July  2021, 14(7): 2497-2515. doi: 10.3934/dcdss.2020399

Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface

a. 

School of Mathematical Sciences, Zhejiang University, Yuquan Campus, Hangzhou 310027, China

b. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

c. 

Department of Computer Engineering, Faculty of Engineering, Ardahan University, Ardahan 75000, Turkey

*Corresponding author: Ghulam Rasool

Received  July 2019 Revised  September 2019 Published  July 2021 Early access  June 2020

Present article aims to investigate the heat and mass transfer developments in boundary layer Jeffery nanofluid flow via Darcy-Forchheimer relation over a stretching surface. A viscous Jeffery naonfluid saturates the porous medium under Darcy-Forchheimer relation. A variable magnetic effect normal to the flow direction is applied to reinforce the electro-magnetic conductivity of the nanofluid. However, small magnetic Reynolds is considered to dismiss the induced magnetic influence. The so-formulated set of governing equations is converted into set of nonlinear ODEs using transformations. Homotopy approach is implemented for convergent relations of velocity field, temperature distribution and the concentration of nanoparticles. Impact of assorted fluid parameters such as local inertial force, Porosity factor, Lewis and Prandtl factors, Brownian diffusion and Thermophoresis on the flow profiles is analyzed diagrammatically. The drag force (skin-friction) and heat-mass flux is especially analyzed through numerical information compiled in tabular form. It has been noticed that the inertial force and porosity factor result in decline of momentum boundary layer but, the scenario is opposite for thermal profile and solute boundary layer. The concentration of nanoparticles increases with increased porosity and inertial effect however, a significant reduction is detected in mass flux.

Citation: Ghulam Rasool, Anum Shafiq, Hülya Durur. Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2497-2515. doi: 10.3934/dcdss.2020399
References:
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K. A. Abro and J. F. Gómez-Aguilar, A comparison of heat and mass transfer on a Walter's-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function, Eur. Phys. J. Plus, 134 (2019). doi: 10.1140/epjp/i2019-12507-4.

[2]

K. A. Abro, I. Khan and J. F. Gómez-Aguilar, Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique, J. Braz. Soc. Mech. Sci. Engrg., 41 (2019). doi: 10.1007/s40430-019-1671-5.

[3]

K. A. AbroM. M. Baig and M. Hussain, Influences of magnetic field in viscoelastic fluid, Int. J. Nonlinear Anal. Appl., 9 (2018), 99-109.  doi: 10.22075/ijnaa.2017.1451.1367.

[4]

K. A. AbroI. KhanK. S. Nisar and A. S. Alsagri, Effects of carbon nanotubes on magnetohydrodynamic flow of methanol based nanofluids via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, Thermal Science, 23 (2019), 883-898.  doi: 10.2298/TSCI180116165A.

[5]

K. A. AbroI. A. AbroS. M. Almani and I. Khan, On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative, J. King Saud Univ. Science, 31 (2019), 973-979.  doi: 10.1016/j.jksus.2018.07.012.

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K. A. AbroA. D. ChandioI. A. Abro and I. Khan, Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives embedded in porous medium, J. Thermal Anal. Calorimetry, 135 (2019), 2197-2207.  doi: 10.1007/s10973-018-7302-z.

[7]

K. A. Abro, M. Hussain and M. M. Baig, An analytic study of molybdenum disulfide nanofluids using the modern approach of Atangana-Baleanu fractional derivatives, Eur. Phys. J. Plus, 132 (2017). doi: 10.1140/epjp/i2017-11689-y.

[8]

K. A. Abro, I. Khan and A. Tassaddiq, Application of Atangana-Baleanu fractional derivative to convection flow of MHD Maxwell fluid in a porous medium over a vertical plate, Math. Model. Nat. Phenom., 13 (2018), 12pp. doi: 10.1051/mmnp/2018007.

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M. I. Afridi, I. Tlili, M. Goodarzi, M. Osman and N. A. Khan, Irreversibility analysis of hybrid nanofluid flow over a thin needle with effects of energy dissipation, Symmetry, 11 (2019). doi: 10.3390/sym11050663.

