Research article Open Access
Numerical Method for the Analysis of Thermal Radiation on Heat Transfer in Nanofluid
J.I. Oahimire1*, F.E Bazuaye2, Taylor S. Harry2
1Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria
2Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria
*Corresponding author: Sergei Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Nigeria, Email: @
Received: July 02, 2016; Accepted: July 23, 2016; Published: July 30, 2016
Citation: Oahimire JI, Bazuaye FE, Harry TS (2016) Numerical Method for the Analysis of Thermal Radiation on Heat Transfer in Nanofluid. Nanosci Technol 3(1): 1-4.
AbstractTop
This present study investigates the effects of thermal radiation on heat transfer in nanofluid. The governing equations are formulated base on already existing model and are transformed to ordinary differential equations using stream function and similarity variables. The resulting dimension less equations are then solved numerically by Runge-Kuta Fehlberg method with shooting techniques using Maple software. With the help of graphs and tables, influences of the governing parameters are discussed. The result shows that the radiation has significance influences on heat transfer.

Keywords: Nanofluid; Runge Kuta-Fehlberg method; Shooting technique; Heat transfer; Radiation
Nomenclature
a = Constant; g = Acceleration due to gravity; k = Thermal Conductivity; Pr = Prandtl Number; T = Fluid Temperature; Tw = Surface Temperature; T = Free Stream Temperature; u,v = Velocity Components; x, y = Cartesian Coordinates; f(x) = Dimensionless Stream Function; Gr = Grashof Number; qr = Heat Flux Radiation; Bo = Magnetic Field of Constant Strength; R = Radiation Parameter; Ks = Rosseland Mean Absorption Coefficient; K = Thermal Conductivity Coefficient
Greek Symbols
β = Thermal Expansion Coefficient; μ = Dynamic Coefficient of Viscosity; θ (η) = Dimensionless Temperature; η = Similarity Variable; ρ = Fluid Density; ψ = Stream Function; σ' = Stefan- Bottzman Constant
Introduction
Nanofluid is formed when Nanoparticles such as Aluminium Oxide (Al2O3), Copper (Cu), Copper Oxide (CuO), Gold (Au), Silver (Ag), Silica particles, e.t.c are mixed with base fluids such as water, oil, acetone, ethylene, etc. The discovery of nanofluids in enhancing heat transfer in industrial processes has drawn the attention of both scholars and industrialists to make researches into this relatively new area.

Wang and Choi, [1] studied the thermal condutivity of nanoparticle fluid mixture containing Al2O3 and CuO nanoparticles and showed that the thermal conductivity of nanofluids increased with increasing volume fraction of the nanoparticles. Steve (2006) found out that a small size radiator filled with well prepared nanofluid containing monosized nanoparticles with 2nm nominal diameter will be okay for heavy vehicle instead of a very big radiator and this will save cost and space. And he also discovered that fuel efficiency will be reduced by 5% in these Heavy Trucks when nanofluids are introduced into their cooling system. John and Baldev, [2] carried out experiment and they concluded that by controlling the linear aggregation length from nano- to macron scales, the thermal conductivity of the nanofluid can be tuned from a low to very high value and that under repeated magnetic cycling, the thermal conductivity is reversible.

Khan and Ipop, [3] studied the boundary-layer flow of a nanofluid past a stretching sheet and their result shows that the heat transfer at fixed values of the Lewis number (Le), the Brownian motion parameter (Nb) and the thermophoresis parameter (Nt) increases with the Prandtl number (Pr). This is because a higher Pr fluid has relatively lower thermal conductivity which reduces conduction and thereby increases the heat transfer rate at the surface of the sheet.

Godson, et al. [4] in separate papers agreed that Nanofluids are the most option to enhance heat transfer and showed that a very small amount of nanoparticles suspended stably in base fluids can provide impressive improvement in the thermal property of such fluids.

Zeinali and Salehi, [5] studied experimentally, the effect of magnetic field of various strengths on the thermal performance of Silver/water nanofluid and their results showed that the thermal efficiency in the presence of magnetic field increased.

