 Research
 Open Access
Influence of thermal radiation on free convective heat and mass transfer past an isothermal vertical oscillating porous plate in the presence of chemical reaction and heat generationabsorption
 Adeyemi Isaiah Fagbade^{1}Email author and
 Adeola John Omowaye^{1}
https://doi.org/10.1186/s136610160601z
© Fagbade and Omowaye 2016
 Received: 9 September 2015
 Accepted: 27 April 2016
 Published: 13 May 2016
Abstract
An approximate analysis of the problem of the transient free convective transfer flow of a Newtonian nongray optically thin fluid past an isothermal vertical oscillating porous plate in the presence of chemical reaction and heat generation/absorption is studied. The dimensionless governing coupled linear partial differential equations are solved using a spectral relaxation method. The essence of the method of the solutionspectral relaxation method SRM is to linearize and decouple the original system of PDEs to form a sequence of independent linear equations that can be solved iteratively. The SRM approach applies the spectral collocation method and a finite different method independently in all underlying independent variables to obtain approximate solutions of the problem. Detailed computations on the influence of the chemical reaction parameter \(A_{2}\), the thermal radiation parameter R, the number Sc, the heat absorption/generation parameter \(Q_{1}\), and the Prandtl number on the flow velocity, temperature, and concentration distributions are illustrated graphically and in table format. It is observed that the flow velocity increases with the increase in either thermal radiation or thermal Grashof number. The temperature profile increases with the increase in either the thermal radiation parameter or the heat absorption/generation parameter. The rate of heat transfer decreases with the increase in the thermal radiation parameter, whereas it increases with increasing value of the heat generation/absorption parameter.
Keywords
 free convection
 thermal radiation
 heat generation
 chemical reaction
 heat and mass transfer
 spectral relaxation method
1 Introduction
The phenomenon of free convection arises in the fluid, when temperature changes cause density variation leading to buoyancy forces acting on the fluid element. This can be seen in our everyday life in atmospheric flow which is driven by temperature differences. Free convection flow is a significant factor in several practical applications, which include, for example, cooling of electronic components, in designs related to thermal insulation, material processing, and geothermal systems, etc.
Extensive research has been conducted on free convection flow past a vertical plate see for instance Ostrach [1] and many others. Free convection at a vertical plate with transpiration was investigated by Kolar and Sastri [2]. Ramanaiah and Malarvizi [3] considered natural convection adjacent to a surface with three thermal boundary conditions. Pop and Soundalgekar [4] investigated the free convection flow past an accelerated infinite plate. Raptis et al. [5] studied the unsteady free convective flow through a porous medium adjacent to a semiinfinite vertical plate using finite difference scheme. Singh and Soundalgekar [6] investigated the problem of transient free convection in cold water past an infinite vertical porous plate. Flows past a vertical plate oscillating in its own plane have many industrial applications. The first exact solution of the NavierStokes equation was given by Stoke [7] which is concerned with flow of viscous incompressible fluid past a horizontal plate oscillating in its own plane. Natural convection effects on the Stokes problem was further investigated by Soundalgekar [8]. The same problem was considered by Senapati et al. [9] for an impulsively started or oscillating plate. Gupta et al. [10] analyzed flow in the Ekman layer on an oscillating plate. An exact solution to the flow of a viscous incompressible unsteady flow past an infinite vertical oscillating plate with variable temperature and mass diffusion by taking into account of the homogeneous chemical reaction of first order was investigated by Muthucumarswamy and Meenakshisundaram [11]. Kishore et al. [12] studied hydromagnetics flow of a viscous incompressible fluid past an oscillating vertical plate embedded in a porous medium with radiation, viscous dissipation and variable heat, and mass diffusion. The governing equations were solved numerically. It is observed that plate oscillation, variable mass diffusion, radiation, viscous dissipation, and a porous medium affect the flow profiles significantly. The process of heat and mass transfer in free convection flow have attracted the attention of a number of scholars due to its application in many branches of science and engineering, viz. in the early stages of melting adjacent to heated surfaces, in chemical engineering processes which are classified as a mass transfer processes, in a cooling device. Gupta and Gupta [13] studied the heat and mass transfer corresponding to the similarity solution for the boundary layer flow over an isothermal stretching sheet subject to blowing or suction. Elbashbeshy [14] investigated heat transfer over a stretching surface with variable and uniform surface heat flux subject to injection and suction.
