July  2021, 14(7): 2517-2533. doi: 10.3934/dcdss.2021059

Marangoni forced convective Casson type nanofluid flow in the presence of Lorentz force generated by Riga plate

1. 

Binjiang College, Nanjing University of Information Science and Technology, Wuxi 214105, Jiangsu, China

2. 

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

3. 

International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

* Corresponding author: Anum Shafiq

Received  June 2019 Revised  December 2020 Published  May 2021

The present communication aims to investigate Marangoni based convective Casson modeled nanofluid flow influenced by the presence of Lorentz forces instigated into the model by an aligned array of magnets in the form of Riga pattern. The exponentially decaying Lorentz force is considered using the Grinberg term. On the liquid - gas or liquid - liquid interface, a realistic temperature and concentration distribution is considered with the assumption that temperature and concentration distributions are variable functions of $ x $. The set of so-formulated governing problems under the umbrella of Navier Stokes equations is transformed into nonlinear ODEs using suitable transformations. Homotopy approach is implemented to achieve convergent series solutions for the said problem. Influence of active fluid parameters such as Casson parameter, Brownian diffusion, Prandtl number, Thermophoresis and others on flow profiles is analyzed graphically. The fluctuation in local physical quantities such as heat and mass flux rates, is noticed to check the significance of current fluid model in many industrial as well as engineering procedures using nanofluids. The outcomes indicate that the effective Lorentz force assists the fluid motion that results in an augmented velocity profile with incremental values of modified Hartman number. Furthermore, incremental data of Casson parameter motivates significant reduction in velocity profile.

Citation: Ghulam Rasool, Anum Shafiq, Chaudry Masood Khalique. Marangoni forced convective Casson type nanofluid flow in the presence of Lorentz force generated by Riga plate. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2517-2533. doi: 10.3934/dcdss.2021059
References:
[1]

A. AdeelA. Saleem and A. Sumaira, Flow of nanofluid past a Riga plate, Journal of Magnetism and Magnetic Materials, 402 (2016), 44-48.   Google Scholar

[2]

R. AhmadM. Mustafa and M. Turkyilmazoglu, Buoyancy effects on nanofluid flow past a convectively heated vertical Riga-plate: A numerical study, Int. J. Heat and Mass. Trans., 111 (2017), 827-835.   Google Scholar

[3]

B. Ali, G. Rasool, S. Hussain, D. Baleanu and S. Bano, Finite Element Study of Magnetohydrodynamics (MHD) and Activation Energy in Darcy-Forchheimer Rotating Flow of Casson Carreau Nanofluid, Processes, 8 (2020), 1185. Google Scholar

[4]

N. A. Asif, Z. Hammouch, M. B. Riaz et al., Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 272. Google Scholar

[5]

A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?, Chaos Solitons Fractals, 136 (2020), 109860, 38 pp. doi: 10.1016/j.chaos.2020.109860.  Google Scholar

[6]

S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, D. App. Non-Newtonian Flows., 231 (1995), 99-105.   Google Scholar

[7]

K. Ganesh Kumar, B. J. Gireesha, B. C. Prasanna umara and O. D. Makinde, Impact of chemical reaction on marangoni boundary layer flow of a casson Nano Liquid in the presence of uniform heat source sink, Diffusion Foundations, 11, 22–32. doi: 10.4028/www.scientific.net/DF.11.22.  Google Scholar

[8]

A. Gailitis and O. Lielausis, On a possibility to reduce the hydrodynamic resistance of a plate in an electrolyte, Appl. Mag. Rep. Phys. Inst., 12 (1961), 143-6.   Google Scholar

[9]

T. HayatS. QayyumA. Alsaedi and A. Shafiq, Inclined magnetic field and heat source/sink aspects in flow of nanofluid with nonlinear thermal radiation, Int. J. H. M. Trans., 103 (2016), 99-107.   Google Scholar

[10]

T. HayatS. A. Shehzad and A. Alsaedi, Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid, Appl. Math. Mech. (English Ed.), 33 (2012), 1301-1312.  doi: 10.1007/s10483-012-1623-6.  Google Scholar

[11]

T. Hayat, U. Shaheen, A. Shafiq, A. Alsaedi and S. Asghar, Marangoni mixed convection flow with Joule heating and nonlinear radiation, AIP Advances, 5 (2015), 077140. doi: 10.1063/1.4927209.  Google Scholar

