Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus An exact solution Engineering Science and Technology, an International Journal xxx (2017)[.]
Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Contents lists available at ScienceDirect Engineering Science and Technology, an International Journal journal homepage: www.elsevier.com/locate/jestch Full Length Article Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution Michael O Oni Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria a r t i c l e i n f o Article history: Received 18 September 2016 Revised December 2016 Accepted 25 December 2016 Available online xxxx Keywords: Heat source Thermal radiation Mixed convection Vertical annulus Porous material Exact solution a b s t r a c t This paper examines the effect of heat source, thermal radiation and porosity on mixed convection flow in a vertical annulus filled with porous material The inner surface of outer cylinder is assumed to be the heated surface Closed-form expression for temperature, velocity, Nusselt number, skin-friction and mass flow rate are obtained in terms of Bessel’s function and modified Bessel’s function of first and second kind Based on depicted graphs, fluid temperature and Nusselt number increase with increase in radiation parameter and heat source parameter while velocity as well as skin-friction decreases with increase in radiation parameter and heat source parameter at the surfaces of the cylinder Ó 2017 Karabuk University Publishing services by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction Over the years, owing to the fact that cylinders have been used in nuclear waste disposal, energy extortion in catalytic beds and undergrounds, convective heat transfer about cylindrical geometries has begun to attract the attention of many researchers since some fluids are good emitter and absorber of thermal radiation, it is of interest to study the effect of heat source on temperature distributions and heat transfer when the fluid is capable of emitting and absorbing thermal radiation This can be attributed to the fact that heat transfer by thermal radiation is becoming of greater importance when we are concerned with space applications (space journey) and higher operating temperatures Several investigations have been carried out on problem of heat transfer by radiation as an important application of space and temperature related problems Greif et al [1] obtained an exact solution for the problem of laminar convective flow in a vertical heated channel in the optically thin limit They concluded that in the optically thin limit, the fluid does not absorb its own emitted radiation which means that there is no self-absorption but the fluid does absorb radiation emitted by the boundaries Viskanta [2] investigated the forced convective flow in a horizontal channel permeated by uniform vertical magnetic field taking radiation into E-mail address: Michaeloni29@yahoo.com account In his work, he studied the effects of magnetic field and radiation on the temperature distribution and the rate of heat transfer in the flow and found that the effect of magnetic field is to decrease fluid velocity Later Gupta and Gupta [3] studied the effect of radiation on the combined free and forced convection of an electrically conducting fluid flowing inside an open-ended vertical channel in the presence of a uniform transverse magnetic field for the case of optically thin limit They found that radiation tends to increase the rate of heat transfer of the fluid there by reducing the effect of natural convection Later, Hossain and Takhar [4] analyzed the effect of radiation using the Rosseland diffusion approximation which leads to nonsimilar solution for the forced and free convection of an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream and uniform surface temperature, while Hossain et al [5] studied the effect of radiation on free convection