In the above mentioned studies the effects of linear heat generation (heat sources/sinks) have not been considered and due to its versatile applicability to ceramic tiles production, the study of heat transfer in the presence of a source/sink has acquired newer dimensions. The study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Possible heat generation effects may alter the temperature distribution; and, consequently, the particle deposition rate in nuclear reactors, electronic chips, and semiconductor wafers. Also, heat generation or absorption effects in moving fluids are important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reactions. Vajravelu and Hadjinicolaou [15] studied the convective heat transfer in an electrically conducting fluid near an isothermal stretching sheet and they studied the effect of internal heat generation or absorption. Chamkha [16] investigated unsteady convective heat and mass transfer past a semiinfinite porous moving plate with heat absorption. Hady et al. [17] studied the problem of free convection flow along a vertical wavy surface embedded in electrically conducting fluid saturated porous media in the presence of an internal heat generation or absorption effect.
The growing need for chemical reaction in industries and engineering requires the study of heat and mass transfer in the presence of different conditions and parameters with a chemical reaction. There are many transfer processes that are governed by the combined action of buoyancy forces due to both thermal and mass diffusion in the presence of a chemical reaction. Chemical reactions can be classified as either homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniformly through a given phase. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. A reaction is said to be of first order if the rate of reaction is directly proportional to the concentration itself, which has many applications in different chemical engineering processes and other industrial applications such as polymer production, manufacturing of ceramics or glassware, and food processing [18]. It has many applications in nuclear reactor and combustion, solar collectors, drying, dehydration operations in chemical and food processing plants, polymer production, etc. The effect of a chemical reaction on a moving isothermal vertical surface with suction has been considered by Muthucumarswamy [19]. Considering this in the study of a chemical reaction, Das et al. [20] considered the effects of a firstorder chemical reaction on the flow past an impulsively started infinite vertical plate with constant heat flux and mass transfer. Muthucumarswamy [21] and [19] studied a firstorder homogeneous chemical reaction on flow past an infinite vertical plate. Anderson et al. in 1994 have studied the diffusion of a chemically reactive species from a linearly stretching sheet. Anjalidevi and Kandasamy [22] investigated the effect of a chemical reaction on the flow along a semiinfinite horizontal plate in the presence of heat transfer. Reference [23] studied the effect of a chemical reaction on the flow in the presence of heat transfer and magnetic field. Muthucumarswamy and Ganesan [24] analyzed the effect of a chemical reaction on the unsteady flow past an impulsively started semiinfinite vertical plate, which is subject to uniform heat flux. On the other hand, radiation heat transfer effects from a porous wall on free convective flow are very important in space technology and high temperature processes, and very little is known about the effects of radiation on the boundary layer of a radiative fluid past a body. The inclusion of radiation effects in the energy equation leads to a highly nonlinear partial differential equation. Actually, many processes in new engineering areas occur at high temperature and knowledge of radiation heat transfer becomes imperative for the design of the pertinent equipment. Nuclear power plants, gas turbines, and the various propulsion devices for aircraft, missiles, satellites, and space vehicles are examples of such engineering areas. The radiation effects of the free convective flow of a gas past a semiinfinite flat plate was studied by Soundalgekar et al. [25] using the CogleyVincentiGiles equilibrium model [26]. Hossain and Takhar [27] studied the effects of radiation of an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream velocity and surface temperature. In similar research work, Makinde [28] investigated the effect of thermal radiation on the free convective flow and mass transfer past a moving vertical porous plate using a superposition method and reported that increase in thermal radiation intensity will definitely enhance the fluid velocity and promote a boundary layer within the flow regime. In this analysis, consideration had been given to gray gases that emit and absorb, but do not scatter thermal radiation; they noted that the Rosseland diffusion approximation provides one of the most straightforward simplifications of the full integropartial differential equations.
