nonlinear radiative heat transfer and hall effects on a viscous fluid in a semi porous curved channel

Nonlinear radiative heat transfer and Hall effects on a viscous fluid in a semi-porous curved channel , Z Abbas, M Naveed , and M Sajid Citation: AIP Advances 5, 107124 (2015); doi: 10.1063/1.4934582 View online: http://dx.doi.org/10.1063/1.4934582 View Table of Contents: http://aip.scitation.org/toc/adv/5/10 Published by the American Institute of Physics AIP ADVANCES 5, 107124 (2015) Nonlinear radiative heat transfer and Hall effects on a viscous fluid in a semi-porous curved channel Z Abbas,1 M Naveed,1,a and M Sajid2 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan Theoretical Physics Division, PINSTECH, P.O Nilore, Islamabad 44000, Pakistan (Received 18 May 2015; accepted October 2015; published online 23 October 2015) In this paper, effects of Hall currents and nonlinear radiative heat transfer in a viscous fluid passing through a semi-porous curved channel coiled in a circle of radius R are analyzed A curvilinear coordinate system is used to develop the mathematical model of the considered problem in the form partial differential equations Similarity solutions of the governing boundary value problems are obtained numerically using shooting method The results are also validated with the well-known finite difference technique known as the Keller-Box method The analysis of the involved pertinent parameters on the velocity and temperature distributions is presented through graphs and tables C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4934582] I INTRODUCTION In past few decades, study of flow and heat transfer phenomena in porous channels or tubes has received considerable attention due to their number of practical applications in the fields of biomedical and mechanical engineering Applications include flow in dialysis of blood in artificial kidney,1 blood oxygenators2 and flow in the capillaries.3 The study of steady incompressible viscous fluid flow between two parallel porous walls was first carried out by Berman.4 He provided the exact solution for the Navier-Stokes equations The research problem considered by Berman4 was extended in many directions for both Newtonian and non-Newtonian fluids For the details interested readers are referred to the articles5–11 and reference therein It is well known that in an ionized fluid with low density the intensity of magnetic field is very strong, the conductivity perpendicular to the magnetic field is low due to the spiraling of electrons and ions about the magnetic lines of force before collision occur and a current induced in the normal direction to both electric and magnetic fields This phenomenon is called the Hall effects.12 The effect of Hall currents on the fluid flow has wide range of applications in astrophysical, meteorological and plasma flow problems Flight magnetic hydrodynamics, MHD power generations, accelerator design, Hall effects sensors, construction of turbines and centrifugal machines13–18 are few examples of such studies The study of thermal radiation on convective heat transfer become a very vital theory in the processes involving high temperature such as nuclear power plant, hypersonic flight missile re-entry, thermal energy storage, rocket combustion chambers, solar power technology, power plant for inter planetary flight and design of pertinent equipment Pantokratoras and Fang20 discussed the Sakiadis flow with nonlinear Rosseland thermal radiation In another paper Pantokratoras and Fang21 studied the effects of nonlinear Rosseland thermal radiation in Blasius flow Cortell22 investigated the fluid flow and nonlinear radiative heat transfer over a stretching sheet A numerical study of nonlinear radiative heat transfer in the flow of a nano