Pr (1) 1 Kr x x Sc () and the boundar conditions (15) become x 1, 0 x, 0, h = 0, 1, 1 at = 0 h, 0, 0 as (3) In equations (19), (0), (1), () and equating coefficient of differential equations 0 x and 1 x, e obtain the coupled non-linear ordinar 1 0 Da f f g f g M f f Gr Gc (4) g gg g M g f g Nh Da 1 0 (5) Gh h g 0 (6) 3 3F 4 1 r 3Pr Fg 1r 1 r 3F Pr Q 3F Pr Ecf 0 (7) Scg KrSc 0 (8) here a prime denotes differentiation ith respect to. In vie of equations (17), (18), the boundar condition (3) reduce to f 0, f 0, g 0, g 1, h 1, 1, 1 at 0 f 1, g 0, h 0, 0, 0 as (9) 3

6 The phsical quantities hich are of importance for this problem are the skin-friction coefficient, Nusselt number and Sherood number, hich are defined b The shear stress at the stretching surface is given b u u k k br 0 = br f (0) xg (0) 0 0 (30) (31) 1 u R The skin-friction coefficient C f (0) xg (0) The all heat flux is given b 0 0 f T br T br q k k k T T (0) u u Nusselt number q br Nu (0) k T T u (3) The all mass flux is given b C br C br m D D D C C (0) u u 0 0 Sherood number m br Sh (0) D C C u (33) Solution of the Problem The shooting method for linear equations is based on replacing the boundar value problem b to initial value problems and the solutions of the boundar value problem is a linear combination beteen the solutions of the to initial value problems. The shooting method for the nonlinear boundar value problem is similar to the linear case, except that the solution of the nonlinear problem cannot be simpl expressed as a linear combination of the solutions of the to initial value problems. Instead, e need to use a sequence of suitable initial values for the derivatives such that the tolerance at the end point of the range is ver small. This sequence of initial values is given b the secand method, and e use the fourth order Runge-Kutta method to solve the initial value problems. The full equations (4) (8) ith the boundar conditions (9) ere solved numericall using the Runge-Kutta method algorithm ith a sstematic guessing f (0), g (0), h(0), (0) and (0) b the shooting technique until the boundar conditions at infinit f ( ) deca exponentiall to one, also g (0), h( ), ( ) and ( ) to zero. The functions f, g, h, and are shon in figures. Results and Discussion As a result of the numerical calculations, the dimensionless velocit, angular velocit, temperature, and concentration distributions for the flo under consideration are obtained and their behavior has been discussed for variations in the governing parameters, namel, the thermal Grashof number Gr, solutal Grashof number Gc, magnetic field parameter, Darc number Da,, porous medium inertia coefficient, vortes viscosit parameter N, microrotation parameter G, Prandtl number Pr, radiation parameter, the parameter of relative difference beteen the temperature of the sheet and temperature far r, heat generation parameter Q, Eckert number Ec, Schmidt number Sc, and chemical reaction parameter Kr. In the present stud, the folloing default parametric values are adopted: Gr =, Gc =, M = 1, Da = 100, = 0.1, N = 0.1, G =.0, Pr = 0.71, F =, r = 5, Q = 0.1, Ec = 5, Sc =, Kr = 0.5. All graphs therefore correspond to these unless specificall indicated on the appropriate graph. In order to assess the accurac of our computed results, the present results have been compared ith Abo-Eldahab and ELaziz [30] for different values of G as shon in Figure 1 ith Gc =, =, =, Ec =, Sc = and Kr =. It is observed that the agreements ith the solution of angular velocit profiles are excellent. 33

