doi: 10.3934/dcdss.2020142

Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation

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

3. 

Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, 44000, Pakistan

* Corresponding author: Asim Aziz

Received  February 2019 Revised  July 2019 Published  November 2019

In this numerical study, researchers explore the flow, heat transfer and entropy of electrically conducting hybrid nanofluid over the horizontal penetrable stretching surface with velocity slip conditions at the interface. The non-Newtonian fluid models lead to better understanding of flow and heat transfer characteristics of nanofluids. Therefore, non-Newtonian Maxwell mathematical model is considered for the hybrid nanofluid and the uniform magnetic field is applied at an angle to the direction of the flow. The Joule heating and thermal radiation impact are also considered in the simplified model. The governing nonlinear partial differential equations for hybrid Maxwell nanofluid flow, heat transfer and entropy generation are simplified by taking boundary layer approximations and then reduced to ordinary differential equations using suitable similarity transformations. The Keller box scheme is then adopted to solve the system of ordinary differential equations. The Ethylene glycol based Copper Ethylene glycol ($ Cu $-$ EG $) nanofluid and Ferro-Copper Ethylene glycol ($ Fe_3O_4-Cu $-$ EG $) hybrid nanofluids are considered to produce the numerical results for velocity, temperature and entropy profiles as well as the skin friction factor and the local Nusselt number. The main findings indicate that hybrid Maxwell nanofluid is better thermal conductor when compared with the conventional nanofluid, the greater angle of inclination of magnetic field offers greater resistance to fluid motion within boundary layer and the heat transfer rate act as descending function of nanoparticles shape factor.

Citation: Asim Aziz, Wasim Jamshed, Yasir Ali, Moniba Shams. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020142
References:
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T. Aziz, A. Aziz and C. M. Khalique, Exact solutions for stokes flow of a non-Newtonian nanofluid model: A lie similarity approach, Zeitschrift fur Naturforschung A, 71 (2016), 621. doi: 10.1515/zna-2016-0031.  Google Scholar

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show all references

References:
[1]

M. S. AbelJ. V. Tawade and M. M. Nandeppanavar, MHD flow and heat transfer for the upper-convected Maxwell fluid over a stretching sheet, Meccanica, 47 (2012), 385-393.  doi: 10.1007/s11012-011-9448-7.  Google Scholar

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M. AfrandD. Toghraie and B. Ruhani, Effects of temperature and nanoparticles concentration on rheological behavior of $Fe_3O_4-Ag/EG$ hybrid nanofluid: An experimental study, Experimental Thermal and Fluid Science, 77 (2016), 38-44.   Google Scholar

[4]

H. I. AnderssonJ. B. Aarseth and B. S. Dandapat, Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, International Journal of Heat and Mass Transfer, 43 (2000), 69-74.   Google Scholar

[5]

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[6]

T. Aziz, A. Aziz and C. M. Khalique, Exact solutions for stokes flow of a non-Newtonian nanofluid model: A lie similarity approach, Zeitschrift fur Naturforschung A, 71 (2016), 621. doi: 10.1515/zna-2016-0031.  Google Scholar

[7]

A. AzizW. Jamshed and T. Aziz, Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity, Open Physics, 16 (2018), 123-136.  doi: 10.1515/phys-2018-0020.  Google Scholar

[8]

A. Aziz and W. Jamshed, Unsteady MHD slip flow of non-Newtonian Power-law nanofluid over a moving surface with temperature dependent thermal conductivity, Discrete and Continuous Dynamical Systems Series S, 11 (2018), 617-630.  doi: 10.3934/dcdss.2018036.  Google Scholar

[9]

M. Bahiraei and N. Mazaheri, Application of a novel hybrid nanofluid containing grapheme-platinum nanoparticles in a chaotic twisted geometry for utilization in miniature devices Thermal and energy efficiency considerations, International Journal of Mechanical Sciences, 138 (2018), 337-349.   Google Scholar

[10]

