Fluid Equations Research Articles

Navier-Stokes: Singularities and Bifurcations

This article presents substantial advances in the analysis of the Navier-Stokes equations for both compressible and incompressible fluids, focusing on the formation of singularities, hypercomplex bifurcations, and regularity in Sobolev and Besov spaces. Through new theorems, we extend the theory of singularities in fluid dynamics and introduce quaternionic bifurcations, representing an innovative extension of classical bifurcation theory. Moreover, we delve into the investigation of the regularity of compressible fluids, exploring the conditions under which solutions remain smooth or develop singularities. These contributions are fundamental to the understanding of global regularity issues, directly linked to the renowned Millennium Prize problem, which seeks definitive answers on the existence and smoothness of solutions to the Navier-Stokes equations. Additionally, we discuss how these theoretical advancements offer new approaches to unresolved problems related to the formation of singularities in turbulent flows and the multiscale behavior of solutions, which are crucial for a comprehensive understanding of fluid dynamics. This work not only broadens the scope of traditional mathematical analysis of the Navier-Stokes equations but also establishes a robust theoretical framework for the investigation of bifurcations and regularity in advanced functional spaces, fostering a deeper understanding of global regularity phenomena and the complex dynamics governing fluid systems.

Electromagnetic Surface Waves in a Relativistic Magnetized Plasma Propagating in a Cylindrical Column

Dispersion relations of electromagnetic surface waves in a magnetized relativistic plasma propagating in a cylindrical column surrounded by vacuum are derived by means of the relativistic fluid equations. The zero order relativistic beam is assumed to stream along the direction of the axial magnetic field. The coupled equations of the E- and B-wave modes are solved analytically in terms of the cylindrical coordinates in weak or intermediate values of the magnetic field in which the cyclotron frequency is much smaller than the relativistic Doppler-shifted frequency. The entangled field components of the E- and B-waves are decoupled by assuming that the flow is incompressible. The surface wave dispersion relations of the E- and B-wave modes show a strong similarity and symmetry. The static magnetic field splits the E- and B-waves, thus doubling the modes in unmagnetized plasma. By taking the radius of the cylinder infinite, the dispersion relations in a semi-infinite plasma are derived.

Generalized Brenier Principle and the Closure Problem of Landgren–Monin–Novikov Hierarchy for Vorticity Field

Brenier’s concept – a representation of solutions to the equations of ideal incompressible fluids in terms of probability measures on the set of Lagrangian trajectories in the case of their stochasticity, is a generalization of Arnold’s principle of least action of finding smooth solutions of Euler’s equations. In this work, the variational generalized Brenier principle (Brenier, J. Am. Math. Soc. 1989) is used to close the infinite chain of Landgren–Monin–Novikov equations for the n-point probability density functions fn of the vortex field of two-dimensional turbulence. In addition, within the framework of the statistical approach, an approximation of the variational problem with conditions at the ends posed by Shnirelman (Mat. Sat. 1985) for the Euler equation is proposed.

Fast simulation strategy for capacitively-coupled plasmas based on fluid model

Fluid simulations are widely used in optimizing the reactor geometry and improving the performance of capacitively coupled plasma (CCP) sources in industry, so high computation speed is very important. In this work, a fast method for CCP fluid simulation based on the framework of Multi-physics Analysis of Plasma Sources (MAPS) is developed, which includes a multi-time-step explicit upwind scheme to solve electron fluid equations, a semi-implicit scheme and an iterative method with in-phase initial value to solve Poisson's equation, an explicit upwind scheme with limited artificial diffusion to solve heavy particle fluid equations, and an acceleration method based on fluid equation modification to reduce the periods required to reach equilibrium. In order to prove the validity and efficiency of the newly developed method, benchmarking against COMSOL and comparison with experimental data have been performed in argon discharges on the Gaseous Electronics Conference (GEC) reactor. Besides, the performance of each acceleration method is tested, and the results indicated that the multi-time-step explicit Euler scheme can effectively decline the computational burden in the bulk plasma and reduce the time cost on the electron fluid equations by half. The in-phase initial value method can greatly decrease the iteration times required to solve linear equations and lower the computational time of Poisson's equation by 77 %. The acceleration method based on equation modification can reduce the periods required to reach equilibrium by two-thirds.

