Received 3 November 2015; accepted 24 December 2015; published 29 December 2015

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1 Open Journal of Fluid Dynamis, 015, 5, Published Online Deember 015 in SiRes. Effets of Thermal Radiation and Radiation Absorption on Flow Past an Impulsively Started Infinite Vertial Plate with Newtonian Heating and Chemial Reation Swetha Ravi 1, Jagdish Prakash, Viswanatha Reddy Gottam 3, Vijaya Kumar Varma Sibyala 3 1 Department of Mathematis, Gudlavalleru Engineering College, Gudlavalleru, India Department of Mathematis, University of Botswana, Gaborone, Botswana 3 Department of Mathematis, S. V. University, Tirupati, India Reeived 3 November 015; aepted 4 Deember 015; published 9 Deember 015 Copyright 015 by authors and Sientifi Researh Publishing In. This work is liensed under the Creative Commons Attribution International Liense (CC BY). Abstrat A perfet solution to the present natural onvetive flow problem of a vertial transfinite plate owing to the impulsive motion in the ubiety of first ordered hemial reation, radiation absorption, radiation, Newtonian heating and speies onentration in its plane is evolved by applying the method of Laplae transforms in losed form at the plate. Exat results for veloity, temperature, onentration fields are prevailed and expressions for heat and mass transfer rates are also found. The effets are analyzed for the respetive invariables for both ammonia and water vapor. Keywords Newtonian Heating, Natural Convetion, Chemial Reation, Inompressible Fluid, Radiation Absorption and Radiation 1. Introdution On hemial reation, the field of mass and heat transfer is of good pragmati importane to applied sientists Corresponding author. How to ite this paper: Ravi, S., Prakash, J., Gottam, V.R. and Sibyala, V.K.V. (015) Effets of Thermal Radiation and Radiation Absorption on Flow Past an Impulsively Started Infinite Vertial Plate with Newtonian Heating and Chemial Reation. Open Journal of Fluid Dynamis, 5,

2 owing to its general ourrene in various fields of engineering and siene. Espeially, the subjet of mass and heat transfer with heat radiation, hemial reation has signifiant role in hydrometallurgial and hemial industries. For a moving plate, a hemial reation takes plae in legion hemial proesses between a fluid and foreign mass. This sue is involved in many industrial usages suh as glassware or eramis manufaturing, food proessing and prodution of polymers. The onvetion study with mass and heat transfer plays a major role in the dispersion and formation of fog, design of hemial proessing equipment s, temperature distribution, and moisture over agriultural fields and in the paper drying proess. Ahmed et al. [1] have identified the analysis for MHD rotating heat or mass transport phenomenon bounded by a vertial osillating surfae in the Mein of Darian porous regime by using Numerial/Laplae transform. Charateristis of the heat and mass transfer in the Mien of hemial reation and thermal radiation for a Newtonian inompressible fluid aross an extending vertial surfae having temperature dependent visosity was studied by Kandasamy et al. []. Makinde [3] examined the free transient onvetion interation of an absorbing, emitting plate with thermal radiation. Mukhopadhyay [4] performed an investigation on the results of heat transfer and thermal radiation on a mixed unsteady onvetive flow aross an extending porous surfae in porous medium. Heat transfer analysis of a fored onvetive flow of the fluid past an embedded plate in a porous medium for an inompressible fluid was examined by Mukhopadhyay and Layek [5]. Muthuumaraswamy and Ganesan [6] looked at the impulsively started transient radiation-onvetion flow with vertial temperature onsequenes. An analysis of the hemial reation, theoretially a result, with variable temperature on a vertial osillating plate was given by Muthuumaraswamy [7]. Reddy et al. [8] investigated the onsequenes of unsteady natural MHD onvetive flow in a porous medium with onstant mass diffusion and Newtonian heating. The effets of MHD radiating and hemially reating fluid past a non-isothermal impulsively started vertial surfae adjaent to a porous regime by using numerial analysis was disussed by Sahin Ahmed [9]. The importane of the present flow problem is to analyze the effets of thermal radiation and radiation absorption on the flow past an impulsively started infinite vertial plate with Newtonian heating and hemial reation.. Mathematial Analysis Free onvetive unsteady flow of the fluid for a vertial transfinite plate with Newtonian heating, past an impulsively started inompressible visous fluid in the Mien of radiation and radiation absorption is studied. Along the plate and in the vertial upward diretion, axis x is hosen and normal to the plate, axis y is onsidered. Initially the fluid and the plate are having same temperature T and the onentration C at all points in a stationary state for time t 0. The oordinate system and the flow model are shown in Figure 1. The plate is fixed with a veloity u 0 in the vertial diretion into impulsive motion versus the gravitational field at time t > 0. We assumed that i) heat transfer rate and the loal surfae temperature T are proportional to one another from the surfae, and near the plate onentration rises to C w and ii) the onsequenes of visous dissipation are negligible in the energy equation. Among the fluid and diffusing speies, there is a first order hemial reation. Sine all the physial quantities are expressed in terms of y, t only and are free from x and in the diretion of x, the plate is onsidered transfinite. Figure 1. A sketh of flowmodel and oordinate system. 365