P. Barnoon and D. Toghraie, Numerical investigation of laminar flow and heat transfer of non-Newtonian nanofluid within a porous medium, Powder Technology, 325 (2018), 78-91.  doi: 10.1016/j.powtec.2017.10.040.  Google Scholar

[11]

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[12]

M. Q. Brewster, Thermal Radiative Transfer and Properties, John Wiley and Sons, 1992. Google Scholar

[13]

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[14]

S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME International Mechanical Engineering Congress and Exposition, 66 (1995), 99-105.   Google Scholar

[15]

R. Cortell, A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet, International Journal of Non-Linear Mechanics, 41 (2006), 78-85.  doi: 10.1016/j.ijnonlinmec.2005.04.008.  Google Scholar

[16]

S. DasS. ChakrabortyR. N. Jana and O. D. Makinde, Entropy analysis of unsteady magneto-nanofluid flow past accelerating stretching sheet with convective boundary condition, Appl. Math. Mech. (English Ed.), 36 (2015), 1593-1610.  doi: 10.1007/s10483-015-2003-6.  Google Scholar

[17]

S. S. U. Devi and S. P. A. Devi, Numerical investigation on three dimensional hybrid $Cu-Al_2O_3/water$ nanofluid flow over a stretching sheet with effecting Lorentz force subject to Newtonian heatings, Canadian Journal of Physics, 94 (2016), 490-496.   Google Scholar

[18]

S. P. A. Devi and S. S. U. Devi, Numerical investigation of hydromagnetic hybrid $Cu-Al_2O_3$ water nanofluid flow over a permeable stretching sheet with suction, Journal of Nonlinear Science and Applications, 17 (2016), 249-257.   Google Scholar

[19]

M. R. EidK. L. MahnyT. Muhammad and M. Sheikholeslami, Numerical treatment for Carreau nanofluid flow over a porous nonlinear stretching surface, Results in Physics, 8 (2018), 1185-1139.  doi: 10.1016/j.rinp.2018.01.070.  Google Scholar

[20]

S. S. GhadikolaeiM. YassariK. H. Hosseinzadeh and D. D. Ganji, Investigation on thermophysical properties of $TiO_2-Cu/H_2O$ hybrid nanofluid transport dependent on shape factor in MHD stagnation point flow, Powder Technology, 322 (2017), 428-438.   Google Scholar

[21]

S. S. GhadikolaeiK. H. HosseinzadehM. YassariH. Sadeghi and D. D. Ganji, Analytical and numerical solution of non-Newtonian second-grade fluid flow on a stretching sheet, Thermal Science and Engineering Progress, 5 (2018), 309-316.  doi: 10.1016/j.tsep.2017.12.010.  Google Scholar

[22]

N. S. GibanovM. A. SheremetH. F. Oztop and N. A. Hamdeh, Mixed convection with entropy generation of nanofluid in a lid-driven cavity under the effects of a heat-conducting solid wall and vertical temperature gradient, Eur. J. Mech. B Fluids, 70 (2018), 148-159.  doi: 10.1016/j.euromechflu.2018.03.002.  Google Scholar

[23]

T. Hayat and S. Nadeem, Heat transfer enhancement with $Ag-CuO/water$ hybrid nanofluid, Results in Physics, 7 (2017), 2317-2324.  doi: 10.1016/j.rinp.2017.06.034.  Google Scholar

[24]

G. Huminic and A. Huminic, The heat transfer performances and entropy generation analysis of hybrid nanofluids in a flattened tube, International Journal of Heat and Mass Transfer, 119 (2018), 813-827.  doi: 10.1016/j.ijheatmasstransfer.2017.11.155.  Google Scholar

[25]

S. HussainS. E. Ahmed and T. Akbar, Investigation on thermophysical properties of $TiO_2-Cu/H_2O$ hybrid nanofluid transport dependent on shape factor in MHD stagnation point flow, International Journal of Heat and Mass Transfer, 114 (2017), 1054-1066.   Google Scholar

[26]