fluid due to solar energy was carried out by Mushtaq et al.23 a Corresponding Author Tel.: +92 62 9255480 e-mail address: rana.m.naveed@gmail.com (M Naveed) 2158-3226/2015/5(10)/107124/11 5, 107124-1 © Author(s) 2015 107124-2 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) The analysis of flow through narrow, curved channel gained significant importance due to its number of application in biological and engineering processes The seeping flow theory cracks and pulmonary alveolar blood flow have low Reynold number and so they can be characterized as a Stokes flow Stokes flow in a curved channel was carried out by Khuri.24 In a couple of papers Nasir et al.25,26 investigated the different aspects of peristaltic flow in curved channel The effects of porous medium on forced convection of a reciprocating curved channel were discussed by Shung Fu et al.27 A literature survey reveals that no study regarding to the Hall effects on flow and heat transfer in a semi porous curved channel by taking nonlinear thermal radiation effects into account has done before So the objective of present analysis is to investigate the Hall effects on a viscous fluid in a semi porous curved channel by incorporating nonlinear thermal radiation Numerical solution for the fluid velocity, pressure and temperature distributions are obtained using shooting method The numerical results obtained through shooting method are also validated using Keller-Box method Numerical results are presented through graphs and tables II MATHEMATICAL FORMULATION Consider the steady, two-dimensional, incompressible flow of an electrically conducting viscous fluid in a semi-porous curved channel of width H coiled in a circle of radius R as shown in Fig Further, the upper wall of channel is porous while the lower wall is impermeable It is also assumed that the wall placing along the s-axis is heated externally with temperature Tw and the electrically conducting viscous fluid is injected uniformly from the other perforated wall to cool the heated wall Flow is caused by suction or blowing A strong magnetic field of uniform strength B0 is applied in the radial direction r and the effects of Hall current are also considered Presence of Hall current generates a force in the z -direction, so the flow becomes three dimensional, however there will be no effects on flow and heat transfer properties in z -direction The generalized Ohm’s law in the absence of electric field E and considering the Hall effects can be written as (see Ref 28)   σ J×B , V×B− J= eη e + m2 where, V is the fluid velocity, B is the magnetic induction vector, J is the current density, m = σB0/eη e is the Hall parameter, σ = e2η e τe/me is the electrical conductivity, e is the electron charge, τe is the electron collision time, η e is the number density of electron me is the mass of electron The influence of electron pressure gradient and the ion slip effects can be neglected for the weakly ionized gases Implementing these assumptions, the boundary layer equations that govern the flow and heat transfer are ∂ ∂u {(r + R) v } + R = 0, ∂r ∂s FIG Geometry of the flow problem (1) 107124-3 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) u2 ∂p = , r+R ρ ∂r (2) Ru ∂u uv R ∂p ∂u + + =− + ∂r r + R ∂s ) r + R ρ r + R ∂s ( 2 σB0 u ∂u ∂u (u + mw) , − ν + − 2 r + R ∂r (r + R) ∂r ρ (1 + m2)     σB02 Ru ∂w ∂ w ∂w ∂w (mu − w) , + =ν + v + ∂r r + R ∂s r + R ∂r ∂r ρ (1 + m2)     ∂T ∂T uR ∂T ∂T ∂ (r + R) qr , ρc p v + + = k1 − (r + R) ∂r ∂r r + R ∂s r + R ∂r ∂r v (3) (4) (5) where u, v and w are the velocity components in s, r and z -directions, respectively, ρ is the density of the fluid, p is the pressure, ν is the kinematics viscosity of fluid, k1 is the thermal conductivity, c p is the specific heat at constant pressure