August 2018, 11(4): 617-630. doi: 10.3934/dcdss.2018036

Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity

1. 

College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi, 46070, Pakistan

2. 

Department of Mathematics, Capital University of Science and Technology, Islamabad 44000, Pakistan

* Corresponding author: Asim Aziz

Received  December 2016 Revised  April 2017 Published  November 2017

In this paper, unsteady magnetohydrodynamic (MHD) boundary layer slip flow and heat transfer of power-law nanofluid over a nonlinear porous stretching sheet is investigated numerically. The thermal conductivity of the nanofluid is assumed as a function of temperature and the partial slip conditions are employed at the boundary. The nonlinear coupled system of partial differential equations governing the flow and heat transfer of a power-law nanofluid is first transformed into a system of nonlinear coupled ordinary differential equations by applying a suitable similarity transformation. The resulting system is then solved numerically using shooting technique. Numerical results are presented in the form of graphs and tables and the effect of the power-law index, velocity and thermal slip parameters, nanofluid volume concentration parameter, applied magnetic field parameter, suction/injection parameter on the velocity and temperature profiles are examined from physical point of view. The boundary layer thickness decreases with increase in strength of applied magnetic field, nanoparticle volume concentration, velocity slip and the unsteadiness of the stretching surface. Whereas thermal boundary layer thickness increase with increasing values of magnetic parameter, nanoparticle volume concentration and velocity slip at the boundary.

Citation: Asim Aziz, Wasim Jamshed. Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 617-630. doi: 10.3934/dcdss.2018036
References:
[1]

F. M. Abbasi, S. A. Shehzad, T. Hayat, A. Alsaedi and M. A. Obid, Influence of heat and mass flux conditions in hydromagnetic flow of Jeffrey nanofluid, AIP Advances, 5 (2015), 037111. doi: 10.1063/1.4914549.

[2]

K. Afzal and A. Aziz, Transport and heat transfer of time dependent MHD slip flow of nanofluids in solar collectors with variable thermal conductivity and thermal radiation, Results in Physics, 6 (2016), 746-753. doi: 10.1016/j.rinp.2016.09.017.

[3]

T. Aziz, A. Aziz and C. M. Khalique, Exact solutions for stokes flow of a non-newtonian nanofluid model: A lie similarity approach, Z. Naturforsch, 71 (2016). doi: 10.1515/zna-2016-0031.

[4]

N. Bhaskar, Reddy, T. poornima and P. Sreenivasulu, Influence of variable thermal conductivity on MHD boundary layar slip flow of ethylene-glycol based CU nanofluids over a stretching sheet with convective boundary condition, International Journal of Engineering Mathematics, 2014.

[5]

K. Bhattacharyya and G. C. Layek, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet , Physics Research International, 2014 (2014), Article ID 592536, 12 pages. doi: 10.1155/2014/592536.

[6]

M. M. Bhatti and M. M. Rashidi, Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet, Journal of Molecular Liquids, 221 (2016), 567-573. doi: 10.1016/j.molliq.2016.05.049.

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R. B. Bird, W. Stewart and E. N. Lightfoot, Transport phenomena, New York: John Wiley, 53 (1960), p879.

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J. Buongiorno, Convective transport in nanofluids, ASME Journal of Heat Transfer, 128 (2005), 240-250. doi: 10.1115/1.2150834.

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S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME International Mechanical Engineering Congress and Exposition, 66 (1995), p99.

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K. DasP. R. Duari and P. K. Kundu, Nanofluid flow over an unsteady stretching surface in presence of thermal radiation, Alexandria Engineering Journal, 53 (2014), 737-745. doi: 10.1016/j.aej.2014.05.002.

[11]

J. A. Eastman, S. U. S. Choi, S. Li, W. Yu and L. J. Thompson, Anomalously inceases effective thermal conductvities of ethylene glycol-bases nanofluids containing copper nanoparticles, Applied Physics Letters, 78 (2001), p6.

[12]

R. EllahiM. Raza and K. Vafai, Series solutions of non-Newtonian nanofluids with Reynolds model and Vogel model by means of the homotopy analysis method, Mathematical and Computer Modelling, 55 (2012), 1876-1891. doi: 10.1016/j.mcm.2011.11.043.