[12]

G. IbanezA. LupezJ. Pantoja and J. Moreira, Entropy generation analysis of a nanofluid flow in MHD porous microchannel with hydrodynamic slip and thermal radiation, Int. J. H. M. Trans., 100 (2016), 89-97.   Google Scholar

[13]

M. A. ImranM. B. RiazN. A. Shah and A. A. Zafar, Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary, Results in Physics, 8 (2018), 1061-1067.   Google Scholar

[14]

Z. IqbalE. AzharZ. Mehmood and E. N. Maraj, Melting heat transport of nanofluidic problem over a Riga plate with erratic thickness: Use of Keller Box scheme, Results in Physics, 7 (2017), 3648-3658.  doi: 10.1016/j.rinp.2017.09.047.  Google Scholar

[15]

A. KamranS. HussainM. Sagheer and N. Akmal, A numerical study of magnetohydrodynamics w in Casson nano id combined with Joule heating and slip boundary conditions, Results in Physics, 7 (2017), 3037-3048.   Google Scholar

[16]

S. J. Liao, Homotopy analysis method: A new analytic method for nonlinear problems, Appl. Math. Mech. (English Ed.), 19 (1998), 957-962.  doi: 10.1007/BF02457955.  Google Scholar

[17]

A. LopezG. IbanezJ. PantojaJ. Moreira and O. Lastres, Entropy generation analysis of MHD nanofluid flow in a porous vertical microchannel with nonlinear thermal radiation, slip flow and convective-radiative boundary conditions, Int. J. H. M. Trans., 107 (2017), 982-994.   Google Scholar

[18]

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. Mag. Mag. Mat., 374 (2015), 569-576.   Google Scholar

[19]

B. Mahanthesh and B. J. Gireesha, Thermal Marangoni convection in two-phase flow of dusty Casson fluid, Results in Physics, 8 (2018), 537-544.   Google Scholar

[20]

B. MahantheshB. J. GireeshaB. C. Prasannakumara and P. B. Sampath Kumar, Magneto-Thermo-Marangoni convective flow of $Cu-H_2O$ nanoliquid past an infinite disk with particle shape and exponential space based heat source effects, Results in Physics, 7 (2017), 2990-2996.   Google Scholar

[21]

B. MahantheshB. J. GireeshaN. S. ShashikumarT. Hayat and A. Alsaedi, Marangoni convection in Casson liquid flow due to an infinite disk with exponential space dependent heat source and cross-diffusion effects, Results in Physics, 9 (2018), 78-85.   Google Scholar

[22]

S. NadeemR. HaqN. S.Akbar and Z. H. Khan, MHD three dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Eng. J., 52 (2013), 577-582.   Google Scholar

[23]

J. R. A. Pearson, On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489-500.   Google Scholar

[24]

M. RamzanM. Bilal and J. D. Chung, Radiative Williamson nanofluid flow over a convectively heated Riga plate with chemical reaction-A numerical approach, Chinese Journal of Physics, 55 (2017), 1663-1673.   Google Scholar

[25]

G. Rasool, W. A. Khan, S. M. Bilal and I. Khan, MHD squeezed Darcy-Forchheimer nanofluid flow between two h-distance apart horizontal plates, Open Physics, 18 (2020). doi: 10.1515/phys-2020-0191.  Google Scholar

[26]

G. Rasool and A. Shafiq, Numerical Exploration of the Features of Thermally Enhanced Chemically Reactive Radiative Powell-Eyring Nanofluid Flow via Darcy Medium over Non-linearly Stretching Surface Affected by a Transverse Magnetic Field and Convective Boundary Conditions, Applied Nanoscience, 2020. doi: 10.1007/s13204-020-01625-2.  Google Scholar

[27]

G. Rasool, A. Shafiq and D. Baleanu, Consequences of Soret-Dufour effects, thermal radiation, and binary chemical reaction on darcy forchheimer flow of nanofluids, Symmetry, 12 (2020), 1421. doi: 10.3390/sym12091421.  Google Scholar

[28]

G. Rasool, A. Shafiq and H. Durur, Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface, Discrete and Continuous Dynamical Systems - Series S, (2019). doi: 10.3934/dcdss.2020399.  Google Scholar

[29]