In view of the significance of the radiation effect as well as the chemical reaction and heat generation effects, we propose in the present paper to investigate the effects of radiation on free convective heat and mass transfer past an isothermal vertical oscillating porous plate in the presence of a chemical reaction and heat generation, using a spectral relaxation method. The governing boundary layer equations are transformed using dimensionless quantities to yield a coupled linear system of partial differential equations. The transformed governing equations are an approach using a spectral relaxation method. The spectral relaxation method is a new numerical method, proposed by Motsa [29], that can be used to solve linear and nonlinear systems of boundary value problems. Our main objectives are to study the effect of the radiation parameter and heat generation on the flow and transport characteristics using SRM. Our work is an extension to the work done by Muthucumaraswamy and Janakiraman [30] by considering the effect of thermal diffusion and heat generation on the flow and heat and mass transfer.
2 Problem formulation
3 Numerical method: spectral relaxation method (SRM)
Here, we discuss the application of the numerical method called the spectral relaxation method (SRM) to obtain an approximate solution of the partial differential equations (14)(16) subject to conditions (17)(18). In the application of SRM, it is fundamental to apply an efficient linearization technique to derive a system of linear equations that can be discretized using basic discretization scheme.
4 Results and discussion
Computational values of skinfriction coefficient, local heat transfer rate (Nusselt number) and Sherwood number at various values of the thermal radiation parameter R when \(\pmb{\mathit{Gr}=\mathit{Gc}=2}\) , \(\pmb{\mathit{Pr}=0.72}\) , \(\pmb{A_{2}=2}\) , \(\pmb{\omega=\pi/6}\) , \(\pmb{\mathit{Sc} =0.62}\) , \(\pmb{Q_{1}=0.5}\) , and \(\pmb{\tau=0.5}\)
R  \(\boldsymbol{\tau^{*} }\)  Nu  Sh 

0.0  2.7973670712  0.4188946992  1.3794604542 
0.5  2.8373439852  0.3592776848  1.3794604542 
1  2.8690666406  0.3198857845  1.3794604542 
1.5  2.8953500265  0.2910196053  1.3794604542 
1.8  2.9091812522  0.2769640603  1.3794604542 
2.0  2.9177411262  0.2686161833  1.3794604542 
Computational values of skinfriction coefficient and local heat transfer rate (Nusselt number) and Sherwood number at various values of heat generation/absorption parameter \(\pmb{Q_{1}}\) when \(\pmb{\mathit{Gr}=\mathit{Gc}=2}\) , \(\pmb{\mathit{Pr}=0.72}\) , \(\pmb{A_{2}=2}\) , \(\pmb{\omega=\pi/6}\) , \(\pmb{\mathit{Sc} =0.62}\) , \(\pmb{R=0.5}\) , and \(\pmb{\tau =0.5}\)
\(\boldsymbol{Q_{1}}\)  \(\boldsymbol{\tau^{*} }\)  Nu  Sh 

0.0  2.7550051157  0.5537767389  1.3794604542 
0.5  2.8373439852  0.3592776848  1.3794604542 
1.0  2.9309483388  0.1351747152  1.3794604542 
1.5  3.0378401707  0.1255127534  1.3794604542 
5 Conclusion
 ⋄:

Buoyancy parameters such as the thermal Grashof number and the mass Grashof number increase the velocity distribution within the momentum boundary layer.
 ⋄:

The presence of heat absorption enhances the temperature distribution and reduces the fluid velocity profile.
 ⋄:

The concentration distribution decreases with increasing value of the chemical reaction parameter.
 ⋄:

Both the velocity and the temperature profiles increase with increasing values of thermal radiation parameter.
