Cu-C_6H_9NaO_7和Ag-C_6H_9NaO_7纳米流体在非线性收缩表面流动的多重解（英文）
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摘要
本文研究了Casson纳米流体在一类非线性收缩表面的流动模型。将Tiwari和Das模型应用于含铜、银的海藻酸钠纳米流体中,通过适当的相似变换,将考虑黏性耗散影响的控制非线性方程组转化为常微分方程的边界值问题(BVPS)。将所得方程用打靶法转化为初值问题,再用四阶Runge-Kutta法求解。为了确定所得到的对偶解的稳定性,对第一(第二)解进行了稳定性分析,发现第一(第二)解是稳定的(不稳定的)和物理可实现的(不可实现的)。在第二解中,随着Casson参数(β)的增加,热边界层的厚度增加,温度也会升高。
Model of Casson nanofluid flow over a nonlinear shrinking surface is considered. Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver. The governing nonlinear equations incorporating the effects of the viscous dissipation are transformed into boundary value problems(BVPs) of ordinary differential equations(ODEs) by using appropriate similarity transformations. The resulting equations are converted into initial value problems(IVPs) using the shooting method which are then solved by Runge-Kutta method of fourth order. In order to determine the stability of the dual solutions obtained, stability analysis is performed and discovered that the first(second) solution is stable(unstable) and physically realizable(unrealizable). Both the thickness of the thermal boundary layer as well as temperature increase when the Casson parameter(β) is increased in the second solution.
Model of Casson nanofluid flow over a nonlinear shrinking surface is considered. Model of Tiwari and Das is applied to nanofluid comprising of sodium alginate with copper and silver. The governing nonlinear equations incorporating the effects of the viscous dissipation are transformed into boundary value problems(BVPs) of ordinary differential equations(ODEs) by using appropriate similarity transformations. The resulting equations are converted into initial value problems(IVPs) using the shooting method which are then solved by Runge-Kutta method of fourth order. In order to determine the stability of the dual solutions obtained, stability analysis is performed and discovered that the first(second) solution is stable(unstable) and physically realizable(unrealizable). Both the thickness of the thermal boundary layer as well as temperature increase when the Casson parameter(β) is increased in the second solution.
引文
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[5]LAAD M,JATTI V K S.Titanium oxide nanoparticles as additives in engine oil[J].Journal of King Saud University-Engineering Sciences,2016,30:116-122.
[6]ROSTAMI M N,DINARVAND S,POP I.Dual solutions for mixed convective stagnation-point flow of an aqueous silica-alumina hybrid nanofluid[J].Chinese Journal of Physics,2018,56:2465-2478.
[7]SHEIKHOLESLAMI M,HAYAT T,ALSAEDI A.On simulation of nanofluid radiation and natural convection in an enclosure with elliptical cylinders[J].International Journal of Heat and Mass Transfer,2017,115:981-991.
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[17]CAPONE F,RIONERO S.Brinkmann viscosity action in porous MHD convection[J].International Journal of NonLinear Mechanics,2016,85:109-117.
[18]BACHOK N,ISHAK A,POP I.Boundary layer stagnationpoint flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid[J].International Journal of Heat and Mass Transfer,2012,55:8122-8128.
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[20]AHMAD R,MUSTAFA M,HAYAT T,ALSAEDI A.Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet[J].Journal of Magnetism and Magnetic Materials,2016,407:69-74.
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[24]SHAHID A,BHATTI M M,BéG O A,KADIR A.Numerical study of radiative Maxwell viscoelastic magnetized flow from a stretching permeable sheet with the Cattaneo-Christov heat flux model[J].Neural Computing and Applications,2017,30:3467-3478.
[25]CASSON N.A flow equation for pigment-oil suspensions of the printing ink type[J].Rheology of Disperse Systems,1959,4:227-233.
[26]SHEHZAD S A,HAYAT T,ALHUTHALI M,ASGHAR S.MHD three-dimensional flow of Jeffrey fluid with Newtonian heating[J].Journal of Central South University,2014,21(4):1428-1433.