[13]

R. Ellahi, M. Hassan and A. Zeeshan, Study on magetohydrodynamic nanofluid by means of single and multi-walled carbon nanotubes suspended in a salt water solution, IEEE Transactions on Nanotechnology, 14 (2015), p726.

[14]

T. Hayat, M. Waqas, S. A. Shehzad and A. Alsaedi, Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects, Journal of Molecular Liquids, 215 (2016), p704.

[15]

T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, A revised model for Jeffrey nanofluid subject to convective condition and heat generation/absorption, PLoS ONE, 12 (2017), e0172518. doi: 10.1371/journal.pone.0172518.

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T. HayatM. Qasim and S. Mesloub, MHD flow and heat transfer over permeable stretching sheet with slip conditions, International Journal for Numerical Methods in Fluids, 66 (2011), 963-975. doi: 10.1002/fld.2294.

[17]

S. Hussain, A. Aziz, T. Aziz and C. M. Khalique, Slip flow and heat transfer of nanofluids over a porous plate embedded in a porous medium with temperature dependent viscosity and thermal conductivity, Applied Sciences, 6 (2016), p376. doi: 10.3390/app6120376.

[18]

W. Ibrahim and B. Shankar, MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions, Computers and Fluids, 75 (2013), 1-10. doi: 10.1016/j.compfluid.2013.01.014.

[19]

R. Kandasamy, P. Loganathan and P. P. Arasu, Scaling group transformation for MHD boundary layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection, Nuclear Engineering and Design, 241 (2011), p2053.

[20]

P. KeblinskiS. R. PhillpotS. U. S. Choi and J. A. Eastman, Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids), International Journal of Heat and Mass Transfer, 45 (2002), 855-863. doi: 10.1016/S0017-9310(01)00175-2.

[21]

P. KeblinskiJ. A. Eastman and D. G. Cahill, Nanofluids for thermal transport, Materials Today, 8 (2005), 36-44. doi: 10.1016/S1369-7021(05)70936-6.

[22]

W. A. Khan and I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer, 53 (2010), 2477-2483. doi: 10.1016/j.ijheatmasstransfer.2010.01.032.

[23]

Y. LinL. ZhengX. ZhangL. Ma and G. Chen, MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation, International Journal of Heat and Mass Transfer, 84 (2015), 903-911. doi: 10.1016/j.ijheatmasstransfer.2015.01.099.

[24]

M. MadhuN. Kishan and A. J. Chamkha, Unsteady flow of a Maxwell nanofluid over a stretching surface in the presence of magnetohydrodynamic and thermal radiation effects, Propulsion and Power Research, 6 (2017), 31-40. doi: 10.1016/j.jppr.2017.01.002.

[25]

O. Makinde and A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, International Journal of Thermal Sciences, 50 (2011), 1326-1332. doi: 10.1016/j.ijthermalsci.2011.02.019.

[26]

T. G. Motsumi and O. D. Makinde, Effects of the thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable strteching flat plate, Physica Scripta, 86 (2012), p4.

[27]

S. Nadeem and N. Muhammad, Impact of stratification and Cattaneo-Christov heat flux in the flow saturated with porous medium, Journal of Molecular Liquids, 224 (2016), 423-430. doi: 10.1016/j.molliq.2016.10.006.

[28]

S. Nadeem, Z. Ahmed and S. Saleem, The effect of variable viscosities on Micropolar flow of two nanofluids, Zeitschrift für Naturforschung A, 71 (2016), p1121. doi: 10.1515/zna-2015-0491.

[29]

S. Nadeem, A. U. Khan and S. Saleem, A comparative analysis on different nanofluid models for the oscillatory stagnation point flow , The European Physical Journal Plus, 131 (2016), p261. doi: 10.1140/epjp/i2016-16261-9.

[30]

S. Nadeem, R. Ul Haq and Z. H. Khan, Model-based analysis of micropolar nanofluid flow over a stretching surface, The European Physical Journal Plus, 45 (2014), p121.

[31]

A. NoghrehabadiR. Pourrajab and M. Ghalambaz, Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature, International Journal of Thermal Sciences, 54 (2012), 253-161. doi: 10.1016/j.ijthermalsci.2011.11.017.

[32]

A. Noghrehabadi, M. Ghalambaz and A. Ghanbarzadeh, Effects of variable viscosity and thermal conductivity on naturalconvection of nanofluids past a vertical plate in porous media, Journal of Mechanics, 30 (2014), p265.

[33]

N. V. Priezjev, Fluid structure and boundary slippage in nanoscale liquid films, Detection of Pathogens in Water Using Micro and Nano-Technology, IWA Publishing, 16 (2012).

[34]

N. Putra, W. Roetzel and S. K. Das, Natural convection of nanofluids, International Journal of Thermal Science, 39 (2003), p775.

[35]

M. Ramzan and M. Bilal, Time dependent MHD nano-second grade fluid flow induced by permeable vertical sheet with mixed convection and thermal radiation, PloS ONE, 10 (2015), e0124929. doi: 10.1371/journal.pone.0124929.

[36]

M. M. RashidiS. BagheriE. Momoniat and N. Freidoonimehr, Entropy analysis of convective MHD flow of third grade non-Newtonian fluid over a stretching sheet, Ain Shams Engineering Journal, 8 (2017), 77-85. doi: 10.1016/j.asej.2015.08.012.

[37]

A. K. Santra, S. Sen and N. Chakraborty, The forced convection of CU-water nanfluid in a channel with both Newtonian and Non-Newtonian models. International Journal of Thermal Science, 48 (2009), p391.

[38]

B. Shankar and Y. Yirga, Unsteady heat and mass transfer in MHD flow of nanofluids over stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Computational, Statistical, Natural and Physical Engineering, 7 (2013), p1248.

[39]

R. Sharma and A. Ishak, Second order slip flow of CU-water nanofluid over a stretching sheet with heat transfer, WSEAS Transactions on Fluid Mechanics, 9 (2014).

[40]

J. SuiL. Zheng and X. Zhang, Boundary layer heat and mass transfer with Cattaneo-Christov double-diffusion in upper-convected Maxwell nanofluid past a stretching sheet with slip velocity, International Journal of Thermal Science, 104 (2016), 461-468. doi: 10.1016/j.ijthermalsci.2016.02.007.

[41]

M. J. Uddin, I. Pop and A. I. M. Ismail, Free convection boundary layer flow of a nanofluid from a convectively heated vertical plate with linear momentum slip boundary condition, Sains Malaysiana, 4 (2012), p1475.

[42]

M. J. Uddin, W. Khan and N. S. Amin, g-jitter mixed convective slip flow of nanofluid past a permeable streatching sheet embedded in a darcian porous media with variable viscosity, PLoS ONE, 9 (2014), p12.

[43]

X. Q. Wang, S. Arun and S. Mujumdar, Heat transfer characteristics of nanofluids, International Journal of Thermal Sciences, 46 (2007), p1.

[44]

X. WangX. Xu and S. U. S. Choi, Thermal conductivity of nanoparticles-fluid mixture, Journal of Thermophysics and Heat Transfer, 13 (1999), 474-480. doi: 10.2514/2.6486.

[45]

L. ZhengJ. NiuX. Zhang and Y. Gao, MHD flow and heat transfer over a porous shrinking surface with velocity slip and temperature jump, Mathematical and Computer Modelling, 56 (2012), 133-144. doi: 10.1016/j.mcm.2011.11.080.

[46]

L. ZhengC. ZhangaX. Zhang and J. Zhang, Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium, Journal of the Franklin Institute, 350 (2013), 990-1007. doi: 10.1016/j.jfranklin.2013.01.022.

[47]

L. C. ZhengY. H. Lin and X. X. Zhang, Marangoni convection of power-law fluids driven by power-law temperature gradient, Journal of The Franklin Institute, 349 (2012), 2585-2597. doi: 10.1016/j.jfranklin.2012.07.004.

show all references

References:
[1]

F. M. Abbasi, S. A. Shehzad, T. Hayat, A. Alsaedi and M. A. Obid, Influence of heat and mass flux conditions in hydromagnetic flow of Jeffrey nanofluid, AIP Advances, 5 (2015), 037111. doi: 10.1063/1.4914549.