Yohannes and Shankar, [6] examine the boundary layer flow of heat and mass transfer in MHD flow of nanofluids through porous media with viscous dissipation and chemical reaction. The governing boundary layer equations were formulated and reduced to a set of ordinary differential equations using similarity transformations and then solved numerically by an explicit finite differential scheme known as the Keller box method. Wan-Mohd, [7] applied Runge-kutta Fehlberg method and shooting technique for solving classical Blasius equation [7]. The well-known Blasius equation is governed by the third order nonlinear ordinary differential equation and then solved numerically using rungekutta Fehlberg method with shooting techniques. Hamad, [8] found analytical solution to the differential equations that model heat transfer in nanofliud. We extended the work of Hamad, [8] by incorporating radiation term to have differential equations that model the effect of thermal radiation on heat transfer of a nanofliud.
Mathematical formulation
Consider the steady laminar two-dimensional flow of an incompressible viscous nanofluid past a linearly semi-infinite stretching sheet under the influence of a constant magnetic field of strength B0 which is applied normally to the sheet. x'and y' are the coordinates along and perpendicular to the sheet while u' and v' are the velocity components in the directions respectively as shown in the figure below:
Figure 1:
A water based nanofluid containing different types of nanoparticles: Al2 O3, Cu, TiO2 and Ag is used with the assumption that both the fluid and the nanoparticles are in thermal equilibrium. Based on the already existing model of Hamad, [8], the governing boundary layer equations of continuity, momentum and heat equations formulated are:
u' x' + v' y' =0           ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyDaiaacEcaa8aabaWdbiabgkGi 2kaadIhacaGGNaaaaiabgUcaRmaalaaapaqaa8qacqGHciITcaWG2b Gaai4jaaWdaeaapeGaeyOaIyRaamyEaiaacEcaaaGaeyypa0JaaGim aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcWdamaabmaabaWdbiaaigdaa8aacaGLOaGa ayzkaaaaaa@53E2@ ρ nf [ u' u x +v' v y ]= μ nf 2 u' y ' 2 σ B 0 u +g β t ( T T )     ( 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHbpGCpaWaaSbaaSqaa8qacaWGUbGaamOzaaWdaeqaaOWdbmaa dmaapaqaa8qacaWG1bGaai4jamaalaaapaqaa8qacqGHciITceWG1b WdayaafaaabaWdbiabgkGi2kqadIhapaGbauaaaaWdbiabgUcaRiaa dAhacaGGNaWaaSaaa8aabaWdbiabgkGi2kqadAhapaGbauaaaeaape GaeyOaIyRabmyEa8aagaqbaaaaa8qacaGLBbGaayzxaaGaeyypa0Ja eqiVd02damaaBaaaleaapeGaamOBaiaadAgaa8aabeaak8qadaWcaa WdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikdaaaGccaWG1bGa ai4jaaWdaeaapeGaeyOaIyRaamyEaiaacEcapaWaaWbaaSqabeaape GaaGOmaaaaaaGccqGHsislcqaHdpWCcaWGcbWdamaaBaaaleaapeGa aGimaaWdaeqaaOWdbiqadwhapaGbauaapeGaey4kaSIaam4zaiabek 7aI9aadaWgaaWcbaWdbiaadshaa8aabeaak8qadaqadaWdaeaapeGa bmiva8aagaqba8qacqGHsislceWGubWdayaafaWaaSbaaSqaa8qacq GHEisPa8aabeaaaOWdbiaawIcacaGLPaaacaGGGcGaaiiOaiaaccka caGGGcGaaiiOa8aadaqadaqaa8qacaaIYaaapaGaayjkaiaawMcaaa aa@711E@ ( ρ c p ) nf [ u' T' x +v' T' y ]= K nf 2 T' y ' 2 q r y'                   ( 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeGaeqyWdiNaam4ya8aadaWgaaWcbaWdbiaadcha a8aabeaaaOWdbiaawIcacaGLPaaapaWaaSbaaSqaa8qacaWGUbGaam OzaaWdaeqaaOWdbmaadmaapaqaa8qacaWG1bGaai4jamaalaaapaqa a8qacqGHciITcaWGubGaai4jaaWdaeaapeGaeyOaIyRabmiEa8aaga qbaaaapeGaey4kaSIaamODaiaacEcadaWcaaWdaeaapeGaeyOaIyRa amivaiaacEcaa8aabaWdbiabgkGi2kqadMhapaGbauaaaaaapeGaay 5waiaaw2faaiabg2da9iaadUeapaWaaSbaaSqaa8qacaWGUbGaamOz aaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaape GaaGOmaaaakiaadsfacaGGNaaapaqaa8qacqGHciITcaWG5bGaai4j a8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgkHiTmaalaaapaqaa8 qacqGHciITcaWGXbWdamaaBaaaleaapeGaamOCaaWdaeqaaaGcbaWd biabgkGi2kaadMhacaGGNaaaaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOa8aadaqadaqaa8qaca aIZaaapaGaayjkaiaawMcaaaaa@7AC9@