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A. AlsaediZ. IqbalM. Mustafa and T. Hayat, Exact solutions for the magnetohydrodynamic flow of a Jeffrey fluid with convective boundary conditions and chemical reaction, Z Naturforsch A., 67 (2012), 517-524.  doi: 10.5560/zna.2012-0054.

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E. H. Aly and A. Ebaid, New analytical and numerical solutions for mixed convection boundary-layer nanofluid flow along an inclined plate embedded in a porous medium, J. Appl. Math., 2013 (2013), 7pp. doi: 10.1155/2013/219486.

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J. A. GbadeyanA. S. IdowuA. W. OgunsolaO. O. Agboola and P. O. Olanrewaju, Heat and mass transfer for Soret and Dufour's effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid in the presence of magnetic field, Glob. J. Sci. Front. Res., 11 (2011), 2249-4626. 

[19]

M. Goodarzi, I. Tlili, Z. Tian and M. Safaei, Efficiency assessment of using graphene nanoplatelets-silver/water nanofluids in microchannel heat sinks with different cross-sections for electronics cooling, Int. J. Numer. Methods Heat Fluid Flow, 30 (2019). doi: 10.1108/HFF-12-2018-0730.

[20]

T. HayatQ. Hussain and T. Javed, The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet, Nonlinear Anal. Real World Appl., 10 (2009), 966-973.  doi: 10.1016/j.nonrwa.2007.11.020.

[21]

T. HayatA. AzizT. Muhammad and B. Ahmad, On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet, J. Magnetism Magnetic Materials, 408 (2016), 99-106.  doi: 10.1016/j.jmmm.2016.02.017.

[22]

T. HayatT. MuhammadS. A. Shehzad and A. Alsaedi, A mathematical study for three-dimensional boundary layer flow of Jeffrey nanofluid, Z. Naturforsch. A., 70 (2015), 225-233. 

[23]

T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, Impact of Cattaneo-Christov heat flux in Jeffrey fluid flow with homogeneous-heterogeneous reactions, PLoS ONE, 11 (2016). doi: 10.1371/journal.pone.0148662.

[24]

T. Hayat, T. Abbas, M. Ayub, T. Muhammad and A. Alsaedi, On squeezed flow of Jeffrey nanofluid between two parallel disks, Appl. Sciences, 6 (2016). doi: 10.3390/app6110346.

[25]

W. Ibrahim and O. D. Makinde, The effect of double stratification on boundary-layer flow and heat transfer of nanofluid over a vertical plate, Comput. & Fluids, 86 (2013), 433-441.  doi: 10.1016/j.compfluid.2013.07.029.

[26]

S. M. Imran, S. Asghar and M. Mushtaq, Mixed convection flow over an unsteady stretching surface in a porous medium with heat source, Math. Prob. Eng., 2012 (2012), 15pp. doi: 10.1155/2012/485418.

[27]

N. S. KhanT. GulS. IslamA. Khan and Z. Shah, Brownian motion and Thermophoresis effects on MHD mixed convective thin film second-grade nanofluid flow with hall effect and heat transfer past a stretching sheet, J. Nanofluids, 6 (2017), 812-829.  doi: 10.1166/jon.2017.1383.

[28]

N. S. KhanT. GulM. A. KhanE. Bonyah and S. Islam, Mixed convection in gravity-driven thin film non-Newtonian nanofluids flow with gyrotactic microorganisms, Results Physics, 7 (2017), 4033-4049.  doi: 10.1016/j.rinp.2017.10.017.

[29]

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

[30]

K. U. RehmanM. AwaisA. HussainN. Kousar and M. Y. Malik, Mathematical analysis on MHD Prandtl-Eyring nanofluid new mass flux conditions, Math. Methods Appl. Sci., 42 (2019), 24-38.  doi: 10.1002/mma.5319.

[31]

K. U. Rehman, I. Shahzadi, M. Y. Malik, Q. M. Al-Mdallal and M. Zahri, On heat transfer in the presence of nano-sized particles suspended in a magnetized rotatory flow field, Case Studies Thermal Engrg., 14 (2019). doi: 10.1016/j.csite.2019.100457.

[32]

M. Kothandapani and S. Srinivas, Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel, Int. J. Non-Linear Mech., 43 (2008), 915-924.  doi: 10.1016/j.ijnonlinmec.2008.06.009.

[33]

L. A. Lund, Z. Omar, I. Khan, J. Raza, M. Bakouri and I. Tlili, Stability analysis of Darcy-Forchheimer flow of casson type nanofluid over an exponential sheet: Investigation of critical points, Symmetry, 11 (2019). doi: 10.3390/sym11030412.

[34]

L. A. LundZ. OmarI. Khan and S. Dero, Multiple solutions of $Cu-C_6 H_9 NaO_7$ and $Ag-C_6 H_9 NaO_7$ nanofluids flow over nonlinear shrinking surface, J. Cent. South Univ., 26 (2019), 1283-1293.  doi: 10.1007/s11771-019-4087-6.

[35]

M. MustafaJ. A. KhanT. Hayat and A. Alsaedi, Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, Int. J. Non-Linear Mech., 71 (2015), 22-29.  doi: 10.1016/j.ijnonlinmec.2015.01.005.

[36]

F. MaboodW. A. Khan and A. I. M. Ismail, MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: A numerical study, J. Magnetism Magnetic Materials, 374 (2015), 569-576.  doi: 10.1016/j.jmmm.2014.09.013.

[37]

G. Rasool, T. Zhang and A. Shafiq, Second grade nanofluidic flow past a convectively heated vertical Riga plate, Physica Scripta, 94 (2019). doi: 10.1088/1402-4896/ab3990.

[38]

G. Rasool and T. Zhang, Characteristics of chemical reaction and convective boundary conditions in Powell-Eyring nanofluid flow along a radiative Riga plate, Heliyon, 5 (2019). doi: 10.1016/j.heliyon.2019.e01479.

[39]

G. Rasool, A. Shafiq, C. M. Khalique and T. Zhang, Magnetohydrodynamic Darcy Forchheimer nanofluid flow over a nonlinear stretching sheet, Physica Scripta, 94 (2019). doi: 10.1088/1402-4896/ab18c8.

[40]

G. RasoolT. ZhangA. Shafiq and H. Durur, Influence of chemical reaction on Marangoni convective flow of nanoliquid in the presence of Lorentz forces and thermal radiation: A numerical investigation, J. Adv. Nanotechnology, 1 (2019), 32-39.  doi: 10.14302/issn.2689-2855.jan-19-2598.

[41]

G. RasoolT. Zhang and A. Shafiq, Marangoni effect in second grade forced convective flow of water based nanofluid, J. Adv. Nanotechnology, 1 (2019), 50-61.  doi: 10.14302/issn.2689-2855.jan-19-2716.

[42]

G. Rasool and T. Zhang, Darcy-Forchheimer nanofluidic flow manifested with Cattaneo-Christov theory of heat and mass flux over non-linearly stretching surface, PLoS ONE, 14 (2019). doi: 10.1371/journal.pone.0221302.

[43]

P. V. Satya Narayana, Effects of variable permeability and radiation absorption on magnetohydrodynamic (MHD) mixed convection flow in a vertical wavy channel with travelling thermal waves, Propuls. Power Res., 4 (2015), 150-160.  doi: 10.1016/j.jppr.2015.07.002.

[44]

I. TliliW. A. Khan and K. Ramadan, MHD flow of nanofluid flow across horizontal circular cylinder: Steady forced convection, J. Nanofluids, 8 (2019), 179-186.  doi: 10.1166/jon.2019.1574.

[45]

I. TliliW. A. Khan and K. Ramadan, Entropy generation due to MHD stagnation point flow of a nanofluid on a stretching surface in the presence of radiation, J. Nanofluids, 7 (2018), 879-890.  doi: 10.1166/jon.2018.1513.

