Flow Past an Exponentially Accelerated Infinite Vertical Plate and Temperature with Variable Mass Diffusion


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1 Internional Journal of Computer Applicions ( ) Volume No, May Flow Past an Exponentially Accelered Infinite Vertical Ple and Temperure with Variable Mass Diffusion KK Asogwa Department of Mhemics Usmanu Danfodiyo University, PMB 8 Sokoto, Nigeria I J Uwanta Department of Mhemics Usmanu Danfodiyo University, PMB 8 Sokoto, Nigeria AA Aliero Department of Biological iences Usmanu Danfodiyo University, PMB 8 Sokoto, Nigeria ABSTRACT An exact solution of flow past an exponentially accelered infinite vertical ple and temperure has been presented in the presence of variable mass diffusion The dimensionless governing equions are solved using Laplace transform technique The velocity, temperure, and concentrion profiles are studied for different physical parameters like radiion parameter R, accelering parameter a, thermal Grashof number Gr, mass Grashof number Gc, chemical reaction parameter K, Prandtl number Pr, hmidt number and time t It is observed th the velocity increases with increase in Gc, Gr, R, a, Pr, and t It is also observed th temperure rise with increasing t, a, R while concentrion increases with increasing t General Terms Asogwa et al Keywords Accelered, vertical ple, radiion, chemical reaction he transfer, variable mass diffusion INTRODUCTION Mixed he and mass transfer plays vital roles, in nuclear reactors, space craft design, design of chemical processing equipment, pollution of the environment, formion and dispersion of fog and moisture of agricultural fields Soundalgekar [9] presented an exact solution to the flow of a viscous fluid past an impulsively started infinite isothermal vertical ple The solution was derived by the Laplacetransform technique and the effects of heing or cooling of the ple on the flow field were discussed Singh and Naveen Kumar [] studied free convection effects on flow past an exponentially accelered vertical ple Gupta et al [8] studied free convection on flow past an linearly accelered vertical ple in the presence of viscous dissipive he using perturbion method Chamkha [] studied the effects of he absorption and thermal radiion on he transfer in a fluid particle flow past a surface in the presence of a gravity field Muthucumaraswamy and Valliamal [6] considered first order chemical reaction on exponentially accelered isothermal vertical ple with mass diffusion, the dimensionless governing equions are solved using laplacetransform technique Chandrakala [] considered analytically thermal radiion effects on moving infinite vertical ple with uniform he flux Muthucumaraswamy et al [] studied the unsteady flow past an accelered infinite vertical ple with variable temperure and uniform mass diffusion Basanth Jha and Ravindra Prasad [] examined Free convection and mass transfer effects on the flow past an accelered vertical ple with he source Das et al [6] studied radiion effects on flow past an impulsively started vertical infinite ple Mass transfer effects on the flow past an exponentially accelered vertical ple with constant he flux were considered by Basanth et al [] Muthucumaraswamy et al [] recently worked on he transfer effects on flow past an exponentially accelered vertical ple with variable temperure Deka and Neog [7] investiged unsteady MHD flow past a vertical oscilling ple with thermal radiion and variable mass diffusion Mazumdar and Deka [] considered MHD flow past an impulsively started infinite vertical ple in presence of thermal radiion Chandrakala and Bhaskar [] studied the effects of he transfer on flow past an exponentially accelered vertical ple with uniform he flux Rajesh and Vijaya [7] presented radiion and mass transfer effects on MHD free convection flow past an exponentially accelered vertical ple with variable temperure Muthucumaraswamy and Ganesan [] analysed the unsteady flow past an impulsively started vertical ple with he and mass transfer Ramana et al [8] considered the effect of critical parameters of MHD flow over a moving isothermal vertical ple with variable mass diffusion Muthucumaraswamy et al [] worked on the diffusion and He transfer effects on exponentially accelered vertical ple with variable temperure Jha et al [9] investiged mass transfer effects on the flow past an exponentially accelered vertical ple with constant he flux Muthucumaraswamy [] studied the interaction of thermal radiion on vertical oscilling ple with variable temperure and mass diffusion This study examines flow past an exponentially accelered infinite vertical ple and temperure with variable mass diffusion The dimensionless governing equions are solved using Laplace transform technique The solutions are obtained in terms of exponential and complementary error functions FORMULATION OF THE PROBLEM Here the flow past an exponentially accelered infinite vertical ple and temperure with variable mass diffusion has been considered The x '  axis is taken along the ple in the vertically upward direction and also the y ' axis is taken normal to the ple At t ' >, the ple is accelered with a velocity u u exp( ) in its own plane and the temperure of the plane is raise a uniform re to velocity,