from a porous vertical plates The role of thermal radiation is of major importance in the design of many advanced energy convection systems operating at high temperature and due to increase in science and technology, radiative heat transfer becomes very important in nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles and space vehicles [6–10] In cylindrical geometry, the studies of heat generation and thermal radiation have been studied by several authors Chamkha [11] analyzed the heat and mass transfer of a MHD flow over a moving http://dx.doi.org/10.1016/j.jestch.2016.12.009 2215-0986/Ó 2017 Karabuk University Publishing services by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Nomenclature a b Da g Gr h H In Jn K Kn N Nu0 Nu1 p P Pr Q0 Re r R T radius of inner cylinder radius of outer cylinder Darcy number gravitational acceleration Grashof number convective heat transfer coefficient dimensionless heat generating (source) parameter modified Bessel’s function of first kind on order n Where n ¼ 0; 1; 2; 3; Bessel’s function of first kind on order n Where n ¼ 0; 1; 2; 3; permeability modified Bessel’s function of second kind on order n Where n ¼ 0; 1; 2; 3; radiation parameter rate of heat transfer at the outer surface of the inner cylinder rate of heat transfer at the inner surface of the outer cylinder dimensional pressure difference dimensionless pressure difference Prandtl number dimensional heat generating (source) parameter Reynold number dimensional coordinate dimensionless coordinate dimensional temperature of the fluid permeable cylinder with heat generation or absorption and chemical reaction He concluded that the role of Hartmann number is to decrease fluid velocity Trujillo et al [12] studied the heat and mass transfer process during the evaporation of water from a circular cylinder through CFD modeling In other related work, Mujtaba and Chamkha [13] discussed the heat and mass transfer from a permeable cylinder in a porous medium with magnetic field and heat generation/absorption Ganesan and Loganathan [14,15] investigated an unsteady natural convective flow past semiinfinite vertical cylinder with heat and mass transfer under different physical situations Hossain et al [16] reported the radiation conduction interaction on mixed convection from a horizontal circular cylinder using an implicit finite-difference scheme Also, Ganesan and Loganathan [17] studied the radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder Gnaneswar and Reddy [18,19] analyzed the radiation and mass transfer effects on an unsteady MHD free convection flow of an incompressible viscous fluid past a moving vertical cylinder Radiation effects on hydromagnetic free convective and mass transfer flow of a gas past a circular cylinder with uniform heat and mass flux was studied by Hakiem [20] He also found that magnetic field parameter retards fluid velocity Yih [21] analyzed the radiation effect on natural convection over a vertical cylinder embedded in a porous media Suneetha and Bhaskar [22] analyzed the radiation and mass transfer effects on MHD free Convection flow past a moving vertical cylinder embedded in a porous medium Other related articles on radiation effect on heat transfer of mixed convection flow for different fluid can be seen in [23–25] The purpose of this paper is to examine theoretically the effects of thermal radiation and porosity on viscous, incompressible and heat generating fluid in a vertical annulus filled with porous material Exact solution for temperature, velocity, skin-friction and Nusselt number are obtained and the effects of governing parameters are discussed with the aid of line graphs T0 Tw u U Yn z Z initial temperature final temperature dimensional velocity of fluid dimensionless velocity of fluid Bessel’s function of second kind on order n Where n ¼ 0; 1; 2; 3; dimensional coordinate dimensionless coordinate Greek alphabets