and qr is the radiative heat flux The corresponding boundary conditions for the flow and heat transfer problem are u = U, v = 0, w = 0, T = Tw at r = 0, v ′R u = 0, v = − , w = 0, T → T∞ as r = H, r+R (6) where v ′ > corresponds to suction, v ′ < corresponds to injection and T∞ is the temperature of the wall lying at a distance H By using Rosseland approximation29 the radiative heat flux is given by qr = − 4σ ∗ ∂T 16σ ∗ ∂T = − T , 3k ∗ ∂r 3k ∗ ∂r (7) where σ ∗ is the Stefan-Boltzman constant and k ∗ is the mean absorption coefficient Eq (7) leads to a highly non-linear energy equation in T and it cannot be solved easily However this problem can be solved by assuming the small temperature difference within the flow (see Refs 17–19) For this, Rosseland approximation can be linearized about the ambient temperature T∞ by replacing T in Eq (7) with T∞3 Therefore, Eq (5) can be written as  )   ( 16σ ∗T∞3 ∂ 2T uR ∂T ∂T ∂T ρc p v + = k1 + + , ∂r r + R ∂s k1 k ∗ ∂r r + R ∂r (8) However, Eq (5) yields in a highly nonlinear radiation expression if we not consider the above assumption which is the aim of the present study So the energy Eq (5) in the presence of non-linear thermal radiation become  ( )  ∂T uR ∂T k1 ∂ 16σ ∗T ∂T (r ρc p v + = 1+ + R) , (9) (r + R) ∂r ∂r r + R ∂s k1 k ∗ ∂r The non-dimensional temperature is defined as θ (η) = T − T∞/Tw − T∞, with T = T∞ [1 + (θ w − 1) θ] and θ w = Tw /T∞ is the temperature parameters So Eq (9) yield v ( )  ∂T uR ∂T α ∂/∂r + Rd(1 + (θ w − 1) θ)3 (r + R)∂T/∂r , + = ∂r r + R ∂s (r + R) (10) where α is the thermal diffusivity and Rd = 16σ ∗T∞3 /3k1 k ∗23 is defined as a radiation parameter For similar solution of the flow equations, we use the following dimensionless variables u= Us ′ −Rv ′ r f (η) , v = f (η) , η = , H r+R H ρU s2 Us p= P (η) , w = g (η) , H H (11) 107124-4 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) Using Eq (11), continuity equation is identically satisfied and Eqs (3), (4), and (10) yields ∂P f ′2 = , ∂η η+K (12)   f′ f ′′ f′ 2K P ′′′ + + = f + − η+K Re η + K (η + K)2 (η + K)3 K f f ′′ M2 K f ′2 Kf f′ ( f ′ + mg) , − + − η + K η + K (η + K) + m2 (13) g′ KRe M 2Re ( f g ′ − g f ′) + (m f − g) = 0, + η+K η+K + m2 ( ) ′ KRe 1 + Rd(1 + (θ w − 1) θ)3 (η + K) θ ′ + f θ ′ = 0, Pr (η + K) (η + K) g ′′ + (14) (15) where K = R/H is the dimensionless radius of curvature, Pr = µc p /k1 is the Prandtl number, M = σB02 H/ρU is the magnetic parameter and Re = HU/ν is the Reynold number The corresponding boundary conditions takes the following form f (0) = 0, f ′(0) = 1, g (0) = 0, θ(0) = 1, f (1) = 1, f ′(1) = 0, g (1) = 0, θ(1) = (16) Eliminating pressure between Eqs (10)and(11), we get ) f ′′′ f ′′ f′ KRe KRe ( ′′ ( f f ′′′ − f ′ f ′′) + − + + f f − f ′2 η + K (η + K) η+K (η + K) (η + K) (17) KRe f f ′ M 2Re M Re ′′ ′ ′ ( ) ( − f + mg = 0, − f + mg) − + m2 (η + K)3 (η + K) (1 + m2) f ′′′′ + Once the fluid velocity f (η) is obtained the pressure can be determined from Eq (13) The physical quantities of interest are the skin-friction coefficient and the rate of heat transfer along the curved wall, which are defined as τrs sqw Cf = , Nu s = , (18) k1 (Tw − T∞) ρUw where τrs is the wall shear stress and qw is the heat flux at the wall along the s -directions, which are given by   ∂u u ∂T τrs = µ − , qw = −k + (qr ) w , (19) ∂r r + R r =0 ∂r r =0 Using Eqs (11) and (19), Eq (18) becomes , K Nu s = − + Rdθ 3w θ ′(0) Re−1/2 s ′′ Re1/2 s C f = f (0) − III NUMERICAL METHOD FOR SOLUTION The non-linear differential equations (14), (15), and (17) subject to boundary conditions (16) is solved numerically by using shooting method along with the fourth order Runge-Kutta integration scheme in the following way f ′ = t, t ′ = p, p′ = z, −2z p t K Re K ( f z − t p) − z′ = + − − f p − t2 η + K (η + K)2 (η + K)3 η + K (η + K)2 KRe M 2Re M 2Re (t + mg) + (p + ml) , + ft + (η + K) (1 + m ) (1 + m2) (η + K)                  (20) 107124-5 Abbas, Naveed, and Sajid g ′ = l, g ′′ = −l KRe M 2Re ( f l − gt) − (m f − g) , − η+K η+K + m2 AIP Advances 5, 107124 (2015) (21) θ ′ = q,        Re Pr K q  2 ′ , ) (θ (1 (θ f q − q = ( −3Rd − 1) + − 1) θ) q − w w η+K η+K  + Rd(1 + (θ w − 1) θ)       (22) with boundary conditions f (0) = 0, t (0) = 1, g (0) = 0, θ (0) = (23) In order to integrate (20), (21), and (22) as an initial value problem we need the value of p (0) i.e f ′′ (0) , z (0) i.e f ′′′ (0) , g ′ (0) i.e l (0) and θ ′ (0) i.e q (0), but no such values are given Choose some suitable values as a guess for f ′′ (0) , f ′′′ (0) , l (0) and θ ′ (0) and then integration is carried out The calculated values of f ′ (1) and θ (1) are compared with the given boundary conditions f (1) = 1, f ′ (1) = 0, g (1) =0 and θ (1) = and the values of f ′′ (0) , f ′′′ (0) , g ′ (0) and θ ′ (0) are adjusted by Newton Raphson’s method to give better approximation for the solution The step size is taken as ∆η = 0.005 The process is repeated until we get the results correct up to the accuracy of 10−5 level To validate the obtained numerical results through the shooting method we have also solved the governing equations using an implicit finite difference scheme known as the Keller-Box method.30 IV RESULTS AND DISCUSSION The nonlinear boundary value problems given in Eqs (14), (15) and (17) subject to boundary conditions (16) are solved numerically using both shooting and Keller-Box methods The fluid velocities f ′(η), g (η), pressure P(η) and temperature θ(η) are plotted in order to see the influence of the several physical involved parameters namely, dimensionless radius of curvature, magnetic parameter, Hall parameter, Reynolds number, Prandtl number, radiation parameter and temperature parameter in Figs 2-11 Furthermore, the magnitude of the skin friction coefficient Re1/2 s C f and the local Nusselt number Re s −1/2 Nu s for different parameters are presented in tables I and II Fig illustrates the effects of dimensionless radius of curvature K on the component of velocity f ′ (η) by keeping M = 1.5, Re = and m = 0.8 fixed It is found that near the permeable plate fluid velocity f ′ (η) is decreased with an increase in the value of K, but after η = 0.4 it starts to increase This is due to the fact that the upper wall of the channel is porous and the suction through it forces the velocity to increase The effects of the magnetic parameter M and the Hall parameter m on the component of velocity f ′ (η) are shown in Fig It is noticed from this Fig that the fluid velocity is FIG Variation of the dimensionless radius of curvature K on the component of velocity f ′ (η) by keeping M = 1.2, Re = 5.0 and m = 0.8 fixed 107124-6 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) TABLE I Numerical values of skin friction coefficient Re1/2 s C f for various values of K, M, Re and m K M Re m Shooting method Keller-Box Method 1.5 3.4029 3.1711 3.0620 2.9159 3.1848 3.2283 3.3933 3.6801 3.5721 4.0999 4.8484 5.4935 3.1711 3.2118 3.2273 3.2362 3.4029 3.1711 3.0620 2.9159 3.1848 3.2283 3.3933 3.6801 3.5721 4.0999 4.8484 5.4935 3.1711 3.2118 3.2273 3.2362 2.5 1.5 10 15 20 1.5 decreased by increasing the value of both the parameters M and m near the permeable wall However the effects of these parameters on the velocity are opposite near the porous wall The magnetic field suppresses the fluid velocity near the permeable wall and this is the expected fact The influence of the Reynolds number Re on the velocity f ′ (η) by keeping other parameters fixed is displayed in Fig It is observed that the velocity of the fluid increased by increasing the value of Re and the same