The boundary conditions of the equations are
u = u w ' ( x )=a x , v =0, T ' = T w ' at y =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWG1bWdayaafaWdbiabg2da9iaadwhapaWaa0baaSqaa8qacaWG 3baapaqaa8qacaGGNaaaaOWaaeWaa8aabaWdbiqadIhapaGbauaaa8 qacaGLOaGaayzkaaGaeyypa0JaamyyaiqadIhapaGbauaapeGaaiil aiqadAhapaGbauaapeGaeyypa0JaaGimaiaacYcacaWGubWdamaaCa aaleqabaWdbiaacEcaaaGccqGH9aqpcaWGubWdamaaDaaaleaapeGa am4DaaWdaeaapeGaai4jaaaakiaadggacaWG0bGabmyEa8aagaqba8 qacqGH9aqpcaaIWaaaaa@4F7A@ u 0, T T ' ,y'                              ( 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qaceWG1bWdayaafaWdbiabgkziUkaaicdacaGGSaGabmiva8aagaqb a8qacqGHsgIRcaWGubWdamaaDaaaleaapeGaeyOhIukapaqaa8qaca GGNaaaaOGaaiilaiaadMhacaGGNaGaeyOKH4QaeyOhIuQaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcWdamaabmaabaWdbiaaisdaa8aa caGLOaGaayzkaaaaaa@6AAD@
Where qr is the radiative heat flux, T′ is the temperature of the fluid, x' and y' are the co-ordinates along and perpendicular to the sheet while u' and v' are the velocity components in the x' and y' directions respectively and a is a constant. The effective density (ρnf), effective dynamic viscosity (μnf), heat capacitance (ρCp)nf and the effective thermal conductivity (knf) of the nanofluid, in that order, are given as
ρ nf =(1A) ρ f +A ρ s μ nf = μ f (1A) 2.5 (ρC ) p nf =(1A) (ρC ) p f +A (ρC ) p s                (  5 ) k = nf k f { k s +2 k f 2A( k f k s ) k s +2 k f +2A( k f k s ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHbp GCdaWgaaWcbaGaamOBaiaadAgaaeqaaOGaeyypa0Jaaiikaiaaigda cqGHsislcaWGbbGaaiykaiabeg8aYnaaBaaaleaacaWGMbaabeaaki abgUcaRiaadgeacqaHbpGCdaWgaaWcbaGaam4CaaqabaaakeaacqaH 8oqBdaWgaaWcbaGaamOBaiaadAgaaeqaaOGaeyypa0ZaaSaaaeaacq aH8oqBdaWgaaWcbaGaamOzaaqabaaakeaacaGGOaGaaGymaiabgkHi TiaadgeacaGGPaWaaWbaaSqabeaacaaIYaGaaiOlaiaaiwdaaaaaaa GcbaGaaiikaiabeg8aYjaadoeadaWgbaWcbaGaamiCaaqabaGccaGG PaWaaSbaaSqaaiaad6gacaWGMbaabeaakiabg2da9iaacIcacaaIXa GaeyOeI0IaamyqaiaacMcacaGGOaGaeqyWdiNaam4qamaaBeaaleaa caWGWbaabeaakiaacMcadaWgaaWcbaGaamOzaaqabaGccqGHRaWkca WGbbGaaiikaiabeg8aYjaadoeadaWgbaWcbaGaamiCaaqabaGccaGG PaWaaSbaaSqaaiaadohaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7qaaaaaaaaaWdbiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOa8aadaqadaqaa8qacaqGGaGaaGynaaWd aiaawIcacaGLPaaaaeaacaWGRbWaaSraaSqaaiaad6gacaWGMbaabe aakiabg2da9iaadUgadaWgaaWcbaGaamOzaaqabaGccaGG7bWaaSaa aeaacaWGRbWaaSbaaSqaaiaadohaaeqaaOGaey4kaSIaaGOmaiaadU gadaWgaaWcbaGaamOzaaqabaGccqGHsislcaaIYaGaamyqaiaacIca caWGRbWaaSbaaSqaaiaadAgaaeqaaOGaeyOeI0Iaam4AamaaBaaale aacaWGZbaabeaakiaacMcaaeaacaWGRbWaaSbaaSqaaiaadohaaeqa aOGaey4kaSIaaGOmaiaadUgadaWgaaWcbaGaamOzaaqabaGccqGHRa WkcaaIYaGaamyqaiaacIcacaWGRbWaaSbaaSqaaiaadAgaaeqaaOGa eyOeI0Iaam4AamaaBaaaleaacaWGZbaabeaakiaacMcaaaGaaiyFaa aaaa@CAA9@
Where A is the solid volume fraction (A ≠ 1), μf is the dynamic viscosity of the base fluid, while ρf and ρs are the densities of the pure fluid and the nanoparticle respectively. The constants kf and ks are the thermal conductivities of the base fluid and the nanoparticle respectively.