7 Figure (a) shos the variation of the dimensionless velocit component for several sets of values of thermal Grashof number Gr. As expected, it is observed that there is a rise in the velocit due to enhancement of thermal buoanc force. The variation of the dimensionless velocit component for several sets of values of solutal Grashof number Gc is depicted in Figure (b). As expected, the fluid velocit increases and the peak value is more distinctive due to the increase in the species buoanc force. It should be mentioned herein that the profiles of g, h, and ere found to be insensible to change in Gr and Gc, therefore, not shon herein for brevit. The effect of variation of the magnetic parameter on the velocit ( and ), angular velocit h, temperature, and concentration profiles is presented in Figures 3(a) 3(e) respectivel. It is ell knon that the application of a uniform magnetic field normal to the flo direction gives rise to a force called Lorentz. This force has the tendenc to slo don the velocit of the fluid and angular velocit of microrotation in the boundar laer and to increase its temperature and concentration. This is obvious from the decreases in the velocit profiles, angular velocit of microrotation profiles, hile temperature and concentration profiles increases, presented in Figures. 3(a) 3(e). Figures 4(a) 4(e) present tpical profiles for the variables of the fluid s -component of velocit ( and ), angular velocit h, temperature, and concentration for different values of Darc number Da. It is noted that values of Da increase the fluid velocities and angular velocit increases, hile temperature and concentration of the fluid decrease. Figures 5(a) 5(e) present the tpical profiles for the variables of the fluid s x-component of velocit ( and ), angular velocit h, temperature and concentration for different values of the porous medium inertia coefficient. Obviousl, the porous medium inertia effects constitute resistance to the flo. Thus as the inertia coefficient increases, the resistance to the flo increases, causing the fluid flo in the porous medium to slo don and the temperature and concentration increase and, therefore, as increases,, and h decreases hile the temperature and concentration increase. Figures 6(a) 6(e) present the tpical profiles for the variables of the fluid s -component of velocit ( and ), angular velocit h, temperature, and concentration for different values of the vortex viscosit parameter. Increases in the values of have a tendenc to increase, h,, and and to decrease. Figure 7 is a plot of the dimensionless angular velocit h profiles for different values of the presence of the microrotation parameter. The curves illustrate that, as the values of increases, the angular velocit h, as expected, decreases ith an increase in the boundar laer thickness as the maximum moves aa from the sheet. Of course, hen the viscosit of the fluid decreases the angular velocit of additive increases. Figure 8(a) Illustrates the dimensionless velocit component for different values of the Prandtl number Pr. The numerical results sho that the effect of increasing values of the Prandtl number results in a decreasing velocit. From Figure 8(b), it is observed that an increase in the Prandtl number results in a decrease of the thermal boundar laer thickness and in general loer average temperature ithin the boundar laer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse aa from the heated plate more rapidl than for higher values of Pr. Hence in the case of smaller Prandtl numbers as the boundar laer is thicker, the rate of heat transfer is reduced. The effect of the radiation parameter on the dimensionless velocit component f and dimensionless temperature is shon in Figures 9 (a) and 9 (b) respectivel. Figure 9 (a) shos that velocit component decreases ith an increase in the radiation parameter. From Figure 9(b) it is seen that the temperature decreases as the radiation parameter increases. This result qualitativel agrees ith expectations, since the effect of radiation is to decrease the rate of energ transport to the fluid, thereb decreasing the temperature of the fluid. The influence of the parameter of relative difference beteen the temperature of the sheet and the temperature far aa from the sheet on dimensionless velocit and temperature profiles are plotted in Figures 10(a) and 10(b), respectivel. Figure 10(a) shos that dimensionless velocit increases ith an increase in. It is observed that the temperature increases ith an increase in r (Figure 10(b)). Figures 11(a) and 11(b) illustrate the respective changes in the profiles of and as the heat generation coefficient is changed. It is clear from Figures 11(a) and 11(b) that increasing in the values of produces increases in the velocit and temperature distributions of the fluid. This is expected since heat generation ( Q 0) causes the thermal boundar laer to become thicker and the fluid to be armer. This enhances the effects of thermal buoanc of the driving bod force due to mass densit variations hich are coupled to the temperature distribution and therefore increasing the fluid velocit distribution. No figures for, h and are presented for the same reason as mentioned before. Figures 1 (a) and 1(b) displa the respective changes in the profiles of and as the Eckert number Ec is changed. The positive Eckert number implies cooling of the plate i.e., loss of heat from the plate to the fluid. Hence, greater viscous dissipative heat causes a rise in the temperature as ell as the velocit, hich is evident from Figures 1(a) and 1(b). The influence of the Schmidt number Sc on the dimensionless velocit f and concentration profiles is plotted in Figures 13(a) and 13(b), respectivel. As the Schmidt number increases the concentration decreases. This causes the concentration buoanc effects to decrease ielding a reduction in the fluid velocit. The reductions in the velocit and concentration profiles are accompanied b simultaneous reductions in the velocit and concentration boundar laers. These behaviors are clear from Figures 13(a) and 13(b). 34

