Numerical simulation of self-similar thermal convection from a spinning cone in anisotropic porous medium
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摘要
Self-similar steady natural convection thermal boundary layer flow from a rotating vertical cone to anisotropic Darcian porous medium is investigated theoretically and numerically. The transformed non-dimensional two-point boundary value problem is reduced to a system of coupled, highly nonlinear ordinary differential equations, which are solved subject to robust surface and free stream boundary conditions with the MAPLE 17 numerical quadrature software. Validation with earlier non-rotating studies is included, and also further verification of rotating solutions is achieved with a variational finite element method(FEM). The rotational(spin) parameter emerges as an inverse function of the Grashof number. The influence of this parameter, primary Darcy number, secondary Darcy number and Prandtl number on tangential velocity and swirl velocity, temperature and heat transfer rate are studied in detail. It is found that the dimensionless tangential velocity increases whilst the dimensionless swirl velocity and temperature decrease with the swirl Darcy number, tangential Darcy number and the rotational parameters. The model finds applications in chemical engineering filtration processing, liquid coating and spinning cone distillation columns.
Self-similar steady natural convection thermal boundary layer flow from a rotating vertical cone to anisotropic Darcian porous medium is investigated theoretically and numerically. The transformed non-dimensional two-point boundary value problem is reduced to a system of coupled, highly nonlinear ordinary differential equations, which are solved subject to robust surface and free stream boundary conditions with the MAPLE 17 numerical quadrature software. Validation with earlier non-rotating studies is included, and also further verification of rotating solutions is achieved with a variational finite element method(FEM). The rotational(spin) parameter emerges as an inverse function of the Grashof number. The influence of this parameter, primary Darcy number, secondary Darcy number and Prandtl number on tangential velocity and swirl velocity, temperature and heat transfer rate are studied in detail. It is found that the dimensionless tangential velocity increases whilst the dimensionless swirl velocity and temperature decrease with the swirl Darcy number, tangential Darcy number and the rotational parameters. The model finds applications in chemical engineering filtration processing, liquid coating and spinning cone distillation columns.
Self-similar steady natural convection thermal boundary layer flow from a rotating vertical cone to anisotropic Darcian porous medium is investigated theoretically and numerically. The transformed non-dimensional two-point boundary value problem is reduced to a system of coupled, highly nonlinear ordinary differential equations, which are solved subject to robust surface and free stream boundary conditions with the MAPLE 17 numerical quadrature software. Validation with earlier non-rotating studies is included, and also further verification of rotating solutions is achieved with a variational finite element method(FEM). The rotational(spin) parameter emerges as an inverse function of the Grashof number. The influence of this parameter, primary Darcy number, secondary Darcy number and Prandtl number on tangential velocity and swirl velocity, temperature and heat transfer rate are studied in detail. It is found that the dimensionless tangential velocity increases whilst the dimensionless swirl velocity and temperature decrease with the swirl Darcy number, tangential Darcy number and the rotational parameters. The model finds applications in chemical engineering filtration processing, liquid coating and spinning cone distillation columns.
引文
[1]KREITH F.Convection heat transfer in rotating systems[J].Advances in Heat Transfer,1968,5:129-251.
[2]MAKARYTCHEV S.V.,LANGRISH T.A.G.and PRINCE R.G.H.Thickness and velocity of wavy liquid films on rotating conical surfaces[J].Chemical Engineering Science,2001,56(1):77-87.
[3]ADACHI T.Oxygen transfer and power consumption in an aeration system using mist and circulation flow generated by a rotating cone[J].Chemical Engineering Science,2015,126:625-632.
[4]CHAMKHA A.J.,RASHAD A.M.Unsteady heat and mass transfer by MHD mixed convection flow from a rotating vertical cone with chemical reaction and Soret and Dufour effects[J].The Canadian Journal of Chemical Engineering,2014,92(4):758-767.
[5]OSALUSI E.,SIDE J.and HARRIS R.et al.The effect of combined viscous dissipation and Joule heating on unsteady mixed convection MHD flow on a rotating cone in a rotating fluid with variable properties in the presence of Hall and ion-slip currents[J].International Communications in Heat and Mass Transfer,2008,35(4):413-429.
[6]NARAYANA M.,AWAD F.G.and SIBANDA P.Free magnetohydrodynamic flow and convection from a vertical spinning cone with cross-diffusion effects[J].Applied Mathematical Modelling,2013,37(5):2662-2678.