Impact of temperature-dependent viscosity on linear and weakly nonlinear stability of double-diffusive convection in viscoelastic fluid

The impact of temperature-dependent viscosity on the double-diffusive viscoelastic convective fluids is investigated, with applications in heat transfer, polymer processing, food industry, biomedical engineering, etc. Both linear and weakly nonlinear analyses are carried out to determine the stability of fluids using the Oldroyd-B model. Analytical criteria for the onset of linear stationary and oscillatory convection are derived. The effects of rheological parameters, linearly and exponentially varying viscosity, salinity Rayleigh number and Prandtl number on the system’s stability are investigated. Linear stability analysis reveals that the interaction between thermal and solute diffusions with rheological parameters favours oscillatory convection over stationary convection. In weakly nonlinear theory, a power series expansion derives a Landau amplitude equation, allowing heat and mass transfer analysis in viscoelastic fluids with temperature-dependent viscosity. Numerical results highlight the effects of variable viscosity on heat and mass transfer rates, represented by Nusselt and Sherwood numbers. Further, the weakly non-linear stability analysis evaluates the impact of the Prandtl number, rheological parameters and salinity Rayleigh number on heat and mass transfer rates. These findings are compared with existing results to validate and enhance our understanding of fluid behaviour under different conditions.

Entropy Generation in Third-Grade Non-Newtonian Fluid Flow and Heat Transport Through Porous Medium in a Horizontal Channel under Heat Generation

In this study, entropy production in flow along with heat transport of non-Newtonian fluid via porous medium, is analyzed. For fluid flow via porous medium, a modified Darcy resistance term, is taken in the momentum equation for third-grade fluid. Temperature-dependent viscosity is considered using Vogel’s model viscosity. Using adequate transformations, the momentum and heat transport equation are reduced to non-dimensional form and solved analytically invoking homotopy analysis method on MATHEMATICA software. Effects of parameters arising in the study are depicted by graphs on velocity distribution, temperature distribution, and entropy production with Bejan number and discussed. For validity of current findings, the values of velocity and temperature are computed for particular values of the parameters and equated with previously published results, excellent agreement achieved. Furthermore, skin-friction coefficient and Nusselt number values are expressed in tabular form for various values of relevant parameters and discussed. It is noticed that slip parameters $\left( \gamma \right)$ and $\left( \beta \right)$ reduce the entropy generation number $\left( NS \right)$. Also noticed that skin friction coefficient upsurges with rising velocity slip parameter $\left( \gamma \right)$ value while, effect of temperature slip parameter $\left( \beta \right)$ is observed to lessen Nusselt number in the absolute sense. HIGHLIGHTS Entropy production in third grade non-Newtonian fluid flow and heat transfer in a channel via porous medium is investigated in the presence of a heat source. Slip boundary conditions are applied. The resulting governing equations are solved analytically using the homotopy analysis method (HAM). Maximum entropy production observed near the lower wall of channel and Bejan number attains maximum value in middle of the channel. GRAPHICAL ABSTRACT

Gyrotactic micro-organism dusty fluid flow over an extended surface

The conventional magnetohydrodynamics (MHD) equations are modeled for the dusty fluid flow past a stretching sheet embedded in a porous medium. These equations are coupled with motile microorganisms, temperature and concentration equations for both fluid and dust particles dynamical equations. The two-phase dusty fluid flow is subjected to magnetic effect, porosity factor, 2nd order chemical reaction, Brownian diffusion, thermophoretic effect and Arrhenius activation energy. An appropriate transformation is used to reset the system of partial differential equations (PDEs) into non-linear system of ODEs (ordinary differential equations). The flow equations are numerically solved by using the parametric continuation method (PCM). For validity of the applied methodology, the results are compared using numerical and graphical results of Matlab built-in package “BVP4c”. Both results show accuracy up to four decimal places. Furthermore, from graphical results, it can be noticed that the fluid particles interaction parameter causes an attractive force among fluid particles which prevents the dust particles phase flow. This technique can be used for biological separations, filters, fluid dust separators and crude oil purification.

A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.

Features of MHD on Secant Curved Annular Circular Plate Lubricant as a Couple-Stress Fluid with Slip Velocity

The purpose of this study is to examine the magneto-hydrodynamic (MHD) and slip velocity effect on secant curved annular circular plates lubricated with couple stress fluid and to derive the modified Reynolds’s equation for non-Newtonian fluid under various operating conditions to obtain the optimum bearing parameters. Based upon the MHD theory and Stokes theory for couple stress fluid, the governing equations relevant to the problem under consideration are derived. This paper analyses the effect of slip velocity on secant curved annular circular plates with an electrically conducting fluid in the presence of a transverse magnetic field. It is found that there is an increase in squeeze film pressure, load carrying capacity and squeeze film time in secant curved annular circular plates due to the presence of magnetic effects with couple stress fluid and slip velocity.

