纳米磁流体在薄片上流动的非线性边界条件:传热和传质的强化（英文）
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摘要
对黏性不可压缩的纳米流体在水平薄板上的稳定边界层及传热传质进行了数值研究。研究过程中,建立了非线性磁场、非线性速度和对流的非线性边界条件。然而,在水动力和热边界条件及磁场中的非线性问题尚未见研究报道。本研究中,同时考虑了布朗运动和热泳扩散,获得了一种相似的解决方案并求解了该常微分方程(非线性)。通过验证找到了影响流体动力学、传热和传质的参数:减少舍伍德和努塞尔数,可降低表面摩擦系数,以及温度和纳米颗粒体积分数的分布。所有这些参数都受到Lewis数、Biot数和Prandtl数,以及拉伸、热泳扩散、布朗运动和磁场参数的影响。详细分析了所观察到的现象,并提出了有用的建议。
In this paper, we have numerically examined the steady boundary layer of a viscous incompressible nanofluid and its heat and mass transfers above a horizontal flat sheet. The boundary conditions considered were a nonlinear magnetic field, a nonlinear velocity and convection. Such nonlinearity in hydrodynamic and heat transfer boundary conditions and also in the magnetic field has not been addressed with the great details in the literature. In this investigation, both the Brownian motion and thermophoretic diffusion have been considered. A similarity solution is achieved and the resulting ordinary differential equations(nonlinear) are worked numerically out. Upon validation, the following hydrodynamic and heat and mass transfers parameters were found: the reduced Sherwood and Nusselt numbers, the reduced skin friction coefficient, and the temperature and nanoparticle volume fraction profiles. All these parameters are found affected by the Lewis, Biot and Prandtl numbers, the stretching, thermophoretic diffusion,Brownian motion and magnetic parameters. The detailed trends observed in this paper are carefully analyzed to provide useful design suggestions.
In this paper, we have numerically examined the steady boundary layer of a viscous incompressible nanofluid and its heat and mass transfers above a horizontal flat sheet. The boundary conditions considered were a nonlinear magnetic field, a nonlinear velocity and convection. Such nonlinearity in hydrodynamic and heat transfer boundary conditions and also in the magnetic field has not been addressed with the great details in the literature. In this investigation, both the Brownian motion and thermophoretic diffusion have been considered. A similarity solution is achieved and the resulting ordinary differential equations(nonlinear) are worked numerically out. Upon validation, the following hydrodynamic and heat and mass transfers parameters were found: the reduced Sherwood and Nusselt numbers, the reduced skin friction coefficient, and the temperature and nanoparticle volume fraction profiles. All these parameters are found affected by the Lewis, Biot and Prandtl numbers, the stretching, thermophoretic diffusion,Brownian motion and magnetic parameters. The detailed trends observed in this paper are carefully analyzed to provide useful design suggestions.
引文
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[2]CORTELL R.A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet[J].International Journal of Nonlinear Mechanics,2006,41:78-85.DOI:https://doi.org/10.1016/j.ijnonlinmec.2005.04.008.
[3]MALEKIAN S,FATHI E,MALEKIAN N,MOGHADASIH,MOGHIMI M.Analytical and numerical investigations of unsteady graphene oxide nanofluid flow between two parallel plates[J].International Journal of Thermophysics,2018,39:100-118.DOI:https://doi.org/10.1007/s10765-018-2422-z.
[4]AHMADI A A,KHODABANDEH E,MOGHADASI H,MALEKIAN N,AKBARI O A,BAHIRAEI M.Numerical study of flow and heat transfer of water-Al2O3 nanofluid inside a channel with an inner cylinder using EulerianLagrangian approach[J].Journal of Thermal Analysis and Calorimetry,2018,132:651-665.DOI:https://doi.org/10.1007/s10973-017-6798-y.
[5]CORTELL R.Viscous flow and heat transfer over a nonlinearly stretching sheet[J].Applied Mathematics and Computation,2007,184:864-873.DOI:https://doi.org/10.1016/j.amc.2006.06.077.
[6]SAJID M,HAYAT T.Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet[J].International Communications in Heat and Mass Transfer,2008,35:347-356.DOI:https://doi.org/10.1016/j.icheatmasstransfer.2007.08.006.
