Rabinowitsch液的可变性对其在多孔通道中对流加热蠕动的影响（英文）
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摘要
考虑液体性质的变化和对流换热的影响,研究了Rabinowitsch液在多孔通道中的蠕动机理。由于倾斜对通道壁的影响,液体的黏度沿通道厚度发生变化,而导热系数则随着温度的变化而变化。在长波长和小雷诺数近似的情况下,利用摄动法求解非线性控制方程,得到轴向速度、温度、热传递系数和流线的表达式并进行了图解分析。运用膨胀、牛顿和假塑性流体模型对上述现象进行了研究。结果表明,可变黏度、多孔参数和滑移参数对膨胀、牛顿和假塑性流体模型的流体流量有很大的影响。研究进一步揭示了由于可变黏度和多孔参数的增加而增强了俘获现象的发生。此外,可以通过适当调节倾斜角来调整所俘获的丸剂尺寸。
The present paper emphasizes the peristaltic mechanism of Rabinowitsch liquid in a complaint porous channel under the influence of variable liquid properties and convective heat transfer. The effect of inclination on the complaint channel walls has been taken into account. The viscosity of the liquid varies across the thickness of the complaint channel, whereas, thermal conductivity varies concerning temperature. The nonlinear governing equations are solved by using perturbation technique under the long wavelength and small Reynold's number approximations. The expressions for axial velocity, temperature, the coefficient of heat transfer and streamlines are obtained and analyzed graphically. The above said physiological phenomena are investigated for a specific set of relevant parameters on dilatant, Newtonian and pseudoplastic fluid models. The results presented here shows that the presence of variable viscosity, porous parameter and slip parameter significantly affects the flow quantities of dilatant, Newtonian and pseudoplastic fluid models. The investigation further reveals that an increase in the value of variable viscosity and porous parameters enhances the occurrence of trapping phenomenon. Moreover, the size of trapped bolus can be eliminated with suitably adjusting the angle of inclination parameter.
The present paper emphasizes the peristaltic mechanism of Rabinowitsch liquid in a complaint porous channel under the influence of variable liquid properties and convective heat transfer. The effect of inclination on the complaint channel walls has been taken into account. The viscosity of the liquid varies across the thickness of the complaint channel, whereas, thermal conductivity varies concerning temperature. The nonlinear governing equations are solved by using perturbation technique under the long wavelength and small Reynold's number approximations. The expressions for axial velocity, temperature, the coefficient of heat transfer and streamlines are obtained and analyzed graphically. The above said physiological phenomena are investigated for a specific set of relevant parameters on dilatant, Newtonian and pseudoplastic fluid models. The results presented here shows that the presence of variable viscosity, porous parameter and slip parameter significantly affects the flow quantities of dilatant, Newtonian and pseudoplastic fluid models. The investigation further reveals that an increase in the value of variable viscosity and porous parameters enhances the occurrence of trapping phenomenon. Moreover, the size of trapped bolus can be eliminated with suitably adjusting the angle of inclination parameter.
引文
[1]EL SHEHAWEY E F,HUSSENY S Z A.Effect of porous boundaries on peristaltic transport through a porous medium[J].Acta Mechanica,2000,143:165-177.DOI:10.1007/BF01170946.
[2]NADEEM S,AKRAM S.Peristaltic flow of a Maxwell model through porous boundaries in a porous medium[J].Transport in Porous Media,2011,56:895-909.DOI:10.1007/s11242-010-9663-z.
[3]TRIPATHI D.Peristaltic hemodynamic flow of a couple-stress fluid through a porous medium with slip effect[J].Transport in Porous Media,2012,92:559-572.DOI:10.1007/s11242-011-9920-9.
[4]ALSAEDI A,ALI N,TRIPATHI D,HAYAT T.Peristaltic flow of couple stress fluid through uniform porous medium[J].Applied Mathematics and Mechanics,2014,35:469-480.DOI:10.1007/s10483-014-1805-8.
[5]SANKAD G C,NAGATHAN P S.Unsteady MHDperistaltic flow of a couple stress fluid through porous medium with wall and slip effects[J].Alexandria Engineering Journal,2016,55:2099-2105.DOI:10.1016/j.aej.2016.06.029.
[6]SHIT G C,RANJIT N K.Role of velocity on peristaltic transport of couple stress fluid through an asymmetric non-uniform channel:Application to digestive system[J].Journal of Molecular Liquid,2016,221:305-315.DOI:10.1016/j.molliq.2016.06.002.