Nonlinear dynamics of micropolar two-phase fluids: Multiple exact solutions

Dual and triple solutions induced by a flexible planar surface for a micropolar two-phase fluid model are studied. The two-phase behavior in the micropolar fluid model occurs due to phase transitions between the fluid phases, influenced by interfacial stresses and heat transfer. The physical implications of these transitions are significant in understanding flow behavior under different mechanical and thermal conditions. This study examines the critical parameters and conditions that lead to these phase transitions, resulting in dual or triple solutions in the flow dynamics. The flow and thermal fields are exact solutions of the steady, two-dimensional two-phase micropolar fluid equations in the form of similarity solution. It is shown that dual and triple exact solutions exist for a highly nonlinear system. Triple solutions exist for the skin friction and temperature gradient identified by the critical numbers ac and μc. It is noted that for sufficiently small values of stretching strength parameter the dual branches for two of the triple solutions exist only in the regions μ≥μc3, and μ≤μc4, where μc3=−5.23 and μc4=−7.72. Numerical results are also provided, validating the model and offering insights into its accuracy and behavior of the model.

Thermodynamic and Kinetic Behavior Study of Gas Bubbles in High‐Temperature Steel Melts

This study provides an exhaustive exploration of the thermal expansion of low‐temperature argon bubbles in high‐temperature molten steel and the resulting dynamic effects. The energy equation of compressible fluid simultaneously with the volume of fluid model under the Euler framework is solved. The generation and rising behavior of bubbles are validated through rigorous physical water model experiments. Findings reveal that, within a heating time of 0.2–0.45 ms, gas bubbles already complete 65–90% of their internal energy increase. The total heat transfer time and heat transfer coefficient of bubbles exhibit distinctive patterns influenced by the initial diameter and temperature of the bubbles. Specifically, under the conditions studied, bubbles exhibit a total heat transfer time of ≈20–60 ms, accompanied by a heat transfer coefficient ranging from 500 to 2600 W (m2 * K)−1. During the thermal expansion process, gas bubbles experience energy transfer and conversion within the two‐phase flow. This article establishes a dynamic model describing bubble thermal expansion in steel melts based on underwater bubble dynamics’ first principles. The model provides a quantitative depiction of instantaneous changes in bubble size, velocity, and kinetic energy under specific conditions.

Invariant Measures for a Class of Stochastic Third-Grade Fluid Equations in 2D and 3D Bounded Domains

Invariant Measures for a Class of Stochastic Third-Grade Fluid Equations in 2D and 3D Bounded Domains

Role of Cairns–Gurevich ion distribution on nonlinear wave propagation in negatively charged dusty plasma

The study of dust-acoustic (DA) nonlinear electrostatic waves in dusty plasma is crucial for understanding plasma behavior in both astrophysical and laboratory environments. Dusty plasmas, characterized by the presence of negatively charged dust particles, exhibit complex dynamics influenced by various factors, including nonthermal electron and ion populations following Cairns–Gurevich distributions. This study addresses the problem of understanding wave propagation under these specific conditions by employing a set of fluid hydrodynamic equations for dust fluid, alongside appropriate electron and Cairns–Gurevich ion distributions, to model the dynamics and propagation of DA nonlinear electrostatic waves under specific plasma conditions with negatively charged dust particles. We derived a nonlinear Zakharov–Kuznetsov (ZK) equation to model the evolution of these waves and further transformed it into the nonlinear Schrödinger (NLS) equation to analyze rogue wave formation and identify unstable and stable zones within the plasma. We focused our analysis on the effects of key parameters, such as the nonthermal index, magnetic field strength, negative dust temperature ratio, polarization force, and density ratio, on the behavior of dust-acoustic waves (DAWs). The results demonstrated that soliton waves, explosive waves, and kink waves exhibit distinct behaviors depending on these parameters and the spatial context of Earth's magnetosphere. These findings provide new insights compared to previous studies into the conditions under which these waveforms emerge and evolve, especially in magnetized environments where earlier models were less effective at accurately characterizing or predicting wave behavior. By applying two different analytical methods, a direct integration approach and a generalized Kudryashov method, we expanded our understanding of nonlinear wave phenomena in dusty plasmas. Our results not only advance theoretical knowledge but also have practical implications for interpreting space plasma observations and guiding experimental design in plasma physics research. This study thus contributes to a more comprehensive understanding of plasma dynamics, with potential applications to astrophysical observation and laboratory plasma experimentation.