Following Rosseland approximation Brewstar [9] the radiative heat flux qr is modeled as
q r = 4 σ 3 k T '4 y                                                   ( 6 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGXbWdamaaBaaaleaapeGaamOCaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacaaI0aGafq4Wdm3dayaafaaabaWdbiaaiodaceWGRb Wdayaafaaaa8qadaWcaaWdaeaapeGaeyOaIyRaamiva8aadaahaaWc beqaa8qacaGGNaGaaGinaaaaaOWdaeaapeGaeyOaIyRabmyEa8aaga qbaaaapeGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckapaWaaeWaaeaapeGaaGOnaaWdaiaawIcacaGL Paaaaaa@7FE7@
Where σ' is the Stefan-Boltzman constant and k' is the mean absorption coefficient. Assuming that the difference in temperature within the flow is such that T '4 can be expressed as a linear combination of the temperature, we expand T '4 in Taylor's series about T as follows:
T '4 = T '4 +4 T '3 ( T T ' )+6 T '2 ( T T ' ) 2 +..           ( 7 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaCaaaleqabaWdbiaacEcacaaI0aaaaOGaeyypa0Ja amiva8aadaqhaaWcbaWdbiabg6HiLcWdaeaapeGaai4jaiaaisdaaa GccqGHRaWkcaaI0aGaamiva8aadaqhaaWcbaWdbiabg6HiLcWdaeaa peGaai4jaiaaiodaaaGcdaqadaWdaeaapeGabmiva8aagaqba8qacq GHsislcaWGubWdamaaDaaaleaapeGaeyOhIukapaqaa8qacaGGNaaa aaGccaGLOaGaayzkaaGaey4kaSIaaGOnaiaadsfapaWaa0baaSqaa8 qacqGHEisPa8aabaWdbiaacEcacaaIYaaaaOWaaeWaa8aabaWdbiqa dsfapaGbauaapeGaeyOeI0Iaamiva8aadaqhaaWcbaWdbiabg6HiLc WdaeaapeGaai4jaaaaaOGaayjkaiaawMcaa8aadaahaaWcbeqaa8qa caaIYaaaaOGaey4kaSIaeyOjGWRaaiOlaiaac6cacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOa8aadaqadaqaa8qacaaI3aaapaGaayjkaiaawMcaaaaa@6AB8@
and neglecting higher order terms beyond the first degree in ( T' - T' ), we have
T '4 3 T ;4 +4 T '3 T                                ( 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGubWdamaaCaaaleqabaWdbiaacEcacaaI0aaaaOGaeyisISRa eyOeI0IaaG4maiaadsfapaWaa0baaSqaa8qacqGHEisPa8aabaWdbi aacUdacaaI0aaaaOGaey4kaSIaaGinaiaadsfapaWaa0baaSqaa8qa cqGHEisPa8aabaWdbiaacEcacaaIZaaaaOGabmiva8aagaqba8qaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcWdamaabmaaba WdbiaaiIdaa8aacaGLOaGaayzkaaaaaa@6C42@
Differentiating equation (6) with respect to y' and using equation (8) to obtain
q r y = 16 T 3 σ 3k' 2 T y '2                                      ( 9 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamyCa8aadaWgaaWcbaWdbiaadkha a8aabeaaaOqaa8qacqGHciITceWG5bWdayaafaaaa8qacqGH9aqpda WcaaWdaeaapeGaeyOeI0IaaGymaiaaiAdacaWGubWdamaaDaaaleaa peGaeyOhIukapaqaa8qacaaIZaaaaOGafq4Wdm3dayaafaaabaWdbi aaiodacaWGRbGaai4jaaaadaWcaaWdaeaapeGaeyOaIy7damaaCaaa leqabaWdbiaaikdaaaGcceWGubWdayaafaaabaWdbiabgkGi2kaadM hapaWaaWbaaSqabeaapeGaai4jaiaaikdaaaaaaOGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOa8aadaqadaqaa8qacaaI5aaapaGaayjkaiaawM caaaaa@7BD1@
Then equation (3) becomes