Z. IqbalN. S. AkbarE. Azhar and E. N. Maraj, Performance of hybrid nanofluid $(Cu-CuO/water)$ on MHD rotating transport in oscillating vertical channel inspired by Hall current and thermal radiation, Alexandria Engineering Journal, 57 (2018), 1943-1954.  doi: 10.1016/j.aej.2017.03.047.  Google Scholar

[27]

A. IshakR. Nazar and I. Pop, Mixed convection on the stagnation point flow towards a vertical, continuously stretching sheet., ASME, Journal of Heat Transfer, 129 (2007), 1087-1090.   Google Scholar

[28]

A. IshakR. Nazar and I. Pop, Boundary layer flow and heat transfer over an unsteady stretching vertical surface, Meccanica, 44 (2009), 369-375.  doi: 10.1007/s11012-008-9176-9.  Google Scholar

[29]

W. Jamshed and A. Aziz, Cattaneo-Christov based study of $TiO_{2}-Cu/H_{2}O$ Casson hybrid nanofluid flow over a stretching surface with entropy generation, Applied Nanoscience, 8 (2008), 1-14.   Google Scholar

[30]

W. Jamshed and A. Aziz, A comparative entropy based analysis of $Cu$ and $Fe_{3}O_{4}$ /methanol Powell-Eyring nanofluid in solar thermal collectors subjected to thermal radiation, variable thermal conductivity and impact of different nanoparticles shape, Result in Physics, 9 (2018), 195-205.   Google Scholar

[31]

P. KeblinskiS. R. PhillpotS. 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.  Google Scholar

[32]

H. B. Keller, A new difference scheme for parabolic problems, 1971 Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970), Academic Press, New York, 2 (1971), 327–350.  Google Scholar

[33]

Y. B. Kho, A. Hussanan, N. M. Sarif, Z. Ismail and M. Z. Salleh, Thermal radiation effects on mhd with flow heat and mass transfer in casson nanofluid over a stretching sheet, MATEC Web of Conferences, 150 (2018), 06036. doi: 10.1051/matecconf/201815006036.  Google Scholar

[34]