G. Rasool, A. Shafiq, C. M. Khalique and T. Zhang, Magnetohydrodynamic Darcy Forchheimer nanofluid flow over nonlinear stretching sheet, Phys. Scr., 94 (2019), 105221. Google Scholar

[30]

G. Rasool and A. Wakif, Numerical spectral examination of EMHD mixed convective flow of second-grade nanofluid towards a vertical Riga plate using an advanced version of the revised Buongiorno's nanofluid model, J. Therm. Anal. Calorim., 143 (2021), 2379-2393.  doi: 10.1007/s10973-020-09865-8.  Google Scholar

[31]

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), e0221302. Google Scholar

[32]

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.  Google Scholar

[33]

G. Rasool, T. Zhang and A. Shafiq, Second grade nanofluidic flow past a convectively heated vertical Riga plate, Physica Scripta., 94 (2019), 125212. Google Scholar

[34]

M. B. Riaz and N. Iftikhar, A comparative study of heat transfer analysis of MHD Maxwell fluid in view of local and nonlocal differential operators, Chaos Solitons Fractals, 132 (2020), 109556, 19 pp. doi: 10.1016/j.chaos.2019.109556.  Google Scholar

[35]

M. B. Riaz and A. A. Zafar, Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives, Math. Model. Nat. Phenom., 13 (2018), Paper No. 8, 12 pp. doi: 10.1051/mmnp/2018005.  Google Scholar

[36]

G. Sarojamma and K. Vendabai, Boundary layer fow of a Casson nanofluid past a vertical exponentially stretching cylinder in the presence of a transverse magnetic feld with internal heat generation/absorption, World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 9 (2015), 138-143.   Google Scholar

[37]

L. E. Scriven and C. V. Sternling, The Marangoni effects, Nature, 187 (1960), 186-188.   Google Scholar

[38]

A. ShafiqZ. Hammouch and A. Turab, Impact of radiation in a stagnation point flow of Walters B fluid towards a Riga plate, Thermal Science and Engineering Progress, 6 (2018), 27-33.   Google Scholar

[39]

M. Sheikholeslami and A. J. Chamkha, Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection, Journal of Molecular Liquids, 225 (2017), 750-757.   Google Scholar

[40]

M. SheikholeslamiM. B. Gerdroodbary and D. D. Ganji, Numerical investigation of forced convective heat transfer of Fe-water nanofluid in the presence of external magnetic source, Comp. Meth. App. Mech. Eng., 315 (2017), 831-845.  doi: 10.1016/j.cma.2016.11.021.  Google Scholar

[41]

A. Uddin, D. Estevez, F. X. Qin and H. X. Peng, Programmable microwire composites: From functional units to material design, J. Phys. D Appl. Phys. Pap., 53 (2020). Google Scholar

[42]

A. S. Uddin, A. Evstigneeva, A. Dzhumazoda, M. M. Salem, M. G. Nematov, A. M. Adam, L. V. Panina and A. T. Marchenko, Temperature effects on the magnetization and magnetoimpedance in ferromagnetic Glass-Covered microwires, Institute of Physics Conference Series, (2017). Google Scholar

[43]

A. Uddin, F. X. Qin, D. Estevez, S. D. Jiang, L. V. Panina and H. X. Peng, Microwave programmable response of Co-based microwire polymer composites through wire microstructure and arrangement optimization, Composites Part B, 176 (2019), 107190. Google Scholar

[44]

Y. L. Xu, A. Uddin, D. Estevez, Y. Luo, H. X. Peng and F. X. Qin, Lightweight microwire/graphene/silicone rubber composites for efficient electromagnetic interference shielding and low microwave reflectivity, Compos. Sci. Technol., 189 (2020). Google Scholar

[45]

S. ZhaoF. X. QinY. LuoY. WangA. UddinX. ZhengD. EstevezH. Wang and H. X. Peng, Responsive left-handed behaviour of ferromagnetic microwire composites by in-situ electric and magnetic fields, Compos. Commun., 19 (2020), 246-252.   Google Scholar

show all references

References:
[1]

A. AdeelA. Saleem and A. Sumaira, Flow of nanofluid past a Riga plate, Journal of Magnetism and Magnetic Materials, 402 (2016), 44-48.   Google Scholar

[2]

R. AhmadM. Mustafa and M. Turkyilmazoglu, Buoyancy effects on nanofluid flow past a convectively heated vertical Riga-plate: A numerical study, Int. J. Heat and Mass. Trans., 111 (2017), 827-835.   Google Scholar