[7]TSAI R,HUANG K H,HUANG J S.Flow and heat transfer over an unsteady stretching surface with non-uniform heat source[J].International Communications in Heat and Mass Transfer,2008,35:1340-1343.DOI:https://doi.org/10.1016/j.icheatmasstransfer.2008.07.001.
[8]NANDEPPANAVAR M,VAJRAVELU K,SUBHAS ABELM,NG Chiu-On.Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature[J].International Journal of Heat and Mass Transfer,2011,54:4960-4965.DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.009.
[9]FANG T,ZHANG J,ZHONG Y.Boundary layer flow over a stretching sheet with variable thickness[J].Applied Mathematics and Computation,2012,218:7241-7252.DOI:https://doi.org/10.1016/j.amc.2011.12.094.
[10]SUBHASHINI S V,SUMATHI R,POP I.Dual solutions in a thermal diffusive flow over a stretching sheet with variable thickness[J].International Communications in Heat and Mass Transfer,2013,48:61-66.DOI:https://doi.org/10.1016/j.icheatmasstransfer.2013.09.007.
[11]YANG Liu,DU Kai,ZHANG Xiao-song.Influence factors on thermal conductivity of ammonia-water nanofluids[J].Journal of Central South University,2012,19:1622-1628.DOI:https://doi.org/10.1007/s11771-012-1185-0.
[12]WUSIMAN Ku-er-ban-jiang,CHUNG H S,NINE M D J,HANDRY A,EOM Y S,KIM J H,JEONG H M.Heat transfer characteristics of nanofluid through circular tube[J].Journal of Central South University,2013,20:142-148.DOI:https://doi.org/10.1007/s11771-013-1469-z.
[13]ZAIB A,BHATTACHARYYA K,SHAFIE S.Unsteady boundary layer flow and heat transfer over an exponentially shrinking sheet with suction in a copper-water nanofluid[J].Journal of Central South University,2015,22:4856-4863.DOI:https://doi.org/10.1007/s11771-015-3037-1
[14]HAQUE A K M,KWON S,KIM J,NOH J,HUH S,CHUNG H,JEONG H.An experimental study on thermal characteristics of nanofluid with graphene and multi-wall carbon nanotubes[J].Journal of Central South University,2015,22:3202-3210.DOI:https://doi.org/10.1007/s11771-015-2857-3.
[15]AZARI A.Thermal conductivity modeling of water containing metal oxide nanoparticles[J].Journal of Central South University,2015,22:1141-1145.DOI:https://doi.org/10.1007/s11771-015-2626-3.
[16]SELIMEFENDIGIL F,?ZTOP H F.Mixed convection in a two-sided elastic walled and Si O2 nanofluid filled cavity with internal heat generation:Effects of inner rotating cylinder and nanoparticle’s shape[J].Journal of Molecular Liquids,2015,212:509-516.DOI:http://dx.doi.org/10.1016/j.molliq.2015.09.037.
[17]SIAVASHI M,JAMALI M.Optimal selection of annulus radius ratio to enhance heat transfer with minimum entropy generation in developing laminar forced convection of water-Al2O3 nanofluid flow[J].Journal of Central South University,2017,24:1850-1865.DOI:https://doi.org/10.1007/s11771-017-3660-0.
[18]SELIMEFENDIGILA F,?ZTOP H F,CHAMKHA A J.Analysis of mixed convection of nanofluid in a 3D lid-driven trapezoidal cavity with flexible side surfaces and inner cylinder[J].International Communications in Heat and Mass Transfer,2017,87:40-51.DOI:http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.06.015.
[19]SELIMEFENDIGIL F,?ZTOP H F.Conjugate natural convection in a nanofluid filled partitioned horizontal annulus formed by two isothermal cylinder surfaces under magnetic field[J].International Journal of Heat and Mass Transfer,2017,108:156-171.DOI:http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.080.
[20]MAGHSOUDI P,SIAVASHI M.Application of nanofluid and optimization of pore size arrangement of heterogeneous porous media to enhance mixed convection inside a two-sided lid-driven cavity[J].Journal of Thermal Analysis and Calorimetry,2018.DOI:https://doi.org/10.1007/s10973-018-7335-3.