2 Internional Journal of Computer Applicions ( ) Volume No, May and the level of concentrion near the ple is raised linearly with time Then under the usual Boussinesq s approximion the unsteady flow equions are momentum equion, energy equion, and mass equion respectively u u g T T g CC t y () C p C T T qr k t y y () C D K C t y () where u is the velocity of the fluid, T is the fluid temperure, C ' is the concentrion, g is gravitional constant, and are the thermal expansions of fluid and concentrion, t is the time, ρ is the fluid density, C p is the specific he capacity, v is the viscosity of the fluid, k is the thermal conductivity, D is the diffusion term, K is chemical reaction parameter and q r is the radiive he flux In this research work the mhemical formulion has chemical reaction, re of radiion he flux which are not included in the work of Muthucumaraswamy et al [] and difference in the boundary conditions for velocity, concentrion and temperure By Rosseland approximion, we assume th the temperure differences within the flow T are such th may be expressed as a linear function of the temperure T This is accomplished by expanding Taylor series about T neglecting higher order terms d The initial and boundary conditions are; t t u u e T T T T e C C C C Bt y :,, u, T, C as y () u Where B, B is a constant The thermal radiion he flux gradient may be expressed as follows q r a T T y ' () Where q r is the radiive he flux, a is the absorption coefficient of the fluid and σis the StefanBoltzmann constant We assumed th the temperure differences within the flow are sufficiently small such th T may be expressed as a linear function of the temperure Expanding T about T in a Taylor s series and neglecting higher order terms, we have T T T T By using equion () and (6)equion () reduces to T k T 6 T T t C y a y p d R On introducing the following nondimensional quantities u u u U t t Y y u T T C C, C, T T C C gt T g C C Gr, Gc, u u C p Td Pr,, R k D ka R (6) (7) (8) Substituting the nondimensional quantities of equion (8) into () to (), leads to U U Gr GcC t y (9) t y () t y C C KC () Where Pr, Gr is the thermal Grashof number, Gc R is the mass Grashof number, is the hmidt number, Pr is the Prandtl number and R is the radiion parameter The initial and boundary conditions reduce: U,, C, for all y, t t : U e, e, C t y U,, C as y () SOLUTION TO THE PROBLEM To solve equions (6) (8), subjected to the boundary conditions of (9), the solutions are obtained for concentrion, temperure and velocity flow in terms of exponential and complementary error function using the Laplace transform technique as follows;
3 t Kt Kt C y e erfc Kt e erfc Kt Kt t e Kt erfc Kt e erfc Kt K ( e y, t e erfc e erfc () e, U y t e erfc e erfc Gre erfc e erfc e erfc a e e erfc e erfc erfc( ) e Gc e erfc dt erfc d dt dt dt e e erfc dt e erfc dt Kt Kt e erfc Kt e erfc Kt Kt Kt dt e erfc Kt e erfc Kt t e Kt erfc kt Kt e erfc Kt K bt bt bt e e erfc bt e erfc bt () y k where, d and b d K t The nusselt number for temperure field is obtain when y= as e aerfc y e aerfc e t t (6) The Sherwood number for concentrion field is obtain when y= as Internional Journal of Computer Applicions ( ) Volume No, May t Kt C Kerfc y Kt e Kerfc Kt e t t Kt Kt K Kerfc Kt e Kerfc Kt e t t (7) The skin friction gives e U aerfc e aerfc e y t t Gre aerfc e aerfc e a t t e t aerfc e aerfc e t t te Gc dt dt dt dt e derfc dt e derfc dt e d t t t Kt Kerfc Kt e Kerfc Kt e t t Kt Kt dt Kt t Kt Kt K Kerfc Kt e Kerfc Kt e Kerfc Kt e Kerfc Kt e t t t t Kt bt bt bt e berfc bt e berfc bt e t (8) RESULTS AND DISCUSSION The problem of flow past an exponentially accelered infinite vertical ple and temperure with variable mass diffusion has been formuled, analysed and solved analytically In order to point out the effects of physical parameters namely; Accelering parameter a, thermal Grashof number Gr, mass Grashof number Gc, Prandtl number Pr, hmidt number, time t, radiion parameter R, and chemical reaction parameter K on the flow pterns, the compution of the flow fields are carried out The value of the Prandtl number Pr is chosen to represent air (Pr = 7) The value of hmidt number is chosen to represent wer vapour ( = 6) The values of velocity, temperure and concentrion are obtained for the physical parameters as mention The velocity profiles has been studied and presented in figure to 8 The effect of velocity for different values of hmidt
