Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet

In this work, we study the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation, included in the energy equation, and variable wall temperature. A similarity transformation was used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme was used to obtain the solution of the boundary value problem. The variations of dimensionless surface temperature, as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include the nanoparticle volume fraction ϕ, the nonlinearly stretching sheet parameter n, the thermal radiation parameter NR, and the viscous dissipation parameter Ec, were graphed and tabulated. Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell for local Nusselt number without taking the effect of nanoparticles

Further results on nonlinearly stretching permeable sheets: analitic solution for MHD flow and mass transfer

The steady magnetohydrodynamic (MHD) flow and mass transfer of an incompressible, viscous, and electrically conducting fluid over a permeable flat surface stretched with nonlinear (quadratic) velocity u(w)(x) = ax + c(0)x(2) and appropriate wall transpiration is investigated. It is shown that the problem permits an analytical solution for the complete set of equations with magnetic field influences when a fictitious presence of a chemical reaction is considered. Velocity and concentration fields are presented through graphs and discussed. The results for both skin friction coefficient f ''(0) and mass transfer gradient c'(0) agree well with numerical results published in the literatureCortell Bataller, R. (2012). Further results on nonlinearly stretching permeable sheets: analitic solution for MHD flow and mass transfer. Mathematical Problems in Engineering. 2012:1-18. doi:10.1155/2012/743130S1182012Cortell, R. (2011). Heat transfer in a fluid through a porous medium over a permeable stretching surface with thermal radiation and variable thermal conductivity. The Canadian Journal of Chemical Engineering, 90(5), 1347-1355. doi:10.1002/cjce.20639Sakiadis, B. C. (1961). Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChE Journal, 7(1), 26-28. doi:10.1002/aic.690070108Crane, L. J. (1970). Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik ZAMP, 21(4), 645-647. doi:10.1007/bf01587695Gupta, P. S., & Gupta, A. S. (1977). Heat and mass transfer on a stretching sheet with suction or blowing. The Canadian Journal of Chemical Engineering, 55(6), 744-746. doi:10.1002/cjce.5450550619Vleggaar, J. (1977). Laminar boundary-layer behaviour on continuous, accelerating surfaces. Chemical Engineering Science, 32(12), 1517-1525. doi:10.1016/0009-2509(77)80249-2Hayat, T., Qasim, M., & Abbas, Z. (2010). Homotopy solution for the unsteady three-dimensional MHD flow and mass transfer in a porous space. Communications in Nonlinear Science and Numerical Simulation, 15(9), 2375-2387. doi:10.1016/j.cnsns.2009.09.013Cortell, R. (2005). Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dynamics Research, 37(4), 231-245. doi:10.1016/j.fluiddyn.2005.05.001Cortell, R. (2007). Toward an understanding of the motion and mass transfer with chemically reactive species for two classes of viscoelastic fluid over a porous stretching sheet. Chemical Engineering and Processing - Process Intensification, 46(10), 982-989. doi:10.1016/j.cep.2007.05.022Ishak, A., Nazar, R., Bachok, N., & Pop, I. (2010). Melting heat transfer in steady laminar flow over a moving surface. Heat and Mass Transfer, 46(4), 463-468. doi:10.1007/s00231-010-0592-8Cortell, R. (2011). Suction, viscous dissipation and thermal radiation effects on the flow and heat transfer of a power-law fluid past an infinite porous plate. Chemical Engineering Research and Design, 89(1), 85-93. doi:10.1016/j.cherd.2010.04.017Takhar, H. S., Raptis, A. A., & Perdikis, C. P. (1987). MHD asymmetric flow past a semi-infinite moving plate. Acta Mechanica, 65(1-4), 287-290. doi:10.1007/bf01176888Kumaran, V., & Ramanaiah, G. (1996). A note on the flow over a stretching sheet. Acta Mechanica, 116(1-4), 229-233. doi:10.1007/bf01171433Weidman, P. D., & Magyari, E. (2009). Generalized Crane flow induced by continuous surfaces stretching with arbitrary velocities. Acta Mechanica, 209(3-4), 353-362. doi:10.1007/s00707-009-0186-zMagyari, E., & Kumaran, V. (2010). Generalized