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Boundary layer flow of magneto-nanomicropolar liquid over an exponentially elongated porous plate with Joule heating and viscous heating: a numerical study

  • Research Article-Mechanical Engineering
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Abstract

Micropolar fluids are used in lubrication theory, thrust bearing technologies, cervical flows, lubricants, paint rheology, and the polymer industry. This study develops the numerical simulation of the magneto-Darcy flow of a polarized nanoliquid with Joule heating and viscous heating mechanisms on an exponentially elongated surface. The effects of linearized Rosseland radiation and temperature-dependent heat generation are considered. The flow is generated by an exponential form of elongation of a flexible sheet. The porous matrix and nanoparticle effects are characterized by the Darcy expression and the two-component Buongiorno model correspondingly. The resulting partial differential systems are solved numerically using the Runge–Kutta-based shooting technique to interpret the importance of key parameters in physical quantities. A direct comparison is made to validate the results. Our results demonstrated that arbitrary movement of the nanoparticles significantly advances the temperature profile by reducing the concentration of nanoparticles. Both Joule heating and viscous heating mechanisms improve the structure of the thermal boundary layer. The porous matrix reduces the velocity of the nanoliquid and thus the width of the velocity boundary layer is reduced.

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Abbreviations

\(u\) and \(v\) :

Velocity along \(x\) and y direction \(\left( {{\rm m\,s}^{{ - 1}} } \right)\)

\(k\) :

Vortex viscosity \(\left( {{\rm m}^{2} ~{\rm s}^{{ - 1}} } \right)\)

\(m\) :

Surface condition parameter \(\left( {0 \le m \le 1} \right)\)

\(B\) :

Variable magnetic field (T)

\(Kp^{*}\) :

Porous matrix

\(C_{\rm f}\) :

Local skin friction coefficient

\(Q\) :

Heat source/sink coefficients

\(v_{w} \left( x \right)\) :

Suction/injection velocity \(\left( {{\rm m\,s}^{{ - 1}} } \right)\)

\(K\) :

Material parameter

\(j\) :

Microinertia per unit mass

\(R\) :

Radiation parameter

\(k^{*}\) :

Absorption coefficient

\(M_{x}\) :

Local couple stress

\(C_{p}\) :

Specific heat \(\left( {{\rm J}~{\rm kg}^{{ - 1}} {\rm K}^{{ - 1}} } \right)\)

\(M\) :

Magnetic field factor

\(C\) :

Concentration

\(T\) :

Temperature (K)

\(T_{\infty }\) :

Ambient temperature (K)

\(D_{\rm B}\) :

Brownian diffusion coefficient

\(D_{\rm T}\) :

Thermophoretic diffusion coefficient

\(Nb\) :

Brownian motion factor

Ec :

Eckert number

N :

Micro-rotation velocity \(\left( {{\rm s}^{{ - 1}} } \right)\)

\(Sh_{x}\) :

Local Sherwood number

\(Nu_{x}\) :

Local Nusselt number

\(Sc\) :

Schmidt number

\(S\) :

Heat source factor

\(Pr\) :

Prandtl number

Nt :

Thermophoresis factor

\(\eta\) :

Similarity variable

\(\kappa\) :

Micropolar fluid material property

\(\sigma\) :

Electrical conductivity \(\left( {{\rm m}^{{ - 1}}\,\Omega ^{{ - 1}} } \right)\)

\(\sigma ^{*}\) :

Stefan–Boltzmann constant

\(\tau\) :

Ratio of the effective heat transfer capacities of particles to liquid

\(\mu\) :

Dynamic viscosity \(\left( {{\rm kg\,m\,s}^{{ - 1}} } \right)\)

\(\rho\) :

Density (kg m−3)

\(\upsilon\) :

Kinematic viscosity \(\left( {{\rm m^{2}\,s^{{ - 1}}}} \right)\)

\(\alpha\) :

Thermal diffusivity \(\left( {{\rm m^{2}\,s^{{ - 1}} }}\right)\)

\(\gamma\) :

Spin gradient viscosity

\(B_{0} ,~K,p_{0} ,~Q_{0} ,~v_{0}\) :

Constants

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Acknowledgements

The authors would also like to thank the Editor and anonymous reviewers for their constructive suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, College of Science and Technology, Wenzhou-Kean University, Wenzhou, 325060, China

    Puneet Rana

  2. Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, Karnataka, 560029, India

    B. Mahanthesh

  3. Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Al- Dawaser, 11991, Saudi Arabia

    Kottakkaran Sooppy Nisar

  4. Department of Mathematics, Gandhi Institute for Technology, Bhubaneswar, Odisha, 752054, India

    K. Swain

  5. Department of Physics, Gandhi Institute for Technology, Bhubaneswar, Odisha, 752054, India

    Manisha Devi

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  1. Puneet Rana
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  2. B. Mahanthesh
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  3. Kottakkaran Sooppy Nisar
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  4. K. Swain
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Correspondence to Puneet Rana or B. Mahanthesh.

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Rana, P., Mahanthesh, B., Nisar, K.S. et al. Boundary layer flow of magneto-nanomicropolar liquid over an exponentially elongated porous plate with Joule heating and viscous heating: a numerical study. Arab J Sci Eng 46, 12405–12415 (2021). https://doi.org/10.1007/s13369-021-05926-8

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  • DOI: https://doi.org/10.1007/s13369-021-05926-8

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