8 - h The effects of the chemical reaction parameter Kr on dimensionless velocit component f and concentration profiles are plotted in figures 14(a) and 14(b). As the chemical reaction parameter increases the velocit and concentration profiles are decreases. These behaviors are clear from Figures 14(a) and 14(b). Finall, in order to verif the proper treatment of the present problem, e ill compare the obtained numerical solution ith results reported b Abo-Eldahab and El Aziz [30]. Table 1 presents comparisons for the velocities of (0) and (0) values for various values. These comparisons sho excellent agreement beteen the results. Table illustrates the missing all functions for velocit, angular velocit, temperature, and concentration functions. These quantities are useful in the evaluation of all shear stresses, gradient of angular velocit, surface heat transfer rate, and mass transfer rate. The results are obtained for = 5 and different values of the thermal Grashof number Gr, solutal Grashof number Gc, magnetic field parameter, Darc number Da, porous medium inertia coefficient, vortes viscosit parameter, microrotation parameter, radiation parameter, the parameter of relative difference beteen the temperature of the sheet and temperature far aa from the sheet, Prandtl number Pr, heat generation parameter Q, Eckert number Ec, and Schmidt number Sc. Table 1 indicates that increasing the values of the Grashof number Gr and solutal Grashof number Gc results in an increase in the values of (0). This is because as Gr and Gc increase, the momentum boundar laer thickness decreases and, therefore, an increase in the values of (0) occurs. The results indicate that a distinct fall in the skin-friction coefficient in the -direction ( (0) and (0)), the surface heat transfer rate (0), and mass transfer rate (0), hile gradient of angular velocit h (0) increases accompanies a rise in the magnetic field parameter. Increases in the values of Da have the effect of increasing the skin-friction function (0), heat transfer rate (0) and mass transfer rate (0) hile the gradient of angular velocit h (0) and the skin-friction function (0) slightl decreases as Da increases. Further, the influence of the porous medium inertia coefficient on the all shear stresses, gradient of angular velocit, surface heat transfer, and surface mass transfer rate is the same as that of the inverse Darc number Da, since it also represents resistance to the flo. Namel, as increases, (0), (0), (0) decrease hile (0), h (0) slightl increases, respectivel. From Table 3 for given values of Gr, Gc,, Da,, Sc and Kr an increase in the values of microrotation parameter leads to reduction in the skin-friction function (0), (0) and (0) hile the skin-friction function (0) and gradient of angular velocit h (0), increase as increases. The skin friction (0) increases and the gradient of angular velocit h (0) is decreased as the microrotation parameter increases, hile the skin-friction coefficient in the x-directions (0) heat transfer rate (0), and mass transfer rate (0) are insensible to change in G. Increasing the values of heat generation parameter or Eckert number Ec results in an increase in the values of (0) and the heat transfer rate (0) decreases. It is observed that the magnitude of the all temperature gradient increases as Prandtl number Pr or radiation parameter increases. From Table 4, the magnitude of the all concentration increases ith an increase in the Schmidt number Sc or chemical reaction parameter Kr. Furthermore, the negative values of the all temperature and concentration gradients, for all values of the dimensionless parameters, are indicative of the phsical fact that the heat flos from the sheet surface to the ambient fluid M=1, Q = 0.1, Da=100,N =0.1,Pr=, Gr=0.5,=1. Present Result Abo-Eldahab & Aziz G = G = G = 6.0 / - h g / / Gr =, 0.5,,.0.0 Gc =, 0.5,, Fig.1: Comparison of angular velocit Profiles. Fig (a) Variation of the velocit component ith Gr Fig (b) Variation of the velocit component ith Gc M =, 0.1, 0.3, 0.5 / M =, 1,, 5 M =, 0.1, 0.5, Fig.3(a) Variation of the velocit component ith Fig. 3(b) Variation of the velocit component ith Fig 3(c) Variation of the velocit component h ith

