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MHD Boundary Layer Flow of a Nano Fluid Past a Porous Shrinking Sheet with Thermal Radiation

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MHD Boundary Layer Flow of a Nano Fluid Past a Porous Shrinking Sheet with Thermal Radiation
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  MHD Boundary Layer Flow of a Nanofluid Passed througha Porous Shrinking Sheet with Thermal Radiation S. Nadeem 1 and Rizwan Ul Haq 2 Abstract:  This study examines the series solution for magnetohydrodynamics (MHD) three-dimensional flow past a porous shrinking sheet.The thermal radiation effects are also taken into account. The model used for the nanofluid incorporated the effect of Brownian motion and thethermophoreses. The involved equations are solved analytically by homotopy analysis method (HAM). Graphical results for the dimension-less velocity, temperature, and nanoparticle fraction are reported and examined for various physical parameters showing the interestingaspects of the obtained solutions. The numerical table for local Nusselt and Sherwood numbers are also computed.  DOI: 10.1061/ (ASCE)AS.1943-5525.0000299.  © 2014 American Society of Civil Engineers. Author keywords:  Shrinking sheet; Nanofluid; Thermal radiation effects; Porous medium; Series solution. Introduction The flow due to shrinking or stretching a sheet has numerous ap-plications due to its practical and cost-related advantages; it hasbeen extensively used in many engineering fields and industriesfor expanding and contracting of surfaces such as shrinking wrap-ping, bundlewrapping, hot rolling, extrusion of sheet material, wirerolling, and glass fiber (Hashim et al. 2008). The main process that converts the loose sleeves are an envelope of plastic that tightlyconforms to the shape of the enclosed contents in shrinking sheets.A shrinking sheet can be made from a variety of materials, eachhaving different strengths, shrinking transparency, and luster.In recent years, flow over a shrinking surface has achieved muchimportance. A few studies on this topic are Sakiadis (1961),Nadeem et al. (2010, 2011), Nadeem and Faraz (2010), Noor et al. (2010), and Hayat et al. (2011a , b). Initially, Sakiadis (1961) dis- cussed the concept of the boundary layer flow over a continuouslystretching surface with a constant speed. Crane (1970) pioneeredwork in the study of linear and exponentially stretching sheet and its idea extended for steady boundary layer flow of an incom-pressible fluid over a linear stretching plate and gave an exact similar solution in closed analytical form.There is another area that has attracted the attention of research-ers known as nanofluids. Nanofluid is mixture of nanosizedparticles suspended in a base fluid that has greater thermal conduc-tivity in comparison with the base fluid. The rate of heat transfer inindustrial processes and the nuclear reactions is due to its higher thermal conductivity (Godson et al. 2010). The heat conductionhas a great importance in many industrial heating or cooling equip-ments. In these days, there is a great advancement in the study of the flow of nanofluids with convective heat transfer. Sebdani et al.