3 The equations for this present flow, by the Boussinesq estimation are as follows u t u = gβ T T + gβ C C + ν ( ) ( ) T t T q ρ r Cp = κ + Q 1 ( C C ) (1) () C t and the onneted onditions for this flow are C = Kr ( C C ) (3) at t 0 : u = 0, T = T, C = C y T h at t > 0 : u = u0, = T, C = C for 0 w y = κ u = 0, T T, C C as y (4) u 0 Here A = υ The term of radiative heat flux by the Rosseland estimation is given by q r 4σ s T = 3k But here with in the flow, presuming that the deviation in temperatures an be showed as a linear ombination 4 of the temperatures and around T, whih is found by expanding T in a Taylor s series as follows: 4 ( ) ( ) T = T + T T T + T T T + (6) and ignoring the higher ordered terms, beyond the first degree, we get Differentiating Equation (5) with respet to T T T T (7) y and applying Equation (6), we get (5) q 3 r 16σ st T = 3k (8) We have inserted the non-dimensional sueeding measures u t u0 yu µ 0 p gβνt U =, t =, y =, Pr =, Gr =, 3 u0 ν ν κ u0 T T νq1 ( Cw C ) gβν ( Cw C ν ) θ =, S =, Q 1 =, Gm =, 3 T ρcut p 0 u0 3 C C Krν 16σ st C =, K =, N r = Cw C u0 3κ k The Equations (1)-(3) are redued into the following forms by using the Equations (8) and (9) as follows U u = Grθ + GC m + t (9) (10) 366

4 θ 1+ N r θ = + QC 1 t Pr C 1 C = KC t S In non-dimensional form, the onditions redue to as follows at t 0 : U = 0, θ = 0, C = 0 y θ at t > 0 : U = 1, = ( 1 + θ ), C = 1 for y = 0 U = 0, θ 0, C 0 as y (11) (1) (13) 3. Solution of the Problem The non-dimensional Equations (10)-(1) assoiated with the onditions given by Equations (13) are evolved by the method of Laplae transforms, and therefore the results of onentration, temperature and veloity are given by C (, t ) = B6 y (14) (, t ) 1 B1 B 1 B3 N1( MB 1 1 MB MB 3 4 MB 4 5 MB 5 3) N( B6 B7) θ y = (15) a a ( y, t ) = ( ) U A C A A A A A A ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] + C A + A + A + A + A + A + A + A + A + A C M m m + M A A + M A A M A A + M A A + M A A M A A + C A A A + A (16) 4. The Rate of Heat Transfer In dimensionless form, heat transfer rate from the temperature gradient is From Equations (15) and (17), we get θ Nu = y y = 0 M b ( t) ( t ) 1 3 ( t) ( t ) Nu = exp a erf a + N M exp b erf i b πat a i b bt + e exp erf πat a b ( M4 M3) M4 ( bt) ( i bt ) 1 ( t) ( t ) ( t ) + M 5 a exp a erf a + M 5 + N KS erf K πt ( t ) ( t) ( ) erf ( ) N exp b S K b K b (17) (18) 5. The Rate of Mass Transfer In dimensionless form, mass transfer rate from the onentration gradient is 367