[46]

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S. ZuhraN. S. Khan and S. Islam, Magnetohydrodynamic second-grade nanofluid flow containing nanoparticles and gyrotactic microorganisms, Comput. Appl. Math., 37 (2018), 6332-6358.  doi: 10.1007/s40314-018-0683-6.

show all references

References:
[1]

K. A. Abro and J. F. Gómez-Aguilar, A comparison of heat and mass transfer on a Walter's-B fluid via Caputo-Fabrizio versus Atangana-Baleanu fractional derivatives using the Fox-H function, Eur. Phys. J. Plus, 134 (2019). doi: 10.1140/epjp/i2019-12507-4.

[2]

K. A. Abro, I. Khan and J. F. Gómez-Aguilar, Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique, J. Braz. Soc. Mech. Sci. Engrg., 41 (2019). doi: 10.1007/s40430-019-1671-5.

[3]

K. A. AbroM. M. Baig and M. Hussain, Influences of magnetic field in viscoelastic fluid, Int. J. Nonlinear Anal. Appl., 9 (2018), 99-109.  doi: 10.22075/ijnaa.2017.1451.1367.

[4]

K. A. AbroI. KhanK. S. Nisar and A. S. Alsagri, Effects of carbon nanotubes on magnetohydrodynamic flow of methanol based nanofluids via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, Thermal Science, 23 (2019), 883-898.  doi: 10.2298/TSCI180116165A.

[5]

K. A. AbroI. A. AbroS. M. Almani and I. Khan, On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative, J. King Saud Univ. Science, 31 (2019), 973-979.  doi: 10.1016/j.jksus.2018.07.012.

[6]

K. A. AbroA. D. ChandioI. A. Abro and I. Khan, Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives embedded in porous medium, J. Thermal Anal. Calorimetry, 135 (2019), 2197-2207.  doi: 10.1007/s10973-018-7302-z.

[7]

K. A. Abro, M. Hussain and M. M. Baig, An analytic study of molybdenum disulfide nanofluids using the modern approach of Atangana-Baleanu fractional derivatives, Eur. Phys. J. Plus, 132 (2017). doi: 10.1140/epjp/i2017-11689-y.

[8]

K. A. Abro, I. Khan and A. Tassaddiq, Application of Atangana-Baleanu fractional derivative to convection flow of MHD Maxwell fluid in a porous medium over a vertical plate, Math. Model. Nat. Phenom., 13 (2018), 12pp. doi: 10.1051/mmnp/2018007.

[9]

M. I. Afridi, I. Tlili, M. Goodarzi, M. Osman and N. A. Khan, Irreversibility analysis of hybrid nanofluid flow over a thin needle with effects of energy dissipation, Symmetry, 11 (2019). doi: 10.3390/sym11050663.

[10]

A. AlsaediZ. IqbalM. Mustafa and T. Hayat, Exact solutions for the magnetohydrodynamic flow of a Jeffrey fluid with convective boundary conditions and chemical reaction, Z Naturforsch A., 67 (2012), 517-524.  doi: 10.5560/zna.2012-0054.

[11]

E. H. Aly and A. Ebaid, New analytical and numerical solutions for mixed convection boundary-layer nanofluid flow along an inclined plate embedded in a porous medium, J. Appl. Math., 2013 (2013), 7pp. doi: 10.1155/2013/219486.

[12]

J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer, 128 (2006), 240-250.  doi: 10.1115/1.2150834.

[13]

J. ChengS. Liao and I. Pop, Analytical series solution for unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium, Transp. Porous Media, 61 (2005), 365-379.  doi: 10.1007/s11242-005-0546-7.

[14]

S. U. S. Choi and J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, ASME International Mechanical Engineering Congress & Exposition, American Society of Mechanical Engineers, San Francisco, CA, 1995.