[3]

B. Ali, G. Rasool, S. Hussain, D. Baleanu and S. Bano, Finite Element Study of Magnetohydrodynamics (MHD) and Activation Energy in Darcy-Forchheimer Rotating Flow of Casson Carreau Nanofluid, Processes, 8 (2020), 1185. Google Scholar

[4]

N. A. Asif, Z. Hammouch, M. B. Riaz et al., Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 272. Google Scholar

[5]

A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?, Chaos Solitons Fractals, 136 (2020), 109860, 38 pp. doi: 10.1016/j.chaos.2020.109860.  Google Scholar

[6]

S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, D. App. Non-Newtonian Flows., 231 (1995), 99-105.   Google Scholar

[7]

K. Ganesh Kumar, B. J. Gireesha, B. C. Prasanna umara and O. D. Makinde, Impact of chemical reaction on marangoni boundary layer flow of a casson Nano Liquid in the presence of uniform heat source sink, Diffusion Foundations, 11, 22–32. doi: 10.4028/www.scientific.net/DF.11.22.  Google Scholar

[8]

A. Gailitis and O. Lielausis, On a possibility to reduce the hydrodynamic resistance of a plate in an electrolyte, Appl. Mag. Rep. Phys. Inst., 12 (1961), 143-6.   Google Scholar

[9]

T. HayatS. QayyumA. Alsaedi and A. Shafiq, Inclined magnetic field and heat source/sink aspects in flow of nanofluid with nonlinear thermal radiation, Int. J. H. M. Trans., 103 (2016), 99-107.   Google Scholar

[10]

T. HayatS. A. Shehzad and A. Alsaedi, Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid, Appl. Math. Mech. (English Ed.), 33 (2012), 1301-1312.  doi: 10.1007/s10483-012-1623-6.  Google Scholar

[11]

T. Hayat, U. Shaheen, A. Shafiq, A. Alsaedi and S. Asghar, Marangoni mixed convection flow with Joule heating and nonlinear radiation, AIP Advances, 5 (2015), 077140. doi: 10.1063/1.4927209.  Google Scholar

[12]

G. IbanezA. LupezJ. Pantoja and J. Moreira, Entropy generation analysis of a nanofluid flow in MHD porous microchannel with hydrodynamic slip and thermal radiation, Int. J. H. M. Trans., 100 (2016), 89-97.   Google Scholar

[13]

M. A. ImranM. B. RiazN. A. Shah and A. A. Zafar, Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary, Results in Physics, 8 (2018), 1061-1067.   Google Scholar

[14]

Z. IqbalE. AzharZ. Mehmood and E. N. Maraj, Melting heat transport of nanofluidic problem over a Riga plate with erratic thickness: Use of Keller Box scheme, Results in Physics, 7 (2017), 3648-3658.  doi: 10.1016/j.rinp.2017.09.047.  Google Scholar

[15]

A. KamranS. HussainM. Sagheer and N. Akmal, A numerical study of magnetohydrodynamics w in Casson nano id combined with Joule heating and slip boundary conditions, Results in Physics, 7 (2017), 3037-3048.   Google Scholar

[16]

S. J. Liao, Homotopy analysis method: A new analytic method for nonlinear problems, Appl. Math. Mech. (English Ed.), 19 (1998), 957-962.  doi: 10.1007/BF02457955.  Google Scholar

[17]

A. LopezG. IbanezJ. PantojaJ. Moreira and O. Lastres, Entropy generation analysis of MHD nanofluid flow in a porous vertical microchannel with nonlinear thermal radiation, slip flow and convective-radiative boundary conditions, Int. J. H. M. Trans., 107 (2017), 982-994.   Google Scholar

[18]

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. Mag. Mag. Mat., 374 (2015), 569-576.   Google Scholar

[19]

B. Mahanthesh and B. J. Gireesha, Thermal Marangoni convection in two-phase flow of dusty Casson fluid, Results in Physics, 8 (2018), 537-544.   Google Scholar

[20]

B. MahantheshB. J. GireeshaB. C. Prasannakumara and P. B. Sampath Kumar, Magneto-Thermo-Marangoni convective flow of $Cu-H_2O$ nanoliquid past an infinite disk with particle shape and exponential space based heat source effects, Results in Physics, 7 (2017), 2990-2996.   Google Scholar