5 Sh = y y = 0 (19) From Equations (14) and (19), we get S ( t ) exp( t) Sh = KS erf K + K πt (0) 6. Dedution The effets of this analysis are in good agreement with the results given by Rajesh [8] in the absene of parameters radiation absorption ( Q 1) and radiation ( N r ), where B 1 y 1 exp, erf 1 erf π 4a B y a A y = = = t t t t B 1 y 3 exp exp ( ) erf π 4a a a a y = + y + t t + t t a t 1 y y y B4 = exp i b exp bt erf i bt + πt 4a a t at i b 1 y y B5 = exp + i b exp b erf i b + π 4a a t y t t t at i b 1 y S y S B6 = erf K t exp y KS + erf + K t exp y KS t t ( ) ( ) exp( b t ) y S B7 = erf K b S K b t ( ) ( ) t exp y ( ) y S + erf + ( K b) t exp( y S ( K b) ) t 1 exp( b t ) y S A = erf K b S K b bb ( d) t y S + erf + ( K b) t exp( y S ( K b) ) t ( ) ( ) t exp y ( ) 1 exp( d t ) y S A3 = erf K d S K d d( d b) t y S + erf + ( K d) t exp( y S ( K d) ) t ( ) ( ) t exp y ( ) 368

6 1 1 y S y S A4 = erf Kt exp y SK + erf + Kt exp y SK bd t t ( b t ) ( ) ( ) 1 exp y y A5 = erf bt exp( y b) + erf + bt exp( y b) bb ( d) t t A 11 1 y A6 = exp πa t 4 t =, A8 = aa16, A9 = aa17,, A7 A1 A 1 1 y 10 exp exp ( ) erf a π 4 a a a a y = + y + t t + t t t 1 1 y y = exp, A1 = erf a π 4a t t a t A = aa, A14 = aa3,, 13 A 1 1 y 15 exp exp ( ) erf a π 4a a a a y = + y + t t + t t at t y y A16 = exp yerf π 4 t t y y t y A17 = t + erf y exp π 4 t t A 1 y 18 = exp exp ( ) erf π 4 i b i b b i b y y t t + t t t A 1 y 19 = exp + exp ( ) erf π 4 i b i b b i b y y t t + t t t A 1 y 0 exp exp ( ) erf π 4 a a a a y = + y + t t +, t t t A t y y y 1 = exp erf 8 1 π 4a a a A = bd A t t y y t y A3 = t + erf y exp a a aπ 4a t t A 1 y 4 exp exp π 4 i b i b y a a b erf i b y = t t + t t at A 1 y 5 exp exp erf π 4 i b a i b y a b i b y = + t t + t t at 369