( ρ c p ) nf [ u' T' x +v' T' y ]= K nf 2 T' y ' 2 + 16 T 3 σ 3k' 2 T y '2             ( 10 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeGaeqyWdiNaam4ya8aadaWgaaWcbaWdbiaadcha a8aabeaaaOWdbiaawIcacaGLPaaapaWaaSbaaSqaa8qacaWGUbGaam OzaaWdaeqaaOWdbmaadmaapaqaa8qacaWG1bGaai4jamaalaaapaqa a8qacqGHciITcaWGubGaai4jaaWdaeaapeGaeyOaIyRabmiEa8aaga qbaaaapeGaey4kaSIaamODaiaacEcadaWcaaWdaeaapeGaeyOaIyRa amivaiaacEcaa8aabaWdbiabgkGi2kqadMhapaGbauaaaaaapeGaay 5waiaaw2faaiabg2da9iaadUeapaWaaSbaaSqaa8qacaWGUbGaamOz aaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaape GaaGOmaaaakiaadsfacaGGNaaapaqaa8qacqGHciITcaWG5bGaai4j a8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRmaalaaapaqaa8 qacaaIXaGaaGOnaiaadsfapaWaa0baaSqaa8qacqGHEisPa8aabaWd biaaiodaaaGccuaHdpWCpaGbauaaaeaapeGaaG4maiaadUgacaGGNa aaamaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaa kiqadsfapaGbauaaaeaapeGaeyOaIyRaamyEa8aadaahaaWcbeqaa8 qacaGGNaGaaGOmaaaaaaGccaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckapaWaae WaaeaapeGaaGymaiaaicdaa8aacaGLOaGaayzkaaaaaa@7ED2@
By the introduction of the following variables
u= u' a v f v= v' a v f ,θ= T T ' T w ' T ' ,x= x' v f a .y= y' v f a              ( 11 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWG1bGaeyypa0ZaaSaaa8aabaWdbiaadwhacaGGNaaapaqaa8qa daGcaaWdaeaapeGaamyyaiaadAhapaWaaSbaaSqaa8qacaWGMbaapa qabaaapeqabaaaaOGaamODaiabg2da9maalaaapaqaa8qacaWG2bGa ai4jaaWdaeaapeWaaOaaa8aabaWdbiaadggacaWG2bWdamaaBaaale aapeGaamOzaaWdaeqaaaWdbeqaaaaakiaacYcacqaH4oqCcqGH9aqp daWcaaWdaeaapeGabmiva8aagaqba8qacqGHsislcaWGubWdamaaDa aaleaapeGaeyOhIukapaqaa8qacaGGNaaaaaGcpaqaa8qacaWGubWd amaaDaaaleaapeGaam4DaaWdaeaapeGaai4jaaaakiabgkHiTiaads fapaWaa0baaSqaa8qacqGHEisPa8aabaWdbiaacEcaaaaaaOGaaiil aiaadIhacqGH9aqpdaWcaaWdaeaapeGaamiEaiaacEcaa8aabaWdbm aakaaapaqaa8qadaWcaaWdaeaapeGaamODa8aadaWgaaWcbaWdbiaa dAgaa8aabeaaaOqaa8qacaWGHbaaaaWcbeaaaaGccaGGUaGaamyEai abg2da9maalaaapaqaa8qacaWG5bGaai4jaaWdaeaapeWaaOaaa8aa baWdbmaalaaapaqaa8qacaWG2bWdamaaBaaaleaapeGaamOzaaWdae qaaaGcbaWdbiaadggaaaaaleqaaaaakiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckapaWaaeWaaeaapeGaaGymaiaaigdaa8aacaGLOaGaayzk aaaaaa@7757@
Equation (11) transform equation (1), (2) and (10) to the followings
u x + v y =0                                          ( 12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWG1baabaGaeyOaIyRaamiEaaaacqGHRaWkdaWcaaqaaiab gkGi2kaadAhaaeaacqGHciITcaWG5baaaiabg2da9iaaicdaqaaaaa aaaaWdbiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOa8aadaqadaqaa8qacaaIXaGaaGOmaaWdaiaawI cacaGLPaaaaaa@74D2@ u u x +v u y = 1 (1A) ρ f + ρ s ( 1 (1A) 2.5 2 u y 2 Mu+ G r θ)          ( 13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaala aabaGaeyOaIyRaamyDaaqaaiabgkGi2kaadIhaaaGaey4kaSIaamOD amaalaaabaGaeyOaIyRaamyDaaqaaiabgkGi2kaadMhaaaGaeyypa0 ZaaSaaaeaacaaIXaaabaGaaiikaiaaigdacqGHsislcaWGbbGaaiyk aiabeg8aYnaaBaaaleaacaWGMbaabeaakiabgUcaRiabeg8aYnaaBa aaleaacaWGZbaabeaaaaGccaGGOaWaaSaaaeaacaaIXaaabaGaaiik aiaaigdacqGHsislcaWGbbGaaiykamaaCaaaleqabaGaaGOmaiaac6 cacaaI1aaaaaaakmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaa aOGaamyDaaqaaiabgkGi2kaadMhadaahaaWcbeqaaiaaikdaaaaaaO GaeyOeI0IaamytaiaadwhacqGHRaWkcaWGhbWaaSbaaSqaaiaadkha aeqaaOGaeqiUdeNaaiykaabaaaaaaaaapeGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOa8aadaqa