[7]ANILKUMAR D.,ROY S.Unsteady mixed convection flow on a rotating cone in a rotating fluid[J].Applied Mathematics and Computation,2004,155(2):545-561.
[8]RAJU S.H.,MALLIKARJUNA B.and VARMA S.V.K.Thermophoretic effect on double diffusive convective flow of a chemically reacting fluid over a rotating cone in porous Medium[J].International Journal of Scientific and Engineering Research,2015,6(1):198-204.
[9]ECE M.C.Free convection flow about a vertical spinning cone under a magnetic field[J].Applied Mathematics and Computation,2006,179(1):231-242.
[10]VAFAI K.Handbook of porous media[M].New York,USA:Marcel Dekker,2005.
[11]BéG O.A.,ZUECO J.and TAKHAR H.S.et al.Transient non-linear optically-thick radiative-convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface:Network simulation solutions[J].Communications in Nonlinear Science and Numerical Simulation,2009,14(11):3856-3866.
[12]MCKIBBIN R.Convection and heat transfer in layered and anisotropic porous media(QUINTARD M.,TODOROVIC M.Editors.Heat and mass transfer in porous media)[M].Amsterdam,The Netherlands:Elsevier,1992,327-336.
[13]WHITE R.E.,SUBRAMANIAN V.R.Computational methods in chemical engineering with maple[M].Berlin,Heidelberg,Germany:Spring-Verlag,2010.
[14]BHARGAVA R.,SHARMA S.and BéG O.A.et al.Finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow[J].Communications in Nonlinear Science and Numerical Simulation,2010,15(5):1210-1223.
[15]BéG O.A.,LIK S.and ZUECO J.et al.Numerical study of magnetohydrodynamic viscous plasma flow in rotating porous media with Hall currents and inclined magnetic field influence[J].Communications in Nonlinear Science and Numerical Simulation,2010,15(2):345-359.
[16]BéG O.A.,UDDIN M.J.and KHAN W.A.Bioconvective non-Newtonian nanofluid transport in porous media containing micro-organisms in a moving free stream[J].Journal of Mechanics in Medicine and Biology,2015,15(5):1550071.
[17]MAKINDE O.D.,BéG O.A.and TAKHAR H.S.Magnetohydrodynamic viscous flow in a rotating porous medium cylindrical annulus with an applied radial magnetic field[J].International Journal of Applied Mathematics and Mechanics,2009,5(6):68-81.
[18]SEMMAH A.,BéG O.A.and MAHMOUD S.R.et al.Thermal buckling properties of zigzag single-walled carbon nanotubes using a refined nonlocal model[J].Advances in materials Research,2014,3(2):77-89.
[19]UDDIN M.J.,YUSOFF N.H.M.and BéG O.A.et al.Lie group analysis and numerical solutions for nonNewtonian nanofluid flow in a porous medium with internal heat generation[J].Physica Scripta,2013,87(2):25401-25414.
[20]CHUNG T.J.The finite element method in fluid flow[M].New York,USA:Wiley,1978.
[21]BéG O.A.,RAWAT S.and ZUECO J.et al.Finite element and network electrical simulation of rotating magnetofluid flow in nonlinear porous media with inclined magnetic field and Hall currents[J].Theoretical and Applied Mechanics,2014,41(1):1-35.
[22]BéG O.A.Numerical methods for multi-physical magnetohydrodynamics,Chapter 1(New developments in hydrodynamics research)[M].New York,USA:Nova Science,2012,1-112.
[23]PRASAD V.R.,GAFFAR S.A.and BéG O.A.Heat and mass transfer of a nanofluid from a horizontal cylinder to a micropolar fluid[J].Journal of Thermophysics and Heat Transfer,2014,29(1):1-13.
[2]MAKARYTCHEV S.V.,LANGRISH T.A.G.and PRINCE R.G.H.Thickness and velocity of wavy liquid films on rotating conical surfaces[J].Chemical Engineering Science,2001,56(1):77-87.
[3]ADACHI T.Oxygen transfer and power consumption in an aeration system using mist and circulation flow generated by a rotating cone[J].Chemical Engineering Science,2015,126:625-632.
[4]CHAMKHA A.J.,RASHAD A.M.Unsteady heat and mass transfer by MHD mixed convection flow from a rotating vertical cone with chemical reaction and Soret and Dufour effects[J].The Canadian Journal of Chemical Engineering,2014,92(4):758-767.