[7]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,2018,135(2):1399-1415.DOI:10.1007/s10973-018-7540-0.
[8]MANJUNATHA G,RAJASHEKHAR C.Slip effects on peristaltic transport of Casson fluid in an inclined elastic tube with porous walls[J].Journal of Advanced Research in Fluid Mechanics and Thermal Sciences,2018,47:201-208.
[9]RAMESH K.Influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel[J].Journal of Molecular Liquid,2016,219:256-271.DOI:10.1016/j.molliq.2016.03.010.
[10]SRINIVAS S,GAYATHRI R,KOTHANDAPANI M.Mixed convective heat and mass transfer in an asymmetric channel with peristalsis[J].Communications in Nonlinear Science and Numerical Simulation,2011,16:1845-1862.DOI:10.1016/j.cnsns.2010.08.004.
[11]ABBASI F M,HAYAT T,AHMAD B.Peristaltic flow in an asymmetric channel with convective boundary conditions and Joule heating[J].Journal Central South University,2014,21:1411-1416.DOI:10.1007/s11771-014-2079-0.
[12]HUSSAIN Q,ASGHAR S,HAYAT T,ALSAEDI A.Heat transfer in a porous saturated wavy channel with asymmetric convective boundary conditions[J].Journal of Central South University,2015,22:392-401.DOI:10.1007/s11771-015-2534-6.
[13]MEBAREK-OUDINA F,BESSAIH R.Oscillatory magnetohydrodynamic natural convection of liquid metal between vertical coaxial cylinders[J].Journal of Applied Fluid Mechanics,2016,9:1655-1665.
[14]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:10.1007/s11771-017-3593-7.
[15]SIAVASHI M,ROSTAMI A.Two-phase simulation of non-Newtonian nanofluid natural convection in a circular annulus partially or completely filled with porous media[J].International Journal of Mechanical Sciences,2017,133:689-703.DOI:10.1016/j.ijmecsci.2017.09.031.
[16]OJJELA O,RAJU A,KAMBHATLA P K.Influence of thermophoresis and induced magnetic field on chemically reacting mixed convective flow of Jeffery fluid between porous parallel plates[J].Journal of Molecular Liquid,2017,232:195-206.DOI:10.1016/j.molliq.2017.02.061.
[17]MEBAREK-OUDINA F.Numerical modelling of the hydrodynamic stability in vertical annulus with heat source of different lengths[J].Engineering Science and Technology,an International Journal,2017,20:1324-1333.DOI:10.1016/j.jestch.2017.08.003.
[18]SIAVASHI M,KARIMI K,XIONG Q,DORANEHGARDM H.Numerical analysis of mixed convection of two-phase non-Newtonian nanofluid flow inside a partially porous square enclosure with a rotating cylinder[J].Journal of Thermal Analysis and Calorimetry,2018:1-21.DOI:10.1007/s10973-018-7945-9.
[19]WAKIF A,BOULAHIA Z,SEHAQUI R.A semianalytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions[J].Results in Physics,2018,9:1438-1454.DOI:10.1016/j.rinp.2018.01.066.
[20]WAKIF A,BOULAHIA Z,ALI F,EID M R,SEHAQUI R.Numerical analysis of the unsteady natural convection MHDCouette nanofluid flow in the presence of thermal radiation using single and two-phase nanofluid models for Cu-water nanofluids[J].International Journal of Applied and Computational Mathematics,2018,4:81.DOI:10.1007/s40819-018-0513-y.
[21]WAKIF A,BOULAHIA Z,MISHRA S R,RASHIDI M M,SEHAQUI R.Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model[J].The European Physical Journal Plus,2018,133:181.DOI:10.1140/epjp/i2018-12037-7.
[22]VAIDYA H,MANJUNATHA G,RAJASHEKHAR C,PRASAD K V.Role of slip and heat transfer on peristaltic transport of Herschel-Bulkley fluid through an elastic tube[J].Multidiscipline Modeling in Materials and Structures,2018,14:940-959.DOI:10.1108/MMMS-11-2017-0144.
[23]MESBAH M,VATANI A,SIAVASHI M,DORANEHGARD M H.Parallel processing of numerical simulation of two-phase flow in fractured reservoirs considering the effect of natural flow barriers using the streamline simulation method[J].International Journal of Heat and Mass Transfer,2019,131:574-583.DOI:10.1016/j.ijheatmasstransfer.2018.11.097.