a thermal diffusivity b coefficient of thermal expansion j thermal conductivity j mean absorption coefficient k aspect ratio ðb=aÞ r Stefan-Boltzmann constant m kinematic viscosity q density h dimensionless temperature of fluid l dynamics viscosity c viscosity ratio Subscript outer surface of inner cylinder k inner surface of outer cylinder Mathematical analysis Consider the steady laminar mixed convection flow of a viscous incompressible heat generating fluid The axis of cylinder is taken along the z-axis, while r-axis is taken in the radial direction The inner surface of the outer cylinder is assumed to be heated to a temperature T w greater than that of surrounding fluid and outer surface of the inner cylinder having temperature T The radius of the inner and outer cylinder walls are a and b respectively as Fig Schematic diagram Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx seen in Fig The following assumptions are made in order to obtain an exact solution: i The fluid motion is fully developed both thermally and hydrodynmaically ii All physical quantities except pressure gradient are independent on z-axis iii Viscous dissipation and displacement currents terms are neglected in the energy equation iv The fluid is considered to be a gray, absorbing emitting radiation but non-scattering medium Under the above assumptions, the governing equations of the flow are respectively the continuity, momentum and energy equations: dv ị ẳ0 dr ð1Þ meff d du0 mu0 dp r ẳ0 ỵ bgT T ị r dr dr K q dz ð2Þ k d dT 1 d Q ðT T ị r rq ị ỵ ẳ0 qC p r dr dr qC p r dr r qC p 3ị u0 ẳ T ẳ T0 at r ¼ a u0 ¼ T ¼ T w at r ẳ b 4ị By using the Rosseland approximation (Brewster [26]), the radiative heat flux is given by qr qr ¼ 4r @T 3k @r ð5Þ Following [27], the function T in Eq (5) can be expressed as a linear function by expanding it in a Taylor series about T w and neglecting higher powers of T as follows: 1.4 H = 0.5 H = 2.5 1.2 θ (R) 0.8 0.6 0.4 N = 0.5, 1.5, 2.5, 3.5 0.2 -0.2 1.1 1.2 1.3 1.4 1.5 R 1.7 1.6 1.8 1.9 Fig Temperature profile versus R varying H and N at k = 2.0 N = 0.5 N = 3.5 θ (R) 0.8 0.6 λ = 2.2, 2.0, 1.8 0.4 0.2 1.2 1.4 1.6 R 1.8 2.2 Fig Temperature profile versus R varying k and N at H = 1.5 Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx T ¼ 4T 2w T 3T 4w ð6Þ Introducing the following dimensionless parameters: U¼ u0 ; u0 h¼ T T0 ; Tw T0 Gr ¼ Re ¼ b k¼ ; a c¼ meff pa ; P¼ ; u0 l m r z R ¼ ;Z ¼ ; a a N¼ jk Q b ; H2 ¼ ; k 4rT u0 a l ; Da ¼ K ; a2 d dh 3H2 N R ỵ hẳ0 R dR dR 3N ỵ U ẳ h ẳ at R ẳ Uẳ0 hẳ1 10ị at R ¼ k The solution of Eqs (8)–(10) is obtained as: hRị ẳ C J m1 Rị þ C Y ðm1 RÞ gbðT T ịa3 7ị m2 URị ẳ C I0 Eqs (2)–(6) in dimensionless form become: c d dU Gr U dP R ỵ h ẳ0 dR Re Da dZ R dR ð11Þ ! ! R R Gr Da ẵC J m1 Rị p ỵ C K p ỵ Re cm21 Da ỵ Dac Dac ỵC Y m1 RÞ Da ð8Þ 1.6 ð9Þ dP dZ ð12Þ H = 0.5 H = 2.5 1.4 1.2 U(R) 0.8 γ = 0.5, 1.0, 1.5 0.6 0.4 0.2 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 R Fig Velocity profile versus R varying H and c at N = 1.5, k = 2.0, Gr/Re = 50, Da = 0.1 1.6 N = 0.5 N = 3.5 1.4 1.2 U(R) Da = 0.001, 0.01, 0.1 0.8 0.6 0.4 0.2 1.1 1.2 1.3 1.4 1.5 R 1.6 1.7 1.8 1.9 Fig Velocity profile versus R varying Da and N at H = 2.5, c = 1.5, k = 2.0, Gr/Re = 50 Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx The rate of heat transfer between the surfaces of the cylinders and the fluid is obtained by differentiating the temperature and is given by: Nu1 ẳ dh ẳ m1 ẵC J m1 ị ỵ C Y m1 ị dR Rẳ1 14ị The skin friction at the outer surface of inner cylinder and inner surface of outer cylinder is respectively given by: " ! !# dU 1 p p p ffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffi ffi C C ¼ ¼ I K dR R¼1 Dac Dac Dac ! !