[2]

K. Afzal and A. Aziz, Transport and heat transfer of time dependent MHD slip flow of nanofluids in solar collectors with variable thermal conductivity and thermal radiation, Results in Physics, 6 (2016), 746-753. doi: 10.1016/j.rinp.2016.09.017.

[3]

T. Aziz, A. Aziz and C. M. Khalique, Exact solutions for stokes flow of a non-newtonian nanofluid model: A lie similarity approach, Z. Naturforsch, 71 (2016). doi: 10.1515/zna-2016-0031.

[4]

N. Bhaskar, Reddy, T. poornima and P. Sreenivasulu, Influence of variable thermal conductivity on MHD boundary layar slip flow of ethylene-glycol based CU nanofluids over a stretching sheet with convective boundary condition, International Journal of Engineering Mathematics, 2014.

[5]

K. Bhattacharyya and G. C. Layek, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet , Physics Research International, 2014 (2014), Article ID 592536, 12 pages. doi: 10.1155/2014/592536.

[6]

M. M. Bhatti and M. M. Rashidi, Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet, Journal of Molecular Liquids, 221 (2016), 567-573. doi: 10.1016/j.molliq.2016.05.049.

[7]

R. B. Bird, W. Stewart and E. N. Lightfoot, Transport phenomena, New York: John Wiley, 53 (1960), p879.

[8]

J. Buongiorno, Convective transport in nanofluids, ASME Journal of Heat Transfer, 128 (2005), 240-250. doi: 10.1115/1.2150834.

[9]

S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME International Mechanical Engineering Congress and Exposition, 66 (1995), p99.

[10]

K. DasP. R. Duari and P. K. Kundu, Nanofluid flow over an unsteady stretching surface in presence of thermal radiation, Alexandria Engineering Journal, 53 (2014), 737-745. doi: 10.1016/j.aej.2014.05.002.

[11]

J. A. Eastman, S. U. S. Choi, S. Li, W. Yu and L. J. Thompson, Anomalously inceases effective thermal conductvities of ethylene glycol-bases nanofluids containing copper nanoparticles, Applied Physics Letters, 78 (2001), p6.

[12]

R. EllahiM. Raza and K. Vafai, Series solutions of non-Newtonian nanofluids with Reynolds model and Vogel model by means of the homotopy analysis method, Mathematical and Computer Modelling, 55 (2012), 1876-1891. doi: 10.1016/j.mcm.2011.11.043.

[13]

R. Ellahi, M. Hassan and A. Zeeshan, Study on magetohydrodynamic nanofluid by means of single and multi-walled carbon nanotubes suspended in a salt water solution, IEEE Transactions on Nanotechnology, 14 (2015), p726.

[14]

T. Hayat, M. Waqas, S. A. Shehzad and A. Alsaedi, Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects, Journal of Molecular Liquids, 215 (2016), p704.

[15]

T. Hayat, A. Aziz, T. Muhammad and A. Alsaedi, A revised model for Jeffrey nanofluid subject to convective condition and heat generation/absorption, PLoS ONE, 12 (2017), e0172518. doi: 10.1371/journal.pone.0172518.

[16]

T. HayatM. Qasim and S. Mesloub, MHD flow and heat transfer over permeable stretching sheet with slip conditions, International Journal for Numerical Methods in Fluids, 66 (2011), 963-975. doi: 10.1002/fld.2294.

[17]

S. Hussain, A. Aziz, T. Aziz and C. M. Khalique, Slip flow and heat transfer of nanofluids over a porous plate embedded in a porous medium with temperature dependent viscosity and thermal conductivity, Applied Sciences, 6 (2016), p376. doi: 10.3390/app6120376.

[18]

W. Ibrahim and B. Shankar, MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions, Computers and Fluids, 75 (2013), 1-10. doi: 10.1016/j.compfluid.2013.01.014.

[19]

R. Kandasamy, P. Loganathan and P. P. Arasu, Scaling group transformation for MHD boundary layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection, Nuclear Engineering and Design, 241 (2011), p2053.

[20]

P. KeblinskiS. R. PhillpotS. U. S. Choi and J. A. Eastman, Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids), International Journal of Heat and Mass Transfer, 45 (2002), 855-863. doi: 10.1016/S0017-9310(01)00175-2.