7 A 1 y 6 exp exp ( ) erf π 4a a a a y = + y + t t + t t a t ( d t ) 1 exp y y A7 = erf dt exp( y d) + erf + dt exp( y d) d( d b) t t ( d t ) exp y y A9 = erf dt exp( y d) + erf + dt exp( y d) t t 1 y S y S A30 = erf K t exp y KS + erf + K t exp y KS t t 7. Results and Disussion ( ) ( ) exp( d t ) y S A31 = erf K d S K d t ( ) ( ) t exp y ( ) y S + erf + ( K d) t exp( y S ( K d) ) t In this field of study, in order to examine the onsequenes of veloity field, temperature profile and onentration profiles by allotting numerial values for several arguments for both water vapor and ammonia, Prandtl number (Pr) and Shmidt number (S) values are onsidered. For several values of different arguments the veloities are analyzed and are presented in Figures -7 at time t = 0.4 respetively, for both the types of heating (Gr < 0, Gm < 0) plate and ooling (Gr > 0, Gm > 0) plate. Figure depits the result of Shmidt number (S) at time t = 0.4 on the flow. With an inrease in S, it is notied that the veloity inreases for heating of the plate and dereases for ooling of the plate and as S inreases at t = 0.4 from and to 0.60, the maximum veloity of the fluid dereases by 5.7% - 10% for ammonia and by 5.6% % for water vapor in the ase of ooling plate and the minimum veloity of the fluid inreases by 10% and 16.58% for ammonia and by 10.6% - 17% for water vapor in the ase of heating plate. Prandtl number (Pr) effets on the flow are expressed in Figure 4 at time t = 0.4. It is identified that there is an inrease in veloity near the plate and then dereases with a point of separation moving far away from the plate in the type of Figure. Veloity profile shows the effet of S and Pr = 0.71, K = 0., Nr = 0.1, Q 1 = 0.1, t =

8 Figure 3. Veloity profile displays the result of S and Pr = 0.71, K = 0., Nr = 0, Q 1 = 0, t = 0.4. Figure 4. Veloity profile shows the effet of Pr and S = 0., K = 0., Nr = 0.1, Q 1 = 0.1, t = 0.4. Figure 5. Veloity profile displays the result of Pr and S = 0., K = 0., Nr = 0, Q 1 = 0, t = 0.4. heating and in the instane of ooling plate, the reverse effet is found with the inrease of Pr. Owing to the variation in parameter of hemial reation (K), Figure 6 reveals the onsequene of veloity profiles at time t = 0.4. As K inreases, it is found that, there is an inrease veloity in the plate of heating type and the veloity dereases in the plate of ooling type and as K inreases at t = 0.4 from 0. - and to 5, the maximum veloity of 371

9 Figure 6. Veloity profile shows the effet of K and S = 0., Pr = 0.71, Nr = 0.1, Q 1 = 0.1, t = 0.4. Figure 7. Veloity profile displays the result of K and S = 0., Pr = 0.71, Nr = 0, Q 1 = 0, t = 0.4. Figure 8. Veloity profile shows the effet of K and S = 0., Pr = 0.71, Nr = 0.1, Q 1 = 0.1, t = 0.4. the fluid dereases by 7.4% - 6.8% for ammonia and by 7.1% - 7.4% for water vapor in the ase of ooling plate and the minimum veloity of the fluid inreases by 8.4% % for ammonia and by 5.56% % for water vapor in the ase of heating plate. 37

10 For various values of Nr (radiation parameter), the veloity profile is shown in Figure 8 at t = 0.4. It is seen that there is a derease in veloity near the plate and then inreases with a point of separation moving away from the plate in the ooling ase and the phenomenon is reversed in the ase of heating type with the derease of Nr. Figure 9 desribes the effets of Q 1 (radiation absorption parameter) at t = 0.4. As Q 1 dereases, it is observed that the veloity inreases for heating plate and dereases for ooling plate. Figure 10 reveals the result of veloity profile at several times (t = 0.4, 0.6, 0.8). It is identified that there is a onsiderable derease in veloity in the heating type and inrease when the plate is ooled as time (t) inreases. The same results are notied in the absene of thermal radiation and radiation absorption for different values of S, Pr, K, t whih are shown in Figure 3, Figure 5, Figure 7, and Figure 11. Hene, these results are in good agreement with the results of Rajesh [10]. The effet of temperature profile for several values of various parameters are studied and shown in Figures 1-17 at time t = 0.. It is observed from Figures 1-14 that the temperature rises with the fall in S and K. In Figures 15-17, it is observed that the temperature rises with the inrease of Nr, Q 1, and t. Moreover Figure 13 shows the results of veloity for different values of Pr. It is notied from the values that, veloity inreases near the Figure 9. Veloity profile displays the result of K and S = 0., Pr = 0.71, Nr = 0, Q1 = 0, t = 0.4. Figure 10. The effet of veloity profile fortand Pr = 0.71, K = 0., Nr = 0.1, S = 0., Q 1 = 0.1. Figure 11. The result of veloity profile for tand S = 0., K = 0., Nr = 0.1, Q 1 = 0.1, Pr =