# dU k k ¼ pffiffiffiffiffiffiffiffiffi C K pffiffiffiffiffiffiffiffiffi C I1 pffiffiffiffiffiffiffiffiffi dR R¼k Dac Dac Dac sk ẳ ỵ 13ị dh Nuk ẳ ẳ m1 ẵC J km1 ị ỵ C Y km1 ị dR Rẳk s1 " Gr Dam1 ẵC J km1 ị ỵ C Y km1 ị Re cm21 Da ỵ 1ị 16ị The amount of fluid passing through the annulus is given by the volume flow rate and defined as: Z V¼ k 17ị RURịdR V ẳ n1 ỵ n2 ỵ n3 þ n4 þ n5 ð18Þ where n1 ; n2 ; n3 ; n4 are constants defined in the Appendix The pressure gradient is obtained using the conservation law as: Gr Dam1 ẵC J m1 ị ỵ C Y m1 ị Re cm21 Da ỵ 1ị 15ị Z k Z RURịdR ẳ k 19ị RdR U(R) 1.5 0.5 Gr/Re = 200, 150, 100, 50, 0, -50, -100, -150, -200 1.1 1.2 1.3 1.4 1.5 R 1.6 1.7 1.8 1.9 Fig Velocity profile versus R varying Gr/Re at H = 1.5, N = 1.5, c = 1.5, k = 2.0 4.55 54 4.5522.533 11.5 52.533 54 4.5 1.5 0.5 3.5 5 2.5 1.5 2.5 4.55 Nu 2.5 H Fig Nusselt number for different values of H and N at k = 2.0, R = Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx dP p1 ¼ dZ p2 Gr t 25 ẳ Re Rẳ1 t 26 20ị there t i for i ¼ 1; 2; 3; are constant defined in Appendix where p1 and p2 are constants defined in Appendix The critical Gr=Re is obtained by: dU ẳ0 dR Rẳ1;k Results and discussion 21ị The solution obtained from Eqs (11)–(23) are seen to be govern by heat source parameter ðHÞ, radiation parameter ðNÞ, Darcy number ðDaÞ, aspect ratio ðkÞ, viscosity ratio ðcÞ and mixed convection parameter ðGr=ReÞ In order to see the effect of these parameters, lines graphs are plotted to capture the physical situation Throughout this work, the heat source parameter is taken over the range 0:5 H 3:0 with reference value of 2.5, thermal radiation over Using Eq (21), the reverse flow occurrence at the outer surface of inner cylinder and inner surface of outer cylinder is respectively given by: Gr t 23 ẳ Re Rẳ1 t 24 23ị 22ị Nu λ 522 533.544.5 11.5 0.5 .5 44 3.5 2.5 1 44 33 5 22 11.5 1.5 44 2.5 5 2.5 H Fig Nusselt number for different values of H and N at k = 2.0, R = k 30 25 N = 0.5 N = 3.5 τ1 20 15 10 0.5 Da = 0.1, 0.01, 0.001 1.5 2.5 H Fig Skin friction for different values of H, Da and N at c = 1.5, k = 2.0, Gr/Re = 50, (R = 1) Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx 0:5 N 3:5, 0:001 Da 0:1 and 0:5 c 1:5 to capture different physical problems Mixed convection parameter 200 ðGr=ReÞ 200 has been chosen to capture cases when the natural convection dominates, ðGr=ReÞ P and otherwise ðGr=ReÞ Fig presents the effect of heat source parameter ðHÞ and thermal radiation parameter ðNÞ on temperature distribution at different points in the annulus, the effect of heat generation and radiation parameter is to increase the fluid temperature It is observed that for small heat generation parameter the effect of thermal radiation is insignificant on fluid temperature in the annulus Fig on the other hand depicts effect of aspect ratio ðkÞ on temperature distribution in the annulus It is found that as k increases, fluid temperature decreases In addition, magnitude of temperature is seen increase with increase in thermal radiation parameter Fig shows combined effect of viscosity ratio ðcÞ and heat source ðHÞ on fluid velocity in the annulus It is observed that fluid velocity is an increasing function of heat source parameter ðHÞ and viscosity ratio ðcÞ at the center of the annulus but the reverse is observed at the region close to the surfaces of the annulus Two points of inflexion are noticed in this figure, at these points, fluid velocity is independent on heat source parameter In similar manner, Fig gives the combined effect of radiation ðNÞ and porosity of porous material ðDaÞ on fluid velocity As expected, as ðDaÞ increases, velocity also increases This can be attributed to the fact that Darcy number ðDaÞ is directly proportional to the permeability which widens the pores of the porous material and as such enhances fluid motion The reverse situation is noticed at the region close to the surfaces of the cylinders In addition, the maximum velocity is reached at the center of the annulus Fig presents the effect of mixed convection parameter ðGr=ReÞ on fluid velocity For positive values of ðGr=ReÞ i.e when natural convection dominates over forced convection, fluid velocity is seen to be higher at the inner surface of the outer cylinder than outer surface of the inner cylinder On the other hand, when 30 25 τλ 20 N = 0.5 N = 3.5 15 10 Da = 0.1, 0.01, 0.001 0.5 1.5 2.5 H Fig 10 Skin friction for different values of H, Da and N at c = 2.0, k = 1.5, Gr/Re = 50 (R = k) 10 τ1 -5 -10 -15 -20 γ =0.5 -25 -30 0.5 γ = 1.0 γ = 1.5 Gr/Re = 150, 100, 50, 1.5 2.5 H Fig 11 Skin friction for different values of H, c and Gr/Re at k = 1.5, N = 1.5, Da = 0.1 Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx ðGr=ReÞ < 0, the reverse result is obtained When ðGr=ReÞ ¼ 0, i.e no natural convection (purely forced convection), the velocity is seen to be symmetric Also, a point of inflexion is noticed at the center of the annulus At this point, fluid velocity is independent whether forced or natural convection Figs and display the rate of heat transfer at the outer surface of the inner cylinder and inner surface of outer cylinder respectively for different values of radiation and heat source parameter It is observed that radiation ðNÞ and heat source ðHÞ enhance rate of heat transfer at the outer surface of inner cylinder This can be attributed to the fact that increase heat source and radiation parameter, enhances fluid temperature which in turn increase the rate at which heat is transfers between the fluid and the surfaces of the cylinders In addition, the rate of heat transfer is seen to be higher at the outer surface of inner cylinder than inner surface of outer cylinder Figs and 10 illustrate combined effect of radiation ðNÞ porosity ðDaÞ and heat source ðHÞ on the skin-friction at the outer surface of inner cylinder and inner surface of outer cylinder At both surfaces of the cylinders, the skin-friction is seen to decrease with increase in Darcy number ðDaÞ, radiation parameter ðNÞ and heat source parameter ðHÞ It is interesting to note that skin-friction is independent on radiation or porosity for small value of heat source parameter This can be explained form the solution obtained, that for small heat source parameter, temperature profile is independent of H and hence the velocity/skin friction Further, skin friction is higher at the inner surface of outer cylinder than outer surface of inner cylinder Figs 11 and 12 depict combined effect of viscosity ratio and mixed convection parameter on the skin-friction at the surfaces of the cylinders The skin-friction at the outer surface of inner cylinder in Fig 11 is seen to decrease with increase in heat source parameter and mixed convection parameter but increase with increase in viscosity ratio The reversed result is noticed for skinfriction at the inner surface of outer cylinder in Fig 12 Also, the maximum skin friction is observed at the inner surface of outer cylinder Table presents the critical values of mixed convection parameter varying radiation parameter, Darcy number and viscosity ratio It is found that the critical Gr=Re increase with increase in Table for different values of N; Da and c at H ¼ 1:5; k ¼ 2:0 Numerical values for critical Gr Re N Da c Gr Re R¼1 Gr Re R¼k 0.5 0.01 0.5 1.0 1.5 0.5 1.0 1.5 219.1270 256.4985 291.0381 52.2403 85.1432 118.0921 363.9459 411.8693 456.1656 75.4014 117.8757 160.4375 0.5 1.0 1.5 0.5 1.0 1.5 208.6325 244.6672 277.9916 50.1094 81.8152 113.5588 394.9476 444.0181 489.7350 80.0711 124.6705 169.4157 0.5 1.0 1.5 0.5 1.0 1.5 200.5026 235.4918 267.8649 48.4508 79.2221 110.0251 426.2374 476.1561 523.0914 84.6400 131.2828 178.1349 0.1 1.5 0.01 0.1 3.5 0.01 0.1 viscosity ratio but decreases with increase in radiation parameter as well as Darcy number Conclusions In this work, an exact solution of combined effects of heat source and thermal radiation on mixed convection flow in a vertical annulus filled with porous material is obtained Closed-form expression for temperature distributions, velocity