[21]

P. KeblinskiJ. A. Eastman and D. G. Cahill, Nanofluids for thermal transport, Materials Today, 8 (2005), 36-44. doi: 10.1016/S1369-7021(05)70936-6.

[22]

W. A. Khan and I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer, 53 (2010), 2477-2483. doi: 10.1016/j.ijheatmasstransfer.2010.01.032.

[23]

Y. LinL. ZhengX. ZhangL. Ma and G. Chen, MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation, International Journal of Heat and Mass Transfer, 84 (2015), 903-911. doi: 10.1016/j.ijheatmasstransfer.2015.01.099.

[24]

M. MadhuN. Kishan and A. J. Chamkha, Unsteady flow of a Maxwell nanofluid over a stretching surface in the presence of magnetohydrodynamic and thermal radiation effects, Propulsion and Power Research, 6 (2017), 31-40. doi: 10.1016/j.jppr.2017.01.002.

[25]

O. Makinde and A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, International Journal of Thermal Sciences, 50 (2011), 1326-1332. doi: 10.1016/j.ijthermalsci.2011.02.019.

[26]

T. G. Motsumi and O. D. Makinde, Effects of the thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable strteching flat plate, Physica Scripta, 86 (2012), p4.

[27]

S. Nadeem and N. Muhammad, Impact of stratification and Cattaneo-Christov heat flux in the flow saturated with porous medium, Journal of Molecular Liquids, 224 (2016), 423-430. doi: 10.1016/j.molliq.2016.10.006.

[28]

S. Nadeem, Z. Ahmed and S. Saleem, The effect of variable viscosities on Micropolar flow of two nanofluids, Zeitschrift für Naturforschung A, 71 (2016), p1121. doi: 10.1515/zna-2015-0491.

[29]

S. Nadeem, A. U. Khan and S. Saleem, A comparative analysis on different nanofluid models for the oscillatory stagnation point flow , The European Physical Journal Plus, 131 (2016), p261. doi: 10.1140/epjp/i2016-16261-9.

[30]

S. Nadeem, R. Ul Haq and Z. H. Khan, Model-based analysis of micropolar nanofluid flow over a stretching surface, The European Physical Journal Plus, 45 (2014), p121.

[31]

A. NoghrehabadiR. Pourrajab and M. Ghalambaz, Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature, International Journal of Thermal Sciences, 54 (2012), 253-161. doi: 10.1016/j.ijthermalsci.2011.11.017.

[32]

A. Noghrehabadi, M. Ghalambaz and A. Ghanbarzadeh, Effects of variable viscosity and thermal conductivity on naturalconvection of nanofluids past a vertical plate in porous media, Journal of Mechanics, 30 (2014), p265.

[33]

N. V. Priezjev, Fluid structure and boundary slippage in nanoscale liquid films, Detection of Pathogens in Water Using Micro and Nano-Technology, IWA Publishing, 16 (2012).

[34]

N. Putra, W. Roetzel and S. K. Das, Natural convection of nanofluids, International Journal of Thermal Science, 39 (2003), p775.

[35]

M. Ramzan and M. Bilal, Time dependent MHD nano-second grade fluid flow induced by permeable vertical sheet with mixed convection and thermal radiation, PloS ONE, 10 (2015), e0124929. doi: 10.1371/journal.pone.0124929.

[36]

M. M. RashidiS. BagheriE. Momoniat and N. Freidoonimehr, Entropy analysis of convective MHD flow of third grade non-Newtonian fluid over a stretching sheet, Ain Shams Engineering Journal, 8 (2017), 77-85. doi: 10.1016/j.asej.2015.08.012.

[37]

A. K. Santra, S. Sen and N. Chakraborty, The forced convection of CU-water nanfluid in a channel with both Newtonian and Non-Newtonian models. International Journal of Thermal Science, 48 (2009), p391.

[38]

B. Shankar and Y. Yirga, Unsteady heat and mass transfer in MHD flow of nanofluids over stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Computational, Statistical, Natural and Physical Engineering, 7 (2013), p1248.

[39]

R. Sharma and A. Ishak, Second order slip flow of CU-water nanofluid over a stretching sheet with heat transfer, WSEAS Transactions on Fluid Mechanics, 9 (2014).