[27]RAMZAN M,FAROOQ M,HAYAT T,ALSAEDI A,CAO J.MHD stagnation point flow by a permeable stretching cylinder with Soret-Dufour effects[J].Journal of Central South University,2015,22(2):707-716.
[28]KUMAR M S,SANDEEP N,KUMAR B R,SALEEM S.Acomparative study of chemically reacting 2D flow of Casson and Maxwell fluids[J].Alexandria Engineering Journal,2017,57:2027-2034.
[29]ABBAS T,BHATTI M M,AYUB M.Aiding and opposing of mixed convection Casson nanofluid flow with chemical reactions through a porous Riga plate[J].Proceedings of the Institution of Mechanical Engineers,Part E:Journal of Process Mechanical Engineering,2018,232(5):519-527.
[30]MAHMOOD A,JAMSHED W,AZIZ A.Entropy and heat transfer analysis using Cattaneo-Christov heat flux model for a boundary layer flow of Casson nanofluid[J].Results in Physics,2018,10:640-649.
[31]ALI F,SHEIKH N A,KHAN I,SAQIB M.Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube:A fractional model[J].Journal of Magnetism and Magnetic Materials,2017,423:327-336.
[32]KHALID A,KHAN I,KHAN A,SHAFIE S,TLILI I.Case study of MHD blood flow in a porous medium with CNTSand thermal analysis[J].Case Studies in Thermal Engineering,2018,12:374-380.
[33]RAZA J,ROHNI A M,OMAR Z.Triple solutions of Casson fluid flow between slowly expanding and contracting walls[J].AIP Conference Proceedings,2017,1905(1):030029.
[34]NAKAMURA M,SAWADA T.Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis[J].Journal of Biomechanical Engineering,1988,110(2):137-143.
[35]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.
[36]RO?CA A V,POP I.Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip[J].International Journal of Heat and Mass Transfer,2013,60:355-364.
[37]WEIDMAN P D,KUBITSCHEK D G,DAVIS A M J.The effect of transpiration on self-similar boundary layer flow over moving surfaces[J].International Journal of Engineering Science,2006,44(11,12):730-737.
[38]ABBAS Z,MASOOD T,OLANREWAJU P O.Dual solutions of MHD stagnation point flow and heat transfer over a stretching/shrinking sheet with generalized slip condition[J].Journal of Central South University,2015,22(6):2376-2384.
[39]MEGAHED A M.Slip flow and variable properties of viscoelastic fluid past a stretching surface embedded in a porous medium with heat generation[J].Journal of Central South University,2016,23(4):991-999.
[40]POSTELNICU A,POP I.Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge[J].Applied Mathematics and Computation,2011,217(9):4359-4368.
[41]HARRIS S D,INGHAM D B,POP I.Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium:Brinkman model with slip[J].Transport in Porous Media,2009,77(2):267-285.
[42]ZHENG Lian-cun,LIANG Chen,ZHANG Xin-xin.Multiple values of skin friction for boundary layer separation flow in power law pseudoplastic fluids[J].Journal of Central South University of Technology,2007,14(1):218-220.
[43]ROHNI A M,AHMAD S,POP I.Note on Cortell’s nonlinearly stretching permeable sheet[J].International Journal of Heat and Mass Transfer,2012,55(21,22):5846-5852.
[44]CORTELL R.Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall[J].Meccanica,2012,47(3):769-781.
[45]HATAMI M,GANJI D D.Natural convection of sodium alginate(SA)non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods[J].Case Studies in Thermal Engineering,2014,2:14-22.
[2]CHOI S U S,ESTMAN J A.Enhancing thermal conductivity of fluids with nanoparticles[J].ASME-Publications,1995,231:99-106.