Variations of heat equation on the half‐line via the Fokas method

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

Horseshoes and spiral waves: capturing the 3D flow induced by a low-mass planet analytically

ABSTRACT The key difficulty faced by 2D models for planet–disc interaction is in appropriately accounting for the impact of the disc’s vertical structure on the dynamics. 3D effects are often mimicked via softening of the planet’s potential; however, the planet-induced flow and torques often depend strongly on the choice of softening length. We show that for a linear adiabatic flow perturbing a vertically isothermal disc, there is a particular vertical average of the 3D equations of motion that exactly reproduces 2D fluid equations for arbitrary adiabatic index. There is a strong connection here with the Lubow–Pringle 2D mode of the disc. Correspondingly, we find a simple, general prescription for the consistent treatment of planetary potentials embedded within ‘2D’ discs. The flow induced by a low-mass planet involves large-scale excited spiral density waves that transport angular momentum radially away from the planet and ‘horseshoe streamlines’ within the coorbital region. We derive simple linear equations governing the flow that locally capture both effects faithfully simultaneously. We present an accurate coorbital flow solution allowing for inexpensive future study of corotation torques, and predict the vertical structure of the coorbital flow and horseshoe region width for different values of adiabatic index, as well as the vertical dependence of the initial shock location. We find strong agreement with the flow computed in 3D numerical simulations, and with 3D one-sided Lindblad torque estimates, which are a factor of 2–3 lower than values from previous 2D simulations.

Numerical and Experimental Study on the Nonlinear Liquid Sloshing in Cassini Tank

Liquid sloshing within propellant tanks of space vehicles has been a major concern in aerospace engineering. The aim of the work in this paper is to develop a flexible computational framework with high precision to simulate three-dimensional large-amplitude liquid sloshing in Cassini tanks. The finite element method is adopted to solve the fluid equations of motion in an arbitrary Lagrangian–Eulerian (ALE) framework, where the characteristic-based split method is combined with a fractional step method for solving the control equations in which the ALE kinematic description is incorporated to track the free liquid surface flexibly and effectively for large-amplitude liquid sloshing in Cassini tanks. In addition, an experimental platform is set up to verify the reliability and effectiveness of the presented method. The numerical results obtained from computer programming by using the ALE finite element method proposed in this paper are compared with the experimental results, proving the model to be successful. The nonlinear phenomena of large-amplitude liquid sloshing, especially rotary sloshing (steady-state swirling and non-steady-state swirling), in Cassini tanks under horizontal harmonic excitation are investigated by both the ground physical experiments and numerical simulations.

Dynamics Modeling and Analysis of Small-Diameter Pipeline Inspection Gauge during Passing Through Elbow

Abstract With small diameters, numerous elbows and complex internal environment, gathering and transportation pipelines are widely applied in the oil and gas pipeline network. To guarantee their safety, a pipeline inspection gauge (PIG) driven by the pressure differential is used. When the PIG collides with the elbows in the pipeline, it exhibits complex nonlinear dynamic behaviors and its reliability and stability may also be greatly affected. To study the vibration response characteristics passing through the elbow, the dynamic modeling of the small-diameter PIG is conducted first in this paper based on the ADAMS software. Then, the fluid-structure coupling is achieved by combining the dynamic model and the numerical calculation program of the fluid equation in Simulink. The simulation results indicate that the axial and swing acceleration amplitudes of the two sections increase with the increasing speed of the double-section detector, while the opposite trend is observed for the vertical acceleration amplitudes. By comparison, it is found that the acceleration response in the second section is generally greater than that in the first section. The results also show that the axial offset of the PIG at the elbow is inversely proportional to the speed and radius of the elbow, which is the main cause of the lift-off values of sensors.