daqaa8qacaaIXaGaaG4maaWdaiaawIcacaGLPaaaaaa@7323@ u θ x +v θ y = 1 p r 1 (1A) (ρ C p ) f +A (ρ C p ) s ( k nf k f + R d ) 2 θ y 2           ( 14 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaala aabaGaeyOaIyRaeqiUdehabaGaeyOaIyRaamiEaaaacqGHRaWkcaWG 2bWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5baaaiabg2 da9maalaaabaGaaGymaaqaaiaadchadaWgaaWcbaGaamOCaaqabaaa aOWaaSaaaeaacaaIXaaabaGaaiikaiaaigdacqGHsislcaWGbbGaai ykaiaacIcacqaHbpGCcaWGdbWaaSbaaSqaaiaadchaaeqaaOGaaiyk amaaBaaaleaacaWGMbaabeaakiabgUcaRiaadgeacaGGOaGaeqyWdi Naam4qamaaBaaaleaacaWGWbaabeaakiaacMcadaWgaaWcbaGaam4C aaqabaaaaOWaaeWaaeaadaWcaaqaaiaadUgadaWgaaWcbaGaamOBai aadAgaaeqaaaGcbaGaam4AamaaBaaaleaacaWGMbaabeaaaaGccqGH RaWkcaWGsbWaaSbaaSqaaiaadsgaaeqaaaGccaGLOaGaayzkaaWaaS aaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaH4oqCaeaacqGH ciITcaWG5bWaaWbaaSqabeaacaaIYaaaaaaakabaaaaaaaaapeGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOa8aadaqadaqaa8qacaaIXaGaaGinaaWdaiaawIcacaGLPa aaaaa@7990@
Where M= σ β 0 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGnbGaeyypa0ZaaSaaa8aabaWdbiabeo8aZjabek7aI9aadaWg aaWcbaWdbiaaicdaa8aabeaaaOqaa8qacaWGHbaaaaaa@3CF2@ is the magnetic field parameter, p r = v f K f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGWbWdamaaBaaaleaapeGaamOCaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacaWG2bWdamaaBaaaleaapeGaamOzaaWdaeqaaaGcba WdbiaadUeapaWaaSbaaSqaa8qacaWGMbaapaqabaaaaaaa@3D77@ is the Prandtl number, R d = 16σ' T '3 3 K * K f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGsbWdamaaBaaaleaapeGaamizaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacaaIXaGaaGOnaiabeo8aZjaacEcacaWGubWdamaaDa aaleaapeGaeyOhIukapaqaa8qacaGGNaGaaG4maaaaaOWdaeaapeGa aG4maiaadUeapaWaaWbaaSqabeaapeGaaiOkaaaakiaadUeapaWaaS baaSqaa8qacaWGMbaapaqabaaaaaaa@45B1@ is the radiation parameter, G r = g β t ( T w ' T ; )x a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGhbWdamaaBaaaleaapeGaamOCaaWdaeqaaOWdbiabg2da9maa laaapaqaa8qacaWGNbGaeqOSdi2damaaBaaaleaapeGaamiDaaWdae qaaOWdbmaabmaapaqaa8qacaWGubWdamaaDaaaleaapeGaam4DaaWd aeaapeGaai4jaaaakiabgkHiTiaadsfapaWaa0baaSqaa8qacqGHEi sPa8aabaWdbiaacUdaaaaakiaawIcacaGLPaaacaWG4baapaqaa8qa caWGHbaaaaaa@47E3@ Is the Grashof number
And the boundary conditions becomes
u=x,v=0,θ=1aty=0 u0,θ0asy(15) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG1b Gaeyypa0JaamiEaiaacYcacaWG2bGaeyypa0JaaGimaiaacYcacaaM c8UaeqiUdeNaeyypa0JaaGymaiaaykW7caaMc8Uaamyyaiaadshaca aMc8UaamyEaiabg2da9iaaicdaaeaacaWG1bGaeyOKH4QaaGimaiaa cYcacaaMc8UaeqiUdeNaeyOKH4QaaGimaiaaykW7caaMc8Uaamyyai aadohacaaMc8UaaGPaVlaadMhacqGHsgIRcqGHEisPcaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaabIcacaqGXaGaaeynaiaabMcaaaaa@E8C5@
By introducing the stream function ψ, which can be defined as
