Investigation of Third-Grade fluid flow in an inclined Microchannel: Utilizing the Hermite wavelet technique for second law analysis

In many heat transfer processes, cooling heat transfer equipment and reducing entropy pose significant challenges for researchers, engineers, and scientists, especially within industrial and applied engineering sectors. The unique characteristics of microchannels make them extremely valuable in applications such as microchannel heat sinks, microprocessors, microchannel heat exchangers, and microwave amplifiers. Microchannel heat sinks were pioneered in 1981 by Tuckerman and Pease (Tuckerman and Pease, (1983, September).), suggesting that employing single-phase forced convective cooling within microchannels could efficiently disperse heat, achieving rates of around 1000 W/m².Since then, numerous researchers in the field of fluid mechanics have focused their studies on these heat sinks.In 1993Peng and Wang (Peng and Wang, 1993) conducted an experiment to explore the thermal conductivity and cooling efficacy of under cooled fluids (water or methanol) circulating through microchannel arrangements. They also observed variations in the thermodynamic properties of the fluid due to the substantial elevation in fluid temperature within the microstructures. Tso and Mahulikar (Tso and Mahulikar, 1998) carried out experimental research in 1998 to look into the transport of liquid heat using microchannel constructions. Chein and Chuang (Chein and Chuang, 2007) studied the use of nanofluids as coolants in microchannel heat sinks (MCHS). Their innovative theoretical analysis showed that adding nanoparticles to the coolants could enhance heat transfer, leading to lower wall temperatures in MCHS and greater energy efficiency. These theoretical predictions were validated by further experiments.Jung et al. (Jung et al., 2006) examined the convective thermal transfer coefficient and resistance factor of a nanofluid in a rectangular microchannel. For this investigation, they constructed an integrated microsystem featuring a sole microchannel on one end and dual localized heaters plus five polysilicon temperature sensors distributed along the channel on the opposite side. In (Ali Ghazwani et al., 2023), numerical computations have been conducted for key engineering quantities such as the skin friction coefficient and mass and heat transfer rates. The study observed that the thermal field increases with the curvature variable, as well as the Eckert and Hartmann numbers.

Entropy generation analysis has also been applied to the optimization of MHD flows relevant to induction devices such as MHD pumps and electric generators (Salas et al., 1999). The effects of thermal radiation are significant in the therapeutic procedure of hyperthermia, particularly in understanding heat transfer in micro-channels in the presence of electric potential, as studied in (Ibáñez et al., 2006). Furthermore, the dimensionless Joule heating parameter has a reducing impact on the Nusselt number for both pseudo-plastic and dilatant fluids; however, its impact on the Nusselt number is more pronounced for dilatant fluids. Viscous dissipation is of interest for many applications, as significant temperature rises are observed in polymer processing flows such as injection molding or extrusion at high rates. The influence of viscous dissipation on heat transfer is particularly important for highly viscous flows, even at moderate velocities. Viscous dissipation transforms kinetic energy into internal energy (heating up the fluid) due to viscosity, thus increasing fluid motion. For this reason, various devices are designed in streambeds to reduce the kinetic energy of flowing water, thereby reducing their erosive potential on banks and river bottoms. Anjali Devi and Ganga (Polat et al., 2022) presented the effects of viscous and Joule dissipation on MHD flow, heat, and mass transfer past a stretching porous surface embedded in a porous medium. Felicita et al. (Devi and Ganga, 2009) explored the influence of thermal and mass exchange on a constant magnetohydrodynamic (MHD) Williamson nanofluid traversing a microchannel, taking into account slip and convective boundary situations. The research also examined the implications of the magnetic field and viscous dissipation on the fluid dynamics. Anita et al. (Felicita et al., 2023) studied entropy generation analysis of carbon nanotubes flowing in a microchannel with varying thermal conductivity. The Buongiorno model was used to explain Brownian motion and thermophoresis during fluid flow, and the Darcy-Forchheimer model was also considered.

Recent technological and industrial advancements have uncovered that some materials used in manufacturing do not behave like Newtonian fluids. These materials include substances such as shampoos, biological materials, drilling muds, soaps, sugar solutions, and many others. They exhibit complex rheological and physical properties that cannot be explained by the Navier-Stokes formulation, indicating they fall under the category of non-Newtonian fluids. Working with non-Newtonian fluids can be challenging for engineers and scientists. To overcome these challenges, various non-Newtonian fluid models have been developed. These models, including the power-law fluid, second, third, and fourth-grade fluids, Casson fluid, Maxwell and Oldroyd-B fluids, and others, serve as essential tools for understanding and analyzing the behavior of non-Newtonian fluids in various industrial and technological domains. There are some materials that don't follow Newton's law of viscosity and can't be described by a single equation. In order to account for their unique characteristics, different fluid models have been developed. One such model is the third-grade fluid model, which explores the effects of shear thinning and shear thickening.

