Recent advancements across numerous scientific and technological disciplines have led researchers to explore boundary layer flow over stretched surfaces as an emerging area of study.

When it comes to engineering problems, the flow behaviour of a physical phenomenon toward stretching problems (for both linear and non-linear cases) is critical. The analysis of fluid flow and its entropy characteristics over a stretching sheet holds significant relevance in various industrial and manufacturing processes. It finds extensive applications in advanced manufacturing techniques such as metal spinning, rubber sheet production, and fiber-reinforced composite manufacturing, including fiberglass fabrication. Additionally, this study is critical in processes like wire drawing, polymer sheet extrusion, and polymer processing, which are integral to the development of high-performance materials. Moreover, the insights gained from such analyses are invaluable in optimizing operations in petroleum industries, where fluid dynamics play a crucial role in processes such as drilling, extraction, and refining. These applications underscore the importance of understanding and controlling fluid behavior for improving efficiency, ensuring product quality, and reducing energy consumption in modern industrial systems. The pace at which the material cools during the technique, as well as the amount of stretching that occurs, determine the desirable qualities of the end product in these conditions. Sakiadis [1–3] analyzed viscous flow over a moving surface using an unsteady incompressible fluid. He investigated the behaviour of boundary layers over continuous surfaces and compared his findings to those of previously published studies [4]. Crane [5] studied continuous viscous flow on a stretched surface in a quiescent fluid with varying velocity. For example, in Hiemenz [6] viscous flow around a stagnation point, the outer flow’s free stream velocity is proportional to distance. Crane [7] expanded the work of Sakiadis [3] to include a stretched cylinder and observed that, despite the boundary layer being significantly thicker than the cylinder, the solution remains stable over a long axial distance despite the thicker boundary layer. A comprehensive investigation into magnetohydrodynamic (MHD) mixed convection heat transfer in a lid-driven wavy enclosure with a fin attached to the bottom was conducted by Fayz-Al-Asad et al. [8]. They employed the Galerkin finite element method to solve the governing equations, yielding valuable insights into the complex interactions between fluid flow and thermal dynamics influenced by magnetic fields. Their parametric analysis revealed that the dimensions of the fin, in addition to the Hartmann and Richardson numbers, play a crucial role in determining flow patterns, temperature distribution, and heat transfer effectiveness. Specifically, it was found that longer fins significantly enhance heat transfer under certain conditions, highlighting the potential of these surfaces to optimize fluid mixing, reduce operational costs, and increase thermal efficiency in enclosure systems.

Furthermore, the study leveraged similarity transformations alongside MATLAB’s bvp4c solver, which indicated that higher Weissenberg numbers enhance flow velocities while concurrently diminishing the concentration boundary layer. These findings have important implications for cooling technologies and various industrial applications. In a separate but related work, Sohail et al. [9] investigated the effects of bio-convection in tree-dimensional Casson Nanofluid subjected to an induced magnetic stretching field. Their findings highlight the relevance of this phenomenon in various applications. Additionally, another research effort [10] focused on the numerical analysis of how a magnetic dipole influences the behavior of a radiative ferromagnetic liquid flowing over a porous stretched sheet. There were a lot of academics who were inspired by Crane’s work and made important contributions by studying the effects of thermal transportation across stretched sheets (see [11–15]).

Rott [16] studied a viscous flow past a moving wall approaching a stagnation point. Danberg et al. [17] recently studied a variation of this problem where the wall is expanded proportionally to Chakrabarti et al. [18] explored an electrically conducting fluid that moved exclusively owing to wall stretching. They observed that temperature increases with the increase of magnetic parameter. The MHD viscoelastic fluid flow across a stretched surface was studied by Andersson [19]. By analytically analyzing the non-linear boundary layer problem, they demonstrate that the influence of viscoelasticity and external magnetic field are the same. Sohail et al. [20] analyzed the effects of magnetic field and viscous dissipation over stretching sheet for the non-Newtonian nanofluid flow. The pace of cooling affects several industrial items’ properties. A magnetic field can also be used to clear molten metals of non-metallic contaminants. Many researchers have addressed the challenge of stretching surfaces using magneto-hydrodynamics [21–24].