Crane flows of micropolar fluids. Communications in Nonlinear Science and Numerical Simulation, 15(11), 3237-3240. doi:10.1016/j.cnsns.2009.12.013Cortell, R. (2007). Flow and heat transfer in a moving fluid over a moving flat surface. Theoretical and Computational Fluid Dynamics, 21(6), 435-446. doi:10.1007/s00162-007-0056-zPalani, G., & Kim, K. Y. (2011). On the diffusion of a chemically reactive species in a convective flow past a vertical plate. Journal of Applied Mechanics and Technical Physics, 52(1), 57-66. doi:10.1134/s0021894411010093Muhaimin, I., & Kandasamy, R. (2010). Local Nonsimilarity Solution for the Impact of a Chemical Reaction in an MHD Mixed Convection Heat and Mass Transfer Flow over a Porous Wedge in the Presence Of Suction/Injection. Journal of Applied Mechanics and Technical Physics, 51(5), 721-731. doi:10.1007/s10808-010-0092-0Abdel-Rahman, G. M. (2010). Thermal-diffusion and MHD for Soret and Dufour’s effects on Hiemenz flow and mass transfer of fluid flow through porous medium onto a stretching surface. Physica B: Condensed Matter, 405(11), 2560-2569. doi:10.1016/j.physb.2010.03.032Rohni, A. M., Ahmad, S., & Pop, I. (2012). Note on Cortell’s non-linearly stretching permeable sheet. International Journal of Heat and Mass Transfer, 55(21-22), 5846-5852. doi:10.1016/j.ijheatmasstransfer.2012.05.080Cortell, R. (2007). Viscous flow and heat transfer over a nonlinearly stretching sheet. Applied Mathematics and Computation, 184(2), 864-873. doi:10.1016/j.amc.2006.06.077Cortell, R. (2008). Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Physics Letters A, 372(5), 631-636. doi:10.1016/j.physleta.2007.08.005Akyildiz, F. T., & Siginer, D. A. (2010). Galerkin–Legendre spectral method for the velocity and thermal boundary layers over a non-linearly stretching sheet. Nonlinear Analysis: Real World Applications, 11(2), 735-741. doi:10.1016/j.nonrwa.2009.01.018Bataller, R. C. (2008). Similarity solutions for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. Journal of Materials Processing Technology, 203(1-3), 176-183. doi:10.1016/j.jmatprotec.2007.09.055Prasad, K. V., & Vajravelu, K. (2009). Heat transfer in the MHD flow of a power law fluid over a non-isothermal stretching sheet. International Journal of Heat and Mass Transfer, 52(21-22), 4956-4965. doi:10.1016/j.ijheatmasstransfer.2009.05.022Raptis, A., & Perdikis, C. (2006). Viscous flow over a non-linearly stretching sheet in the presence of a chemical reaction and magnetic field. International Journal of Non-Linear Mechanics, 41(4), 527-529. doi:10.1016/j.ijnonlinmec.2005.12.003Kelson, N. A. (2011). Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet. International Journal of Non-Linear Mechanics, 46(8), 1090-1091. doi:10.1016/j.ijnonlinmec.2011.04.025Ahmad, A., & Asghar, S. (2011). Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field. Applied Mathematics Letters, 24(11), 1905-1909. doi:10.1016/j.aml.2011.05.016Cortell, R. (2007). MHD flow and mass transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet with chemically reactive species. Chemical Engineering and Processing: Process Intensification, 46(8), 721-728. doi:10.1016/j.cep.2006.09.008Andersson, H. I., Bech, K. H., & Dandapat, B. S. (1992). Magnetohydrodynamic flow of a power-law fluid over a stretching sheet. International Journal of Non-Linear Mechanics, 27(6), 929-936. doi:10.1016/0020-7462(92)90045-9Vajravelu, K., Prasad, K. V., & Prasanna Rao, N. S. (2011). Diffusion of a chemically reactive species of a power-law fluid past a stretching surface. Computers & Mathematics with Applications, 62(1), 93-108. doi:10.1016/j.camwa.2011.04.055Akyildiz, F. T., Bellout, H., & Vajravelu, K. (2006). Diffusion of chemically reactive species in a porous medium over a stretching sheet. Journal of Mathematical Analysis and Applications, 320(1), 322-339. doi:10.1016/j.jmaa.2005.06.095Andersson, H. I., Hansen, O. R., & Holmedal, B. (1994). Diffusion of a chemically reactive species from a stretching sheet. International Journal of Heat and Mass Transfer, 37(4), 659-664. doi:10.1016/0017-9310(94)90137-6Makinde, O. D. (2010). On MHD heat and mass transfer over a moving vertical plate with a convective surface boundary condition. The Canadian Journal of Chemical Engineering, 88(6), 983-990. doi:10.1002/cjce.2036

Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature

In this paper we study the flow and heat transfer characteristics of a viscous fluid over a nonlinearly stretching sheet in the presence of non-uniform heat source and variable wall temperature. A similarity transformation is used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme is used to obtain the solution of the boundary value problem. The effects of various parameters (such as the power law index n, the Prandtl number Pr, the wall temperature parameter λ, the space dependent heat source parameter A* and the temperature dependent heat source parameter B*) on the heat transfer characteristics are analyzed. The numerical results for the heat transfer coefficient (the Nusselt number) are presented for several sets of values of the parameters and are discussed. The results reveal many interesting behaviors that warrant further study on the effects of non-uniform heat source and the variable wall temperature on the heat transfer phenomena at the nonlinear stretching sheet. © 2011 Elsevier Ltd. All rights reserved.postprin

Soret and Dufour effects on unsteady mixed convection slip flow of Casson fluid over a nonlinearly stretching sheet with convective boundary condition

Unsteady mixed convection flow of Casson fluid towards a nonlinearly stretching sheet with the slip and convective boundary conditions is analyzed in this work. The effects of Soret Dufour, viscous dissipation and heat generation/absorption are also investigated. After using some suitable transformations, the unsteady nonlinear problem is solved by using Keller-box method. Numerical solutions for wall shear stress and high temperature transfer rate are calculated and compared with previously published work, an excellent arrangement is followed. It is noticed that fluid velocity reduces for both local unsteadiness and Casson parameters. It is likewise noticed that the influence of a Dufour number of dimensionless temperature is more prominent as compared to species concentration. Furthermore, the temperature was found to be increased in the case of nonlinear thermal radiation

Similarity solutions for the stagnation-point flow and heat transfer over a nonlinearly stretching/shrinking sheet

This paper presents a numerical analysis of a stagnation-point flow towards a nonlinearly stretching/shrinking sheet immersed in a viscous fluid. The stretching/shrinking velocity and the external flow velocity impinges normal to the stretching/shrinking sheet are assumed to be in the form U ~ xm, where m is a constant and x is the distance from the stagnation point. The governing partial differential equations are converted into ordinary ones by a similarity transformation, before being solved numerically. The variations of the skin friction coefficient and the heat transfer rate at the surface with the governing parameters are graphed and tabulated. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique for m > 1/3

Chemically radiative dissipative MHD Casson nanofluid flow on a non-linear elongating stretched sheet with numerous slip and convective boundary conditions

The flow of a Casson nanofluid across a nonlinear stretching surface with a velocity slip and a convective boundary condition is investigated in this work in the magnetohydrodynamic (MHD) domain. This technique emphasizes a variety of effects, including chemical reaction, viscosity dissipation, and velocity ratio. In this study, Brownian motion and thermophoresis are also illustrated. It is assumed that suction exists while a magnetic field is uniform. The governing nonlinear partial differential equations are converted into a set of nonlinear ordinary differential equations using the required similarity transformations, and the Runge-Kutta-Fehlberg fourth-fifth method is then used to solve the system. The updated results are fairly similar to the earlier ones. The graphs and tables examine how various variables affect the speeds, temperatures, concentrations of substances, skin friction values, Sherwood numbers and Nusselt numbers

Stagnation point flow of a MHD Powell-Eyring fluid over a nonlinearly stretching sheet in the presence of heat source/sink

This study investigates the stagnation point flow of a MHD Powell-Eyring fluid over a nonlinearly stretching sheet in the presence of heat source/sink. Similarity transformations are used to convert highly non-linear partial differential equations into ordinary differential equations. The transformed nonlinear boundary layer equations are then solved numerically using Keller Box method. The effects of various physical parameters on the dimensionless velocity and temperature profiles are depicted graphically. Present results are compared with previously published work and the results are found to be in very good agreement. Numerical results for local skin-friction and local Nusselt number are tabulated for different physical parameters