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A steady boundary layer flow of Powell-Eyring nanofluid past a stretching sheet with variable thickness in the presence of thermal radiation and viscous dissipation is studied numerically. The model is used for the nanofluid incorporates the effects
   www.tjprc.org editor@tjprc.org EFFECTS OF THERMAL RADIATION AND VISCOUS DISSIPATION ON POWELL-EYRING NANOFLUID WITH VARIABLE THICKNESS D. VIDYANADHA BABU 1  & M. SURYANARAYANA REDDY 2   1  Department of mathematics, Priyadarshani College of Engineering, Sullurpet, Nellore Dt, A.P., India 2  Department of mathematics, JNTUA College of engineering, Pulivendula, A.P, India    ABSTRACT  A steady boundary layer flow of Powell- Eyring nanofluid past a stretching sheet with variable thickness in the  presence of thermal radiation and viscous dissipation is studied numerically. The model is used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The suitable transformations are applied to convert the  governing partial differential equations into a set of nonlinear coupled ordinary differential equations. Runge-Kutta-based  shooting technique is employed to yield the numerical solutions for the model. The obtained results for the velocity,  temperature and concentration are analyzed graphically for several physical parameters. It is found that an increment in wall thickness parameter results in decrease of velocity, temperature and concentration profiles. Further, in tabular form  the numerical values are given for the local skin friction coefficient, local Nusselt number and Sherwood number. A  remarkable agreement is noticed by comparing the present results with the results reported in the literature as a special  case.  KEYWORDS: Thermal Radiation, Powell – Eyring Fluid, Heat Viscous Dissipation & MHD Received:  Jul 07, 2017; Accepted:  Jul 31, 2017; Published:  Aug 09, 2017; Paper Id.:  IJMPERDAUG201739   INTRODUCTION  Viscous boundary layer flow due to a stretching/shrinking sheet is of significant importance due to its vast applications. Aerodynamic extrusion of plastic sheets, glass fiber production, paper production, heat treated materials travelling between a feed roll and a wind-up roll, cooling of an infinite metallic plate in a cooling bath and manufacturing of polymeric sheets are some examples for practical applications of non-Newtonian fluid flow over a stretching/shrinking surface. The quality of the final product depends on the rate of heat transfer at the stretching surface. This stretching/shrinking may not necessarily be linear. It may be quadratic, power-law, exponential and so on. Bilal Asharf et.al. [1] Studied the three dimensional boundary layer flow of Eyrng-Powell nanofluid by convectively heated exponentially stretching sheet. Hayat et.al. [2] Investigated the axisymmetric Powell-Eyring fluid flow with convective boundary condition. Javed et.at. [3] Analyzed the boundary layer flow of an Eyring -Powell non-Newtonian fluid over a stretching sheet. Hayat et.al. [4] Studied the effect of MHD boundary layer flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness. Nazar et.al. [5] Investigated the mixed convection boundary layer flow an isothermal horizontal circular cylinder embedded in a porous medium filled with a nanofluid for both cases of a heated and cooled cylinder. Solar energy is probably the most suitable source of renewable energy that can meet the current energy requirements. The energy obtained from nature in the form of solar radiations can be directly transformed into heat and electricity. The idea of using small particles to collect solar energy was first investigated by Hunt [6] in the  Or i   gi  n al  Ar  t  i   c l   e  International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249-6890; ISSN (E): 2249-8001 Vol. 7, Issue 4, Aug 2017, 389-402 © TJPRC Pvt. Ltd     390    D. Vidyanadha Babu & M. Suryanarayana Reddy    Impact Factor (JCC): 6.8765 NAAS Rating: 3.11   1970s. Researchers concluded that with the addition of nanoparticles in the base fluids, heat transfer and the solar collection processes can be improved. Masuda et al. [7] discussed the alteration of thermal conductivity