[40]

J. SuiL. Zheng and X. Zhang, Boundary layer heat and mass transfer with Cattaneo-Christov double-diffusion in upper-convected Maxwell nanofluid past a stretching sheet with slip velocity, International Journal of Thermal Science, 104 (2016), 461-468. doi: 10.1016/j.ijthermalsci.2016.02.007.

[41]

M. J. Uddin, I. Pop and A. I. M. Ismail, Free convection boundary layer flow of a nanofluid from a convectively heated vertical plate with linear momentum slip boundary condition, Sains Malaysiana, 4 (2012), p1475.

[42]

M. J. Uddin, W. Khan and N. S. Amin, g-jitter mixed convective slip flow of nanofluid past a permeable streatching sheet embedded in a darcian porous media with variable viscosity, PLoS ONE, 9 (2014), p12.

[43]

X. Q. Wang, S. Arun and S. Mujumdar, Heat transfer characteristics of nanofluids, International Journal of Thermal Sciences, 46 (2007), p1.

[44]

X. WangX. Xu and S. U. S. Choi, Thermal conductivity of nanoparticles-fluid mixture, Journal of Thermophysics and Heat Transfer, 13 (1999), 474-480. doi: 10.2514/2.6486.

[45]

L. ZhengJ. NiuX. Zhang and Y. Gao, MHD flow and heat transfer over a porous shrinking surface with velocity slip and temperature jump, Mathematical and Computer Modelling, 56 (2012), 133-144. doi: 10.1016/j.mcm.2011.11.080.

[46]

L. ZhengC. ZhangaX. Zhang and J. Zhang, Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium, Journal of the Franklin Institute, 350 (2013), 990-1007. doi: 10.1016/j.jfranklin.2013.01.022.

[47]

L. C. ZhengY. H. Lin and X. X. Zhang, Marangoni convection of power-law fluids driven by power-law temperature gradient, Journal of The Franklin Institute, 349 (2012), 2585-2597. doi: 10.1016/j.jfranklin.2012.07.004.

Figure 1.  Geometry of the problem
Figure 2.  Velocity profiles for different values of parameter $A$
Figure 3.  Temperature profiles for different values of parameter $A$
Figure 4.  Velocity profiles for different values of parameter $M$
Figure 5.  Temperature profiles for different values of parameter $M$
Figure 6.  Velocity profiles for different values of parameter $\phi$
Figure 7.  Temperature profiles for different values of parameter $\phi$
Figure 8.  Velocity profiles for different values of parameter $\delta$
Figure 9.  Temperature profiles for different values of parameter $\delta$
Figure 10.  Velocity profiles for different values of parameter $S$
Figure 11.  Temperature profiles for different values of parameter $S$
Figure 12.  Velocity profiles for different values of parameter $S$
Figure 13.  Temperature profiles for different values of parameter $S$
Table 1.  Values of $-f''(0)$ for the variation of parameters and fixed $Pr= 6.2$, $\Delta = 1.0$ and $\phi = 0.0$
$M$$S$$\delta$ $A$$-f''(0)$ $-f''(0)$ $-f''(0)$
T.HayatKhadeejahPresent
0.251.01.00.20.601570.601570.60160
1.00.21.00.20.575630.575630.57560
1.00.51.00.20.6022850.6022650.60228
$M$$S$$\delta$ $A$$-f''(0)$ $-f''(0)$ $-f''(0)$
T.HayatKhadeejahPresent
0.251.01.00.20.601570.601570.60160
1.00.21.00.20.575630.575630.57560
1.00.51.00.20.6022850.6022650.60228
Table 2.  Thermophysical properties of the base fluid and nanoparticles
Physical properties Base fluidNanoparticles
WaterCu
$C_{p}(J/kgK)$4179385
$\rho(kg/m^{3})$997.18933
$k(W/mK)$ 0.613400
$\sigma (\Omega.m)^{-1}$0.05$5.96\times10^{7}$
Physical properties Base fluidNanoparticles
WaterCu
$C_{p}(J/kgK)$4179385
$\rho(kg/m^{3})$997.18933
$k(W/mK)$ 0.613400
$\sigma (\Omega.m)^{-1}$0.05$5.96\times10^{7}$
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