[21]

B. MahantheshB. J. GireeshaN. S. ShashikumarT. Hayat and A. Alsaedi, Marangoni convection in Casson liquid flow due to an infinite disk with exponential space dependent heat source and cross-diffusion effects, Results in Physics, 9 (2018), 78-85.   Google Scholar

[22]

S. NadeemR. HaqN. S.Akbar and Z. H. Khan, MHD three dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Eng. J., 52 (2013), 577-582.   Google Scholar

[23]

J. R. A. Pearson, On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489-500.   Google Scholar

[24]

M. RamzanM. Bilal and J. D. Chung, Radiative Williamson nanofluid flow over a convectively heated Riga plate with chemical reaction-A numerical approach, Chinese Journal of Physics, 55 (2017), 1663-1673.   Google Scholar

[25]

G. Rasool, W. A. Khan, S. M. Bilal and I. Khan, MHD squeezed Darcy-Forchheimer nanofluid flow between two h-distance apart horizontal plates, Open Physics, 18 (2020). doi: 10.1515/phys-2020-0191.  Google Scholar

[26]

G. Rasool and A. Shafiq, Numerical Exploration of the Features of Thermally Enhanced Chemically Reactive Radiative Powell-Eyring Nanofluid Flow via Darcy Medium over Non-linearly Stretching Surface Affected by a Transverse Magnetic Field and Convective Boundary Conditions, Applied Nanoscience, 2020. doi: 10.1007/s13204-020-01625-2.  Google Scholar

[27]

G. Rasool, A. Shafiq and D. Baleanu, Consequences of Soret-Dufour effects, thermal radiation, and binary chemical reaction on darcy forchheimer flow of nanofluids, Symmetry, 12 (2020), 1421. doi: 10.3390/sym12091421.  Google Scholar

[28]

G. Rasool, A. Shafiq and H. Durur, Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface, Discrete and Continuous Dynamical Systems - Series S, (2019). doi: 10.3934/dcdss.2020399.  Google Scholar

[29]

G. Rasool, A. Shafiq, C. M. Khalique and T. Zhang, Magnetohydrodynamic Darcy Forchheimer nanofluid flow over nonlinear stretching sheet, Phys. Scr., 94 (2019), 105221. Google Scholar

[30]

G. Rasool and A. Wakif, Numerical spectral examination of EMHD mixed convective flow of second-grade nanofluid towards a vertical Riga plate using an advanced version of the revised Buongiorno's nanofluid model, J. Therm. Anal. Calorim., 143 (2021), 2379-2393.  doi: 10.1007/s10973-020-09865-8.  Google Scholar

[31]

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), e0221302. Google Scholar

[32]

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.  Google Scholar

[33]

G. Rasool, T. Zhang and A. Shafiq, Second grade nanofluidic flow past a convectively heated vertical Riga plate, Physica Scripta., 94 (2019), 125212. Google Scholar

[34]

M. B. Riaz and N. Iftikhar, A comparative study of heat transfer analysis of MHD Maxwell fluid in view of local and nonlocal differential operators, Chaos Solitons Fractals, 132 (2020), 109556, 19 pp. doi: 10.1016/j.chaos.2019.109556.  Google Scholar

[35]

M. B. Riaz and A. A. Zafar, Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives, Math. Model. Nat. Phenom., 13 (2018), Paper No. 8, 12 pp. doi: 10.1051/mmnp/2018005.  Google Scholar

[36]

G. Sarojamma and K. Vendabai, Boundary layer fow of a Casson nanofluid past a vertical exponentially stretching cylinder in the presence of a transverse magnetic feld with internal heat generation/absorption, World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 9 (2015), 138-143.   Google Scholar

[37]

L. E. Scriven and C. V. Sternling, The Marangoni effects, Nature, 187 (1960), 186-188.   Google Scholar

[38]

A. ShafiqZ. Hammouch and A. Turab, Impact of radiation in a stagnation point flow of Walters B fluid towards a Riga plate, Thermal Science and Engineering Progress, 6 (2018), 27-33.   Google Scholar

[39]

M. Sheikholeslami and A. J. Chamkha, Influence of Lorentz forces on nanofluid forced convection considering Marangoni convection, Journal of Molecular Liquids, 225 (2017), 750-757.   Google Scholar