[24]NADEEM S,AKBAR N S.Influence of heat transfer and variable viscosity in vertical porous annulus with peristalsis[J].Journal of Porous Media,2011,14:849-863.DOI:10.1615/JPor Media.v14.i10.20.
[25]KHAN A M,ELLAHI R,USMAN M.The effects of variable viscosity on the peristaltic flow of non-Newtonian fluid through porous medium in an inclined channel with slip boundary conditions[J].Journal of Porous Media,2013,16:59-67.DOI:10.1016/j.aej.2015.03.030.
[26]SINHA A,SHIT G C,RANJIT N K.Peristaltic transport of MHD flow and heat transfer in an asymmetric channel:Effects of variable viscosity,velocity-slip and temperature jump[J].Alexandria Engineering Journal,2015,54:691-704.
[27]VAJRAVELU K,PRASAD K V,CHIU-ON N G,VAIDYA H.MHD squeeze flow and heat transfer of a nanofluid between parallel disks with variable fluid properties and transpiration[J].International Journal of Mechanical and Materials Engineering,2019,12:9.DOI:10.1186/s40712-017-0076-4.
[28]PRASAD K V,VAJRAVELU K,VAIDYA H,BASHA N Z,UMESH V.Thermal and species concentration of MHDCasson fluid at a vertical sheet in the presence of variable fluid properties[J].Ain Shams Engineering Journal,2018,9:1763-1779.DOI:10.1016/j.asej.2016.08.017.
[29]PRASAD K V,VAIDYA H,VAJRAVELU K.MHD mixed convection heat transfer over a non-linear slender elastic sheet with variable fluid properties[J].Applied Mathematics and Nonlinear Sciences,2017,2:351-366.DOI:10.21042/AMNS.2017.2.00029.
[30]HAYAT T,FAROOQ S,AHMAD B,ALSAEDI A.Consequences of variable thermal conductivity and activation energy on peristalsis in curved configuration[J].Journal of Molecular Liquid,2018,263:258-267.DOI:10.1016/j.molliq.2018.04.109.
[31]RAJASHEKHAR C,MANJUNATHA G,VAIDYA H,DIVYA B B,PRASAD K V.Peristaltic flow of Casson liquid in an inclined porous tube with convective boundary conditions and variable liquid properties[J].Frontiers in Heat and Mass Transfer,2018,11:35.
[32]HAYAT T,JAVED M,ALI N.MHD peristaltic of a Jeffery fluid in a channel with compliant walls and porous space[J].Transport in Porous Media,2008,74:259-274.DOI:10.1007/s11242-007-9196-2.
[33]JAVED M,HAYAT T,ALSAEDI A.Peristaltic flow of Burgers’fluid with compliant walls and heat transfer[J].Applied Mathematics and Computation,2014,244:654-671.DOI:10.1016/j.amc.2014.07.009.
[34]BHATTI M M,ELLAHI R,ZEESHAN A.Study of variable magnetic field on the peristaltic flow of Jeffery fluid in a non-uniform rectangular duct having compliant walls[J].Journal of Molecular Liquid,2016,222:101-108.DOI:10.1016/j.molliq.2016.07.013.
[35]TANVEER A,HAYAT T,ALSAEDI A,AHMAD B.On modified Darcy’s law utilization in peristalsis of Sisko fluid[J].Journal of Molecular Liquid,2017,236:290-297.DOI:10.1016/j.molliq.2017.04.041.
[36]HAYAT T,NAWAZ S,ALSAEDI A,RAFIQ M.Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel[J].Results in Physics,2017,7:982-990.DOI:10.1016/j.rinp.2017.02.022.
[37]AKBAR N S,BUTT A W.Heat transfer analysis of Rabinowitsch fluid flow due to metachronal wave of cilia[J].Results in Physics,2015,5:92-98.DOI:10.1016/j.rinp.2015.03.005.
[38]ALI N,SAJID M,JAVID K,AHMED R.Peristaltic flow of Rabinowitsch fluid in a curved channel:Mathematical analysis revisited[J].Z Naturforsch,2016,72:245-251.DOI:10.1515/zna-2016-0334.
[39]SADAF H,NADEEM S.Analysis of combined convective and viscous dissipation effects for peristaltic flow Rabinowitsch fluid model[J].Journal of Bionic Engineering,2017,14:182-190.DOI:10.1016/S1672-6529(16)60389-X.
[40]SINGH U P,MEDHAVI A,GUPTA R S,BHATT S S.Analysis of peristaltic of non-Newtonian fluids through nonuniform tubes:Rabinowitsch fluid model[J].ZNaturforsch,2017,72:601-608.DOI:10.1515/zna-2017-0033.