[3]MAHIAN O,KOLSI L,AMANI M,ESTELLéP,AHMADIG,KLEINSTREUER C,MARSHALL J S,SIAVASHI M,TAYLOR R A,NIAZMAND H,WONGWISES S.Recent advances in modeling and simulation of nanofluid flows-Part I:Fundamental and theory[J].Physics Reports,2018,790:1-48.
[4]MAHIAN O,KOLSI L,AMANI M,ESTELLéP,AHMADIG,KLEINSTREUER C,MARSHALL J S,TAYLOR R A,ABU-NADA E,RASHIDI S,NIAZMAND H.Recent advances in modeling and simulation of nanofluid flows-part II:Applications[J].Physics Reports,2018,791:1-59.
[5]LAAD M,JATTI V K S.Titanium oxide nanoparticles as additives in engine oil[J].Journal of King Saud University-Engineering Sciences,2016,30:116-122.
[6]ROSTAMI M N,DINARVAND S,POP I.Dual solutions for mixed convective stagnation-point flow of an aqueous silica-alumina hybrid nanofluid[J].Chinese Journal of Physics,2018,56:2465-2478.
[7]SHEIKHOLESLAMI M,HAYAT T,ALSAEDI A.On simulation of nanofluid radiation and natural convection in an enclosure with elliptical cylinders[J].International Journal of Heat and Mass Transfer,2017,115:981-991.
[8]BHATTI M M,RASHIDI M M.Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet[J].Journal of Molecular Liquids,2016,221:567-573.
[9]MISHRA S R,BHATTI M M.Simultaneous effects of chemical reaction and Ohmic heating with heat and mass transfer over a stretching surface:A numerical study[J].Chinese Journal of Chemical Engineering,2017,25(9):1137-1142.
[10]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,2018:1-7.
[11]NAKHCHI M E,ESFAHANI J A.Cu-water nanofluid flow and heat transfer in a heat exchanger tube equipped with cross-cut twisted tape[J].Powder Technology,2018,339:1399-1415.
[12]HAYAT T,RASHID M,ALSAEDI A,AHMAD B.Flow of nanofluid by nonlinear stretching velocity[J].Results in Physics,2018,8:1104-1109.
[13]BUONGIORNO J.Convective transport in nanofluids[J].Journal of Heat Transfer,2006,128(3):240-250.
[14]KHANAFER K,VAFAI K,LIGHTSTONE M.Buoyancydriven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids[J].International Journal of Heat and Mass Transfer,2003,46(19):3639-3653.
[15]NIELD D A,KUZNETSOV A V.The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid[J].International Journal of Heat and Mass Transfer,2009,52(25,26):5792-5795.
[16]TIWARI R K,DAS M K.Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids[J].International Journal of Heat and Mass Transfer,2007,50(9,10):2002-2018.
[17]CAPONE F,RIONERO S.Brinkmann viscosity action in porous MHD convection[J].International Journal of NonLinear Mechanics,2016,85:109-117.
[18]BACHOK N,ISHAK A,POP I.Boundary layer stagnationpoint flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid[J].International Journal of Heat and Mass Transfer,2012,55:8122-8128.
[19]RAHMAN M M,RO?CA 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.
[20]AHMAD R,MUSTAFA M,HAYAT T,ALSAEDI A.Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet[J].Journal of Magnetism and Magnetic Materials,2016,407:69-74.
[21]NIELD D A,KUZNETSOV A V.The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid[J].International Journal of Heat and Mass Transfer,2011,54(1-3):374-378.
[22]MATIN M H,POP I.Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall[J].International Communications in Heat and Mass Transfer,2013,46:134-141.
[23]AHMED T N,KHAN,I.Mixed convection flow of sodium alginate(SA-NaAlg)based molybdenum disulphide(MoS2)nanofluids:Maxwell Garnetts and Brinkman models[J].Results in Physics,2018,8:752-757.
[24]SHAHID A,BHATTI M M,BéG O A,KADIR A.Numerical study of radiative Maxwell viscoelastic magnetized flow from a stretching permeable sheet with the Cattaneo-Christov heat flux model[J].Neural Computing and Applications,2017,30:3467-3478.
