Entropy generation and activation energy on nonlinearly thermo-radiative and magnetohydrodynamic heat transfer of Jeffrey nanofluid with non-uniform heat source/sink using SQLM

ABSTRACT

Entropy generation and activation energy analysis on MHD heat and mass transfer of Jeffrey nanofluid under the impact of nonlinear thermal radiation and heat source/sink (non-uniform) over a linearly stretching sheet has been investigated numerically using Spectral Quasilinearization Method (SQLM). The Brownian motion with thermophoretic effects has been utilized. The basic equations concerned with the problem are solved numerically. The impacts of different governing physical parameters are analyzed on the velocity, temperature, and concentration fields, along with entropy generation, and Bejan number profiles are analyzed graphically. It is found that the concentration profiles have a mixed tendency with magnetic parameters. In addition, the temperature profiles increase in the presence of Brownian motion with thermophoresis effects, whereas the composite behavior is noticed on the entropy generation profiles. It is also found that both the temperature and concentration profiles diminish for various values of Deborah number while the velocity profile increases. Also, it is noticed that both the temperature profiles increase for enhanced values of the space-dependent and time-dependent parameters. Due to activation energy, the Bejan number profiles decrease, whereas the entropy generation profiles increase far away from the expandable sheet but decrease closer to the sheet.

KEYWORDS:

• Entropy generation
• activation energy
• magnetohydrodynamic (MHD)
• heat generation/sink
• thermal radiation (nonlinear)
• jeffrey nanofluid
• spectral quasilinearization method (SQLM)

Acknowledgments

We are profoundly grateful to the Editor-in-Chief and the Reviewers for their valuable comments and insightful recommendations, significantly augmenting the excellence of this paper.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.

Data availability statement

No data was used for the research described in the article.

Nomenclature

$\mathit{a}$	=	stretching rate of the sheet (unit: $s^{-1}$)
$A^{\ast }$	=	space-dependent parameter
$B^{\ast }$	=	time-dependent parameter
$B_0$	=	strength of-magnetic field
Br	=	Brinkman Number
$\mathit{C}$	=	nanoparticles concentration
$C_p$	=	specific heat at constant pressure (unit: $\mathit{Jk}{g}^{-1}{K}^{-1}$)
$D_T$	=	thermophoretic diffusion coefficient (unit: $m^2{s}^{-1}$)
$D_B$	=	Brownian diffusion coefficient (unit: $m^2{s}^{-1}$)
$E_c$	=	Eckert number
fw	=	suction/injection or mass flux parameter
$\mathit{Ha}$	=	Hartmann number
$\mathit{Mn}$	=	magnetic parameter
Nb	=	Brownian motion parameter
$\mathit{Le}$	=	Lewis number
$\mathit{k}$	=	thermal conductivity (unit: W $m^{-1}{K}^{-1}$)
$K^{\ast }$	=	mean absorption coeffi cient (unit: $m^{-1}$)
Nt	=	thermophoresis parameter
NG	=	entropy generation number
$S_{\mathit{gen}}$	=	local volumetric rate of entropy generation (unit: W$m^{-3}{K}^{-1}$)
$\mathit{R}$	=	nonlinear thermal radiation parameter
$R_1$	=	chemical reaction parameter
Pr	=	Prandtl number
$\mathit{T}$	=	temperature variable (unit: K)
$\mathit{u}$	=	x-direction velocity (unit: m $s^{-1}$)
$u_w$	=	stretching sheet velocity (unit: m $s^{-1}$)
$\mathit{v}$	=	y-direction velocity (unit: m $s^{-1}$)
$\mathit{x}$	=	x-axis along the stretching sheet (unit: m)
$\mathit{y}$	=	y-axis normal to the stretching sheet (unit: m)
$\mathit{f}$	=	dimensionless velocity variable
$\mathit{Be}$	=	Bejan number
$\mathit{E}$	=	non-dimensional activation energy
$\mathit{n}$	=	dimensionless constant exponent
Greek letters	=
$\mathit{\alpha }$	=	thermal diffusivity (unit: $m^2{s}^{-1}$)
$\mathit{\beta }$	=	Deborah number
$\mathit{\tau }$	=	ratio of the nanoparticles heat capacity to fluid heat capacity
$\mathit{\nu }$	=	kinematic viscosity (unit: $m^2{s}^{-1}$)
$\mathit{\eta }$	=	similarity variable
$\mathit{\epsilon }$	=	dimensionless constant
$\mathit{\kappa }$	=	thermal conductivity of the fluid
$\mathit{\rho }$	=	density of the fluid (unit: kg $m^{-3}$)
$\mathit{\mu }$	=	dynamic viscosity (unit: N s$m^{-2}$)
$\bar{R}$	=	universal gas constant
${\sigma }_1$	=	electric conductivity (unit: mho$s^{-1}$)
${\sigma }^{\ast }$	=	Stefan – Boltzmann constant
${\rho }_p$	=	nanoparticles density (unit: kg $m^{-3}$)
$\mathit{\theta }$	=	non-dimensional temperature
$\mathit{\varphi }$	=	non-dimensional nanoparticles volume fraction
${\theta }_w$	=	non-dimensional temperature ratio parameter ($=\frac{T_W}{T_{\mathrm{\infty }}}$)
$\mathit{\lambda }$	=	ratio of relaxation to retardation parameter
${\lambda }_1$	=	relaxation time (unit: s)
$\mathrm{\Omega }$	=	non-dimensional difference in concentration field
$\mathrm{\Sigma }$	=	non-dimensional difference in temperature field
$\mathit{\gamma }$	=	temperature relative parameter
Subscripts	=
$\mathrm{\infty }$	=	infinity
$\mathit{f}$	=	fluid
$\mathit{p}$	=	particle
$\mathit{w}$	=	sheet surface

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/02286203.2024.2349504

Additional information

Notes on contributors

Dulal Pal

Prof. Dulal Pal has obtained his Ph.D. degrees from Bangalore University, India. He has served as Professor & Head of Mathematics Department and Dean of Science at Visva-Bharati University, Santiniketan, West Bengal, India. For over 44 years, Prof. Pal has been actively engaged in research in the field of Computational Fluid Dynamics. His research interests span various topics, including heat and mass transfer in Newtonian/non-Newtonian fluids and nanofluids, flow through and past porous/non-porous media, stability analysis for convective boundary layer flow over stretching/shrinking sheet, and flow and heat transfer in a lid-driven cavity flow with thermal radiation. Prof. Pal has supervised numerous postgraduate students for their project work and has mentored several Ph.D. students, contributing significantly to their academic and research development. He was awarded a prestigious DAAD German fellowship for pursuing post-doctoral research work at the Technical University of Munich, Germany, reflecting his international recognition and collaboration with foreign University. His research contributions are evident in his publication record, with over 176 research papers published in reputed national and international journals. Additionally, he has served as a reviewer for many important and reputable international journals. Prof. Pal’s name appeared in the “World Top 2% most influential Scientists List 2023,” prepared by Stanford University, United Kingdom. He has 5533 citations with h-index 44 and i10 index 105.

Sagar Mondal

Dr. Sagar Mondal serves as an Assistant Professor in the Department of Mathematics at Ananda Mohan College in Kolkata, West Bengal, India, affiliated with Calcutta University. He holds an M.Sc. degree from the Indian Institute of Technology, New Delhi, and completed his Ph.D. at Visva-Bharati, (a Central University) located in Santiniketan, Birbhum, West Bengal, India. His research interests encompass Computational Fluid Dynamics, Numerical Analysis, and Differential Equations. Dr. Mondal has contributed to numerous publications in internationally renowned journals.