(2012) studied the mixed convection flow of nanofluid withvariable properties in a square cavity. A finite-volume method isutilized to analyze the results. Boundary layer flow of nanofluidover a moving surface in a flowing fluid was discussed by Bachok et al. (2010). In this study, the authors have observed the influenceof heat generation and absorption and suction and blowing.Makinde and Aziz (2011) carried out a study to see the influenceof convective boundary conditions on the flow of nanofluid past a stretching surface. They employed the fourth-fifth order Runge-Kutta method to obtain velocity, temperature, and concentrationprofiles. Khan and Aziz (2011a , b), numerically discussed the natu- ral convection flow of nanofluid with uniform surface heat fluxover a vertical plate. In another attempt, Khan and Aziz (2011a , b) discussed the double-diffusive natural convection flow saturatedover a vertical plate. They also considered the effects of prescribedsurface heat, solute, and nanoparticle fluxes in this study. Nadeem and Lee (2012) analyzed the boundary layer flow of nanofluid over an exponentially stretched surface. They compute series solutionswith the help of homotopy analysis method (HAM). Recently,many authors have investigated the nanofluid for different flowgeometries, e.g., Khan and Pop (2010), Hamad and Ferdows(2012), Khanafer et al. (2003), Noghrehabadi et al. (2012), and Hady et al. (2012).The magnetohydrodynamic (MHD) flow has numerous appli-cations in physics, chemistry, and engineering processes. It hasseveral applications in the polymer industry in which there arestretched plastic sheets and in metallurgy in which hydromagnetictechniques are employed. Most of the metallurgical processes in-volve the cooling of continuous strips or filaments by drawingthem. A large number of articles exist in the literature. A few recent studies on this type of flows are Sheikholeslami et al. (2012), Noor et al. (2010), Nadeem and Hussain (2009), and Yazdi et al. (2011). The aim of the present investigation is to study the effects of magnetic fields and thermal radiation on three-dimensional flow of nanofluid. The flow is confined over a porous shrinking surface.Series solutions for velocity, temperature, and concentration arecomputed by employing HAM; this method is a powerful analyti-cal tool and already several investigators used this method for dif-ferent flow problems and differential equations (Zheng et al. 2012;Liao 2003, 2004; Hayat et al. 2011a , b; Bataineh et al. 2009; Abbasbandyand Shivanian 2011; Hayat and Qasim 2010; Dehghan and Salehi 2011; Ariel 2007). Results are discussed by plotting graphs and by computing numerical values of local Nusselt andSherwood numbers. 