[25]CASSON N.A flow equation for pigment-oil suspensions of the printing ink type[J].Rheology of Disperse Systems,1959,4:227-233.
[26]SHEHZAD S A,HAYAT T,ALHUTHALI M,ASGHAR S.MHD three-dimensional flow of Jeffrey fluid with Newtonian heating[J].Journal of Central South University,2014,21(4):1428-1433.
[27]RAMZAN M,FAROOQ M,HAYAT T,ALSAEDI A,CAO J.MHD stagnation point flow by a permeable stretching cylinder with Soret-Dufour effects[J].Journal of Central South University,2015,22(2):707-716.
[28]KUMAR M S,SANDEEP N,KUMAR B R,SALEEM S.Acomparative study of chemically reacting 2D flow of Casson and Maxwell fluids[J].Alexandria Engineering Journal,2017,57:2027-2034.
[29]ABBAS T,BHATTI M M,AYUB M.Aiding and opposing of mixed convection Casson nanofluid flow with chemical reactions through a porous Riga plate[J].Proceedings of the Institution of Mechanical Engineers,Part E:Journal of Process Mechanical Engineering,2018,232(5):519-527.
[30]MAHMOOD A,JAMSHED W,AZIZ A.Entropy and heat transfer analysis using Cattaneo-Christov heat flux model for a boundary layer flow of Casson nanofluid[J].Results in Physics,2018,10:640-649.
[31]ALI F,SHEIKH N A,KHAN I,SAQIB M.Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube:A fractional model[J].Journal of Magnetism and Magnetic Materials,2017,423:327-336.
[32]KHALID A,KHAN I,KHAN A,SHAFIE S,TLILI I.Case study of MHD blood flow in a porous medium with CNTSand thermal analysis[J].Case Studies in Thermal Engineering,2018,12:374-380.
[33]RAZA J,ROHNI A M,OMAR Z.Triple solutions of Casson fluid flow between slowly expanding and contracting walls[J].AIP Conference Proceedings,2017,1905(1):030029.
[34]NAKAMURA M,SAWADA T.Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis[J].Journal of Biomechanical Engineering,1988,110(2):137-143.
[35]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.
[36]RO?CA A V,POP I.Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip[J].International Journal of Heat and Mass Transfer,2013,60:355-364.
[37]WEIDMAN P D,KUBITSCHEK D G,DAVIS A M J.The effect of transpiration on self-similar boundary layer flow over moving surfaces[J].International Journal of Engineering Science,2006,44(11,12):730-737.
[38]ABBAS Z,MASOOD T,OLANREWAJU P O.Dual solutions of MHD stagnation point flow and heat transfer over a stretching/shrinking sheet with generalized slip condition[J].Journal of Central South University,2015,22(6):2376-2384.
[39]MEGAHED A M.Slip flow and variable properties of viscoelastic fluid past a stretching surface embedded in a porous medium with heat generation[J].Journal of Central South University,2016,23(4):991-999.
[40]POSTELNICU A,POP I.Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge[J].Applied Mathematics and Computation,2011,217(9):4359-4368.
[41]HARRIS S D,INGHAM D B,POP I.Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium:Brinkman model with slip[J].Transport in Porous Media,2009,77(2):267-285.
[42]ZHENG Lian-cun,LIANG Chen,ZHANG Xin-xin.Multiple values of skin friction for boundary layer separation flow in power law pseudoplastic fluids[J].Journal of Central South University of Technology,2007,14(1):218-220.
[43]ROHNI A M,AHMAD S,POP I.Note on Cortell’s nonlinearly stretching permeable sheet[J].International Journal of Heat and Mass Transfer,2012,55(21,22):5846-5852.
[44]CORTELL R.Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall[J].Meccanica,2012,47(3):769-781.
[45]HATAMI M,GANJI D D.Natural convection of sodium alginate(SA)non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods[J].Case Studies in Thermal Engineering,2014,2:14-22.