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Impact of Entropy Generation and Nonlinear Thermal Radiation on Darcy–Forchheimer Flow of MnFe2O4-Casson/Water Nanofluid due to a Rotating Disk: Application to Brain Dynamics

  • Research Article-Mechanical Engineering
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Abstract

The present article candidly states the incremental impact of nonlinear thermal radiation on heat transfer enhancement due to Darcy–Forchheimer flow of spinel-type MnFe2O4-Casson/water nanofluids due to a stretched rotating disk. In present contest, the entropy generation approach is highlighted specially as a powerful tool for the analysis of the brain function, in accordance with the theological and philosophical approach of Saint Thomas Aquinas. The some of the results of the present study that strengthening of permeability and Casson parameter contribute to the diminution of radial and tangential velocity profiles and yield shrinkage of the related boundary layers. An increase in thermal radiation leading to more heat propagating into the fluid thereby improves the TBL. Fluids with non-Newtonian behavior contribute greater entropy generation rate compared to Newtonian fluids. The most significant outcome is that the entropy generation makes a real contribution to the brain function or analysis of the function of the brain.

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Abbreviations

\( (u,v,w) \) :

Velocity components in increasing \( \left( {r,\,\phi ,\,z} \right) \) directions \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

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

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

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

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

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

Heat capacitance of base fluid \( \left( {{\text{J}}\,{\text{kg}}^{2} \,{\text{m}}^{3} \,{\text{K}}^{ - 1} } \right) \)

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

Heat capacitance of solid nanoparticles \( \left( {{\text{J}}\,{\text{kg}}^{2} \,{\text{m}}^{3} \,{\text{K}}^{ - 1} } \right) \)

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

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

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

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

\( \mu_{\text{nf}} \) :

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

\( \mu_{\text{f}} \) :

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

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

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

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

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

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

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

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

TC of base fluid \( \left( {{\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right) \)

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

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

\( T_{w} \) :

Surface temperature (K)

\( T \) :

Fluid temperature (K)

\( k* \) :

Mean absorption coefficient

\( \sigma * \) :

Stefan Boltzmann constant

\( p \) :

Pressure (Pa)

\( \alpha_{\text{f}} \) :

Thermal diffusivity of base fluid \( \left( {{\text{m}}^{2} \,{\text{s}}^{ - 1} } \right) \)

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

Thermal Grassof number

\( {\text{Rd}} = \frac{{4\sigma^{*} T_{\infty }^{3} }}{{k^{*} k_{\text{f}} }} \) :

Radiation parameter

\( F^{\prime}\left( \eta \right) \) :

Radial velocity

\( F\left( \eta \right) \) :

Axial velocity

\( K^{*} \) :

Permeability of porous medium

\( \text{Re} = r\left( {\frac{r\varOmega }{{\upsilon_{\text{f}} }}} \right) \) :

Rotational Reynolds number

\( \delta = \frac{a}{\varOmega } \) :

Stretching-strength parameter

Fr:

Inertia coefficient

\( F^{*} \left( { = \frac{{C_{\text{d}} }}{{rK^{{*^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }} }}} \right) \) :

Non-uniform inertia coefficient

\( C_{\text{d}} \) :

Drag coefficient

\( S = \frac{W}{{\sqrt {2\varOmega \upsilon_{\text{f}} } }} \) :

Suction parameter

\( \alpha = \frac{\Delta T}{{T_{\infty } }} \) :

Thermal ratio parameter

\( {\text{Br}} = \frac{{\mu_{\text{nf}} \varOmega^{2} R^{2} }}{{k_{\text{nf}} \Delta T}} \) :

Rotational Brinkman number

\( \tilde{r} = \frac{r}{R} \) :

Dimensionless radial coordinate

\( N_{\text{G}} = \frac{{\dot{S}^{\prime\prime\prime}_{\text{gen}} }}{{\left( {{{k_{\text{nf}} \Delta T\varOmega } \mathord{\left/ {\vphantom {{k_{\text{nf}} \Delta T\varOmega } {T_{\infty } \upsilon_{\text{f}} }}} \right. \kern-0pt} {T_{\infty } \upsilon_{\text{f}} }}} \right)}} \) :

Dimensionless EG rate

\( T_{\text{f}} \) :

Temperature of heated fluid (K)

\( T_{\infty } \) :

Ambient fluid temperature (K)

\( K = \frac{{\upsilon_{\text{f}} }}{{\delta K^{*} }} \) :

Porosity parameter

\( h_{\text{f}} \) :

Heat transfer coefficient \( \left( {{\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 1} } \right) \)

\( \beta \) :

Casson parameter

\( \phi \) :

Solid volume fraction

\( G\left( \eta \right) \) :

Tangential velocity

\( \theta \left( \eta \right) \) :

Non-dimensional temperature

\( \Pr = \frac{{v_{\text{f}} }}{{\alpha_{\text{f}} }} \) :

Prandtl number

\( {\text{Ec}} = \frac{{r^{2} \varOmega^{2} }}{{\left( {C_{p} } \right)_{\text{f}} T_{\infty } \left( {\theta_{\text{f}} - 1} \right)}} \) :

Eckert number

\( {\text{Bi}} = \frac{{h_{\text{f}} }}{{k_{\text{f}} }}\sqrt {\frac{{\upsilon_{\text{f}} }}{2\varOmega }} \) :

Biot number

\( \theta_{\text{f}} = \frac{{T_{\text{f}} }}{{T_{\infty } }} \) :

Temperature ratio parameter

\( \varOmega \) :

Constant angular velocity

\( \Delta T = T_{\text{f}} - T_{\infty } \) :

Temperature difference

\( \text{Re} = \frac{{\varOmega R^{2} }}{{\upsilon_{\text{f}} }} \) :

Rotational Reynolds number

EG:

Entropy generation

BLT:

Boundary layer thickness

TC:

Thermal conductivity

RD:

Rotating disk

TR:

Thermal radiation

DFF:

Darcy–Forchheimer flow

NLTR:

Nonlinear thermal radiation

TBL:

Thermal boundary layer

f:

Base fluid

nf:

Nanofluids

s:

Nano solid particles

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Authors and Affiliations

  1. Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana

    Sachin Shaw

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

    A. S. Dogonchi

  3. Department of Physics, IHSE, Siksha “O” Anusandhan Deemed to be University, Bhubaneswar, Odisha, 751003, India

    M. K. Nayak

  4. Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha, 7395, South Africa

    O. D. Makinde

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  1. Sachin Shaw
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  4. O. D. Makinde
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Correspondence to M. K. Nayak.

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Shaw, S., Dogonchi, A.S., Nayak, M.K. et al. Impact of Entropy Generation and Nonlinear Thermal Radiation on Darcy–Forchheimer Flow of MnFe2O4-Casson/Water Nanofluid due to a Rotating Disk: Application to Brain Dynamics. Arab J Sci Eng 45, 5471–5490 (2020). https://doi.org/10.1007/s13369-020-04453-2

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