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Entropy optimization analysis on nonlinear thermal radiative electromagnetic Darcy–Forchheimer flow of SWCNT/MWCNT nanomaterials

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Abstract

Carbon nanotubes (CNTs) are inevitable due to its tremendously high thermal and electrical conductivities, strength, stiffness, and toughness characteristics. The utilization of both porous media and nanofluids with CNTs as nanoparticles can augment the thermal efficiency of typical physical systems significantly. In view of such advantages, the present study is intended to convey the influence of entropy minimization and nonlinear thermal radiation on the electromagnetic flow of nanofluids with single wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs) nanoparticles suspensions past the surface of thin needle. In addition, the famous Darcy Forchheimer flow and Cattaneo-Christov heat flux models are implemented. The required numerical solution is devised pragmatically via bvp4c in MATLAB for the system of highly nonlinear ordinary differential equations. It is found that the porous matrix and local Forchheimer parameter are detrimental to the regular flow of nanofluids. Thermal fields magnify in respect of hiked porosity and temperature ratio parameters and diminish due to rise in electric and thermal relaxation parameters. Entropy minimization due to porous irreversibility is prominent for MWCNTs than SWCNTs. Bejan number upsurges due to rise in volume fraction and porosity parameter for both SWCNT-water and MWCNT-water nanofluids.

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Abbreviations

\((u,v)\) :

Velocity components in the axial and radial directions \(\left( {{\text{m}}\,{\text{s}}^{ - 1} } \right)\)

\(\rho_{nf}\) :

Effective density of the nf \(\left( {{\text{kg}}\,{\text{m}}^{ - 3} } \right)\)

hc:

Heat capacitance

nf:

Nanofluid

bf:

Base fluid

tc:

Thermal conductivity

HT:

Heat transfer

HTR:

Heat transfer rate

CNT:

Carbon nanotube

DFF:

Darcy–Forchheimer flow

CCHF:

Cattaneo–Christov Heat Flux

NLTR:

Nonlinear thermal radiation

CCDDT:

Cattaneo–Christov double diffusion theory

MBL:

Momentum boundary layer

TBL:

Thermal boundary layer

\(\left( {\rho C_{p} } \right)_{nf}\) :

hc of the nf \(\left( {{\text{J}}\,{\text{kg}}^{2} \,{\text{m}}^{3} {\text{K}}^{ - 1} } \right)\)

\(\left( {\rho C_{p} } \right)_{f}\) :

hc of bf \(\left( {{\text{J}}\,{\text{kg}}^{2} \,{\text{m}}^{3} {\text{K}}^{ - 1} } \right)\)

\(\left( {\rho C_{p} } \right)_{{{\text{CNT}}}}\) :

hc of carbon nanotubes \(\left( {{\text{J}}\,{\text{kg}}^{2} \,{\text{m}}^{3} {\text{K}}^{ - 1} } \right)\)

\(\rho_{{{\text{CNT}}}}\) :

Density of carbon nanotubes \(\left( {{\text{kg}}\,{\text{m}}^{ - 3} } \right)\)

\(\rho_{f}\) :

Density of bf \(\left( {{\text{kg}}\,{\text{m}}^{ - 3} } \right)\)

\(\mu_{nf}\) :

Effective dynamic viscosity of the nf \(\left( {{\text{kg}}\,{\text{m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\mu_{f}\) :

Effective dynamic viscosity of bf \(\left( {{\text{kg}}\,{\text{m}}^{ - 1} {\text{s}}^{ - 1} } \right)\)

\(\beta_{nf}\) :

Thermal expansion of nf \(\left( {{\text{K}}^{ - 1} } \right)\)

\(\beta_{f}\) :

Thermal expansion of bf \(\left( {{\text{K}}^{ - 1} } \right)\)

\(\beta_{{{\text{CNT}}}}\) :

Thermal expansion of carbon nanotubes \(\left( {{\text{K}}^{ - 1} } \right)\)

\(k_{nf}\) :

tc of nf \(\left( {{\text{W}}\,\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right)\)

\(k_{f}\) :

tc of bf \(\left( {{\text{W}}\,\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right)\)

\(k_{{{\text{CNT}}}}\) :

tc of carbon nanotubes \(\left( {{\text{W}}\,\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right)\)

\( \upsilon _{{nf}} \) :

Kinematic viscosity of nf \(\left( {{\text{m}}^{2} {\text{s}}^{ - 1} } \right)\)