behavior is noted as for K Fig elucidate the variation in transverse velocity field g (η) for different values of magnetic parameter M and dimensionless radius of curvature K It can be seen from this Fig that influence of increasing the value of M is to increase the transverse velocity g (η) However, it has the reverse behavior for K as the transverse velocity g (η) decreased by increasing the value of K The effects of Hall parameter m and Re on transverse velocity field are shown in Fig It is evident from this Fig that g (η) is increased for higher values of Re and m Fig depict the variation of Re and M on pressure distribution P (η) It is evident from this Fig that the magnitude of P (η) is decreased by an increase in Re However, it increases for large values of M The variation of Hall parameter m and dimensionless radius of curvature K on the pressure N u s for various values of K, Re, Pr, Rd and θ w TABLE II Numerical values of local Nusselt number Re−1/2 s K Re Pr Rd θw Shooting method Keller-Box method 5 0.3 2.5 10.5892 10.1546 9.8679 11.9386 14.1779 17.2507 11.9965 13.0601 14.4899 12.4491 14.3952 16.9004 12.2839 14.6966 17.3319 10.5892 10.1546 9.8679 11.9386 14.1779 17.2507 11.9965 13.0601 14.4899 12.4491 14.3952 16.9004 12.2839 14.6966 17.3319 10 15 10 12 15 0.5 0.7 1.0 0.3 3.5 107124-7 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) FIG Variation of the magnetic parameter M and for two values of Hall parameter m on the component of velocity f ′ (η) by keeping K = andRe = fixed FIG Variation of the Reynold number Re on the component of velocity f ′ (η) by keeping K = 2, M = 0.5 and m = 1.5 fixed FIG Variation of the magnetic parameter M and for the two values of dimensionless radius of curvature K on the transverse velocity profile g (η) by keeping Re = and m = 0.5 fixed distribution P (η) is shown in Fig It is found from this Fig that the magnitude of pressure distribution P (η) is a decreasing function for both m and K Fig illustrate the variation in the temperature profile θ (η) for various values of the Prandtl number Pr and Re From this Fig it is evident that for high values of Pr and Re, the temperature and the thickness of the thermal boundary layer are decreased with an increase in the values of Pr and Re The effects of the nonlinear radiation parameter Rd on the temperature distribution θ (η) are displayed in Fig 10 It is observed that the temperature and thermal boundary layer thickness are increased for 107124-8 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) FIG Variation of the Reynold number Re and Hall current parameter m on the transverse velocity profile g (η) by keeping K = and M = 1.5 fixed FIG Variation of the Reynold number Re and magnetic parameter M on the pressure distribution P (η) by keeping K = and m = 0.8 fixed FIG Variation of the Hall current parameter m and two values of dimensionless radius of curvature K on the pressure distribution P (η) by keeping Re = and M = fixed higher values of radiation parameter Rd Fig 11 demonstrates the influence of temperature parameter θ w on the temperature profile θ (η) It is noticed from this Fig that the temperature and the thermal boundary layer thickness are increased by increasing the values of θ w It is also noticed from this Fig that when θ r 1, the temperature profile of the nonlinear Rosseland approximation leads to a linear Rosseland approximation and for the higher values of θ w , represents the higher wall temperature as compared to ambient temperature Also the temperature profile become broader and S-shaped as discussed by Pantokratoras and Fang20 representing the existence of the adiabatic case for higher values of θ w 107124-9 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) FIG Variation of the Prandtl number Pr and