1 Professor, Dept. of Mathematics, Quaid-I-Azam Univ., Islamabad44000, Pakistan. E-mail: snqau@hotmail.com  2 Dept. of Mathematics, Quaid-I-Azam Univ., Islamabad 44000,Pakistan (corresponding author). E-mail: ideal_riz@hotmail.com Note. This manuscript was submitted on July 4, 2012; approved onNovember 16, 2012; published online on November 19, 2012. Discussionperiod open until November 26, 2014; separate discussions must be sub-mitted for individual papers. This paper is part of the  Journal of Aero- space Engineering , © ASCE, ISSN 0893-1321/04014061(9)/$25.00.  © ASCE 04014061-1 J. Aerosp. Eng. J. Aerosp. Eng.    D  o  w  n   l  o  a   d  e   d   f  r  o  m  a  s  c  e   l   i   b  r  a  r  y .  o  r  g   b  y   U  n   i  v  e  r  s   i   t  y  o   f   W  e  s   t  e  r  n   O  n   t  a  r   i  o  o  n   0   7   /   1   0   /   1   4 .   C  o  p  y  r   i  g   h   t   A   S   C   E .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y  ;  a   l   l  r   i  g   h   t  s  r  e  s  e  r  v  e   d .  Mathematical Model The balance of mass, momentum, temperature, and nanoparticlevolume fraction are given by (Khanafer et al. 2003)div  V   ¼  0  ð 1 Þ ρ  f  dV dt ¼  −∇ P  þ  μ ∇ 2 V   þ  J  × B −  μ K V   ð 2 Þð ρ c Þ  f  dT dt ¼  k ∇ 2 T   þ ð ρ c Þ p ½ D B ∇ C  ·  ∇ T   þ ð DT  = T  ∞ Þ ∇ T   ·  ∇ T  ð 3 Þ dCdt ¼  D B ∇ 2 C  þ ð D T  = T  ∞ Þ ∇ 2 T   ð 4 Þ where  ρ  f   = density of the base fluid;  ρ p  = density of the particle; c  = volumetric volume expansion coefficient;  V   = velocity vector; d  = dt  = material time derivative;  P  = pressure;  C  = nanoparticlefraction;  D B  = Brownian diffusion coefficient; and  D T   = thermo-phoretic diffusion coefficient. Mathematical Formulation Consider the three-dimensional steady incompressible viscousfluid past a horizontal shrinking sheet. In addition, porosity andthermal radiation effect have been considered when fluid is passingthrough the shrinking sheet. The continuity, momentum, energy,and nanoparticle fraction for the present boundary layer flow arereduced to the following equations (Hashim et al. 2008; Khan and Pop 2010): ∂  u ∂   x þ ∂  v ∂  y þ ∂  w ∂  z ¼  0  ð 5 Þ u ∂  u ∂   x þ  v ∂  u ∂  y þ  w ∂  u ∂  z ¼  −  1 ρ  f  ∂  p ∂   x þ  ν   ∂  2 u ∂   x 2  þ ∂  2 u ∂  y 2  þ ∂  2 u ∂  z 2  − σ  B 20 ρ  u −  ν  K u  ð 6 Þ u ∂  v ∂   x þ  v ∂  v ∂  y þ  w ∂  v ∂  z ¼  −  1 ρ  f  ∂  p ∂  y þ ν   ∂  2 v ∂   x 2  þ ∂  2 v ∂  y 2  þ ∂  2 v ∂  z 2  − σ  B 20 ρ  v −  ν  K v  ð 7 Þ u ∂  w ∂   x þ  v ∂  w ∂  y þ  w ∂  w ∂  z ¼  −  1 ρ  f  ∂  p ∂  z þ  ν   ∂  2 w ∂   x 2  þ ∂  2 w ∂  y 2  þ ∂  2 w ∂  z 2   ð 8 Þ u ∂  T  ∂   x þ v ∂  T  ∂  y þ w ∂  T  ∂  z ¼ α  ∂  2 T  ∂   x 2  þ ∂  2 T  ∂  y 2  þ ∂  2 T  ∂  z 2  þ τ   D B  ∂  C ∂   x ∂  T  ∂   x þ ∂  C ∂  y ∂  T  ∂  y þ ∂  C ∂  z ∂  T  ∂  z  þ  D T  T  ∞  ×  ∂  T  ∂   x  2 þ  ∂  T  ∂  y  2 þ  ∂  T  ∂  z  2  −∂  q r ∂  z ð 9 Þ u ∂  C ∂   x þ  v ∂  C ∂  y þ  w ∂  C ∂  z ¼  D B  ∂  2 C ∂   x 2  þ ∂  2 C ∂  y 2  þ ∂  2 C ∂  z 2  þ  D T  T  ∞  ∂  2 T  ∂   x 2  þ ∂  2 T  ∂  y 2  þ ∂  2 T  ∂  z 2   ð 10 Þ where  u ,  v , and  w  = respective velocities in the  x -,  y -, and z -directions, respectively;  ρ  f   = density of the base fluid;  ν   = kin-ematic viscosity of the fluid; σ   = electrical conductivity; ρ = densityof the fluid;  B 0  = magnetic induction;  K   = porous medium per-meability; α = thermal diffusivity;  T   = fluid temperature;  C = nano-particle fraction;  T  w  and C w  = temperature offluid and nanoparticlefraction at wall, respectively;  D B  = Brownian diffusion cofficent; D T   = thermophoretic diffusion coefficient;  τ   ¼ ð ρ c Þ p = ð ρ c Þ  f   is theratio between the effective heat capacity of the nanoparticlematerial and heat capacity of the fluid;  c  = volumetric volume ex-pansion coefficient; and  ρ p  = density of the particles. When  z  tendsto infinity, the ambient values of   T   and  C  are denoted by  T  ∞  and C ∞ . The associated boundary conditions of Eqs. (6) – (10) are u  ¼  − U   ¼  − ax ;  v  ¼  − a ð m − 1 Þ y ;  w  ¼  − W  ; T   ¼  T  w ;  C  ¼  C w  at   z  ¼  0  u  →  0 ;  v  →  0 ; T   →  T  ∞ ;  C  →  C ∞  as  z  →  ∞  ð 11 Þ in which  a  >  0  is the shrinking constant;  W   >  0  is the suctionvelocity;  m  ¼  1  when sheets shrink in the  x -direction only; and m  ¼  2  when sheets shrink asymmetrically. Using Rosselandapproximation of radiation for an optically thick layer, i.e., the op-tical thick layer, is a dimensionless quantity that characterizes theattenuation of optical radiation in the medium. The attenuation re-sults from the joint action of light absorption and light scattering.The effects of radiation amplification caused by multiple scatteringare not taken into account [for more detail see Viskanta and Grosh(1962) and Bhargava and Rana (2010)] q r  ¼  − 4 σ   3 k  ∂  T  4 ∂  z ð 12 Þ where  k   = mean absorption coefficient; and  σ    = Stefan-Boltmannconstant.  T  4 , expressed as a linear function of temperature by usingTaylor series expansion about   T  ∞ , is T  4 ¼  4 T  3 ∞ T   − 3 T  4 ∞  ð 13 Þ Introducing the following similarity transformations (Hashim et al. 2008): η   ¼  ffiffiffi a ν  r   z ;  u  ¼  axf  0 ð η  Þ ;  v  ¼  a ð m − 1 Þ yf  0 ð η  Þ ; w  ¼  −  ffiffiffiffiffiffiffiffiffi a ν  m p   f  ð η  Þ  θ  ¼  T   − T  ∞ T  w  − T  ∞ ;  ϕ  ¼  C − C ∞ C w  − C ∞ ð 14 Þ Making use of Eq. (14), the equation of continuity is identicallysatisfied and Eqs. (6) – (10) along with Eqs. (11) and (12) take the following form:  f  ‴ − ð M  2 þ  λ Þ  f  0  − ð  f  0 Þ 2 þ  mff  0 0  ¼  0  ð 15 Þ  1  þ  43 N  r  θ 0  þ  Pr  ½  f  θ 0  þ  N  b ð θ 0 ϕ 0 Þ þ  N  t ð θ 0 Þ 2  ¼  0  ð 16 Þ ϕ 0  þ  L e Pr  ð  f  ϕ 0 Þ þ  N  t N  b θ 0  ¼  0  ð 17 Þ  f  ð 0 Þ ¼  S ;  f  0 ð 0 Þ ¼  − 1 ;  f  0 ð ∞ Þ ¼  0  ð 18 Þ  © ASCE 04014061-2 J. Aerosp. Eng. J. Aerosp. Eng.    