[21]SIAVASHI M,RASAM H,IZADI A.Similarity solution of air and nanofluid impingement cooling of a cylindrical porous heat sink[J].Journal of Thermal Analysis and Calorimetry,2019.DOI:https://doi.org/10.1007/s10973-018-7540-0.
[22]BUONGIORNO J.Convective transport in nanofluids[J].ASME Journal of Heat Transfer,2006,128:240-250.DOI:10.1115/1.2150834.
[23]KHAN W A,POP I.Boundary-layer flow of a nanofluid past a stretching sheet[J].International Journal of Heat and Mass Transfer,2010,53:2477-2483.DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2010.01.032.
[24]MAKINDE O D,AZIZ A.Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition[J].International Journal of Thermal Sciences,2011,50:1326-1332.DOI:https://doi.org/10.1016/j.ijthermalsci.2011.02.019.
[25]RANA P,BHARGAVA R.Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet:A numerical study[J].Communications in Nonlinear Science and Numerical Simulation,2012,17:212-226.DOI:https://doi.org/10.1016/j.cnsns.2011.05.009.
[26]MUSTAFAA M,HAYAT T,OBAIDAT S.Boundary layer flow of a nanofluid over an exponentially stretching sheet with convective boundary conditions[J].International Journal of Numerical Methods for Heat and Fluid Flow,2013,23:945-959.DOI:https://doi.org/10.1108/HFF-09-2011-0179.
[27]RAHMAN M,ROSCA A V,POP I.Boundary layer flow of a nanofluid past a permeable exponentially shrinking/stretching surface with second order slip using Buongiorno’s model[J].International Journal of Heat and Mass Transfer,2014,77:1133-1143.DOI:https://doi.org/10.1016/j.ijheat masstransfer.2014.06.013.
[28]MUKHOPADHYAY S,LAYEK G C,SAMAD Sk.Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity[J].International Journal of Heat and Mass Transfer,2005,48:4460-4466.DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2005.05.027.
[29]ANJALI DEVI S P,THIYAGARAJAN M.Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature[J].Heat and Mass Transfer,2006,42:671-677.DOI:https://doi.org/10.1007/s00231-005-0640-y.
[30]ABEL M S,NANDEPPANAVAR M.Heat transfer in MHDviscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink[J].Communications in Nonlinear Science and Numerical Simulation,2009,14:2120-2131.DOI:https://doi.org/10.1016/j.cnsns.2008.06.004.
[31]PRASAD K V,PAL D,UMESH V,RAO N S P.The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet[J].Communications in Nonlinear Science and Numerical Simulation,2010,15:331-344.DOI:https://doi.org/10.1016/j.cnsns.2009.04.003.
[32]IBRAHIM W,SHANKER B.Unsteady MHD boundarylayer flow and heat transfer due to stretching sheet in the presence of heat source or sink[J].Computers and Fluids,2012,70:21-28.DOI:https://doi.org/10.1016/j.compfluid.2012.08.019.
[33]CORTELL R.MHD(magneto-hydrodynamic)flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption[J].Energy,2014,74:896-905.DOI:https://doi.org/10.1016/j.energy.2014.07.069.
[34]GILL S.A process for the step-by-step integration of differential equations in an automatic digital computing machine[J].Proceedings of the Cambridge Philosophical Society,1951,47(1):96-108.DOI:https://doi.org/10.1017/S0305004100026414.
[2]CORTELL R.A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet[J].International Journal of Nonlinear Mechanics,2006,41:78-85.DOI:https://doi.org/10.1016/j.ijnonlinmec.2005.04.008.
[3]MALEKIAN S,FATHI E,MALEKIAN N,MOGHADASIH,MOGHIMI M.Analytical and numerical investigations of unsteady graphene oxide nanofluid flow between two parallel plates[J].International Journal of Thermophysics,2018,39:100-118.DOI:https://doi.org/10.1007/s10765-018-2422-z.
[4]AHMADI A A,KHODABANDEH E,MOGHADASI H,MALEKIAN N,AKBARI O A,BAHIRAEI M.Numerical study of flow and heat transfer of water-Al2O3 nanofluid inside a channel with an inner cylinder using EulerianLagrangian approach[J].Journal of Thermal Analysis and Calorimetry,2018,132:651-665.DOI:https://doi.org/10.1007/s10973-017-6798-y.