[5]OSALUSI E.,SIDE J.and HARRIS R.et al.The effect of combined viscous dissipation and Joule heating on unsteady mixed convection MHD flow on a rotating cone in a rotating fluid with variable properties in the presence of Hall and ion-slip currents[J].International Communications in Heat and Mass Transfer,2008,35(4):413-429.
[6]NARAYANA M.,AWAD F.G.and SIBANDA P.Free magnetohydrodynamic flow and convection from a vertical spinning cone with cross-diffusion effects[J].Applied Mathematical Modelling,2013,37(5):2662-2678.
[7]ANILKUMAR D.,ROY S.Unsteady mixed convection flow on a rotating cone in a rotating fluid[J].Applied Mathematics and Computation,2004,155(2):545-561.
[8]RAJU S.H.,MALLIKARJUNA B.and VARMA S.V.K.Thermophoretic effect on double diffusive convective flow of a chemically reacting fluid over a rotating cone in porous Medium[J].International Journal of Scientific and Engineering Research,2015,6(1):198-204.
[9]ECE M.C.Free convection flow about a vertical spinning cone under a magnetic field[J].Applied Mathematics and Computation,2006,179(1):231-242.
[10]VAFAI K.Handbook of porous media[M].New York,USA:Marcel Dekker,2005.
[11]BéG O.A.,ZUECO J.and TAKHAR H.S.et al.Transient non-linear optically-thick radiative-convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface:Network simulation solutions[J].Communications in Nonlinear Science and Numerical Simulation,2009,14(11):3856-3866.
[12]MCKIBBIN R.Convection and heat transfer in layered and anisotropic porous media(QUINTARD M.,TODOROVIC M.Editors.Heat and mass transfer in porous media)[M].Amsterdam,The Netherlands:Elsevier,1992,327-336.
[13]WHITE R.E.,SUBRAMANIAN V.R.Computational methods in chemical engineering with maple[M].Berlin,Heidelberg,Germany:Spring-Verlag,2010.
[14]BHARGAVA R.,SHARMA S.and BéG O.A.et al.Finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow[J].Communications in Nonlinear Science and Numerical Simulation,2010,15(5):1210-1223.
[15]BéG O.A.,LIK S.and ZUECO J.et al.Numerical study of magnetohydrodynamic viscous plasma flow in rotating porous media with Hall currents and inclined magnetic field influence[J].Communications in Nonlinear Science and Numerical Simulation,2010,15(2):345-359.
[16]BéG O.A.,UDDIN M.J.and KHAN W.A.Bioconvective non-Newtonian nanofluid transport in porous media containing micro-organisms in a moving free stream[J].Journal of Mechanics in Medicine and Biology,2015,15(5):1550071.
[17]MAKINDE O.D.,BéG O.A.and TAKHAR H.S.Magnetohydrodynamic viscous flow in a rotating porous medium cylindrical annulus with an applied radial magnetic field[J].International Journal of Applied Mathematics and Mechanics,2009,5(6):68-81.
[18]SEMMAH A.,BéG O.A.and MAHMOUD S.R.et al.Thermal buckling properties of zigzag single-walled carbon nanotubes using a refined nonlocal model[J].Advances in materials Research,2014,3(2):77-89.
[19]UDDIN M.J.,YUSOFF N.H.M.and BéG O.A.et al.Lie group analysis and numerical solutions for nonNewtonian nanofluid flow in a porous medium with internal heat generation[J].Physica Scripta,2013,87(2):25401-25414.
[20]CHUNG T.J.The finite element method in fluid flow[M].New York,USA:Wiley,1978.
[21]BéG O.A.,RAWAT S.and ZUECO J.et al.Finite element and network electrical simulation of rotating magnetofluid flow in nonlinear porous media with inclined magnetic field and Hall currents[J].Theoretical and Applied Mechanics,2014,41(1):1-35.
[22]BéG O.A.Numerical methods for multi-physical magnetohydrodynamics,Chapter 1(New developments in hydrodynamics research)[M].New York,USA:Nova Science,2012,1-112.
[23]PRASAD V.R.,GAFFAR S.A.and BéG O.A.Heat and mass transfer of a nanofluid from a horizontal cylinder to a micropolar fluid[J].Journal of Thermophysics and Heat Transfer,2014,29(1):1-13.