4 Internional Journal of Computer Applicions ( ) Volume No, May number ( = 6,,, 6) is presented in figure The trend shows th the velocity increases with increasing hmidt number The effect of velocity for different values of time (t =,, 6, 8) is also presented in figure It is then observed th the velocity increases with increasing values of time The effect of velocity profiles again have been studied for different values of thermal Grashof number (Gr =,, ) and mass Grashof number (Gc =,, ) is studied and then presented in figure and respectively The results are here observed th the increase in the values of velocity increases with increasing values of Gr and Gc respectively =6,,, 6 Figure profiles for different values of 6 t=,, 6, Figure profiles for different values of t The velocity profiles for different values of thermal Grashof number (Gr =,, ) is seen in Figure It is observed th velocity increases with increasing Gr The velocity profiles for different values of mass Grashof number (Gc =,, ) is presented in Figure It is observed th velocity increases with increasing Gc Gr=,, Figure velocity profiles for different values of Gr 8 6 Gc=,, Figure profiles for different values of Gc The velocity profiles for different values of accelering parameter (a =,,, ) is seen in Figure It is observed th velocity increases with increasing a The velocity profiles for different values of radiion parameter (R =,,, ) is presented in Figure 6 It is observed th velocity increases with increasing R 8 6 Figure profiles for different values of a a =,,, R =,,, Figure 6 profiles for different values of R The velocity profiles for different values of Prandtl number (Pr = 7, 8,, 7) is seen in Figure 7 It is observed th velocity increases with decreasing Pr The velocity profiles for different values of chemical reaction parameter (K=,, 6, 8) is presented in Figure 8 It is observed th velocity increases with decreasing K
5 Temperure Temperure Temperure Temperure Internional Journal of Computer Applicions ( ) Volume No, May Pr= 7, 8,, 7 Pr=7, 8,, 7 Figure 7 profiles for different values of Pr Figure Temperure profiles for different values of Pr Pr= 7, 8,, 7 The temperure profiles for different values of accelering parameter (a =,, 8, ) is presented in Figure It shows th temperure rises with increasing a The temperure profiles for different values radiion parameter (R =,,, ) is presented figure It is observed th temperure increases with increasing R Figure 8 profiles for different values of K The temperure profiles has been studied and presented in figure 9 to The effect of temperure for different values of time (t =, 6,, ) is presented in Figure 9 It shows th temperure rises with increasing t The temperure profiles for different values Prandtl number (Pr = 7, 8,, 7) is presented in figure It is observed th increases in Prandtl number Pr decreases the temperure a =,, 8, Figure Temperure profiles for different values of a 8 R =,,, 6 t=, 6,, Figure 9 Temperure profiles for different values of t Figure Temperure profiles for different values of R The concentrion profiles has been studied and presented in figure to The effect of concentrion for different values of time (t =,, 6, 8) is presented in Figure It is observed th concentrion increases with increasing t The concentrion profiles for different values of hmidt number ( = 6,,, 6) is presented in Figure It is observed th the concentrion increase with increasing The effect of concentrion for different values of chemical reaction parameter (K=,,, ) is presented in Figure It is observed th the concentrion increases with a decrease in the values of K
6 Concentrion Concentrion Concentrion Internional Journal of Computer Applicions ( ) Volume No, May 8 Table Skin friction Gr Gc t R a Pr K t =,, 6, Figure Concentrion profiles for different values of t =6,,, Figure Concentrion profiles for different values of K=,,, Table Nusselt number Nu t R a Pr Nu Figure Concentrion profiles for different values of K Tables to gives the Skin friction, Nusselt number and Sherwood number respectively Table shows the effect of physical parameters Gr, Gc, R, a, t, and Pr on the skin frictionits observed th the shear stress increases when Gr, Gc, R, a, t, and Pr increases Table represents the variion of physical parameters Pr,a,t,R on the nusselt number which determines the re of he transferits observed th re of he transfer increases when R, a, t, Pr increases Table Sherwood number Sh K t Sh Table gives the effect of physical parameters, K and t on the Sherwood numberits observed th mass transfer increases when K,t, increases 6