[15]

R. Cortell, Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math. Comput., 184 (2007), 864-873.  doi: 10.1016/j.amc.2006.06.077.

[16]

A. S. DogonchiF. Selimefendigil and D. D. Ganji, Magneto-hydrodynamic natural convection of CuO-water nanofluid in complex shaped enclosure considering various nanoparticle shapes, Int. J. Numer. Methods Heat Fluid Flow, 29 (2019), 1663-1679.  doi: 10.1108/HFF-06-2018-0294.

[17]

A. S. DogonchiM. A. SheremetD. D. Ganji and I. Pop, Free convection of copper-water nanofluid in a porous gap between hot rectangular cylinder and cold circular cylinder under the effect of inclined magnetic field, J. Thermal Anal. Calorimetry, 135 (2019), 1171-1184.  doi: 10.1007/s10973-018-7396-3.

[18]

J. A. GbadeyanA. S. IdowuA. W. OgunsolaO. O. Agboola and P. O. Olanrewaju, Heat and mass transfer for Soret and Dufour's effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid in the presence of magnetic field, Glob. J. Sci. Front. Res., 11 (2011), 2249-4626. 

[19]

M. Goodarzi, I. Tlili, Z. Tian and M. Safaei, Efficiency assessment of using graphene nanoplatelets-silver/water nanofluids in microchannel heat sinks with different cross-sections for electronics cooling, Int. J. Numer. Methods Heat Fluid Flow, 30 (2019). doi: 10.1108/HFF-12-2018-0730.

[20]

T. HayatQ. Hussain and T. Javed, The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet, Nonlinear Anal. Real World Appl., 10 (2009), 966-973.  doi: 10.1016/j.nonrwa.2007.11.020.

[21]

T. HayatA. AzizT. Muhammad and B. Ahmad, On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet, J. Magnetism Magnetic Materials, 408 (2016), 99-106.  doi: 10.1016/j.jmmm.2016.02.017.

[22]

T. HayatT. MuhammadS. A. Shehzad and A. Alsaedi, A mathematical study for three-dimensional boundary layer flow of Jeffrey nanofluid, Z. Naturforsch. A., 70 (2015), 225-233. 

[23]

T. Hayat, S. Qayyum, M. Imtiaz and A. Alsaedi, Impact of Cattaneo-Christov heat flux in Jeffrey fluid flow with homogeneous-heterogeneous reactions, PLoS ONE, 11 (2016). doi: 10.1371/journal.pone.0148662.

[24]

T. Hayat, T. Abbas, M. Ayub, T. Muhammad and A. Alsaedi, On squeezed flow of Jeffrey nanofluid between two parallel disks, Appl. Sciences, 6 (2016). doi: 10.3390/app6110346.

[25]

W. Ibrahim and O. D. Makinde, The effect of double stratification on boundary-layer flow and heat transfer of nanofluid over a vertical plate, Comput. & Fluids, 86 (2013), 433-441.  doi: 10.1016/j.compfluid.2013.07.029.

[26]

S. M. Imran, S. Asghar and M. Mushtaq, Mixed convection flow over an unsteady stretching surface in a porous medium with heat source, Math. Prob. Eng., 2012 (2012), 15pp. doi: 10.1155/2012/485418.

[27]

N. S. KhanT. GulS. IslamA. Khan and Z. Shah, Brownian motion and Thermophoresis effects on MHD mixed convective thin film second-grade nanofluid flow with hall effect and heat transfer past a stretching sheet, J. Nanofluids, 6 (2017), 812-829.  doi: 10.1166/jon.2017.1383.

[28]

N. S. KhanT. GulM. A. KhanE. Bonyah and S. Islam, Mixed convection in gravity-driven thin film non-Newtonian nanofluids flow with gyrotactic microorganisms, Results Physics, 7 (2017), 4033-4049.  doi: 10.1016/j.rinp.2017.10.017.