[5]CORTELL R.Viscous flow and heat transfer over a nonlinearly stretching sheet[J].Applied Mathematics and Computation,2007,184:864-873.DOI:https://doi.org/10.1016/j.amc.2006.06.077.
[6]SAJID M,HAYAT T.Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet[J].International Communications in Heat and Mass Transfer,2008,35:347-356.DOI:https://doi.org/10.1016/j.icheatmasstransfer.2007.08.006.
[7]TSAI R,HUANG K H,HUANG J S.Flow and heat transfer over an unsteady stretching surface with non-uniform heat source[J].International Communications in Heat and Mass Transfer,2008,35:1340-1343.DOI:https://doi.org/10.1016/j.icheatmasstransfer.2008.07.001.
[8]NANDEPPANAVAR M,VAJRAVELU K,SUBHAS ABELM,NG Chiu-On.Heat transfer over a nonlinearly stretching sheet with non-uniform heat source and variable wall temperature[J].International Journal of Heat and Mass Transfer,2011,54:4960-4965.DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.009.
[9]FANG T,ZHANG J,ZHONG Y.Boundary layer flow over a stretching sheet with variable thickness[J].Applied Mathematics and Computation,2012,218:7241-7252.DOI:https://doi.org/10.1016/j.amc.2011.12.094.
[10]SUBHASHINI S V,SUMATHI R,POP I.Dual solutions in a thermal diffusive flow over a stretching sheet with variable thickness[J].International Communications in Heat and Mass Transfer,2013,48:61-66.DOI:https://doi.org/10.1016/j.icheatmasstransfer.2013.09.007.
[11]YANG Liu,DU Kai,ZHANG Xiao-song.Influence factors on thermal conductivity of ammonia-water nanofluids[J].Journal of Central South University,2012,19:1622-1628.DOI:https://doi.org/10.1007/s11771-012-1185-0.
[12]WUSIMAN Ku-er-ban-jiang,CHUNG H S,NINE M D J,HANDRY A,EOM Y S,KIM J H,JEONG H M.Heat transfer characteristics of nanofluid through circular tube[J].Journal of Central South University,2013,20:142-148.DOI:https://doi.org/10.1007/s11771-013-1469-z.
[13]ZAIB A,BHATTACHARYYA K,SHAFIE S.Unsteady boundary layer flow and heat transfer over an exponentially shrinking sheet with suction in a copper-water nanofluid[J].Journal of Central South University,2015,22:4856-4863.DOI:https://doi.org/10.1007/s11771-015-3037-1
[14]HAQUE A K M,KWON S,KIM J,NOH J,HUH S,CHUNG H,JEONG H.An experimental study on thermal characteristics of nanofluid with graphene and multi-wall carbon nanotubes[J].Journal of Central South University,2015,22:3202-3210.DOI:https://doi.org/10.1007/s11771-015-2857-3.
[15]AZARI A.Thermal conductivity modeling of water containing metal oxide nanoparticles[J].Journal of Central South University,2015,22:1141-1145.DOI:https://doi.org/10.1007/s11771-015-2626-3.
[16]SELIMEFENDIGIL F,?ZTOP H F.Mixed convection in a two-sided elastic walled and Si O2 nanofluid filled cavity with internal heat generation:Effects of inner rotating cylinder and nanoparticle’s shape[J].Journal of Molecular Liquids,2015,212:509-516.DOI:http://dx.doi.org/10.1016/j.molliq.2015.09.037.
[17]SIAVASHI M,JAMALI M.Optimal selection of annulus radius ratio to enhance heat transfer with minimum entropy generation in developing laminar forced convection of water-Al2O3 nanofluid flow[J].Journal of Central South University,2017,24:1850-1865.DOI:https://doi.org/10.1007/s11771-017-3660-0.
[18]SELIMEFENDIGILA F,?ZTOP H F,CHAMKHA A J.Analysis of mixed convection of nanofluid in a 3D lid-driven trapezoidal cavity with flexible side surfaces and inner cylinder[J].International Communications in Heat and Mass Transfer,2017,87:40-51.DOI:http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.06.015.