D  o  w  n   l  o  a   d  e   d   f  r  o  m  a  s  c  e   l   i   b  r  a  r  y .  o  r  g   b  y   U  n   i  v  e  r  s   i   t  y  o   f   W  e  s   t  e  r  n   O  n   t  a  r   i  o  o  n   0   7   /   1   0   /   1   4 .   C  o  p  y  r   i  g   h   t   A   S   C   E .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y  ;  a   l   l  r   i  g   h   t  s  r  e  s  e  r  v  e   d .  θ ð 0 Þ ¼  1 ;  θ ð ∞ Þ ¼  0  ð 19 Þ ϕ ð 0 Þ ¼  1 ;  ϕ ð ∞ Þ ¼  0  ð 20 Þ In these expressions  M  2 ¼ ð σ  B 2 o Þ = ð ρ a Þ  is the magneticReynolds,  S  is the shrinking parameter, λ  ¼  α = ð aK  Þ  is the porosityparameter,  Pr   ¼  ν  = α  is the Prandtl number,  N  r  ¼ ð 4 σ   T  3 ∞ Þ = ð k  k Þ is the thermal radiation effect,  N  b  ¼ ½ð ρ c Þ P D B ð C w  − C ∞ Þ = ½ ν  ð ρ c Þ P   is the Brownian motion,  N  t  ¼ ½ð ρ c Þ P D T  ð T  w  − T  ∞ Þ = ½ ν  ð ρ c Þ P   is the thermophoresis parameter, and  L e  ¼  α = D B  is theLewis number. Expressions for the local Nusselt number Nuand the local Sherwood number Sh areNu  ¼  xq w α ð T  w  − T  ∞ Þ ;  Sh  ¼  xq m D B ð C w  − C ∞ Þ ð 21 Þ where  q w  and  q m  = heat flux and mass flux, respectively. q w  ¼  − α  ∂  T  ∂  z  z ¼ 0 ;  q m  ¼  − D B  ∂  C ∂  z  z ¼ 0 ð 22 Þ The dimensionless form of Eq. (22) takes the form  R − 1 = 2  x  Nu  ¼  −  1  þ  43 N  r  θ 0 ð 0 Þ ;  R − 1 = 2  x  Sh  ¼  − ϕ 0 ð 0 Þ ð 23 Þ where  R  x  ¼  u w ð  x Þ  x = ν   is the local Reynolds number based on theshrinking velocity  u w ð  x Þ . Homotopy Analysis Solutions The governing equations with the subjected boundary conditionshave been solved by a homotopy analysis technique. First, the func-tions  f  ð η  Þ ,  θ ð η  Þ , and  ϕ ð η  Þ  by a set of base functions f η  k exp ð − n η  Þj k  ≥  0 ; n  ≥  0 g ð 24 Þ are expressed into the forms  f  m ð η  Þ ¼ X ∞ n ¼ 0 X ∞ k ¼ 0 a km ; n η  k exp ð − n η  Þ θ m ð η  Þ ¼ X ∞ n ¼ 0 X ∞ k ¼ 0 b km ; n η  k exp ð − n η  Þ ϕ m ð η  Þ ¼ X ∞ n ¼ 0 X ∞ k ¼ 0 c km ; n η  k exp ð − n η  Þ ð 25 Þ  f  0 ð η  Þ ¼  S − 1  þ  exp ð − η  Þ  θ 0 ð η  Þ ¼  exp ð − η  Þ ϕ 0 ð η  Þ ¼  exp ð − η  Þ ð 26 Þ L  f   ¼  d  3  f d  η  3  − df d  η   L θ  ¼  d  2 θ d  η  2  þ  d  θ d  η   L ϕ  ¼  d  2 ϕ d  η  2  þ  d  ϕ d  η   ð 27 Þ L  f  ½ C 1  þ C 2 exp ð η  Þþ C 3 exp ð − η  Þ¼ 0  L θ ½ C 7  þ C 8 exp ð − η  Þ¼ 0 L ϕ ½ C 9  þ C 10 exp ð − η  Þ¼ 0  ð 28 Þ Here  f  0 ,  θ 0 , and  ϕ 0  stand for the initial guesses of   f  ,  θ ,and  ϕ , respectively,  L  f  ,  L θ , and  L ϕ  the linear operators, and C i ð i  ¼  1 − 10 Þ  the arbitrary constants.Denoting the embedding and nonzero auxiliary parameters by p  ∈  ½ 0 ; 1   and  h  f  ,  h g  ,  h θ , and  h ϕ , respectively, the zeroth and  m thorder problems are  B 0  B 0  B 0  B 0  zu = -ax x y (a) (b) Fig. 1.  (a) Geometry of the problem; (b)  ℏ -curve when  M   ¼  1 ,  m  ¼  1 ,  S  ¼  1 ,  Pr   ¼  2 ,  N  r  ¼  0 . 2 ,  N  t  ¼  0 . 2 ,  N  b  ¼  0 . 1 ,  L e  ¼  2 , and  λ  ¼  0 . 5 Table 1.  Convergence of the HAM Solutions for Different Order of Approximation When  S  ¼  1 ,  M   ¼  1 ,  m  ¼  1 ,  Pr   ¼  1 ,  N  b  ¼  0 . 1 , N  t  ¼  0 . 2 ,  N  r  ¼  0 . 1 ,  L e  ¼  2 . 0 , and  λ  ¼  1 Order of approximation  f  0 0 ð 0 Þ  − θ 0 ð 0 Þ  − ϕ 0 ð 0 Þ 1 1.75000 0.573333 0.06132310 2.30179 0.506052 0.21618220 2.30278 0.506025 0.21611325 2.30278 0.506026 0.21611030 2.30278 0.506026 0.21611035 2.30278 0.506026 0.21611040 2.30278 0.506026 0.216110  © ASCE 04014061-3 J. Aerosp. Eng. J. Aerosp. Eng.    D  o  w  n   l  o  a   d  e   d   f  r  o  m  a  s  c  e   l   i   b  r  a  r  y .  o  r  g   b  y   U  n   i  v  e  r  s   i   t  y  o   f   W  e  s   t  e  r  n   O  n   t  a  r   i  o  o  n   0   7   /   1   0   /   1   4 .   C  o  p  y  r   i  g   h   t   A   S   C   E .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y  ;  a   l   l  r   i  g   h   t  s  r  e  s  e  r  v  e   d .  ð 1 − p Þ L  f  ½ ˆ  f  ð η  ; p Þ −  f  0 ð η  Þ ¼  ph  f  N  f  ½ ˆ  f  ð η  ; p Þ ;  ˆ g  ð η  ; p Þð 1 − p Þ L θ ½ ˆ θ ð η  ; p Þ − θ 0 ð η  Þ ¼  ph θ N θ ½ ˆ  f  ð η  ; p Þ ;  ˆ g  ð η  ; p Þ ;  ˆ θ ð η  ; p Þð 1 − p Þ L ϕ ½ ˆ ϕ ð η  ; p Þ − ϕ 0 ð η  Þ ¼  ph ϕ N ϕ ½ ˆ  f  ð η  ; p Þ ;  ˆ θ ð η  ; p Þ ;  ˆ ϕ ð η  ; p Þð 29 Þ ˆ  f  ð η  ; p Þj η  ¼ 0  ¼  S ;  ∂  ˆ  f  ð η  ; p Þ ∂  η   η  ¼ 0 ¼  − 1 ; ∂  ˆ  f  ð η  ; p Þ ∂  η   η  ¼ ∞ ¼  0  ˆ θ ð η  ; p Þj η  ¼ 0  ¼  1 ;  ˆ θ ð η  ; p Þj η  ¼ ∞  ¼  0 ; ˆ ϕ ð η  ; p Þj η  ¼ 0  ¼  1 ;  ˆ ϕ ð η  ; p Þj η  ¼ ∞  ¼  0  ð 30 Þ L  f  ½  f  m ð η  Þ − χ m  f  m − 1 ð η  Þ ¼  h  f  R  f m ð η  Þ L θ ½ θ m ð η  Þ − χ m θ m − 1 ð η  Þ ¼  h θ R θ m ð η  Þ L ϕ ½ ϕ m ð η  Þ − χ m ϕ m − 1 ð η  Þ ¼  h ϕ R ϕ m ð η  Þ ð 31 Þ  f  m ð 0 Þ ¼  f  0 m ð 0 Þ ¼  f  0 m ð ∞ Þ ¼  0  θ m ð 0 Þ ¼  θ m ð ∞ Þ ¼  0 ; ϕ m ð 0 Þ ¼  0  ¼  ϕ m ð ∞ Þ ð 32 Þ N  f  ½ ˆ  f  ð η  ; p Þ ;  ˆ g  ð η  ; p Þ ¼  ∂  3  ˆ  f  ð η  ; p Þ ∂  η  3  −  ∂  ˆ  f  ð η  ; p Þ ∂  η   2 þ  ˆ  f  ð η  ; p Þ ∂  2  ˆ  f  ð η  ; p Þ ∂  η  2  − ð M  2 þ λ Þ  ∂  ˆ  f  ð η  ; p Þ ∂  η    ð 33 Þ 0 1 2 3 4-1-0.8-0.6-0.4-0.20 S = 1S = 2S = 3S = 4S = 5 M =1.0= 0.5m =1.0N b =0.5L e  =2.0N t  =0.5N r  =0.5Pr =6.0 Fig. 2.  Variation of   f  0 ð η  Þ  for various values of   S 0 1 2 3 4-1-0.8-0.6-0.4-0.20 M = 1M = 2M = 3M = 4M = 5 s = 2.0= 0.5m =1.0N b =0.2L e  =2.0N t  =0.7N r  =0.5Pr =6.0 Fig. 3.  Variation of   f  0 ð η  Þ  for various values of   λ       f      '      (     e      t     a      ) 0 1 2 3 4-1-0.8-0.6-0.4-0.20 = 1= 2= 3= 4= 5 S = 2.0M = 1.0m =1.0N b =0.2L e  =2.0N t  =0.7N r  = 0.5Pr =6.0 Fig. 4.  Variation of   f  0 ð η  Þ  for various values of   M  0 2 4 6 8 1000.20.40.60.81 N b  = 0.1N b  = 0.3N b  = 0.5N b  = 0.7N b  = 0.9 ( )( ) = 0.6m = 0.5M = 0.5L e  = 0.5S = 1.5N t  = 0.2N r  = 0.2Pr = 2.0 Fig. 5.  