u   =    ψ  y  ,    v  =  -    ψ x . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhcba9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciqacaqabeaadaqaaqaaaOqaaiaabwhacaqGGa GaaeiiaiaabccacaqG9aGaaeiiaiaabccadaWcaaqaaiabgkGi2kaa bccacaqGipaabaGaeyOaIyRaaeiiaiaabMhaaaGaaeiiaiaabYcaca qGGaGaaeiiaiaabccacaqGGaGaaeODaiaabccacaqGGaGaaeypaiaa bccacaqGGaGaaeylaiaabccadaWcaaqaaiaabccacqGHciITcaqGGa GaaeiYdaqaaiabgkGi2kaaysW7caqG4baaaiaac6caaaa@5310@
and using the similarity transforma t i o n s
η=y,ψ=xf(η),θ=θ(η)                           ( 16 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGMaey ypa0JaamyEaiaacYcacqaHipqEcqGH9aqpcaWG4bGaamOzaiaacIca cqaH3oaAcaGGPaGaaiilaiabeI7aXjabg2da9iabeI7aXjaacIcacq aH3oaAcaGGPaaeaaaaaaaaa8qacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckapaWaaeWaae aapeGaaGymaiaaiAdaa8aacaGLOaGaayzkaaaaaa@6BB8@
We have
f + (1A) 2.5 { f f (f') 2 }[ (1A) ρ f + ρ s )(M f + G r θ) ]=0                   ( 17 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaasa Gaey4kaSIaaiikaiaaigdacqGHsislcaWGbbGaaiykamaaCaaaleqa baGaaGOmaiaac6cacaaI1aaaaOWaaiWaaeaacaWGMbGabmOzayaaga GaeyOeI0IaaiikaiaadAgacaGGNaGaaiykamaaCaaaleqabaGaaGOm aaaaaOGaay5Eaiaaw2haamaadmaabaGaaiikaiaaigdacqGHsislca WGbbGaaiykaiabeg8aYnaaBaaaleaacaWGMbaabeaakiabgUcaRiab eg8aYnaaBaaaleaacaWGZbaabeaakiaacMcacqGHsislcaGGOaGaam ytaiqadAgagaqbaiabgUcaRiaadEeadaWgaaWcbaGaamOCaaqabaGc cqaH4oqCcaGGPaaacaGLBbGaayzxaaGaeyypa0JaaGimaabaaaaaaa aapeGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOa8aadaqadaqaa8qacaaIXaGaaG4naaWd aiaawIcacaGLPaaaaaa@76BC@ 1 P r 1 (1A) (ρ C p ) f +A (ρ C p ) s [ k nf k f + R d ] θ (η)+f(η) θ (η)=0                        ( 18 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamiuamaaBaaaleaacaWGYbaabeaaaaGcdaWcaaqaaiaa igdaaeaacaGGOaGaaGymaiabgkHiTiaadgeacaGGPaGaaiikaiabeg 8aYjaadoeadaWgaaWcbaGaamiCaaqabaGccaGGPaWaaSbaaSqaaiaa dAgaaeqaaOGaey4kaSIaamyqaiaacIcacqaHbpGCcaWGdbWaaSbaaS qaaiaadchaaeqaaOGaaiykamaaBaaaleaacaWGZbaabeaaaaGcdaWa daqaamaalaaabaGaam4AamaaBaaaleaacaWGUbGaamOzaaqabaaake aacaWGRbWaaSbaaSqaaiaadAgaaeqaaaaakiabgUcaRmaavababeWc baGaamizaaqab0qaaiaadkfaaaaakiaawUfacaGLDbaacuaH4oqCga GbaiaacIcacqaH3oaAcaGGPaGaey4kaSIaamOzaiaacIcacqaH3oaA caGGPaGafqiUdeNbauaacaGGOaGaeq4TdGMaaiykaiabg2da9iaaic daqaaaaaaaaaWdbiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOa8aadaqadaqaa8qacaaIXaGaaGioaaWdaiaawIcaca GLPaaaaaa@839E@
And the boundary conditions now becomes
f(0)=0, f (0)=1atη=0 f0atη. θ(0)=1andθ()=0, }(19) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe qaaiaadAgacaGGOaGaaGimaiaacMcacqGH9aqpcaaIWaGaaiilaiaa ygW7caaMc8UaaGPaVlaaykW7ceWGMbGbauaacaGGOaGaaGimaiaacM cacqGH9aqpcaaIXaGaaGPaVlaaykW7caWGHbGaamiDaiaaykW7caaM c8UaaGPaVlabeE7aOjabg2da9iaaicdacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7aeaacaWGMbGaeyOKH4QaaGimaiaaykW7caaM c8UaamyyaiaadshacaaMc8UaaGPaVlabeE7aOjabgkziUkabg6HiLk aac6caaeaacqaH4oqCcaGGOaGaaGimaiaacMcacqGH9aqpcaaIXaGa aGPaVlaaykW7caWGHbGaamOBaiaadsgacaaMc8UaaGPaVlabeI7aXj aacIcacqGHEisPcaGGPaGaeyypa0JaaGimaiaacYcaaaGaayzFaaGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caGG OaGaaGymaiaaiMdacaGGPaaaaa@B817@
Numerical Solution
Table 1: Thermo physical properties of water and nanoparticles Hamad [8].