[41]SARAVANA R,VAJRAVELU K,SREENADH S.Influence of compliant walls and heat transfer on the peristaltic transport of a Rabinowitsch fluid in an inclined channel[J].Z Naturforsch,2018,73:833-842.DOI:10.1515/zna-2018-0181.
[42]VAIDYA H,RAJASHEKHAR C,MANJUNATHA G,PRASAD K V.Rheological properties and peristalsis of Rabinowitsch fluid through complaint porous walls in an inclined channel[J].Journal of Nanofluids,2019,8:970-979.
[43]AAMIR A,ASGHAR S,AWAIS A.Thermophoresis and concentration effects in a fourth grade peristaltic flow with convective walls[J].Journal of Central South University,2017,24:1654-1662.DOI:10.1007/s11771-017-3571-0.
[44]BEAVERS G S,JOSEPH D D.Boundary conditions at a naturally permeable wall[J].Journal of Fluid Mechanics,1961,30:197-207.DOI:10.1017/S0022112067001375.
[45]SAFFMAN P G.On the Boundary conditions at the surface of a porous medium,Studies in Applied Mathematics,1971,1:93-101.DOI:10.1002/sapm197150293.
[46]ZOLFAGHARIAN A,DARZI M,GHASEMI S E.Analysis of nano droplet dynamics with various sphericities using efficient computational techniques[J].Journal of Central South University,2017,24:2353-2359.DOI:10.1007/s11771-017-3647-x.
[2]NADEEM S,AKRAM S.Peristaltic flow of a Maxwell model through porous boundaries in a porous medium[J].Transport in Porous Media,2011,56:895-909.DOI:10.1007/s11242-010-9663-z.
[3]TRIPATHI D.Peristaltic hemodynamic flow of a couple-stress fluid through a porous medium with slip effect[J].Transport in Porous Media,2012,92:559-572.DOI:10.1007/s11242-011-9920-9.
[4]ALSAEDI A,ALI N,TRIPATHI D,HAYAT T.Peristaltic flow of couple stress fluid through uniform porous medium[J].Applied Mathematics and Mechanics,2014,35:469-480.DOI:10.1007/s10483-014-1805-8.
[5]SANKAD G C,NAGATHAN P S.Unsteady MHDperistaltic flow of a couple stress fluid through porous medium with wall and slip effects[J].Alexandria Engineering Journal,2016,55:2099-2105.DOI:10.1016/j.aej.2016.06.029.
[6]SHIT G C,RANJIT N K.Role of velocity on peristaltic transport of couple stress fluid through an asymmetric non-uniform channel:Application to digestive system[J].Journal of Molecular Liquid,2016,221:305-315.DOI:10.1016/j.molliq.2016.06.002.
[7]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,2018,135(2):1399-1415.DOI:10.1007/s10973-018-7540-0.
[8]MANJUNATHA G,RAJASHEKHAR C.Slip effects on peristaltic transport of Casson fluid in an inclined elastic tube with porous walls[J].Journal of Advanced Research in Fluid Mechanics and Thermal Sciences,2018,47:201-208.
[9]RAMESH K.Influence of heat and mass transfer on peristaltic flow of a couple stress fluid through porous medium in the presence of inclined magnetic field in an inclined asymmetric channel[J].Journal of Molecular Liquid,2016,219:256-271.DOI:10.1016/j.molliq.2016.03.010.
[10]SRINIVAS S,GAYATHRI R,KOTHANDAPANI M.Mixed convective heat and mass transfer in an asymmetric channel with peristalsis[J].Communications in Nonlinear Science and Numerical Simulation,2011,16:1845-1862.DOI:10.1016/j.cnsns.2010.08.004.
[11]ABBASI F M,HAYAT T,AHMAD B.Peristaltic flow in an asymmetric channel with convective boundary conditions and Joule heating[J].Journal Central South University,2014,21:1411-1416.DOI:10.1007/s11771-014-2079-0.
[12]HUSSAIN Q,ASGHAR S,HAYAT T,ALSAEDI A.Heat transfer in a porous saturated wavy channel with asymmetric convective boundary conditions[J].Journal of Central South University,2015,22:392-401.DOI:10.1007/s11771-015-2534-6.
[13]MEBAREK-OUDINA F,BESSAIH R.Oscillatory magnetohydrodynamic natural convection of liquid metal between vertical coaxial cylinders[J].Journal of Applied Fluid Mechanics,2016,9:1655-1665.