Many specialists [25–28] have recently taken an interest in the study of thermal transportation over radiated material through the stretch surface as a result of its extensive use in engineering and industrial activities. These include rubber manufacturing, colloidal suspension and glass sock manufacture, metal spinning and plastic film drawing, paper and textile production, as well as the food processing and geothermal energy sectors. Radiation occurs often in engineering difficulties. Li et al. [29] discussed the usage of radioactive nanofluid flow in light of its applications. Several researchers have recently looked at heat transport issues. Kumam et al. [30] conducted an in-depth study of Casson fluid dynamics within a rotating channel, focusing on magnetohydrodynamic (MHD) radiative flow and the effects of an internal heat source. The research provides a comprehensive analysis of how these combined influences impact the fluid’s behavior, including its velocity, temperature distribution, and energy transfer mechanisms. By addressing the interplay of magnetic fields, radiation, and heat generation, the study offers valuable insights into the applications of Casson fluid in engineering and industrial processes, such as in cooling systems, chemical re-actors, and material processing under complex flow conditions.

The method of minimizing entropy generation is utilized in order to optimize thermal engineering devices with the aim of achieving greater energy efficiency. The availability of engineering equipment is diminished as a result of irreversibilities. The idea of entropy generation serves as a measure to evaluate the degree of irreversibility that is inherent in a particular process. It is crucial to emphasize that the second law of thermodynamics exhibits higher dependability compared to the first law of thermo-dynamics, due to the efficacy limitations of the latter in the domain of heat transfer engineering systems. In recent years, there has been a growing focus among researchers on exploring the practical applications and implications of the second law of thermodynamics within thermal engineering systems. This trend highlights the importance of entropy analysis, energy efficiency, and irreversibility minimization as critical tools for optimizing the performance of engineering systems. By studying the second law, scholars aim to identify and quantify sources of energy loss, thereby enabling the design of more efficient systems across various applications, including power generation, refrigeration, air conditioning, and industrial heat exchangers.

The interest also stems from the increasing demand for sustainable and energy-efficient technologies to address global energy challenges. Understanding the second law facilitates the development of advanced methodologies for energy recovery, waste heat utilization, and process optimization, which are vital for reducing carbon footprints. Furthermore, the integration of the second law’s principles into modern computational tools has allowed for more precise simulations and real-world applications, reinforcing its role as a cornerstone in the advancement of thermal engineering. Bejan [31] conducted an investigation into the entropy analysis in a process of convective heat transfer. Shit et al. [32] analyzed an entropy generation in an unsteady two-dimensional magnetohydrodynamic (MHD) nanofluid flow over porous exponentially radiated stretching surface. Building on this work, Shit and Mandal [33] investigated entropy generation in an unsteady MHD flow of Casson nanofluid over a vertically radiated stretching plate using Buongiorno’s model. Their results revealed that the Casson parameter significantly increases the entropy number, while thermal radiation intensifies the entropy number near the plate. Within this particular context, several pertinent and noteworthy inquiries are expounded upon within the aforementioned articles [34–37].

The literature review highlights a notable gap in research focusing on axisymmetric flow, particularly over radially stretched surfaces. Shahzad et al. [38] addressed this gap by investigating the unsteady axisymmetric flow and heat transfer phenomena under the combined influences of Ohmic heating, viscosity, magnetic fields, and radiation over a stretched surface, with a particular emphasis on irreversibility effects.

The study stands out for its novel approach of employing a similarity transformation to convert the governing equations into nonlinear coupled ordinary differential equations, which were subsequently, solved numerically using the MATLAB BVP4C solver. The analysis meticulously examines the impact of various physical parameters on velocity and temperature profiles, with results presented both graphically and in tabular form for clarity and comprehensiveness.

The findings not only provide fresh insights into the interplay of multiple physical effects on axisymmetric flow but also demonstrate strong agreement with existing literature, thereby validating the robustness and accuracy of the proposed model. This work contributes significantly to the field by advancing the understanding of complex flow and heat transfer phenomena in radially stretched systems, a topic previously underexplored in the literature.

Think about the time-dependent heat transfer and radially extended viscous fluid flow represented in Fig. 1 as an example. The Reynolds number (Re) was reportedly low. Consequently, the induced magnetic field can be neglected. The fluid begins to flow as the surface stretches (radially) at a velocity Uw=ar1−ct. Tw=T∞+br1−ct, where T∞ is the ambient temperature, and Tw > T∞; moreover, a > 0, b ≥ 0, and c ≥ 0 are all constants (with ct < 1), and a and c have dimension t−1. Tw is assumed to be the wall temperature. The mathematical representation of the aforementioned physical phenomena of fluid flow and heat transfer is described by the following coupled partial differential equations (PDEs) [38]:

(1)∂u∂r+ur+∂w∂z=0, (2)∂u∂t+u∂u∂r+w∂u∂z=ν(∂2u∂z2)−σρ(B(t)2u), (3)∂T∂t+u∂T∂r+w∂T∂z=kρCp(∂2T∂z2)+μρCp(∂u∂z)2+σρCp(B(t)2u2)−1σρCp∂∂zqrad,where qrad presents the radiative heat flux and described by Rosseland approximation [39] as: qrad=−43σske∂T4∂z=−43σskeT3∂T∂z,here, ke and σs, denote the mean absorption coefficient and the Stefan-Boltzmann constant, respectively. Assuming the temperature variations within the flow are relatively small, the term T⁴ can be approximated using a Taylor series expansion around the ambient temperature T_∞ with higher-order terms being neglected for simplification. As a result, the radiative heat flux qrad can be linearized and expressed in the following form: T4 ≅ 4T∞3T−3T∞.with the corresponding boundary conditions (4){u=Uw=ar1−ct,w=W0,T=TW=T∞+br1−ctat z=0,u⟶o,T⟶T∞as z⟶∞.},where W0 stands for surface mass transfer for injection (W0 > 0) and suction (W0 < 0) and kinematic viscosity are defined as: W0=−2(vUwr)12and ν=μρ.

The dimensionless governing equations are subjected to the similarity transformation given as: (5)Ψ(r,z)=−r2UwRe−12f(η),η=zrRe12,and θ(η)=T−T∞Tw−T∞.

The Stokes streams function is defined u=−1r∂Ψ∂z and w=1r∂Ψ∂r, Uw being the stretching velocity. The velocity component along r and z-axes can be simply computed as follows: (6)u=Uwf′(η) and w=−2UwRe−12f(η),by using the above defined transformation Eqs. (2)–(4) takes the form: (7)f′′′+2ff′′−f′2−A(f′+η2f′′)−Mf′=0 (8)1pr(1+Rd)θ′′+2fθ′−f′θ+A(θ+12ηθ′)+MEcf′2+1prEcf′′2=0as long as the transform boundary conditions are: (9)f(η)=S,f′(η)=1 and θ(η)=1 at η=0, (10)f′(η)=0 and θ(η)=0 as η⟶∞,

A = a/c is unsteadiness parameter, Pr=μcρk Prandtl number, and mass transfer is denoted by S, where S > 0 indicates mass suction and S > 0 indicates mass injection. The formulae for the physical quantities Nu and C_f, which are local Nusselt number and skin friction respectively, are given as: (11)Cf=τw12ρU2 and Nu=rqwk(Tw−T∞),where τ_w and q_w are, respectively, known as shear stress (wall) and heat flux (wall) which are mathematically described as: (12)τw=μ∂u∂z|z=0 and qw=−k(∂T∂z)|z=0.hence Eq. (11) takes the form: (13)12Re12Cf=f″(0),Re−12Nu=−θ′(0).

The mathematical model in this study is numerically analyzed to evaluate the effects of various factors, including magnetohydrodynamics (MHD), radiation, dissipation, Joule heating, entropy generation number, and Bejan number. The analysis incorporates appropriate boundary conditions. The physical phenomena, involving momentum and energy, are described by nonlinear partial differential equations (PDEs) in both time and space. These PDEs are transformed into a system of nonlinear ordinary differential equations (ODEs) using a suitable similarity transformation. The resulting ODEs are solved numerically using the BVP4C method in MATLAB, ensuring adherence to the relevant boundary conditions. The influence of parameters such as A, (unsteadiness parameter), S > 0, (suction parameter), S < 0, (injection parameter), and the magnetic parameter on velocity and temperature is analyzed and presented graphically. Additional graphs illustrate the effects of the Prandtl number, Eckert number, and magnetic parameter on the temperature profile. Furthermore, the entropy generation number and Bejan number are graphically depicted. Default parameter values are specified in the descriptions accompanying each figure.

This study analyzed entropy generation and thermophysical properties in radiative materials using a radially stretching disk as the model. Tables 1 and 2 detail the effects of various physical parameters on skin friction and the local Nusselt number (wall heat flux and transfer rate). A comparison with previous studies revealed excellent agreement, affirming the validity of the results. The influence of thermophysical properties on flow behavior, entropy generation, and the Bejan number was illustrated and interpreted graphically. The key findings of the analysis are summarized below: –

Increasing the radiation parameter raises the temperature, with a similar trend observed for the magnetic parameter.

Additionally, the utilization of entropy generation analysis in unsteady processes is a more intricate task, as it requires identifying an optimal time history to minimize the entropy generated within a finite time span. For this reason, entropy generation is infrequently applied in addressing transient operations and off-design conditions. These results provide valuable insights into the thermal management of radiative systems and the optimization of energy processes in various engineering applications.

This study is constrained by its focus on a simplified model, which may not fully represent the complexities found in practical applications. Additionally, the investigation did not account for the effects of varying material properties or complex geometries, which could significantly impact the results.