[29]

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

[30]

K. U. RehmanM. AwaisA. HussainN. Kousar and M. Y. Malik, Mathematical analysis on MHD Prandtl-Eyring nanofluid new mass flux conditions, Math. Methods Appl. Sci., 42 (2019), 24-38.  doi: 10.1002/mma.5319.

[31]

K. U. Rehman, I. Shahzadi, M. Y. Malik, Q. M. Al-Mdallal and M. Zahri, On heat transfer in the presence of nano-sized particles suspended in a magnetized rotatory flow field, Case Studies Thermal Engrg., 14 (2019). doi: 10.1016/j.csite.2019.100457.

[32]

M. Kothandapani and S. Srinivas, Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel, Int. J. Non-Linear Mech., 43 (2008), 915-924.  doi: 10.1016/j.ijnonlinmec.2008.06.009.

[33]

L. A. Lund, Z. Omar, I. Khan, J. Raza, M. Bakouri and I. Tlili, Stability analysis of Darcy-Forchheimer flow of casson type nanofluid over an exponential sheet: Investigation of critical points, Symmetry, 11 (2019). doi: 10.3390/sym11030412.

[34]

L. A. LundZ. OmarI. Khan and S. Dero, Multiple solutions of $Cu-C_6 H_9 NaO_7$ and $Ag-C_6 H_9 NaO_7$ nanofluids flow over nonlinear shrinking surface, J. Cent. South Univ., 26 (2019), 1283-1293.  doi: 10.1007/s11771-019-4087-6.

[35]

M. MustafaJ. A. KhanT. Hayat and A. Alsaedi, Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet, Int. J. Non-Linear Mech., 71 (2015), 22-29.  doi: 10.1016/j.ijnonlinmec.2015.01.005.

[36]

F. MaboodW. A. Khan and A. I. M. Ismail, MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: A numerical study, J. Magnetism Magnetic Materials, 374 (2015), 569-576.  doi: 10.1016/j.jmmm.2014.09.013.

[37]

G. Rasool, T. Zhang and A. Shafiq, Second grade nanofluidic flow past a convectively heated vertical Riga plate, Physica Scripta, 94 (2019). doi: 10.1088/1402-4896/ab3990.

[38]

G. Rasool and T. Zhang, Characteristics of chemical reaction and convective boundary conditions in Powell-Eyring nanofluid flow along a radiative Riga plate, Heliyon, 5 (2019). doi: 10.1016/j.heliyon.2019.e01479.

[39]

G. Rasool, A. Shafiq, C. M. Khalique and T. Zhang, Magnetohydrodynamic Darcy Forchheimer nanofluid flow over a nonlinear stretching sheet, Physica Scripta, 94 (2019). doi: 10.1088/1402-4896/ab18c8.

[40]

G. RasoolT. ZhangA. Shafiq and H. Durur, Influence of chemical reaction on Marangoni convective flow of nanoliquid in the presence of Lorentz forces and thermal radiation: A numerical investigation, J. Adv. Nanotechnology, 1 (2019), 32-39.  doi: 10.14302/issn.2689-2855.jan-19-2598.

[41]

G. RasoolT. Zhang and A. Shafiq, Marangoni effect in second grade forced convective flow of water based nanofluid, J. Adv. Nanotechnology, 1 (2019), 50-61.  doi: 10.14302/issn.2689-2855.jan-19-2716.

[42]

G. Rasool and T. Zhang, Darcy-Forchheimer nanofluidic flow manifested with Cattaneo-Christov theory of heat and mass flux over non-linearly stretching surface, PLoS ONE, 14 (2019). doi: 10.1371/journal.pone.0221302.

[43]

P. V. Satya Narayana, Effects of variable permeability and radiation absorption on magnetohydrodynamic (MHD) mixed convection flow in a vertical wavy channel with travelling thermal waves, Propuls. Power Res., 4 (2015), 150-160.  doi: 10.1016/j.jppr.2015.07.002.

[44]

I. TliliW. A. Khan and K. Ramadan, MHD flow of nanofluid flow across horizontal circular cylinder: Steady forced convection, J. Nanofluids, 8 (2019), 179-186.  doi: 10.1166/jon.2019.1574.