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Figure 1.  Flow geometry
Figure 2.  Flow Sheet of Keller Box method
Figure 3.  Velocity distribution against $ \beta $
Figure 4.  Temperature distribution against $ \beta $
Figure 5.  Entropy generation against $ \beta $
Figure 6.  Velocity distribution against $ M $
Figure 7.  Temperature distribution against $ M $
Figure 8.  Entropy generation against $ M $
Figure 9.  Velocity distribution against $ \Gamma $
Figure 10.  Temperature distribution against $ \Gamma $
Figure 11.  Entropy generation against $ \Gamma $
Figure 12.  Velocity distribution against $ \phi, \phi_{hnf} $
Figure 13.  Temp distribution against $ \phi, \phi_{hnf} $
Figure 14.  Entropy generation against $ \phi, \phi_{hnf} $
Figure 15.  Temperature distribution against $ Nr $
Figure 16.  Entropy generation against $ Nr $
Figure 17.  Temperature distribution against $ Ec $
Figure 18.  Entropy generation against $ Ec $
Figure 19.  Temperature distribution against $ Bi $
Figure 20.  Entropy generation against $ Bi $
Figure 21.  Temperature distribution against $ m $
Figure 22.  Entropy generation distribution against $ m $
Figure 23.  Entropy generation distribution against the parameter $ Re $
Figure 24.  Entropy generation distribution against the parameter $ Br $
Table 1.  Comparison results of -$ \theta^{'}(0) $ for different Prandtl number when $ \beta = 0 $, $ A = 0 $, $ M = 0 $, $ \Gamma = 0 $, $ \phi_{1} = 0 $, $ \phi_{2} = 0 $, $ \Lambda = 0 $, $ Nr = 0 $, $ Ec = 0 $, $ S = 0 $, $ m = 3 $ and $ B_i = 0 $
$ Pr $ $ Ishak $ $ Nazar $ $ Abolbashari $ $ Das $ $ Present $
Results[27] Results[28] Results[2] Results[16] Results
0.72 0.8086 0.8086 0.80863135 0.80876122 0.80876181
1.0 1.0000 1.0000 1.00000000 1.00000000 1.00000000
3.0 1.9237 1.9236 1.92368259 1.92357431 1.92357420
7.0 3.0723 3.0722 3.07225021 3.07314679 3.07314651
10 3.7207 3.7006 3.72067390 3.72055436 3.72055429
$ Pr $ $ Ishak $ $ Nazar $ $ Abolbashari $ $ Das $ $ Present $
Results[27] Results[28] Results[2] Results[16] Results
0.72 0.8086 0.8086 0.80863135 0.80876122 0.80876181
1.0 1.0000 1.0000 1.00000000 1.00000000 1.00000000
3.0 1.9237 1.9236 1.92368259 1.92357431 1.92357420
7.0 3.0723 3.0722 3.07225021 3.07314679 3.07314651
10 3.7207 3.7006 3.72067390 3.72055436 3.72055429
Table 2.  Thermo-physical properties of Base Fluid and Nanoparticles
Thermo-physical $ \rho({kg}/{m^{3}}) $ $ c_p({J}/{kgK}) $ $ k({W}/{mK}) $ $ \sigma({S}/{m}) $
Ethylene glycol $ (EG) $ 1114 2415 0.252 $ 5.5 \times 10^{-6} $
Pure water $ (H_2O) $ 997.1 4179 0.613 $ 0.05 $
Copper $ (Cu) $ 8933 385.0 401.00 $ 5.96 \times 10^{7} $
Ferro $ (Fe_3O_4) $ 5180 670 9.7 $ 0.74 \times 10^{6} $
Copper oxide $ (Cu_O) $ 6510 540 18 $ 5.96 \times 10^{7} $
Alumina $ (Al_{2}O_{3}) $ 3970 765.0 40.000 $ 3.5 \times 10^{7} $
Titanium oxide $ (T_iO_{2}) $ 4250 686.2 8.9538 $ 2.38 \times 10^{6} $
Thermo-physical $ \rho({kg}/{m^{3}}) $ $ c_p({J}/{kgK}) $ $ k({W}/{mK}) $ $ \sigma({S}/{m}) $
Ethylene glycol $ (EG) $ 1114 2415 0.252 $ 5.5 \times 10^{-6} $
