International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)

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International Association of ientific Innovation and Research (IASIR) (An Association Unifying the iences, Engineering, and Alied Research) International Journal of Emerging Technologies in

net ISSN (Print): 79-0047 ISSN (Online): 79-0055 Newtonian Heating and Mass Transfer Effects of Chemical Reactive Fluid on Natural Convection Flow Past a Vertical Surface with Radiation Arita Jain

1 International Association of ientific Innovation and Research (IASIR) (An Association Unifying the iences, Engineering, and Alied Research) International Journal of Emerging Technologies in Comutational and Alied iences (IJETCAS) ISSN (Print): ISSN (Online): Newtonian Heating and Mass Transfer Effects of Chemical Reactive Fluid on Natural Convection Flow Past a Vertical Surface with Radiation Arita Jain Head, Deartment of Mathematics, JECRC UDML College of Engineering, Jaiur- 3008, Rajasthan, INDIA Abstract: The aim of this aer is to investigate the influence of chemical reaction and the combined effects of Newtonian heating and mass transfer on laminar boundary layer flow over an accelerated vertical late in the resence of radiation. The governing equations are solved analytically by Lalace-transform technique. Grah results are resented for temerature, velocity, skin friction. The effects of various arameters on flow variables are illustrated grahically and the hysical asect of the roblem is discussed. Keywords: Chemical reaction; Newtonian heating; radiation; accelerated late. I. INTRODUCTION In many cases in the rocess of free convection, chemical reaction takes lace due to the resence of foreign masses (as imurities) in fluid. This tye of chemical reaction may change the temerature and the heat content of the fluid and may affect the free convection rocess. Flow ast a vertical late with chemical reaction is analyzed by Fayez [1] and Sarada et. al. [] under different hysical conditions. Bhaben et. al. [3] analyzed chemical reaction effects on flow ast a vertical late with variable temerature. In many ractical situations where the heat transfer from the surface is taken to be roortional to the local surface temerature. Such tyes of roortionally condition of the heat transfer to the local surface temerature is termed as Newtonian heating. Unsteady boundary layer flow ast a vertical late with Newtonian heating is elucidated by Chaudharyet.al.[4]. Chemical reaction and mass transfer effects on flow ast a surface with Newtonian heating is analyzed by Rajesh [5]. In the above mentioned studies, the effects of radiation on flow has not been considered. Actually, many rocesses in new engineering areas occur at high temerature and knowledge of radiative heat transfer becomes imerative for the design of the ertinent equiment. Nuclear ower lants, gas turbines and the various roulsion devices for aircraft, missiles, satellites, and sace vehicles are examles of such engineering areas. Thermal radiation effect on flow ast a vertical late with mass transfer is elucidated by Rajut et. al. [6]. Natural convective flow ast a late in the resence of radiation is studied by Chaudhary et. al. [7].Radiation and Newtonian heating effects on flow ast a vertical late under different hysical conditions are analyzed by Narahari et. al. [8] and Das et.al. [9]. The aim of the resent work is to rovide an exact solution for the roblem of chemically reactive fluid flow over a moving vertical late in resence of radiation with Newtonian heating. II. MATHEMATICAL ANALYSIS Consider the unsteady two-dimensional flow of an incomressible and viscous fluid along an infinite vertical late. The x-axis is taken on the infinite late and arallel to the free stream velocity and y-axis normal to it. Initially, the late and the fluid are at same temerature T with concentration level C at all oints. At time 3 t > 0. It accelerates with a velocity t in its own lane. At the same time, the heat transfer from late to the fluid is directly roortional to the local surface temerature T and the late concentration is changed to C.It is assumed that there exist a homogeneous chemical reaction of first order with w constant rate K l between the diffusing secies and the fluid. Since the late is infinite in extent therefore the flow variables are the functions of y and t only. The fluid is considered to be gray absorbing-emitting radiation but non scattering medium. The radiation heat flux in the x-direction is considered negligible in comarison that of y -direction. Then neglecting viscous dissiation and assuming variation of density in the body force term (Boussinesq s aroximation), the roblem can be governed by the following set of equations: T T q r 1 (1) t C y C y IJETCAS ; 014, IJETCAS All Rights Reserved Page 4