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

For a compact Riemannian m-manifold (Mm,g),m>1, endowed with a nontrivial conformal vector field ζ with a conformal factor σ, there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ=ζ¯+∇ρ, where ζ¯ satisfies divζ¯=0, which is called the Hodge vector, and ρ is the Hodge potential of ζ. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on Mm. The first result of this article states that a compact Riemannian m-manifold Mm is an m-sphere Sm(c) if and only if (1) for a nonzero constant c, the function −σ/c is a solution of the Poisson equation Δρ=mσ, and (2) the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2. The second result states that if Mm has constant scalar curvature τ=m(m−1)c>0, then it is an Sm(c) if and only if the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2 and the Hodge potential ρ satisfies a certain static perfect fluid equation. The third result provides another new characterization of Sm(c) using the affinity tensor of the Hodge vector ζ¯ of a conformal vector field ζ on a compact Riemannian manifold Mm with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold Mm, m>2, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field ζ whose affinity tensor vanishes identically and ζ annihilates its associated tensor φ.

The freeze-drying performance under pressures from 1.325 to 101.325 kPa

The low drying rate is the main limitation of atmospheric freeze drying. The drying rate depends on the balance between heat and mass transfer. Among the process variables, air pressure changes have the opposite effects on heat and mass transfer simultaneously. The increase in pressure favors heat transfer but hinders mass transfer. Therefore, a conjugate model was proposed and validated to quantitatively investigate the heat and mass transport characteristics of lamellar products during freeze-drying in a tunnel under different pressures (1.325–101.325 kPa). To describe the effects of air pressure, the Navier-Stokes equations of the external fluid were coupled with the adsorption-sublimation model. Results show the regulation of transfer resistance values and drying rate with moisture content during the drying process, indicating that freeze-drying is the external heat transfer control at first, then the internal mass transfer control, and eventually, the heat transfer promotion can accelerate the process. The optimal constant pressure value was found to be 5.325 kPa, with the shortest drying time (17.3 h). It has been demonstrated that freeze drying employing a specific laddering-controlled strategy, offers the benefit of reducing drying time in comparison to the use of optimal constant pressure; however, the impact was not remarkable.

Buoyancy-driven heat transfer and entropy analysis of a hydromagnetic GO-Fe3O4/H2O hybrid nanofluid in an energy storage enclosure partially filled with non-Darcy porous medium under an oblique magnetic field

Purpose Thermal energy storage systems use thermal energy to elevate the temperature of a storage substance, enabling the release of energy during a discharge cycle. The storage or retrieval of energy occurs through the heating or cooling of either a liquid or a solid, without undergoing a phase change, within a sensible heat storage system. In a sensible packed bed thermal energy storage system, the structure comprises porous media that form the packed solid material, while fluid occupies the voids. Thus, a cavity, partially filled with a fluid layer and partially with a saturated porous layer, has become important in the investigation of natural convection heat transfer, carrying significant relevance within thermal energy storage systems. Motivated by these insights, the current investigation delves into the convection heat transfer driven by buoyancy and entropy generation within a partially porous cavity that is differentially heated, vertically layered and filled with a hybrid nanofluid. Design/methodology/approach The investigation encompasses two distinct scenarios. In the first instance, the porous layer is positioned next to the heated wall, while the opposite region consists of a fluid layer. In the second case, the layers switch places, with the fluid layer adjacent to the heated wall. The system of equations for fluid and porous media, along with appropriate initial and boundary conditions, is addressed using the finite difference method. The Tiwari–Das model is used in this investigation, and the viscosity and thermal conductivity are determined using correlations specific to spherical nanoparticles. Findings Comprehensive numerical simulations have been performed, considering controlling factors such as the Darcy number, nanoparticle volume fraction, Rayleigh number, bottom slit position and Hartmann number. The visual representation of the numerical findings includes streamlines, isotherms and entropy lines, as well as plots illustrating average entropy generation and the average Nusselt number. These representations aim to provide insight into the influence of these parameters across a spectrum of scenarios. Originality/value The computational outcomes indicate that with an increase in the Darcy number, the addition of 2.5% magnetite nanoparticles to the GO nanofluid results in an enhanced heat transfer rate, showing increases of 0.567% in Case 1 and 3.894% in Case 2. Compared with Case 2, Case 1 exhibits a 59.90% enhancement in heat transfer within the enclosure. Positioning the porous layer next to the partially cooled wall significantly boosts the average total entropy production, showing a substantial increase of 11.36% at an elevated Rayleigh number value. Positioning the hot slit near the bottom wall leads to a reduction in total entropy generation by 33.20% compared to its placement at the center and by 33.32% in comparison to its proximity to the top wall.