Variation of   θ ð η  Þ  and  ϕ ð η  Þ  for various values of   Pr   © ASCE 04014061-4 J. Aerosp. Eng. J. Aerosp. Eng.    D  o  w  n   l  o  a   d  e   d   f  r  o  m  a  s  c  e   l   i   b  r  a  r  y .  o  r  g   b  y   U  n   i  v  e  r  s   i   t  y  o   f   W  e  s   t  e  r  n   O  n   t  a  r   i  o  o  n   0   7   /   1   0   /   1   4 .   C  o  p  y  r   i  g   h   t   A   S   C   E .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y  ;  a   l   l  r   i  g   h   t  s  r  e  s  e  r  v  e   d .  N θ ½ ˆ  f  ð η  ; p Þ ;  ˆ g  ð η  ; p Þ ;  ˆ θ ð η  ; p Þ ;  ˆ ϕ ð η  ; p Þ ¼  1  þ  43 N  r  ∂  2 ˆ θ ð η  ; p Þ ∂  η  2 þ  Pr   ˆ  f  ð η  ; p Þ ∂  ˆ θ ð η  ; p Þ ∂  η   þ  N  b ∂  ˆ θ ð η  ; p Þ ∂  η  ∂  ˆ ϕ ð η  ; p Þ ∂  η  þ  N  t  ∂  ˆ θ ð η  ; p Þ ∂  η   2   ð 34 Þ N ϕ ½ ˆ  f  ð η  ; p Þ ;  ˆ θ ð η  ; p Þ ;  ˆ ϕ ð η  ; p Þ ¼  ∂  2  ˆ ϕ ð η  ; p Þ ∂  η  2 þ  L e Pr   ˆ  f  ð η  ; p Þ ∂  ˆ ϕ ð η  ; p Þ ∂  η   þ  N  t N  b ∂  2 ˆ θ ð η  ; p Þ ∂  η  2  ð 35 Þ R  f m ð η  Þ ¼  f  ‴ m − 1 − ð M  2 þ λ Þð  f  0 m − 1 Þþ X m − 1 k ¼ 0  f  m − 1 − k  f  00 k  − X m − 1 k ¼ 0  f  0 m − 1 − k  f  0 k ð 36 Þ R θ m ð η  Þ ¼  1  þ  43 N  r  θ 0 m − 1  þ Pr  X m − 1 k ¼ 0 ½  f  m − 1 − k θ 0 k  þ  N  b ð ϕ 0 m − 1 − k θ 0 k Þþ  N  t ð θ 0 m − 1 − k θ 0 k Þ ð 37 Þ R ϕ m ð η  Þ ¼  ϕ 0 m − 1  þ  Pr  L e X m − 1 k ¼ 0  f  m − 1 − k ϕ 0 k  þ  N  t N  b θ 0 m − 1  ð 38 Þ χ m  ¼  0 ;  m  ≤  11 ;  m  >  1  ð 39 Þ 0 1 2 3 4 500.20.40.60.811.2 N t  = 0.1N t  = 0.3N t  = 0.5N t  = 0.7N t  = 0.9 = 0.6m = 0.5Le =1.0Pr = 5.0S = 1.5N b  = 0.2N r  =0.2 ( )( ) Fig. 6.  Variation of   θ ð η  Þ  and  ϕ ð η  Þ  for various values of   L e 0 1 2 3 4 500.20.40.60.81 N r  = 0.0N r  = 0.2N r  = 0.4N r  = 0.6N r  = 0.8 = 0.6m =0.5L e = 1.0Pr= 5.0S = 1.5N b =0.2N t  =0.2M =0.5 ( )( ) Fig. 7.  Variation of   θ ð η  Þ  and  ϕ ð η  Þ  for various values of   N  b 0 5 1000.511.52 Pr = 2Pr = 3Pr = 4Pr = 5Pr = 6 ( )( ) = 1.0s = 1.0M = 1.0N b = 0.2L e  = 2.0N t  = 0.7N r  = 0.5 Fig. 8.  Variation of   θ ð η  Þ  and  ϕ ð η  Þ  for various values of   N  t 1.8 2 2.2 2.40.320.340.360.380.4 0 2 4 6 800.20.40.60.81   L e  = 2.0L e  = 2.5L e  = 3.0L e  = 3.5L e  = 4.0 ( )( ) S = 1.5= 0.6m = 0.5N b = 0.8N t  = 0.2N r  = 0.5M = 0.5Pr=1.0 Fig. 9.  Variation of   θ ð η  Þ  and  ϕ ð η  Þ  for various values of   N  r  © ASCE 04014061-5 J. Aerosp. Eng. J. Aerosp. Eng.    D  o  w  n   l  o  a   d  e   d   f  r  o  m  a  s  c  e   l   i   b  r  a  r  y .  o  r  g   b  y   U  n   i  v  e  r  s   i   t  y  o   f   W  e  s   t  e  r  n   O  n   t  a  r   i  o  o  n   0   7   /   1   0   /   1   4 .   C  o  p  y  r   i  g   h   t   A   S   C   E .   F  o  r  p  e  r  s  o  n  a   l  u  s  e  o  n   l  y  ;  a   l   l  r   i  g   h   t  s  r  e  s  e  r  v  e   d .
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