\(\sigma_{nf}\) :

Electrical conductivity of nf

\(E_{0}\) :

Uniform strength of electric field

\(B_{0}\) :

Uniform strength of magnetic field

\(\sigma^{*}\) :

Stefan–Boltzmann constant

\(k^{*}\) :

Mean absorption coefficient

\(K\) :

Porous medium permeability

\(\phi\) :

Solid volume fraction

\(T\) :

Fluid temperature in the boundary layer (K)

\(T_{w}\) :

Temperature on the surface of thin needle (K)

\(T_{\infty }\) :

Ambient fluid temperature (K)

\(\alpha_{f}\) :

Thermal diffusivity of the bf \(\left( {{\text{m}}^{2} {\text{s}}^{ - 1} } \right)\)

\(\alpha_{f}\) :

Thermal diffusivity of the bf \(\left( {{\text{m}}^{2} {\text{s}}^{ - 1} } \right)\)

\(F\left( { = \frac{{c_{b} }}{{\sqrt {K^{*} } }}} \right)\) :

Inertia coefficient in the porous medium

\(c_{b}\) :

Drag coefficient

\(F_{r} = \frac{{xc_{b} }}{{\sqrt {K^{*} } }}\) :

Local inertia coefficient

\(\Gamma\) :

Relaxation time for heat flux

\(Gr = \frac{{g\beta T_{\infty } \left( {\theta_{r} - 1} \right)x^{3} }}{{\upsilon_{f}^{2} }}\) :

Thermal Grassof number

\({\text{Re}} = \frac{Ux}{{\upsilon_{f} }}\) :

Reynolds number

\(M = \frac{{\sigma_{f} B_{0}^{2} x}}{{\rho_{f} U}}\) :

Hartmann number

\(\beta = \frac{{\upsilon_{f} x}}{{K^{*} U}}\) :

Porosity parameter

\(Pr = \frac{{\upsilon_{f} }}{{\alpha_{f} }}\) :

Prandtl number

\(Nr = \,\frac{{k^{*} k_{f} }}{{4\sigma^{*} T_{\infty } }}\) :

Radiation parameter

\(E_{c} = \frac{{U^{2} }}{{T_{\infty } \left( {\theta_{r} - 1} \right)\left( {c_{p} } \right)_{f} }}\) :

Eckert number

\(\Omega = \frac{U\Gamma }{x}\) :

Thermal relaxation parameter

\(\lambda = \frac{{U_{w} }}{U}\) :

Velocity ratio parameter

\(Br = \Pr \cdot Ec = \frac{{\mu_{f} U^{2} }}{{k_{f} T_{\infty } \left( {\theta_{r} - 1} \right)}}\) :

Brinkman number

\(\theta_{r} = \frac{{T_{w} }}{{T_{\infty } }}\) :

Temperature ratio parameter

\(S^{\prime\prime\prime}_{0} = \frac{{4k_{f} \left( {T_{w} - T_{\infty } } \right)^{2} }}{{T_{\infty }^{2} x^{2} }}\) :

Nondimensional characteristic EG rate

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Authors and Affiliations

  1. Department of Physics, IHSE, Siksha ‘O’ Anusandhan Deemed to be University, Bhubaneswar, 751003, India

    M. K. Nayak

  2. Department of Information Technology, Fanshawe College, London, ON, Canada

    F. Mabood

  3. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Iskander Tlili

  4. Faculty of Civil Engineering, Duy Tan University, Da Nang, 550000, Vietnam

    Iskander Tlili

  5. Department of Mechanical Engineering, Aliabad Katoul Branch, Islamic Azad University, Aliabad Katoul, Iran

    A. S. Dogonchi

  6. Department of Mechanical Engineering, College of Engineering, Prince Mohammad Bin Fahd University, Al Khobar, 31952, Kingdom of Saudi Arabia

    W. A. Khan

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  1. M. K. Nayak
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  2. F. Mabood
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  4. A. S. Dogonchi
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Correspondence to Iskander Tlili.

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Nayak, M.K., Mabood, F., Tlili, I. et al. Entropy optimization analysis on nonlinear thermal radiative electromagnetic Darcy–Forchheimer flow of SWCNT/MWCNT nanomaterials. Appl Nanosci 11, 399–418 (2021). https://doi.org/10.1007/s13204-020-01611-8

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  • DOI: https://doi.org/10.1007/s13204-020-01611-8

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