[14]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:10.1007/s11771-017-3593-7.
[15]SIAVASHI M,ROSTAMI A.Two-phase simulation of non-Newtonian nanofluid natural convection in a circular annulus partially or completely filled with porous media[J].International Journal of Mechanical Sciences,2017,133:689-703.DOI:10.1016/j.ijmecsci.2017.09.031.
[16]OJJELA O,RAJU A,KAMBHATLA P K.Influence of thermophoresis and induced magnetic field on chemically reacting mixed convective flow of Jeffery fluid between porous parallel plates[J].Journal of Molecular Liquid,2017,232:195-206.DOI:10.1016/j.molliq.2017.02.061.
[17]MEBAREK-OUDINA F.Numerical modelling of the hydrodynamic stability in vertical annulus with heat source of different lengths[J].Engineering Science and Technology,an International Journal,2017,20:1324-1333.DOI:10.1016/j.jestch.2017.08.003.
[18]SIAVASHI M,KARIMI K,XIONG Q,DORANEHGARDM H.Numerical analysis of mixed convection of two-phase non-Newtonian nanofluid flow inside a partially porous square enclosure with a rotating cylinder[J].Journal of Thermal Analysis and Calorimetry,2018:1-21.DOI:10.1007/s10973-018-7945-9.
[19]WAKIF A,BOULAHIA Z,SEHAQUI R.A semianalytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions[J].Results in Physics,2018,9:1438-1454.DOI:10.1016/j.rinp.2018.01.066.
[20]WAKIF A,BOULAHIA Z,ALI F,EID M R,SEHAQUI R.Numerical analysis of the unsteady natural convection MHDCouette nanofluid flow in the presence of thermal radiation using single and two-phase nanofluid models for Cu-water nanofluids[J].International Journal of Applied and Computational Mathematics,2018,4:81.DOI:10.1007/s40819-018-0513-y.
[21]WAKIF A,BOULAHIA Z,MISHRA S R,RASHIDI M M,SEHAQUI R.Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model[J].The European Physical Journal Plus,2018,133:181.DOI:10.1140/epjp/i2018-12037-7.
[22]VAIDYA H,MANJUNATHA G,RAJASHEKHAR C,PRASAD K V.Role of slip and heat transfer on peristaltic transport of Herschel-Bulkley fluid through an elastic tube[J].Multidiscipline Modeling in Materials and Structures,2018,14:940-959.DOI:10.1108/MMMS-11-2017-0144.
[23]MESBAH M,VATANI A,SIAVASHI M,DORANEHGARD M H.Parallel processing of numerical simulation of two-phase flow in fractured reservoirs considering the effect of natural flow barriers using the streamline simulation method[J].International Journal of Heat and Mass Transfer,2019,131:574-583.DOI:10.1016/j.ijheatmasstransfer.2018.11.097.
[24]NADEEM S,AKBAR N S.Influence of heat transfer and variable viscosity in vertical porous annulus with peristalsis[J].Journal of Porous Media,2011,14:849-863.DOI:10.1615/JPor Media.v14.i10.20.
[25]KHAN A M,ELLAHI R,USMAN M.The effects of variable viscosity on the peristaltic flow of non-Newtonian fluid through porous medium in an inclined channel with slip boundary conditions[J].Journal of Porous Media,2013,16:59-67.DOI:10.1016/j.aej.2015.03.030.
[26]SINHA A,SHIT G C,RANJIT N K.Peristaltic transport of MHD flow and heat transfer in an asymmetric channel:Effects of variable viscosity,velocity-slip and temperature jump[J].Alexandria Engineering Journal,2015,54:691-704.
[27]VAJRAVELU K,PRASAD K V,CHIU-ON N G,VAIDYA H.MHD squeeze flow and heat transfer of a nanofluid between parallel disks with variable fluid properties and transpiration[J].International Journal of Mechanical and Materials Engineering,2019,12:9.DOI:10.1186/s40712-017-0076-4.
[28]PRASAD K V,VAJRAVELU K,VAIDYA H,BASHA N Z,UMESH V.Thermal and species concentration of MHDCasson fluid at a vertical sheet in the presence of variable fluid properties[J].Ain Shams Engineering Journal,2018,9:1763-1779.DOI:10.1016/j.asej.2016.08.017.
[29]PRASAD K V,VAIDYA H,VAJRAVELU K.MHD mixed convection heat transfer over a non-linear slender elastic sheet with variable fluid properties[J].Applied Mathematics and Nonlinear Sciences,2017,2:351-366.DOI:10.21042/AMNS.2017.2.00029.