[40]

M. SheikholeslamiM. B. Gerdroodbary and D. D. Ganji, Numerical investigation of forced convective heat transfer of Fe-water nanofluid in the presence of external magnetic source, Comp. Meth. App. Mech. Eng., 315 (2017), 831-845.  doi: 10.1016/j.cma.2016.11.021.  Google Scholar

[41]

A. Uddin, D. Estevez, F. X. Qin and H. X. Peng, Programmable microwire composites: From functional units to material design, J. Phys. D Appl. Phys. Pap., 53 (2020). Google Scholar

[42]

A. S. Uddin, A. Evstigneeva, A. Dzhumazoda, M. M. Salem, M. G. Nematov, A. M. Adam, L. V. Panina and A. T. Marchenko, Temperature effects on the magnetization and magnetoimpedance in ferromagnetic Glass-Covered microwires, Institute of Physics Conference Series, (2017). Google Scholar

[43]

A. Uddin, F. X. Qin, D. Estevez, S. D. Jiang, L. V. Panina and H. X. Peng, Microwave programmable response of Co-based microwire polymer composites through wire microstructure and arrangement optimization, Composites Part B, 176 (2019), 107190. Google Scholar

[44]

Y. L. Xu, A. Uddin, D. Estevez, Y. Luo, H. X. Peng and F. X. Qin, Lightweight microwire/graphene/silicone rubber composites for efficient electromagnetic interference shielding and low microwave reflectivity, Compos. Sci. Technol., 189 (2020). Google Scholar

[45]

S. ZhaoF. X. QinY. LuoY. WangA. UddinX. ZhengD. EstevezH. Wang and H. X. Peng, Responsive left-handed behaviour of ferromagnetic microwire composites by in-situ electric and magnetic fields, Compos. Commun., 19 (2020), 246-252.   Google Scholar