Reynold number Re on the temperature distribution θ (η) by keeping K = 2, M = 1.2, m = 0.5, Rd = and θ w = fixed FIG 10 Variation of the radiation parameter Rd on the temperature distribution θ (η) by keeping K = 2, M = 1.3, Re = 5, m = 0.5, Pr = and θ w = fixed FIG 11 Variation of the temperature parameter θ w on the temperature distribution θ (η) by keeping K = 2, M = 1.5, Re = 5, m = 0.5, Pr = and Rd = 0.3 fixed Table I is given to show the numerical values of the skin friction coefficient Re1/2 s C f for various values of K, M, Re and m It is found that the magnitude of Re1/2 s C f is increased by increasing the value of M, Re and m,but it has the opposite trend for K Table II is made to show the numerical values of the local Nusselt number Re−1/2 Nu s for various values of K, Re, Pr, Rd and θ w From this table it is s observed that the magnitude of Re−1/2 Nu s is increased by increasing the value of Re, Pr, Rd and θ w s but it has the opposite behavior for K as the magnitude of local Nusselt number decreased by increasing the value of K 107124-10 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) V CONCLUDING REMARKS Hall effects on flow of a viscous fluid in a semi-porous curved channel by incorporating nonlinear Rosseland thermal radiation is discussed in this paper Numerical solutions are obtained by using shooting and Keller-Box methods to compute the fluid velocity, pressure, temperature, skin friction coefficient and local Nusselt number for different set of parameters Following findings have been made on the basis of graphical results: • Flow velocity and the transverse velocity are increased by increasing the value of m and Re, whilst the magnitude of pressure distribution is decreased with an increase in m and Re Also the temperature and thermal boundary layer thickness of the fluid is decreased by Re • Flow velocity is decreased by increasing the value of K and M However, the transverse velocity and magnitude of pressure distribution are increased with an increment inM, but it decreased with an increase in K • The temperature of the fluid is increased by increasing the non-linearized radiation parameter Rd Also the thermal boundary layer thickness increasing as Rd increased • Temperature and as well as the thermal boundary layer thickness decreased with an increase in Pr • Temperature of the fluid is increased with an increase in θ w and become s -shaped for large value of θ w • Absolute value of skin friction coefficient is decreased with an increase in K, however it increased by increasing the value of M, Re and m • Absolute value of Nusselt number is decreased with an increase in K, however it increased by increasing the value of Re, Pr, Rd and θ w • The numerical results are in excellent agreement with both the shooting and Keller-Box methods ACKNOWLEDGMENT One of the authors M Sajid acknowledges the support provided by AS-ICTP V Wernert, O Schaf, H Ghobarkar, and R Denoyal, “Adsorption properties of zeolites for artificial kidney applications,” Micro.Mesop.Mater 83(1-3), 101-113 (2005) A R Geoerke, J Leung, and S R Wickramasinghe, “Mass and momentum transfer in blood oxygenators,” Chem Eng Sci 57, 2035-2046 (2002) A Jafari, P Zamankhan, S M Mousavi, and P Kolari, “Numerical investigation of blood flow part II: in capillaries,” Commun Nonlinear Sci Numer.Simul 14, 1396-1402 (2009) A S Berman, “Laminar flow in channel with porous walls,” J Appl Phys 24, 1232-1235 (1953) M Layeghi and H R Seyf, “Fluid flow in an annular microchannel subject to uniform wall injections,” J Fluids Eng 132, 054502-054505 (2008) J P Kumar, J C Umavathi, Ali J Chamkha, and I Pop, “Fully-developed force-convective flow of micropolar and viscous fluids in vertical channel,” Appl Maths Modell 34, 1175-1186 (2010) M Sajid, Z Abbas, and