profiles, Nusselt number (representing the rate of heat transfer), skin-friction and mass flow rate are obtained in term of Bessel’s function and modified Bessel’s function of first and second kinds respectively The effects of governing parameters such as heat source parameter ðHÞ, radiation parameter ðNÞ, mixed convection parameter ðGr=ReÞ, Darcy number ðDaÞ, aspect ratio ðkÞ and viscosity ratio ðcÞ on temperature, velocity, Nusselt number and skin-friction are illustrated with the use of line graphs Based on the figures depicted, the following conclusions can be drawn: 24 22 Gr/Re = 150, 100, 50, γ = 0.5 γ = 1.0 γ = 1.5 20 18 τλ 16 14 12 10 0.5 1.5 2.5 H Fig 12 Skin friction for different values of H, c and Gr/Re at k = 2.0, N = 1.5 (R = k) Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2016.12.009 M.O Oni / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Da C1 C2 kJ km1 ị J m1 ịị ỵ ðkY ðkm1 Þ Y ðm1 ÞÞ m1 cm21 Da ỵ 1ị m1 1 Thermal radiation as well as heat source parameter increase fluid temperature Fluid velocity decreases with increase in heat source, viscosity, radiation and mixed convection parameter but increases with increase in Darcy number at the outer surface of inner cylinder while the reverse trend is occurs at the inner surface of outer cylinder The rate of heat transfer at the surfaces of the cylinders increases with increase in radiation and heat source The skin-friction at the surfaces of the cylinders is reduced by porosity (Darcy number), heat source parameter and radiation parameter Reverse flow occurrence at the surfaces of the cylinder decreases with increase in porosity and radiation parameter, but increases with increase in viscosity ratio Reverse flow in the annulus can be avoided by choosing appropriate values for heat source parameter, radiation parameter, Darcy number and viscosity ratio When Gr=Re ¼ 0, this work reduces to pressure driven flow and the effect of N and H on velocity profile and skin friction is suppressed t12 ¼ Acknowledgements t23 ¼ t17 t 21 ; The author is thankful to his supervisors Prof B.K Jha and Prof A.O Ajibade for their support throughout the compilation of this article and their fatherly impartations Appendix s 3NH2 m1 ẳ ; 3N ỵ sffiffiffiffiffiffiffiffiffi ; m2 ¼ cDa m3 ¼ m2 Y m1 ị ; ẵY km1 ịJ ðm1 Þ J ðkm1 ÞY ðm1 ị J m1 ị C2 ẳ ẵY km1 ÞJ ðm1 Þ J ðkm1 ÞY m1 ị C1 ẳ dP Gr dP Gr ỵ t5 ; C ẳ t8 ỵ t9 ; dZ Re dZ Re n1 ẳ C m3 ẵkI1 km2 ị I1 m2 ị C ẳ t4 n2 ẳ C m3 ẵK m2 ị kK km2 ị; C DaGr ẵkJ km1 ị J m1 ị n3 ẳ Recm21 Da ỵ 1ịm1 C DaGr ẵkY km1 ị Y m1 ị; Recm21 Da ỵ 1ịm1 n4 ẳ n5 ẳ Da dP k2 1ị dZ t ẳ DaẵK km2 ị K m2 ị; t2 ẳ DaK m2 ị cm21 Da ỵ 1ị t ẳ ẵK km2 ịI0 m2 ị I0 km2 ịK m2 ị; t ẳ DaẵI0 km2 Þ I0 ðm2 Þ; t9 ¼ t7 ¼ t7 t3 t 10 ẳ Dak2 1ị ; t 11 ẳ k2 1ị t4 ẳ t1 ; t3 DaI0 m2 ị ; cm21 Da ỵ 1ị t5 ¼ t2 t3 t8 ¼ t6 ; t3 t13 ¼ kI1 ðkm2 Þ I1 ðm2 Þ; t 14 ẳ kK km2 ị K m2 Þ t15 ¼ m3 ðt9 t 14 t5 t13 ị t12 ; p1 ẳ t 11 ỵ Gr ½t 15 ; Re p2 ¼ m3 ðt4 t13 t t 14 ị t10 t16 ẳ Dam1 ; cm21 Da ỵ 1ị t17 ẳ t8 m2 K ðm2 Þ t m2 I1 ðm2 Þ; t18 ẳ t5 m2 I1 m2 ị t9 m2 K ðm2 Þ t16 ðC J m1 ị ỵ C Y m1 ịị t19 ¼ t8 m2 K ðkm2 Þ t4 m2 I1 km2 ị; t 22 ẳ p2 p1 t20 ẳ t5 m2 I1 ðkm2 Þ t m2 K ðkm2 Þ t16 ðC J ðkm1 ị t 11 ỵ C Y km1 ịị; t 21 ¼ p1 t24 ¼ t18 t 17 t 22 ; t 25 ¼ t 19 t 21 ; t24 ¼ t20 t 19 t 22 : References [1] R Greif, I.S Habib, J.C Lin, Laminar convection of a radiating gas in a vertical channel, J Fluid Dyn 46 (1971) 513–520 [2] R Viskanta, Effect of transverse magnetic field on heat transfer to an electrically conducting and thermal radiating fluid flowing in a parallel 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H and N at k = 2.0, R = Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, ... Loganathan [17] studied the radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder Gnaneswar and Reddy [18,19] analyzed the radiation and mass... Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: An exact solution, Eng Sci Tech., Int