Figure 1.  Geometry of the problem
Figure 2.  HAM Curves
Figure 3.  Variations noted in $ f'(\eta) $ for incremental $ \beta $
Figure 4.  Variations noted in $ f'(\eta) $ for incremental $ Q_1 $
Figure 5.  Variations noted in $ f'(\eta) $ for incremental $ r $
Figure 6.  Variations noted in $ \theta(\eta) $ for incremental $ Q_1 $
Figure 7.  Variations noted in $ \theta(\eta) $ for incremental $ N_b $
Figure 8.  Variations noted in $ \theta(\eta) $ for incremental $ Nt $
Figure 9.  Variations noted in $ \theta(\eta) $ for incremental $ Ec $
Figure 10.  Variations noted in $ \phi(\eta) $ for incremental $ Sc $
Figure 11.  Variations noted in $ \phi(\eta) $ for incremental $ Nt $
Figure 12.  Variations noted in Nusselt number for incremental $ Nt $
Figure 13.  Variations noted in Sherwood number for incremental $ Sc $
Table Ⅰ.  Convergence
Orders $-f^{\prime \prime }$ $-\theta ^{\prime }$ $-\phi ^{\prime }$
$1$ $1.141912$ $0.8995612$ $1.5211265$
$5$ $1.256923$ $0.9566101$ $1.3211122$
$10$ $1.300122$ $1.1223114$ $1.1886111$
$15$ $1.300222$ $1.2623532$ $0.9956332$
$20$ $1.300222$ $1.3112112$ $0.8915622$
$25$ $1.300222$ $1.3112211$ $0.8915512$
$30$ $1.300222$ $1.3112211$ $0.8915512$
$35$ $1.300222$ $1.3112211$ $0.8915512$
$40$ $1.300222$ $1.3112211$ $0.8915512$
$50$ $1.300222$ $1.3112211$ $0.8915512$
Orders $-f^{\prime \prime }$ $-\theta ^{\prime }$ $-\phi ^{\prime }$
$1$ $1.141912$ $0.8995612$ $1.5211265$
$5$ $1.256923$ $0.9566101$ $1.3211122$
$10$ $1.300122$ $1.1223114$ $1.1886111$
$15$ $1.300222$ $1.2623532$ $0.9956332$
$20$ $1.300222$ $1.3112112$ $0.8915622$
$25$ $1.300222$ $1.3112211$ $0.8915512$
$30$ $1.300222$ $1.3112211$ $0.8915512$
$35$ $1.300222$ $1.3112211$ $0.8915512$
$40$ $1.300222$ $1.3112211$ $0.8915512$
$50$ $1.300222$ $1.3112211$ $0.8915512$
Table Ⅱ.  Comparison of Nusselt and Sherwood numbers with Hayat et al. [11]
$\beta$ $Q_1 $ $Sc$ $Nb$ $Nt$ $Pr$ $Ec$ ${Re}_{x}^{-1/2}Nu_{x}$ ${Re}_{x}^{-1/2}Nu_{x} $ ${Re}_{x}^{-1/2}Sh_{x}$
Current Hayat et al. [11]
$0.35$ $0.1$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7801$ $0.7799$ $0.6232$
$0.4$ $0.7810$ $0.7816$ $0.6200$
$0.6$ $0.7820$ $0.7833$ $0.6155$
$0.6$ $0.0$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7790$ $0.7855$ $0.5662$
$0.2$ $0.7770$ $0.7829$ $0.5524$
$0.4$ $0.7720$ $0.7778$ $0.5412$
$0.6$ $0.1$ $0.0$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7901$ $0.8100$ $0.7100$
$0.2$ $0.7799$ $0.7836$ $0.7100$
$0.4$ $0.7400$ $0.7566$ $0.7101$
$0.6$ $0.1$ $0.1$ $0.1$ $0.5$ $0.3$ $0.2$ $0.8101$ $--$ $0.6525$
$0.3$ $0.8002$ $--$ $0.6580$
$1.5$ $0.7800$ $--$ $0.6612$
$0.6$ $0.1$ $0.1$ $0.5$ $0.1$ $0.3$ $0.2$ $0.8536$ $--$ $0.5412$
$0.3$ $0.8825$ $--$ $0.5300$
$0.5$ $0.9122$ $--$ $0.5121$
$0.6$ $0.1$ $0.1$ $0.5$ $0.5$ $1.2$ $0.2$ $1.1121$ $1.1532$ $1.0021$
$2.2$ $1.2112$ $1.4241$ $1.1121$
$3.2$ $1.3001$ $1.6521$ $1.2231$
$0.6$ $0.1$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7921$ $0.8081$ $0.7101$
$0.3$ $0.7600$ $0.7833$ $0.7200$
$0.4$ $0.7401$ $0.7586$ $0.7405$
$\beta$ $Q_1 $ $Sc$ $Nb$ $Nt$ $Pr$ $Ec$ ${Re}_{x}^{-1/2}Nu_{x}$ ${Re}_{x}^{-1/2}Nu_{x} $ ${Re}_{x}^{-1/2}Sh_{x}$
Current Hayat et al. [11]
$0.35$ $0.1$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7801$ $0.7799$ $0.6232$
$0.4$ $0.7810$ $0.7816$ $0.6200$
$0.6$ $0.7820$ $0.7833$ $0.6155$
$0.6$ $0.0$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7790$ $0.7855$ $0.5662$
$0.2$ $0.7770$ $0.7829$ $0.5524$
$0.4$ $0.7720$ $0.7778$ $0.5412$
$0.6$ $0.1$ $0.0$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7901$ $0.8100$ $0.7100$
$0.2$ $0.7799$ $0.7836$ $0.7100$
$0.4$ $0.7400$ $0.7566$ $0.7101$
$0.6$ $0.1$ $0.1$ $0.1$ $0.5$ $0.3$ $0.2$ $0.8101$ $--$ $0.6525$
$0.3$ $0.8002$ $--$ $0.6580$
$1.5$ $0.7800$ $--$ $0.6612$
$0.6$ $0.1$ $0.1$ $0.5$ $0.1$ $0.3$ $0.2$ $0.8536$ $--$ $0.5412$
$0.3$ $0.8825$ $--$ $0.5300$
$0.5$ $0.9122$ $--$ $0.5121$
$0.6$ $0.1$ $0.1$ $0.5$ $0.5$ $1.2$ $0.2$ $1.1121$ $1.1532$ $1.0021$
$2.2$ $1.2112$ $1.4241$ $1.1121$
$3.2$ $1.3001$ $1.6521$ $1.2231$
$0.6$ $0.1$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7921$ $0.8081$ $0.7101$
$0.3$ $0.7600$ $0.7833$ $0.7200$
$0.4$ $0.7401$ $0.7586$ $0.7405$
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