Pure water $ (H_2O) $ 997.1 4179 0.613 $ 0.05 $
Copper $ (Cu) $ 8933 385.0 401.00 $ 5.96 \times 10^{7} $
Ferro $ (Fe_3O_4) $ 5180 670 9.7 $ 0.74 \times 10^{6} $
Copper oxide $ (Cu_O) $ 6510 540 18 $ 5.96 \times 10^{7} $
Alumina $ (Al_{2}O_{3}) $ 3970 765.0 40.000 $ 3.5 \times 10^{7} $
Titanium oxide $ (T_iO_{2}) $ 4250 686.2 8.9538 $ 2.38 \times 10^{6} $
Table 3.  Values of Skin Friction $ = C_fRe_x^\frac{1}{2} $ and Nusselt Number $ = N_uRe_x^\frac{-1}{2} $ for $ P_{r} = 6.2 $ and $ m = 3 $
$ \beta $ $ A $ $ M $ $ \Gamma $ $ \phi $ $ \phi_{2} $ $ \Lambda $ $ Nr $ $ Ec $ $ Bi $ $ C_fRe_x^\frac{1}{2} $ $ C_fRe_x^\frac{1}{2} $ $ N_uRe_x^\frac{-1}{2} $ $ N_uRe_x^\frac{-1}{2} $
$ Cu $-$ EG $ $ Fe_3O_4-Cu/EG $ $ Cu $-$ EG $ $ Fe_3O_4-Cu/EG $
0.01 0.6 0.6 $ \pi/4 $ 0.18 0.09 0.1 0.2 0.2 0.1 1.1918 1.8311 0.0818 0.0824
0.1 2.0011 2.0673 0.0813 0.0819
0.3 2.1039 2.0894 0.0804 0.0808
0.6 1.1918 1.8311 0.0818 0.0824
1.6 2.2308 2.1736 0.0839 0.0879
2.6 2.4621 2.3211 0.0896 0.0899
0.6 1.1918 1.8311 0.0818 0.0824
1.6 2.2661 2.1068 0.0809 0.0820
2.6 2.4149 2.2922 0.0800 0.0815
$ \pi/4 $ 1.1918 1.8311 0.0818 0.0824
$ \pi/3 $ 1.2100 2.0138 0.0810 0.0821
$ \pi/2 $ 1.8610 2.0771 0.0802 0.0820
0.09 2.1314 - 0.0801 -
0.15 2.0229 - 0.0810 -
0.18 1.1912 - 0.0818 -
0.0 - 1.9597 - 0.0811
0.06 - 1.9021 - 0.0818
0.09 - 1.8311 - 0.0824
0.0 2.0121 2.0065 0.0852 0.0858
0.1 1.1918 1.8311 0.0818 0.0824
0.2 1.1728 1.6216 0.0810 0.0820
0.0 1.1918 1.8311 0.0818 0.0824
0.2 1.1918 1.8311 0.0921 0.1269
0.4 1.1918 1.8311 0.0996 0.2106
0.2 1.1918 1.8311 0.0706 0.0580
0.4 1.1918 1.8311 0.0818 0.0824
0.6 1.1918 1.8311 0.0881 0.0829
0.1 1.1918 1.8311 0.0828 0.0824
0.2 1.1918 1.8311 0.013 0.0821
0.6 1.1918 1.8311 0.0806 0.0815
$ \beta $ $ A $ $ M $ $ \Gamma $ $ \phi $ $ \phi_{2} $ $ \Lambda $ $ Nr $ $ Ec $ $ Bi $ $ C_fRe_x^\frac{1}{2} $ $ C_fRe_x^\frac{1}{2} $ $ N_uRe_x^\frac{-1}{2} $ $ N_uRe_x^\frac{-1}{2} $
$ Cu $-$ EG $ $ Fe_3O_4-Cu/EG $ $ Cu $-$ EG $ $ Fe_3O_4-Cu/EG $
0.01 0.6 0.6 $ \pi/4 $ 0.18 0.09 0.1 0.2 0.2 0.1 1.1918 1.8311 0.0818 0.0824
0.1 2.0011 2.0673 0.0813 0.0819
0.3 2.1039 2.0894 0.0804 0.0808
0.6 1.1918 1.8311 0.0818 0.0824
1.6 2.2308 2.1736 0.0839 0.0879
2.6 2.4621 2.3211 0.0896 0.0899
0.6 1.1918 1.8311 0.0818 0.0824
1.6 2.2661 2.1068 0.0809 0.0820
2.6 2.4149 2.2922 0.0800 0.0815
$ \pi/4 $ 1.1918 1.8311 0.0818 0.0824
$ \pi/3 $ 1.2100 2.0138 0.0810 0.0821
$ \pi/2 $ 1.8610 2.0771 0.0802 0.0820
0.09 2.1314 - 0.0801 -
0.15 2.0229 - 0.0810 -
0.18 1.1912 - 0.0818 -
0.0 - 1.9597 - 0.0811
0.06 - 1.9021 - 0.0818
0.09 - 1.8311 - 0.0824
0.0 2.0121 2.0065 0.0852 0.0858
0.1 1.1918 1.8311 0.0818 0.0824
0.2 1.1728 1.6216 0.0810 0.0820
0.0 1.1918 1.8311 0.0818 0.0824
0.2 1.1918 1.8311 0.0921 0.1269
0.4 1.1918 1.8311 0.0996 0.2106
0.2 1.1918 1.8311 0.0706 0.0580
0.4 1.1918 1.8311 0.0818 0.0824
0.6 1.1918 1.8311 0.0881 0.0829
0.1 1.1918 1.8311 0.0828 0.0824
0.2 1.1918 1.8311 0.013 0.0821
0.6 1.1918 1.8311 0.0806 0.0815
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