11 Figure 1. Temperature profile presents the effet of S and Pr = 0.71, K = 0., Nr = 0.1, Q 1 = 0.1, t = 0.. Figure 13. Temperature profile shows the resultof Pr and S = 0., K = 0., Nr = 0.1, Q 1 = 0.1, t = 0.. Figure 14. Temperature profile displays ensue of K and S = 0., Pr = 0.71, Nr = 0.1, Q 1 = 0.1, t = 0.. Figure 15. Temperature profile indiates the effet of Nr and 0., K = 0., Q 1 = 0.1, Pr = 0.71, t =

12 Figure 16. The effet of temperature profile for Q 1 and Pr = 0.71, S = 0., Nr = 0.1, K = 0., t = 0.. Figure 17. Theresult of temperature profile for tand S = 0., K = 0., Q 1 = 0.1, Pr = 0.1, Nr = 0.1. Figure 18. Conentration profile displays the effet of S and K = 0., t = 0.. plate and falls far away from the plate with a point of separation. The effet of onentration profile for several values of various parameters is studied and is presented in Figures 18-0 at times 0.4. From Figure 18 and Figure 19, it is identified that with the derease in S and K, the onentration inreases. And from Figure 0, it is found that there is a rise in onentration with the rise in time t. For several values of various arguments, the Sherwood number versus time is presented in Figure 1 and Figure. It is identified that, there is an inrease in Sherwood number for both hydrogen and water vapor with the inrease of S and K. For different values of various arguments for both hydrogen and water vapor, the Nusselt number versus time is shown in Figure 3. From this figure, it is observed that, there is a rise in Nusselt number with the fall in Pr. 375

13 Figure 19. Conentration profile shows theresult of K and S = 0., t = 0.. Figure 0. Conentration profile presents theensue of t and S = 0., K = 0.. Figure 1. Effet of S on the real part of Sherwood number and K = 0.. Figure. Result of K on the real part of Sherwood number and S =

14 Figure 3. Effet of Pr on the real part of nusselt number and K = 0., S = 0., Nr = 0.1, Q 1 = 0.1. Figure 4. Result of Pr on the real part of nusselt number and K = 0., S = 0., Nr = 0, Q 1 = 0. And the same results are notied in the absene of thermal radiation and radiation absorption for different values of Pr whih is shown in Figure 4. Hene, these results are in good agreement with the results of Rajesh [10]. Referenes [1] Ahmed, S., Batin, A. and Chamka, A.J. (015) Numerial/Laplae Transform Analysis for MHD Rotating Heat/ Mass Transport in a Darian Porous Regime Bounded by an Osillating Vertial Surfae. Alexandria Engineering Journal, 54, [] Kandasamy, R., Muhaimin, I. and Saim, H.B. (010) Group Analysis for the Effets of the Temperature Dependent Fluid Visosity and Chemial Reation on Free Convetive Heat and Mass Transfer. Journal of Applied Mehanis and Tehnial Physis, 51, [3] Makinde, O.D. (005) Free Convetion Flow with Thermal Radiation and Mass Transfer Past a Moving Vertial Porous Plate. International Communiations in Heat and Mass Transfer, 3, [4] Mukhopadhyay, S. (009) Effet of Thermal Radiation on Unsteady Mixed Convetion Flow and Heat Transfer over a Porous Strething Surfae in Porous Medium. International Journal of Heat and Mass Transfer, 5, [5] Mukhopadhyay, S. and Layek, G.C. (009) Radiation Effet on Fored Convetive Flow and Heat Transfer over a Porous Plate in a Porous Medium. Meania, 44, [6] Muthuumaraswamy, R. and Ganesan, P. (003) Radiation Effets on Flow Past an Impulsively Started Infinite Vertial Plate with Variable Temperature. International Journal of Applied Mehanis and Engineering, 8, [7] Muthuumaraswamy, R. (010) Chemial Reation Effets on Vertial Osillating Plate with Variable Temperature. Chemial Industry & Chemial Engineering Quarterly, 16, [8] Hussanan, A., Ismail, Z., Khan, I., Hussein, A.G. and Shafie, S. (014) Unsteady Boundary Layer MHD Free Convetion Flow in a Porous Medium with Constant Mass Diffusion and Newtonian Heating. The European Physial Journal Plus, 19, 46. [9] Ahmed, S. (014) Numerial Analysis for MHD Chemially Reating and Radiating Fluid Past a Non Isothermal Im- 377