2 Arita Jain, International Journal of Emerging Technologies in Comutational and Alied iences, 7(3), December 013-February, 014, t C C D k lc y u u g (T T ) g c (C C ) t y with following initial and boundary conditions u 0, T T, C C for all y, t 0 (4) 3 t T u, -h st, C C w at y 0, t 0 y u 0, T T,C C as y, t 0 The radiation heat flux term, by using the Rosseland s aroximation is given by 4 4 T q (5) r 3 y where U R is reference velocity, g is gravitational acceleration, C is secific heat at constant ressure, D is mass diffusivity, is thermal exansion coefficient, C is concentration exansion coefficient, is density, is thermal conductivity of fluid, is mean absortion coefficient, is kinematic viscosity and q r is radiation heat flux, σ is Stefan-Boltzmann constant. Using equation (5) in (1) 3 T T 16T T C (6) t y 3 y Introducing the following dimensionless quantities t y u t, y, u t L U R R R C T T gtt Pr,,,G 3 D T C C g C(Cw C ) kl C, Gm,k 3 Cw C 1 3 gt 3 13 L t gt) R R, R 4 T 3 T T T,U ( g T), 1/3 w R (7) where L R is reference length, t R is reference time, Gm is modified Grash of number, Pr is Prandtl number, is hmidt number and u is dimensionless velocity comonent, is dimensionless temerature, C is dimensionless concentration, is viscosity of fluid, t is time in dimensionless coordinate, R is radiation arameter and k is chemical reaction arameter. The governing equations (1) to (3) reduce to the following non-dimensional form 4 Pr ( 1 ) t 3 R y C 1 C kc t y (9) u u G Gm C t y (10) with the following initial and boundary conditions u 0, 0, C 0 forall y, t 0 (11) u t, (1 ), C 1 at y 0, t 0 y u 0, 0, C 0 as y, t 0 () (3) (8) (1) IJETCAS ; 014, IJETCAS All Rights Reserved Page 43

3 Arita Jain, International Journal of Emerging Technologies in Comutational and Alied iences, 7(3), December 013-February, 014, h heating from the late exists. On solving equations (8) to (10) by Lalace-transform technique, we get s is Newtonian heating arameter. Equation (1) gives θ=0 when ϒ=0 which hysically means that no ex( t b t)erfc ( a-b t ) erfc a (13) 1 C {ex( k t )erfc ( kt) ex( k t )erfc ( kt)} (14) For 1 u ex ( ) t (1+ )erfc( ) - Gb 1 ex(- ) +bex(b t b t)erfc( b t) a14 t Gb 1 ex(- a) +bex(b t t)erfc( a b t) a14 t Gm + erfc( ) k Gm kt kt kt kt kt ex( ) (ex( )erfc( ) ex( )erfc( )) k Gm (erfc( k t) ex( kt erfc( k t)) k Gm kt kt k t kt k t ex( )(ex( )erfc( ) ex( )erfc( )) k Where Pr y b, a, a 1 R t (15) III. SKIN-FRICTION From velocity field, skin-friction at the late in non dimensional form is exressed as: u y y0 t Gb b +b ex(b t) erfc( b t ) t a1 t Gb b a +b ex(b t)erfc( b t ) a 1 t Gm k t k t k ex( ) erf ( ) Gm k k k erf kt Gm k t k t k ex( ) erf ( ) k (16) IV. DISCUSSION Figure 1 elucidates the effects of Pr on temerature rofile. It is observed that temerature is maximum at the late then tend to zero far away from the late. Further, thickness of thermal boundary layer decreases as Pr increases. This is due to the fact that thermal conductivity of fluid decreases with increasing Pr, resulting a decrease in thermal boundary layer thickness It is also seen that it decreases steely for Pr = 7 than that of Pr = IJETCAS ; 014, IJETCAS All Rights Reserved Page 44

4 Arita Jain, International Journal of Emerging Technologies in Comutational and Alied iences, 7(3), December 013-February, 014, Figure 1: Temerature rofile ϒ=0.01, R=5, t=0. Figure illustrates the influences of ϒ and t on the velocity against η for Pr=0.71. It is noticed that at the late, fluid velocity is equal to time then it increases and attains maximum velocity in the vicinity of the late(η<1) after that it decreases and vanish far away from the late. Further, it also increases with an increase in time at each oint in the flow field. Moreover, with an increase in ϒ the velocity increases. Figure 3 elucidates the effects of G, Gm on velocity rofile. It is noticed that thickness of velocity boundary layer increases with η then attains its maximum value then decreases to asymtotic value. It is observed that there is an increase in velocity with an increase in the value of G and Gm. The reason is that the values of Grashof number and modified Grashof number has the tendency to increase the mass buoyancy effect. The increase in velocity due to increase in G and Gm is more near the late than away from the late. Figure : Velocity rofile for Pr=0.71, R=1, k=0., G=5, Gm=, =0. Figure 3: Velocity rofile for Pr=0.71,R=1,ϒ=1,t=0., k=0., =0. IJETCAS ; 014, IJETCAS All Rights Reserved Page 45