Compound

ρ (kg/ m3)

Cp(J/ kgK)

k (W/ mK

Pure water

997.1

4179

0.613

Copper (Cu)

8933

385

401

Alumina (Al2O3 )

3970

765

40

Silver (Ag)

10500

235

429

Titanium Oxide (TiO2)

4250

686.2

8.9538

An algorithm was written and run with Maple codes to plot graphs and generate tables for analysis.
The thermo physical properties of pure water and those of the nanoparticles as given in table 1 by Hamad [8] will now be substituted in the transformed coupled ordinary differential equations. These equations will now be solved using Runge- Kutta-Fehlberg (RKF) method with shooting technique. The algorithm of Runge-Kutta-Fehlberg (RKF) method is order 5 and it gives a better approximation of the solution than the general Runge – Kutta method of order 4.

An algorithm was written and run with Maple codes to plot graphs and generate tables for analysis.
Results
Results are shown in figures 2- 3 and table 2
From derivation, the the Nuselt number can be written as follows;
Nu= θ (0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepu0dbbc8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadw hacqGH9aqpcqGHsislcuaH4oqCgaqbaiaacIcacaaIWaGaaiykaiaa ykW7caaMc8oaaa@3FFD@
Discussion of Results
Figure 2:Effect of M on velocity distribution profiles.
Figure 3:Effect of Rd on Temperature distribution profiles.
Figure 4:
Table 2: Values of −θ′(0) for various Rd, A When Gr = 0.1 , M = 0.5 and Pr = 6.2 .

Rd

A

-θ'(O)

Cu

Ag

Al2CO3

TiO2

0.5

0.30

1.31430

1.23109

1.28489

1.28961

 

0.35

1.33022

1.23834

1.29732

1.30086

0.40

1.34938

1.24650

1.31206

1.31409

0.45

1.37892

1.25349

1.33263

1.33158

2

0.30

0.85353

0.79483

0.83119

0.82915

 

0.35

0.86695

0.80360

0.84275

0.84011

0.40

0.87137

0.80646

0.84654

0.84371

0.45

0.87576

0.80929

0.85030

0.84727

4

0.30

0.60425

0.55941

0.58668

0.58351

We have formulated the differential equations modeling the effect of radiation on heat transfer in nanofluid based on the already existing model. Numerical evaluation of the numerical solutions reported in the previous sections was performed and the results are presented in graphical and tabular forms. This was done to illustrate the influence of some parameter involved.

The effect of magnetic field parameter on velocity distribution profiles across the boundary layer is presented in Figure 1. It is obvious that the effect of increasing values of the magnetic field parameter M results in a decreasing velocity distribution. This is due to the fact that the introduction of transverse magnetic field normal to the flow direction has a tendency to create a drag force due to Lorentz force and hence results in retarding the velocity profiles Figure 2 shows the influence of the radiation parameter on the temperature distribution profiles. The temperature increases as the radiation increases. Figure 3 presents the temperature distribution profiles for different values of the Prandtl number (Pr). The results show that the effect of increasing values of the Prandtl number results in a decrease in the temperature.

In table 2, as the radiation parameter increases, the nuselt number of the nanoparticles decreases, that is to say the effect of increasing the radiation parameters is to decrease the rate of heat transfer of the nanoparticles.Also, as the fraction volume parameter (A) increases, nuselt number increases.
Conclusion
Numerical solution was applied to the differential equation that model the effect of thermal radiation on heat transfer on nanofluid using runge-kutta Fehlberg method. It was found that the effect of high radiation is to decrease the rate of heat transfer.
ReferencesTop
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  3. Khan WA and Pop I. Boundary- Layer flow of a nanofluid past a streching sheet. International Journal of Heat and Mass Transfer. 2010;53(11-12):2477-2483.
  4. Godson L, Raja B, Mohan LD and Wongwises S. Enhancement of Heat Transfer Using nanofluids. Renewable and Sustainable energy Reviews. 2010; 14(2): 629 – 641.
  5. Zeinali SH, Salehi H, Noie SH. The effect of Magnetic field and nanofluid on thermal performance of two-phase closed Thermosyphon (TPCT). Int. Journal of Physical Sciences. 2012;7(4):534 – 543. DOI: 10.5897/ IJPS11.1019.
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