[45]

I. TliliW. A. Khan and K. Ramadan, Entropy generation due to MHD stagnation point flow of a nanofluid on a stretching surface in the presence of radiation, J. Nanofluids, 7 (2018), 879-890.  doi: 10.1166/jon.2018.1513.

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M. Turkyilmazoglu and I. Pop, Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect, Int. J. Heat Mass Transfer, 59 (2013), 167-171.  doi: 10.1016/j.ijheatmasstransfer.2012.12.009.

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M. Turkyilmazoglu and I. Pop, Exact analytical solutions for the flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid, Int. J. Heat Mass Transfer, 57 (2013), 82-88.  doi: 10.1016/j.ijheatmasstransfer.2012.10.006.

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S. ZuhraN. S. Khan and S. Islam, Magnetohydrodynamic second-grade nanofluid flow containing nanoparticles and gyrotactic microorganisms, Comput. Appl. Math., 37 (2018), 6332-6358.  doi: 10.1007/s40314-018-0683-6.

Figure 1.  Physical model and coordinate system
Figure 2.  H-curves
Figure 3.  Graph of $ f'(\eta) $ against $ \beta $
Figure 4.  Graph of $ f'(\eta) $ against $ F_r $
Figure 5.  Graph of $ f'(\eta) $ against $ \lambda_1 $
Figure 6.  Graph of $ f'(\eta) $ against $ \lambda $
Figure 7.  Graph of $ \theta(\eta) $ against $ \lambda_1 $
Figure 8.  Graph of $ \theta(\eta) $ against $ N_b $
Figure 9.  Graph of $ \theta(\eta) $ against $ N_t $
Figure 10.  Graph of $ \theta(\eta) $ against $ \lambda $
Figure 11.  Graph of $ \theta(\eta) $ against $ Pr $
Figure 12.  Graph of $ \phi(\eta) $ against $ \beta $
Figure 13.  Graph of $ \phi(\eta) $ against $ F_r $
Figure 14.  Graph of $ \phi(\eta) $ against $ \lambda_1 $
Figure 15.  Graph of $ \phi(\eta) $ against $ Le $
Figure 16.  Graph of $ \phi(\eta) $ against $ Nt $
Table 1.  Convergence
Order$-\frac{\partial^2f}{\partial\eta^2}(0)$$-\frac{\partial \theta}{\partial \eta}(0)$$-\frac{\partial \phi}{\partial \eta}(0)$
$1$$1.13462$$0.60003$$0.49222$
$5$$1.15525$$0.48633$$0.43226$
$10$$1.15526$$0.47564$$0.38262$
$15$$1.15526$$0.47404$$0.37521$
$25$$1.15526$$0.46484$$0.36552$
$35$$1.15526$$0.46484$$0.36552$
$50$$1.15526$$0.46484$$0.36552$
Order$-\frac{\partial^2f}{\partial\eta^2}(0)$$-\frac{\partial \theta}{\partial \eta}(0)$$-\frac{\partial \phi}{\partial \eta}(0)$