5 Arita Jain, International Journal of Emerging Technologies in Comutational and Alied iences, 7(3), December 013-February, 014, Skin- friction for different arameters against time t is resented in Figure 4. For smaller values of time the maximum value of magnitude of skin friction occurs and then it decreases raidly with an increase in t. Moreover, the Figure reveals that magnitude of skin-friction increases with an increase of Gm and ϒ. It is observed from figure that magnitude of skin friction is lower for water vaor (=0.60) in comarison to for Hydrogen gas (=0.). Physically, it is correct since an increase in serves to increase momentum boundary layer thickness. Moreover, magnitude of skin friction decreases with an increase in the value of chemical reaction arameter k, the reason is that increasing value of chemical reaction arameter reflects decrease in kinematic viscosity or viscosity of fluid which results decrease in the value of magnitude of skin friction. Figure 4: Skin-friction for R=1, G=5, Pr=7 V. The results of the flow roblem indicates: 1. Increasing Prandtl Number the temerature decreases. 3. Fluid velocity increases with an increase in Grashof number, modified Grashof number, time and Newtonian heating Parameter. 4. There is a fall in the value of skin friction with an increase in hmidt number, chemical reaction arameter and rise with an increase in value of modified Grashof number and Newtonian heating arameter. REFERENCES [1] Fayza Mohammed Nasser El-Fayez, Effects of chemical reaction on the unsteady free convection flow ast an infinite vertical ermeable moving late with variable temerature, JSEMAT, vol., , 01. [] Sarada K. and Shanker B., The effect of chemical reaction on an unsteady MHD free convection flow ast an infinite vertical orous late with variable suction, International Journal of Engineering Modern Research, vol. 3, , 013.tio [3] Bhaben Ch. Neog, Das Rudra Kr, Unsteady Free Convection MHD Flow ast a vertical late with variable temerature and chemical reaction, International Journal of Engineering Research & Technology, vol. 1,. 1-5, 01. [4] Chaudhary R. C. and Jain P., Unsteady free convection boundary layer flow ast an imulsively started vertical surface with Newtonian heating, Romanian Journal of Physics, vol. 51, , 006. [5] Rajesh V., Effects of mass transfer on flow ast an imulsively started infinite vertical late with Newtonian heating and chemical reaction, Journal of Engineering Physics and Thermohysics, vol. 85,.1 8, 01. [6] Rajut U. S. and Kumar S., Radiation Effects on MHD flow ast an imulsively started vertical late with variable heat and mass transfer, Int. J. of Al. Math. and Mech., vol. 8, , 01. [7] Chaudhary R. C. and Jain Arita, Unsteady free convection flow ast an oscillating late with constant mass flux in the resence of radiation, Acta Technica CSAV, vol. 5, , 007. [8] Narahari M. and Yunus Nayan M., Free convection flow ast an imulsively started infinite vertical Plate with Newtonian heating in the resence of thermal radiation and mass diffusion, Turkish Journal of Engineering and Environmental iences, vol. 35, , 011. [9] Das S., Mandal C. and Jana R. N., Radiation effects on unsteady free convection flow ast a vertical late with Newtonian heating, International Journal of Comuter Alications, vol. 41, , 01. IJETCAS ; 014, IJETCAS All Rights Reserved Page 46

u Velocity in x-direction Velocity in y-direction g- Acceleration due to gravity c - Specific heat at constant presure

u Velocity in x-direction Velocity in y-direction g- Acceleration due to gravity c - Specific heat at constant presure International Journal of Engineering Research and Develoment ISSN: 78-67X, Volume 1, Issue 8 (June 1), PP.-6.ijerd.com Boundary Layer Flo in Porous Medium Past a Moving Vertical Plate ith Variable Thermal