7 Internional Journal of Computer Applicions ( ) Volume No, May SUMMARY AND CONCLUSION Flow past an exponentially accelered infinite vertical ple and temperure with variable mass diffusion has been studied The dimensional governing equions are solved by Laplace transform technique The effect of different parameters like accelering parameter, radiion parameter, chemical reaction parameter, hmidt number, Prandtl number, mass Grashof number, thermal Grashof number, and time are presented graphically It is observed th velocity profile increases with increasing parameter namely R, t, Gc, Gr and a while, Pr and K decreases with increasing velocity It also observed th temperure rise with increasing t, a, R while Pr decreases with increasing temperure The concentrion profiles increases with increasing t while and K decreases with increase in concentrion 6 REFERENCES [] Basanth Kumar Jha and Ravindra Prasad (99) Free convection and mass transfer effects on the flow past an accelered vertical ple with he source, Mechanics Research Communicions, Vol 7, Pp 8 [] Basanth Kumar Jha, Ravindra Prasad, Surendra Raj (99) Mass transfer effects on the flow past an exponentially accelered vertical ple with constant he flux, Astrophysics and Space ience, Vol 8, Pp  [] Chandrakala P () Thermal Radiion effects on moving infinite vertical ple with uniform he flux, Internional Journal of dynamics fluids, Vol Pp 9 [] Chandrakala P and Bhaskar P() Effects of he transfer on flow past an exponentially accelered vertical ple with uniform he flux, Internional Journal of dynamics fluids, Vol 7(), Pp 96 [] Chamkha AJ () Effects of he absorption and thermal radiion on he transfer in a fluid particle flow past a surface in the presence of a gravity field, Internional Journal thermal sciences, Vol 9, Pp 66 [6] Das, UN,Deka RK and Soundalgekar VM (996) Radiion effects on flow past an impulsively started vertical infinite ple, Journal of Theoretical Mechanics, Vol, Pp  [7] Deka RK and Neog BC (9) Unsteady MHD flow past a vertical oscilling plewith thermal radiion and variable mass diffusion, Chamchuri Journal of Mhemics, Vol (), Pp 799 [8] Gupta AS, Pop I and Sounldalgekar VM (979) Free convective effects on the flow past an accelered vertical ple in an incompressible dissipive fluid, Rev Roum ience and Applied Technology, Vol, Pp 668 [9] Jha BK,Prasad R and Rai S(99) Mass transfer effects on the flow past an exponentially accelered vertical ple with constanthe flux, Astrophysics and Space ience, Vol 8, Pp [] Mazumdar MK and Deka RK (7) MHD flow past an impulsively started infinite vertical ple in presence of thermal radiion, Romanian Journal of Physics, Vol (6), Pp 9 [] Muthucumaraswamy R(6) The interaction of thermal radiion on vertical oscilling ple with variable temperure and mass diffusion, Theoretical Applied Mechanics, Vol (), Pp 7 [] Muthucumaraswamy R and Ganesan P(998) Unsteady flow past an impulsively started vertical ple with he and mass transfer, He and Mass Transfer, Vol, Pp 879 [] Muthucumaraswamy R,Shappan KE and Naragan R(8) He transfer effects on flow past an exponentially accelered vertical ple with variable temperure, Theoretical Applied Mechanics, Vol (), Pp  [] Muthucumaraswamy R,Shappan KE and Naragan R() Diffusion and He transfer effects on exponentially accelered vertical ple with variable temperure, Theoretical Applied Mechanics, Vol (), Pp 777 [] Muthucumaraswamy R,Sundar Raj M and Subramanian VSA (9) Unsteady flow past an accelered infinite vertical ple with variable temperure and mass diffusion, Internional Journal of Applied Mhemics and Mechanics, Vol (6), Pp 6 [6] Muthucumaraswamy R and Valliamal V (9) First order chemical reaction on exponentially accelered isothermal vertical ple with mass diffusion, Internional Journal of Engineering, TUME VII, fascicule I (ISSN 866) [7] Rajesh V and Vijaya Kumar V (9) Radiion and mass transfer effects on MHD free convection flow past an exponentially accelered vertical ple with variable temperure, ARPN Journal of Engineering and Applied iences, Vol (6), (ISSN ) [8] Ramana Reddy G,Ramana Murthy V and Bhaskar Reddy N() Effect of critical parameters of MHD flow over a moving isothermal vertical ple with variable mass diffusion Internional Journal of dynamics fluids, Vol, Pp  [9] Soundalgekar VM (977) Free convection effects on the stokes problem for an infinite vertical ple, ASME Journal of He Transfer, Vol 99, Pp 99 [] Singh AK and Naveen Kumar (98) Free convection flow past an exponentially accelered vertical ple, Astrophysics and space, Vol 98, Pp 8 7
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