Declarations
Acknowledgements
We are grateful to SS Motsa for his kind assistance with spectral relaxation method (SRM), and Boneze Chika who moderated this paper and in that sense improved the manuscript significantly.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Ostrach, S: Analysis of laminar free convective flow and heat transfer about a flat plate parallel to direction of the generating body force. NACA technical report 1111 (1952) Google Scholar
 Kolar, AK, Sastri, VM: Free convective transpiration over a vertical plate: a numerical study. Heat Mass Transf. 23, 327336 (1988) Google Scholar
 Ramanaiah, G, Malarvizi, G: Unified treatment of free convection adjacent to a vertical plate with three thermal boundary conditions. Heat Mass Transf. 27, 393396 (1992) Google Scholar
 Pop, I, Soundalgekar, VM: Free convection flow past an accelerated infinite plate. Z. Angew. Math. Mech. 60, 167168 (1980) View ArticleMATHGoogle Scholar
 Raptis, A, Singh, AK, Rai, KD: The unsteady free convective flow through a porous medium adjacent to a semiinfinite vertical plate using finite difference scheme. Mech. Res. Commun. 14, 916 (1987) View ArticleMATHGoogle Scholar
 Singh, AK, Soundalgekar, VM: Transient free convection in cold water past an infinite vertical porous plate. Int. J. Energy Res. 14, 413420 (1990) View ArticleGoogle Scholar
 Stokes, GG: On the effect of the internal friction of fluid on the motion of pendulum. Trans. Camb. Philos. Soc. 9, 8106 (1851) Google Scholar
 Soundalgekar, VM: Free convection effects on the flow past a vertical oscillating plate. Astrophys. Space Sci. 64, 165172 (1979) View ArticleGoogle Scholar
 Senapati, N, Dhal, RK, Das, TK: Effects of chemical reaction on free convection MHD flow through porous medium bounded by vertical surface with slip flow region. Am. J. Comput. Appl. Math. 2(3), 124135 (2012). doi:10.5923/j.ajcam.20120203.10 View ArticleGoogle Scholar
 Gupta, AS, Misra, JC, Reza, M, Soundalgekar, VM: Flow in the Ekman layer on an oscillating porous plate. Acta Mech. 165, 116 (2003) View ArticleMATHGoogle Scholar
 Muthucumaraswamy, R, Meenakshisundaram, S: Theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature. Theor. Appl. Mech. 33(3), 245257 (2006) View ArticleMATHGoogle Scholar
 Kishore, PM, Rajesh, V, Verma, SV: The effects of thermal radiation and viscous dissipation on MHD heat and mass diffusion flow past an oscillating vertical plate embedded in a porous medium with variable surface conditions. Theor. Appl. Mech. 39(2), 99125 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Gupta, PS, Gupta, AS: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55, 744746 (1977) View ArticleGoogle Scholar
 Elbashbeshy, EMA: Heat transfer over a stretching surface with variable surface heat flux. J. Phys. D, Appl. Phys. 31, 19511958 (1988) View ArticleGoogle Scholar
 Vajravelu, K, Hadjinicolaou, A: Convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. Int. J. Eng. Sci. 35(1213), 12371244 (1997) View ArticleMATHGoogle Scholar
 Chamkha, AJ: Unsteady MHD convective heat and mass transfer past a semiinfinite vertical permeable moving plate with heat absorption. Int. J. Eng. Sci. 24, 217230 (2004) View ArticleMATHGoogle Scholar
 Hady, FM, Mohamed, RA, Mahdy, A: MHD free convection flow along a vertical wavy surface with heat generation or absorption effect. Int. Commun. Heat Mass Transf. 33, 12531263 (2006) View ArticleGoogle Scholar