9 M =, 0.5,,.0 1. M =, 0.5,, / g / -h -h g / g / -h 0.9 Da = 10, 5, 50, Fig 3(d) Variation of the velocit component ith. Fig 3(e) Variation of the velocit component ith. Fig. 4(a) Variation of the velocit component ith Da Da 10, 5, 50, 100 Da = 10, 5, 50, 100 Da = 10, 5, 50, Fig. 4(b) Variation of the velocit component ith Da Fig 4(c) Variation of the velocit component h ith Da Fig. 4(d) Variation of the velocit component ith Da.0 1. Da = 10, 5, 50, 100 / =, 1, 5, 0.1 =, 1, 5, Fig 4(e) Variation of the velocit component ith Da Fig. 5(a) Variation of the velocit component ith Fig 5(b) Variation of the velocit component ith =, 1, 5, 0.1 =, 1, 5, =, 1, 5, Fig. 5(c) Variation of the velocit component - h ith Fig 5(d) Variation of the velocit component ith Fig 5(e) Variation of the velocit component ith N =, 0.5,, / N =, 0.5,,.0 N =, 0.5,, Fig 6(a) Variation of the velocit component ith Fig 6(b) Variation of the velocit component ith Fig 6(c) Variation of the velocit component - h ith.

10 -h G =,.0, 5.0, 1 N =, 0.5,,.0 N =, 0.5,, Fig 6(d) Variation of the velocit component ith Fig 6(e) Variation of the velocit component ith Fig. 7 Variation of the velocit component - h ith / Pr =, 0.71,, / 0.9 F =,, 5.0, 1 Pr =, 0.71,, Fig 8(a) Variation of the velocit component ith Pr Fig 8(b) Variation of the velocit component ith Pr Fig 9(a) Variation of the velocit component ith F =,, 5.0, 1 1. r =, 0.1, 0.3, 0.5 / 0.9 r =, 0.1, 0.3, Fig 9(b) Variation of the velocit component ith Fig 10(a) Variation of the velocit component ith Fig 10(b) Variation of the velocit component ith Q =, 0.1, 0.3, / Q =, 0.1, 0.3, / 0.9 Ec =, 0.1, 0.3, Fig 11(a) Variation of the velocit component ith Q Fig 11(b) Variation of the velocit component ith Q Fig 1(a) Variation of the velocit component f ith Ec.0 Ec =, 0.1, 0.3, / Sc = 0.,, 0.78, 0.94 Sc = 0.,, 0.78, Fig 1(b) Variation of the velocit component ith Ec Fig 13(a) Variation of the velocit component f ith Sc Fig 13(b) Variation of the velocit component ith Sc. 37

11 .0 1. / Kr =, 0.5,,.0 Kr =, 0.5,, Fig 14(a) Variation of the velocit component f ith Kr Fig 14(b) Variation of the velocit component ith Kr Table 1 Comparison of the present results ith the literature results given b Abo-Eldahab E M and Aziz M A El [3], Gc =, F =, r =, Sc =. Pr =, Da = 100, = 1, N1 = 0.1, G =, Gr = 0.5, Q = 0.1. M f (0) g (0) Present Abo-Eldahab [30 ] Present Abo-Eldahab[30] Table Variation of f, g, h, and at the plate ith Gr, Gc, M, Da and for N = 0.1, G =.0, Pr = 0.71, F =, r = 5, Q = 0.1, Ec = 5, Sc =, Kr = 0.5. Gr Gc M Da f (0) g (0) h (0) (0) (0) Table 3 Variation of f, g, h, and at the plate ith G, Pr, N, F, Q, Ec for Gr =, Gc =, M = 1, Da = 100, Sc =, Kr = 0.5. G Pr N F Q Ec f (0) g (0) h (0) (0) (0)

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