and viscosity by dispersing ultra-fine particles in the liquid. Choi and Eastman [8] were the first to introduce the terminology of nanofluids when they experimentally discovered an effective way of controlling heat transfer rate using nanoparticles. Buongiorno [9] developed the nonhomogeneous equilibrium mathematical model for convective transport of nanofluids. He concluded that Brownian motion and thermophoretic diffusion of nanoparticles are the most important mechanisms for the abnormal convective heat transfer enhancement. The relevant processes are briefly described in [10–12]. Investigations in the nanofluid flows have received remarkable popularity in research community in last couple of decades primarily due to their variety of applications in power generation, in transportation where nanofluid may be utilized in vehicles as coolant, shock absorber, fuel additives etc., in cooling and heating problems which may involve the use of nanofluids for cooling of microchips in computer processors, in improving performance efficiency of refrigerant/air-conditioners etc. and in biomedical applications in which magnetic nanoparticles may be used in medicine, cancer therapy and tumor analysis. Recently the researchers have proposed the idea of using solar collector based nanofluids for optimal utilization of solar energy radiation [13, 14]. Buongiorno and Hu [15] discussed the heat transfer enhancement via nanoparticles for nuclear reactor application. Huminic and Huminic [16] showed that use of nanofluids in heat exchangers has advantage in the energy efficiency and it leads to better system performance. Ramzan et.al. [17] Discussed effects radiation and MHD on Powell-Eyring nanofluid over a stretching cylinder with chemical reaction. Ullah et.al. [18] Studied theoretically the influence of thermal radiation and MHD on natural convective flow of casson nanofluid over a non linearly stretching sheet. Mustafa et.al. [19] Runge-Kutta fourth –fifth order method with shooting technique to explore the steady boundary layer flow of nanofluid past a vertical plate with nonlinear radiation. Numerical solution for steady boundary layer flow of dusty fluid over a radiating stretching surface embedded in a thermally stratified porous medium in the presence of uniform heat source was studied by Gireesha et.al. [20]. Recently, Ramzan et.al. [21] Analyzed the radiative Jeffery nanofluid with convective heat and mass conditions. To the best of the author’s knowledge, there seems no existing document about steady boundary layer flow of Powell- Eyring nanofluid past a stretching sheet with variable thickness in the presence of thermal radiation and heat viscous dissipation. Hence the aim of the present investigation is to examine the steady boundary layer flow of Powell- Eyring nanofluid past a stretching sheet with variable thickness in the presence of thermal radiation and viscous dissipation. MATHEMATICAL FORMULATION Radiative powell-Eyring nanofluid over an impermeable nonlinear heated stretching sheet with variable thickness is considered. An incompressible fluid is selected electrically conducting. A non-uniform magnetic field 210 )()( − += n b x B x B  is imposed transverse to the stretching sheet. Magnetic Reynolds number is chosen small. Induced magnetic and electric fields are not accounted. Brownian and thermophoresis in the nanofluid are considered. Newly developed condition for mass flux is imposed. The fluid formation is such that the  x- axis is presumed along the sheet while  y -axis is transverse to it. The temperature of the sheet is different from that of the ambient medium. The fluid viscosity is assumed to vary with the temperature while the other fluid properties are assumed constant. The stretching surface has the nonlinear velocity ( ) nw  b xU U   += 0 where 0 U   is the reference velocity and b is   Effects of Thermal Radiation and Viscous Dissipation on Powell-Eyring Nanofluid with Variable Thickness 391 www.tjprc.org editor@tjprc.org the dimensional constant. Further it is assumed that sheet at ( ) 21  n b x A y − += is not flat (where n  is the velocity power index and  A  is assumed very small so that the sheet