Sahoo and Poncet (Anitha et al., 2024) studied non-Newtonian flow and heat transfer over a stretching sheet with partial slip boundary condition under a magnetic field. They found that third grade fluid parameter affects the momentum boundary layer thickness and thermal boundary layer thickness. Rising β results in the augmentation of the momentum boundary layer's thickness and the reduction of the thermal boundary layer's thickness. Abbasbandy and Hayat (Sahoo and Poncet, 2011) used a specific third-grade fluid model to formulate the problem of time-dependent boundary layer flow over a moving porous surface. They obtained an analytical solution for this problem by applying the homotopy analysis method (HAM). In (Haq et al., 2022), the flow characteristics of a tangent hyperbolic nanofluid over a permeable stretched surface of a cylinder have been studied. The Cattaneo–Christov (C–C) double diffusion model is employed instead of the classical Fourier and Fick's laws to model the energy and concentration expressions. Additionally, the effects of a magnetic field and activation energy are also considered.

Hussain et al. (Hussain et al., 2015) explored a flow scenario arising from the stretching of a surface under convective circumstances in a magnetohydrodynamic nanofluid subjected to solar irradiation. The analysis covered both heat and nanoparticle mass transfer under convective conditions. For the study, the researchers used an incompressible third-grade fluid as the base fluid which is characterized by both shear thinning and shear thickening behavior. Anitha and Gireesha (Anitha and Gireesha, 2024) investigated the thermal performance of a non-Newtonian substance within a permeable slanted microchannel. They examined the influence of heat production, thermal radiation, and Hall phenomena on the fluid's characteristics. The results provided insights into the thermal management of non-Newtonian fluids in microchannels. In (Haq et al., 2021), a theoretical analysis of the magnetized flow of Williamson nanomaterial over a permeable surface of a cylinder is conducted. Gyrotactic microorganisms are introduced to stabilize the suspended nanoparticles in the Williamson liquid. The analysis incorporates the Darcy-Forchheimer model along with porosity effects in the flow.

The utilization of the second law of thermodynamics enhances the efficiency of engineering processes and aids in the design of thermal systems. By assessing entropy generation, this law enables the quantification of heat transfer and identification of irreversible processes within a system. Entropy generation can arise from various mechanisms. Researchers strive to minimize entropy generation to optimize energy efficiency, applying their insights across diverse applications like pumps, heat exchangers, pipe networks, and turbines. Challenges related to fluid friction irreversibility often arise in energy-related tasks such as cooling systems for electronics and geothermal or solar power collectors. Furthermore, the generation of entropy in thermal systems results in a reduction of available work. Therefore, managing fluid flow and heat transfer irreversibilities is essential for effectively controlling entropy generation.

Eegunjobi and Makinde (Haddad et al., 2004) conducted research to assess the influence of the second law of thermodynamics on the continuous motion of an incompressible fluid with varying viscosity and electrical conductivity inside a duct featuring permeable boundaries and convective conditions on the surface. They found that it is possible to minimize entropy generation by adjusting the values of the thermophysical parameters that govern the flow system in a suitable manner. Shashikumar et al. (Eegunjobi and Makinde, 2013) investigated thermal energy exchange and the production of disorder in a magnetohydrodynamic flow featuring a Casson-type fluid passing through a porous microchannel. Their results demonstrated that the generation of entropy rises with heightened values of the radiation parameter and Biot number. Gireesha and Roja (Shashikumar et al., 2018) conducted a numerical exploration of disorder production in a Casson fluid with electrical conductivity flowing through an inclined microchannel. The investigation encompassed the consideration of fluid slippage and convective circumstances, along with factors such as dissipation of viscosity, natural convective currents, generation of heat due to Joule effect, the influence of magnetic fields, and a uniform supply or removal of heat.

Wavelet theory has become a prominent method in applied mathematics, and it is widely used in various fields such as computer science, signal analysis, image processing, mathematical modeling, and other applied sciences. Many mathematicians have contributed significantly to the development of wavelet-based numerical methods, taking advantage of the versatility and effectiveness of this framework to solve complex problems in different domain (Gireesha and Roja, 2020, Oruç, 2018). In this study, to investigate nonlinear ordinary differential equations, the Hermite wavelets operational matrix method is used. Unlike perturbation methods that depend on small or large parameters, such as the Homotopy analysis method which relies on $C_0$, the approach used here does not require any such conditions. This is a significant advantage of the proposed method.

To the best of the authors knowledge, no previous literature has examined how different factors affect entropy generation in a third-grade fluid flow through an inclined channel with a uniform heat source/sink. The parameters examined in this investigation encompassed the Hartmann value, Grashof coefficient, Reynolds value, proportion parameter for the heat source, Biot coefficient, and incline angle. To explore this, sets of nonlinear ordinary differential equations were derived and solved through a semi-analytical approach employing the Hermite wavelet technique. This is the first time this technique has been applied in this context. The research provides visual representations that comprehensively examine the impact of different relevant factors concerning the motion of the fluid, temperature distribution, generation of disorder, Bejan coefficient, and rate of thermal energy exchange. These outcomes hold significance for advancements in small-scale engineering applications, including micro-mixing, and offer insights into addressing a range of issues related to the flow of substances and the transfer of heat.