15 pulsively Started Vertial Surfae Adjaent to a Porous Regime. Ain Shams Engineering Journal, 5, [10] Rajesh, V. (01) Effets of Mass Transfer on Flow Past an Impulsively Started Infinite Vertial Plate with Newtonian Heating and Chemial Reation. Journal of Engineering Physis and Thermophysis, 85, Nomenlature C : onentration far away from the plate; C : onentration; C : non-dimensional speies onentration; C w : onentration at the plate; D : moleular diffusivity; C p : speifi heat at onstant pressure; G r : thermal grashoff number; G m : mass grashoff number; h : oeffiient of heat transfer; g : aeleration due to gravity; K r : parameter of hemial reation; K : non-dimensional parameter of hemial reation; q r : radiative heat flux; Q 1: parameter of radiation absorption; k : mean absorption oeffiient; σ s : Stefan-Boltzmann onstant; N r : parameter of radiation; Pr : Prandtl number; S : Shmidt number; Sh : Sherwood number; Nu : Nusselt number; T : temperature; T, ambient temperature; t : non-dimensional time; t : time; U : non-dimensional veloity of the in the u : veloity of the fluid; x, diretion; u 0 : plate veloity; x and y : Cartesian oordinates along the plate and normal to the plate; y : non-dimensional oordinate; β : volumetri oeffiient of thermal expansion; β : volumetri oeffiient of expansion with onentration; µ : visosity; ρ : density; θ : non-dimensional temperature; κ : thermal ondutivity; ν : kinemati visosity. 378

16 Appendix Qb 1 Q1 KS 1+ Nr KS N1 =, N =, d =, a =, b = K as aks 1 1 S Pr S a E E E E M = E, M = E, M = E, M = E, S S S S E M = E, M = P P, M = P P, M = P P, S M = P P, M = P P, M = P P, M = P P, E1 =, E 3 =, E 3 3 =, E4 =, ib i b a ib i b a a a b b a ( + ) ( ) ( + ) K K b E = E E E, E =, E = ( ) b a ib i b a K b K + a E8 =, E 3 9 =, E10 = E8 E7 E9, ib i b a ( + ) a( a+ b) ( ) ( ) ( ) KS K b S K b S P1 =, P =, P 3 = b a ib b i b a ib b i b + a ( + ) a ( a+ b), ( ) K a S P = P = P P P P = ap + ap + i bp + ap i bp, 4, 5 3 4, P = bp + ap i abp + i abp bp, P =, P =, ( ) b a ib b i b a 1 1 P =, P =, P = P P P, 10 ib b i b a ( + ) a ( a+ b) P = ap + ap + i bp + ap i bp, P = bp + ap i abp + i abp bp, G Q bd, G, G Q ab, Gm C = C = C = C = KS r 1 r r ak S 1 a KS ( 1 a) a., 379