 Cussler, EL: Diffusion: Mass Transfer in Fluid Systems, 2nd edn. Cambridge University Press, Cambridge (1997) Google Scholar
 Muthucumarswamy, R: Effects of a chemical reaction on moving isothermal vertical surface with suction. Acta Mech. 155(12), 6570 (2002). doi:10.1007/BF01170840 View ArticleGoogle Scholar
 Das, UN, Deka, RK, Soundalgekar, VM: Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction. Forsch. Ingenieurwes. 60(10), 284287 (1994). doi:10.1007/BF02601318 View ArticleGoogle Scholar
 Muthucumaraswamy, R, Ganesan, P: On impulsive motion of a vertical plate with heat flux and diffusion of chemically reactive species. Forsch. Ingenieurwes. 66, 1723 (2000) View ArticleGoogle Scholar
 Anjalidevi, SP, Kandasamy, R: Effects of chemical reaction, heat and mass transfer on laminar flow along a semi infinite horizontal plate. Heat Mass Transf. 35, 465467 (1999) View ArticleGoogle Scholar
 Anjalidevi, SP, Kandasamy, R: Effects of chemical reaction, heat and mass transfer on MHD flow past a semi infinite plate. Z. Angew. Math. Mech. 80, 697700 (2000) View ArticleMATHGoogle Scholar
 Muthucumarswamy, R, Ganesan, P: First order chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux. Acta Mech. 147(14), 4557 (2001). doi:10.1007/BF01182351 View ArticleMATHGoogle Scholar
 Soundalgekar, VM, Takhar, HS, Vighnesam, NV: The combined free and forced convection flow past a semiinfinite plate with variable surface temperature. Nucl. Eng. Des. 110, 9598 (1960) View ArticleGoogle Scholar
 Cogely, AC, Vinceti, WG, Giles, SE: Differential approximation for radiative heat transfer in a nongray gas near equilibrium. AIAA J. 6, 551553 (1968) View ArticleGoogle Scholar
 Hossain, MA, Takhar, HS: Radiation effect on mixed convection along a vertical plate with uniform surface temperature. Heat Mass Transf. 31, 243248 (1996) View ArticleGoogle Scholar
 Makinde, OD: Free convective flow with thermal radiation and mass transfer past a moving vertical porous plate. Int. Commun. Heat Mass Transf. 32, 14111419 (2005) View ArticleGoogle Scholar
 Motsa, SS: A new spectral relaxation method fora similarity variable nonlinear boundary layer flow systems. Chem. Eng. Commun. 201, 241256 (2014) View ArticleGoogle Scholar
 Muthucumaraswamy, R, Janakiraman, B: Mass transfer effects on isothermal vertical oscillating plate in the presence of chemical reaction. Int. J. Appl. Math. Mech. 4(1), 6674 (2008) Google Scholar
 Aboeldahab, EM, El Gendy, MS: Radiation effect on MHD freeconvective flow of a gas past a semiinfinite vertical plate with variable thermophysical properties for hightemperature differences. Can. J. Phys. 80, 16091619 (2002) View ArticleGoogle Scholar
 Cogley, AC, Vincenty, WG, Gilles, SE: Differential approximation for radiative transfer in a nongrey gas near equilibrium. AIAA J. 6, 551553 (1968) View ArticleGoogle Scholar
 Raptis, A: Radiation and free convection flow through a porous medium. Int. Commun. Heat Mass Transf. 25, 289295 (1998) View ArticleGoogle Scholar
 Sparrow, EM, Cess, RD: Radiation Heat Transfer: Augmented Edition, Chapters 1 and 7. Hemisphere, Washington (1978) Google Scholar
 Motsa, SS, Dlamini, PG, Khumalo, M: Solving hyperchaotic systems using the spectral relaxation method. Abstr. Appl. Anal. 2012, Article ID 203461 (2012). doi:10.1155/2012/203461 MathSciNetView ArticleMATHGoogle Scholar
 Motsa, SS, Dlamini, PG, Khumalo, M: A new multistage spectral relaxation method for solving chaotic initial value systems. Nonlinear Dyn. 72(12), 265283 (2013) MathSciNetView ArticleMATHGoogle Scholar