$1$$1.13462$$0.60003$$0.49222$
$5$$1.15525$$0.48633$$0.43226$
$10$$1.15526$$0.47564$$0.38262$
$15$$1.15526$$0.47404$$0.37521$
$25$$1.15526$$0.46484$$0.36552$
$35$$1.15526$$0.46484$$0.36552$
$50$$1.15526$$0.46484$$0.36552$
Table 2.  Variation in Skin - friction (Drag force)
$M$$F_{r}$$\lambda $$-\text{Re}_{x}^{1/2}C_{fx}$
$0.1$$0.1$$0.2$$1.2551$
$0.5$$1.3559$
$1.0$$1.8553$
$0.2$$0.1$$0.3$$1.1532$
$0.5$$1.3452$
$1.0$$1.5028$
$0.2$$0.1$$0.1$$1.0732$
$0.5$$1.3132$
$1.0$$1.5532$
$M$$F_{r}$$\lambda $$-\text{Re}_{x}^{1/2}C_{fx}$
$0.1$$0.1$$0.2$$1.2551$
$0.5$$1.3559$
$1.0$$1.8553$
$0.2$$0.1$$0.3$$1.1532$
$0.5$$1.3452$
$1.0$$1.5028$
$0.2$$0.1$$0.1$$1.0732$
$0.5$$1.3132$
$1.0$$1.5532$
Table 3.  Variation in Nusslt and Sherwood numbers
$M$$F_{r}$$\lambda $$Pr$$Nb$$Nt$$Le$$-\mathrm{{Re}}_{x}^{-1/2}Nu_{x}$$-\mathrm{{Re}}_{x}^{-1/2}Sh_{x}$
$0.1$$0.1$$0.2$$1.0$$0.2$$0.1$$1.0$$0.4998$$0.3992$
$0.5$$0.4772$$0.3658$
$1.0$$0.4320$$0.3324$
$0.2$$0.1$$0.2$$1.0$$0.2$$0.1$$1.0$$0.5014$$0.4158$
$0.5$$0.4882$$0.3852$
$1.0$$0.4724$$0.3682$
$0.2$$0.1$$0.1$$1.0$$0.2$$0.1$$1.0$$0.5656$$0.4442$
$0.5$$0.4965$$0.4012$
$1.0$$0.4647$$0.3687$
$0.2$$0.1$$0.2$$0.1$$0.2$$0.1$$1.0$$0.4254$$0.3220$
$0.5$$0.5185$$0.4132$
$1.3$$0.6067$$0.5568$
$0.2$$0.1$$0.2$$1.0$$0.4$$0.1$$1.0$$0.4232$$0.5158$
$0.8$$0.3637$$0.5370$
$1.2$$0.3175$$0.5701$
$0.2$$0.1$$0.2$$1.0$$0.2$$0.1$$1.0$$0.4344$$0.5598$
$0.5$$0.3988$$0.4007$
$1.0$$0.3655$$0.3202$
$0.2$$0.1$$0.2$$1.0$$0.2$$0.1$$0.4$$0.4656$$0.2331$
$0.8$$0.5112$$0.4125$
$1.2$$0.5551$$0.6365$
$M$$F_{r}$$\lambda $$Pr$$Nb$$Nt$$Le$$-\mathrm{{Re}}_{x}^{-1/2}Nu_{x}$$-\mathrm{{Re}}_{x}^{-1/2}Sh_{x}$
$0.1$$0.1$$0.2$$1.0$$0.2$$0.1$$1.0$$0.4998$$0.3992$
$0.5$$0.4772$$0.3658$
$1.0$$0.4320$$0.3324$
$0.2$$0.1$$0.2$$1.0$$0.2$$0.1$$1.0$$0.5014$$0.4158$
$0.5$$0.4882$$0.3852$
$1.0$$0.4724$$0.3682$
$0.2$$0.1$$0.1$$1.0$$0.2$$0.1$$1.0$$0.5656$$0.4442$
$0.5$$0.4965$$0.4012$
$1.0$$0.4647$$0.3687$
$0.2$$0.1$$0.2$$0.1$$0.2$$0.1$$1.0$$0.4254$$0.3220$
$0.5$$0.5185$$0.4132$
$1.3$$0.6067$$0.5568$
$0.2$$0.1$$0.2$$1.0$$0.4$$0.1$$1.0$$0.4232$$0.5158$
$0.8$$0.3637$$0.5370$
$1.2$$0.3175$$0.5701$
$0.2$$0.1$$0.2$$1.0$$0.2$$0.1$$1.0$$0.4344$$0.5598$
$0.5$$0.3988$$0.4007$
$1.0$$0.3655$$0.3202$
$0.2$$0.1$$0.2$$1.0$$0.2$$0.1$$0.4$$0.4656$$0.2331$
$0.8$$0.5112$$0.4125$
$1.2$$0.5551$$0.6365$
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