T Hayat, “Homotopy analysis for boundary layer flow of a micropolar fluid through a porous channel,” 33, 4120-4125 (2009) S Srinivas, A Gupta, S Gulati, and A S Reddy, “Flow and mass transfer effects on viscous fluid in a porous channel with moving/stationary walls in presence of chemical reaction,” I Commun Heat Mass Transf (2013) (in Press) Z Abbas, B Ahmad, and S Ali, “Chemically reactive hydromagnetic flow of a second grade fluid in a semi-porous channel,” J Appl Mech Tech Phys (2014) In Press 10 Z Abbas, M Sajid, and T Hayat, “MHD boundary layer flow of an upper convected Maxwell fluid in a porous channel,” Theor Comput.Fluid Dyn 20, 229-238 (2006) 11 T Hayat and Z Abbas, “Heat transfer analysis on the MHD flow of a second grade fluid in a channel with porous medium,” Chaos, Solut Frac 38, 556-567 (2008) 12 G W Sutton and A Sherman, Engineering Magnetohydrodynamics (McGraw-Hill, New York, 1965) 13 M V Krishna, E Neeraja, and G S Babu, “Hall effects on MHD convection flow through a medium in a rotating parallel plate channel,” Asian J Curr Eng Maths 181-189 (2013) 14 H Sato, “The Hall effect in the viscous flow of ionized gas between two parallel plates under transverse magnetic field,” J Phys Soc Japan 27, 1051-1059 (1961) 15 T Yamanishi, “Hall effect in the viscous flow of ionized gas through straight channels,” in 17th Annual Meeting Phys Soc Japan (1962), Vol 5, p 29 16 D Pal and B Talukdar, “Influence of Hall current and thermal radiation on MHD convective heat and mass transfer in a rotating porous channel with chemical reaction,” Hind Publ Corp Int J Eng Maths 367064 (2013) 17 S Purkayastha and R Choudhury, “Hall current and thermal radiation effects on MHD convective flow of an elastic-viscous fluid in a rotating porous channel,” WSEAS Trans Appl Theor Mech 9, 196-205 (2014) 107124-11 18 Abbas, Naveed, and Sajid AIP Advances 5, 107124 (2015) K D Singh and R Kumar, “Combined effects of Hall current and rMuthuraj, Effects of thermal radiation and space porosity on MHD mixed otation on free convection MHD flow in a porous channel,” Ind J Pure Appl Phys 47(9), 617-623 (2009) 19 M Naveed, Z Abbas, and M Sajid, “MHD flow of a micropolar fluid due to a curved stretching sheet with thermal radiation,” J Appl Fluid Mech (In Press) 20 A Pantokratoras and T Fang, “Sakiadis flow with nonlinear Rosseland thermal radiation,” Phys Scrip 87, 015703 21 A Pantokratoras and T Fang, “Blasius flow with nonlinear Rosseland thermal radiation,” Meccanica 49(6), 1539-1545 (2014) 22 R Cortell, “Fluid flow and radiative nonlinear heat transfer over a stretching sheet,” J King Saud Uni Sci 26 (2013) ).08.004 23 M Mustafa, A Mushtaq, T Hayat, and B Ahmad, “Nonlinear radiation heat treansfer effects in the natural convective boundary layer flow of nanofluid past a vertical plate: A numerical study,” Plos One 9(9), 1-10 (2014) 24 A Khuri, “Stokes flow in curved channel,” J Compt Appl Maths 187(2), 171-191 (2006) 25 N Ali, M Sajid, and T Hayat, “Long wavelength flow analysis in a curved channel,” ZNA 65a, 191-196 (2010) 26 N Ali, M Sajid, Z Abbas, and T Javed, “Non-Newtonian fluid flow induced by peristaltic waves in a curved channel,” Eurp J Mech.-B/Fluids 29(5), 387-394 (2010) 27 Wu-Shung Fu, Chung-Jen Chen, Yu-Chih Lai, and Shang-Hao Huang, “Effects of a porous medium on forced convection of a reciprocating curved channel,” Int Commun Heat Mass Transf 58, 63-67 (2014) 28 R C Meyer, “On reducing aerodynamic heat transfer rates by manetohydrodynamic techniques,” J Aerospace Sci 25, 561 (1958) 29 S Rosseland, Astrophysik und atom-theoretischeGrundlagen (Springer-Verlag, Berlin, 1931), pp 41-44 30 H B Keller and T Cebeci, “Accurate numerical methods for boundary layer flows II: two dimensional turbulent flows,” AIAA Journal 10, 1193-1199 (1972)