[30]HAYAT T,FAROOQ S,AHMAD B,ALSAEDI A.Consequences of variable thermal conductivity and activation energy on peristalsis in curved configuration[J].Journal of Molecular Liquid,2018,263:258-267.DOI:10.1016/j.molliq.2018.04.109.
[31]RAJASHEKHAR C,MANJUNATHA G,VAIDYA H,DIVYA B B,PRASAD K V.Peristaltic flow of Casson liquid in an inclined porous tube with convective boundary conditions and variable liquid properties[J].Frontiers in Heat and Mass Transfer,2018,11:35.
[32]HAYAT T,JAVED M,ALI N.MHD peristaltic of a Jeffery fluid in a channel with compliant walls and porous space[J].Transport in Porous Media,2008,74:259-274.DOI:10.1007/s11242-007-9196-2.
[33]JAVED M,HAYAT T,ALSAEDI A.Peristaltic flow of Burgers’fluid with compliant walls and heat transfer[J].Applied Mathematics and Computation,2014,244:654-671.DOI:10.1016/j.amc.2014.07.009.
[34]BHATTI M M,ELLAHI R,ZEESHAN A.Study of variable magnetic field on the peristaltic flow of Jeffery fluid in a non-uniform rectangular duct having compliant walls[J].Journal of Molecular Liquid,2016,222:101-108.DOI:10.1016/j.molliq.2016.07.013.
[35]TANVEER A,HAYAT T,ALSAEDI A,AHMAD B.On modified Darcy’s law utilization in peristalsis of Sisko fluid[J].Journal of Molecular Liquid,2017,236:290-297.DOI:10.1016/j.molliq.2017.04.041.
[36]HAYAT T,NAWAZ S,ALSAEDI A,RAFIQ M.Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel[J].Results in Physics,2017,7:982-990.DOI:10.1016/j.rinp.2017.02.022.
[37]AKBAR N S,BUTT A W.Heat transfer analysis of Rabinowitsch fluid flow due to metachronal wave of cilia[J].Results in Physics,2015,5:92-98.DOI:10.1016/j.rinp.2015.03.005.
[38]ALI N,SAJID M,JAVID K,AHMED R.Peristaltic flow of Rabinowitsch fluid in a curved channel:Mathematical analysis revisited[J].Z Naturforsch,2016,72:245-251.DOI:10.1515/zna-2016-0334.
[39]SADAF H,NADEEM S.Analysis of combined convective and viscous dissipation effects for peristaltic flow Rabinowitsch fluid model[J].Journal of Bionic Engineering,2017,14:182-190.DOI:10.1016/S1672-6529(16)60389-X.
[40]SINGH U P,MEDHAVI A,GUPTA R S,BHATT S S.Analysis of peristaltic of non-Newtonian fluids through nonuniform tubes:Rabinowitsch fluid model[J].ZNaturforsch,2017,72:601-608.DOI:10.1515/zna-2017-0033.
[41]SARAVANA R,VAJRAVELU K,SREENADH S.Influence of compliant walls and heat transfer on the peristaltic transport of a Rabinowitsch fluid in an inclined channel[J].Z Naturforsch,2018,73:833-842.DOI:10.1515/zna-2018-0181.
[42]VAIDYA H,RAJASHEKHAR C,MANJUNATHA G,PRASAD K V.Rheological properties and peristalsis of Rabinowitsch fluid through complaint porous walls in an inclined channel[J].Journal of Nanofluids,2019,8:970-979.
[43]AAMIR A,ASGHAR S,AWAIS A.Thermophoresis and concentration effects in a fourth grade peristaltic flow with convective walls[J].Journal of Central South University,2017,24:1654-1662.DOI:10.1007/s11771-017-3571-0.
[44]BEAVERS G S,JOSEPH D D.Boundary conditions at a naturally permeable wall[J].Journal of Fluid Mechanics,1961,30:197-207.DOI:10.1017/S0022112067001375.
[45]SAFFMAN P G.On the Boundary conditions at the surface of a porous medium,Studies in Applied Mathematics,1971,1:93-101.DOI:10.1002/sapm197150293.
[46]ZOLFAGHARIAN A,DARZI M,GHASEMI S E.Analysis of nano droplet dynamics with various sphericities using efficient computational techniques[J].Journal of Central South University,2017,24:2353-2359.DOI:10.1007/s11771-017-3647-x.