retain adequately thin). We also noticed that for n=1  the problem reduces to a flat sheet. The governing expressions for considered flow are given by: 0 =∂∂+∂∂  yv xu  (1) ( ) u x B yu yud  yud v yuv xuu  ρ σ  ρβ  ρβ  2222322 211 −∂∂      ∂∂−∂∂      +=∂∂+∂∂  (2) ( ) 2222 1      ∂∂+∂∂−          ∂∂+∂∂∂∂+∂∂=∂∂+∂∂ ∞  yuc yqc yT T  D yC  yT  D yT  yT v xT u  f r  f T  B f  υ  ρ τ α   (3)      ∂∂+∂∂=∂∂+∂∂ ∞ 2222  yT T  D yC  D yC v xC u  T  B  (4) With boundary conditions ( ) ( ) ( ) 21,0 ,0, nwwnw  b x A yat C C T T vb xU  xU u − +====+==  (5) ∞→→→→→ ∞∞  yasC C T T vu , ,0,0  (6) Here u and v are the corresponding velocity components parallel to  x - and  y -directions respectively,  µ   the dynamic viscosity,  ρ  µ  = v designates the kinematic viscosity,  ρ  the fluid density, d   and  β  are the material liquid parameters of Powell-Eyring model, k   the thermal conductivity, ( )       =  f nf  ck   ρ α  the thermal diffusivity of liquid, τ  the ratio of the heat capacity of liquid of the nanoparticles material to the effective heat capacity of the base fluid,  B  D indicates the Brownian diffusion coefficient, T   D represents the thermophoretic diffusion, T   the temperature of the fluid, C   the nanoparticles concentration, w T   and ∞ T  are the sheet and ambient fluid temperatures and ∞ C  the ambient fluid nanoparticles concentration.   392    D. Vidyanadha Babu & M. Suryanarayana Reddy    Impact Factor (JCC): 6.8765 NAAS Rating: 3.11   Figure 1: Physical Model and Coordinate System   Using the Rossel and approximations, the radiative heat flux is given by  zT K q r  ∂∂−= 4 *3*4 σ   (7) Where, * σ  and * K  are, respectively, the Stephan – Boltzman constant and the mean absorption coefficient. Assume that the temperature differences within the flow are small such that 4 T   in a Taylor series about ∞ T  and neglecting higher order terms. We get 44 34 ∞∞  −≅  T T T T   (8) Substituting eq.’s (7) and (8) in eq. (3). We have ( ) 2222 *3*16      ∂∂+          ∂∂+∂∂∂∂+∂∂      +=∂∂+∂∂ ∞∞  yuc yT T  D yC  yT  D yT cK T  yT v xT u  f T  B f  f  υ τ  ρ σ α   (9) Transformations are expressed as follows: ( ) ( ) ( ) ( ) ( )( )( ) ( ) ∞∞∞∞−− −−=Φ−−=Θ+       +=      +−′++       +−=′+= C C C C T T T T vb xU n ynnF F b xvU  nvF b xU u wwnnn η η η η η η η  ,,211121, 10100  (10) Incompressibility condition is satisfied identically and Eqs.(2), (4)-(6) and (9) take the following forms ( ) 01221121 22 =′      +−′′′′′       +−′      +−′′+′′′+  F  M nF F n N F nnF F F  N   λ   (11) 0Pr1 22 =′′+Θ′+Φ′Θ′′+Θ′+Θ′′       +  f  Ec Nt b N F   Nr   (12)   Effects of Thermal Radiation and Viscous Dissipation on Powell-Eyring Nanofluid with Variable Thickness 393 www.tjprc.org editor@tjprc.org 0Pr  =Θ′′      +Φ′+Φ′′  Nb Nt  LeF   (13) ( ) ( ) ( ) ( ) 1,1,1, 11 =Φ=Θ= ′      +−=  α α α α α   F nnF   (14) ( ) ( ) ( ) 0,0,0  →∞Φ→∞Θ→∞ ′ F   (15) In the above expressions primes designate differentiation with respect to η  . Where vU n A 0 21       += α   is the wall thickness parameter and vU n A 0 21       +== α η   shows the plate surface. Upon using ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ξ φ α η φ η ξ θ α η θ η ξ α η η   =−=Φ=−=Θ=−= ,,  f  f F   Eqs.(11)-(15) yield (See fig.1). ( ) 01221121 22 =′      +−′′′′′       +−′      +−′′+′′′+  f  M n f  f n N  f nn f  f  f  N   λ   (16) 0Pr1 22 =′′+′+′′+′+′′+  f  Ec Nt  Nb f   Nr  θ φ θ θ θ   (17) 0Pr  =′′      +′+′′  θ φ φ   Nb Nt  Lef   (18) ( ) ( ) ( ) 1)0(,10,10, 110  ===′      +−=  φ θ α   f nn f   (19)  ( ) ( ) ( ) 0,0,0  =∞=∞=∞ ′  φ θ   f   (20) where  N   and λ  are the fluid parameters,  M   represents the magnetic parameter,  Nb  the Brownian motion parameter , Pr   the Prfzandtl number,  Nt  the thermophoresis parameter,  Le  presents Lewis number,  Nr   the radiation parameter,  Ec  the Eckert number and prime indicates differentiation with respect to ξ  . The non-dimensional parameters are ( ) ( ) )(,**16,,)(,Pr,, 4,1 302013230 ∞∞∞∞∞− −===−=−===+== T T cU  EckK T  Nr  D LevT T T  D Nt vC C  D NbvU  B M b xvd U d  N  w f w B f wT w B f n σ α τ τ α  ρ σ λ  βµ   Surface drag coefficient and surface heat transfer are expressed as follows:
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