[19]SELIMEFENDIGIL F,?ZTOP H F.Conjugate natural convection in a nanofluid filled partitioned horizontal annulus formed by two isothermal cylinder surfaces under magnetic field[J].International Journal of Heat and Mass Transfer,2017,108:156-171.DOI:http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.080.
[20]MAGHSOUDI P,SIAVASHI M.Application of nanofluid and optimization of pore size arrangement of heterogeneous porous media to enhance mixed convection inside a two-sided lid-driven cavity[J].Journal of Thermal Analysis and Calorimetry,2018.DOI:https://doi.org/10.1007/s10973-018-7335-3.
[21]SIAVASHI M,RASAM H,IZADI A.Similarity solution of air and nanofluid impingement cooling of a cylindrical porous heat sink[J].Journal of Thermal Analysis and Calorimetry,2019.DOI:https://doi.org/10.1007/s10973-018-7540-0.
[22]BUONGIORNO J.Convective transport in nanofluids[J].ASME Journal of Heat Transfer,2006,128:240-250.DOI:10.1115/1.2150834.
[23]KHAN W A,POP I.Boundary-layer flow of a nanofluid past a stretching sheet[J].International Journal of Heat and Mass Transfer,2010,53:2477-2483.DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2010.01.032.
[24]MAKINDE O D,AZIZ A.Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition[J].International Journal of Thermal Sciences,2011,50:1326-1332.DOI:https://doi.org/10.1016/j.ijthermalsci.2011.02.019.
[25]RANA P,BHARGAVA R.Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet:A numerical study[J].Communications in Nonlinear Science and Numerical Simulation,2012,17:212-226.DOI:https://doi.org/10.1016/j.cnsns.2011.05.009.
[26]MUSTAFAA M,HAYAT T,OBAIDAT S.Boundary layer flow of a nanofluid over an exponentially stretching sheet with convective boundary conditions[J].International Journal of Numerical Methods for Heat and Fluid Flow,2013,23:945-959.DOI:https://doi.org/10.1108/HFF-09-2011-0179.
[27]RAHMAN M,ROSCA A V,POP I.Boundary layer flow of a nanofluid past a permeable exponentially shrinking/stretching surface with second order slip using Buongiorno’s model[J].International Journal of Heat and Mass Transfer,2014,77:1133-1143.DOI:https://doi.org/10.1016/j.ijheat masstransfer.2014.06.013.
[28]MUKHOPADHYAY S,LAYEK G C,SAMAD Sk.Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity[J].International Journal of Heat and Mass Transfer,2005,48:4460-4466.DOI:https://doi.org/10.1016/j.ijheatmasstransfer.2005.05.027.
[29]ANJALI DEVI S P,THIYAGARAJAN M.Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature[J].Heat and Mass Transfer,2006,42:671-677.DOI:https://doi.org/10.1007/s00231-005-0640-y.
[30]ABEL M S,NANDEPPANAVAR M.Heat transfer in MHDviscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink[J].Communications in Nonlinear Science and Numerical Simulation,2009,14:2120-2131.DOI:https://doi.org/10.1016/j.cnsns.2008.06.004.
[31]PRASAD K V,PAL D,UMESH V,RAO N S P.The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet[J].Communications in Nonlinear Science and Numerical Simulation,2010,15:331-344.DOI:https://doi.org/10.1016/j.cnsns.2009.04.003.
[32]IBRAHIM W,SHANKER B.Unsteady MHD boundarylayer flow and heat transfer due to stretching sheet in the presence of heat source or sink[J].Computers and Fluids,2012,70:21-28.DOI:https://doi.org/10.1016/j.compfluid.2012.08.019.
[33]CORTELL R.MHD(magneto-hydrodynamic)flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption[J].Energy,2014,74:896-905.DOI:https://doi.org/10.1016/j.energy.2014.07.069.
[34]GILL S.A process for the step-by-step integration of differential equations in an automatic digital computing machine[J].Proceedings of the Cambridge Philosophical Society,1951,47(1):96-108.DOI:https://doi.org/10.1017/S0305004100026414.