Momentum Principle Research Articles

Analytic Determination of the Roller Length of a Hydraulic Jump in an Open Channel Flow Using a Bouncing Ball

A hydraulic jump is a natural occurrence that occurs in spillways, rivers, and other open channel flows when water or other liquid flowing with a high velocity discharges into a region of lower velocity with an attendant abrupt rise in the liquid surface. Such a phenomenon, known as hydraulic jump, is normally accompanied by substantial dissipation of energy. Many researchers, in the past, focus attention in the numerical study of the hydraulic jump, under varied working situations. Few attempts are made to study the occurrence analytically. In this paper, the author studied the incident analytically using a bouncing ball to develop a model to examine the lengths of hydraulic jumps in a horizontal open channel flow. The model development is based on the laws of motion, the principles of impulse and momentum, and the classical hydraulic jump formula. The model was later verified with the roller length obtained from the series of experiments conducted in a large-size facility. The roller length that the new model estimated compared well with the experimental results conducted within the Froude number ranges of 2.00 and 16.00. The model is ease to use and its accuracy as determined by the Pearson correlation coefficient is between 0.93 and 1.00.

Quantum Theory of Electron Spin Based on the Extended Least Action Principle and Information Metrics of Spin Orientations

Quantum theory of electron spin is developed here based on the extended least action principle and assumptions of intrinsic angular momentum of an electron with random orientations. The novelty of the formulation is the introduction of relative entropy for the random orientations of intrinsic angular momentum when extremizing the total actions. Applying recursively this extended least action principle, we show that the quantization of electron spin is a mathematical consequence when the order of relative entropy approaches a limit. In addition, the formulation of the measurement probability when a second Stern-Gerlach apparatus is rotated relative to the first Stern-Gerlach apparatus, and the Schrödinger-Pauli equation, are recovered successfully. Furthermore, the principle allows us to provide an intuitive physical model and formulation to explain the entanglement phenomenon between two electron spins. In this model, spin entanglement is the consequence of the correlation between the random orientations of the intrinsic angular momenta of the two electrons. Since spin orientation is an intrinsic local property of the electron, the correlation of spin orientations can be preserved and manifested even when the two electrons are remotely separated. The entanglement of a spin singlet state is represented by two joint probability density functions that reflect the orientation correlation. Using these joint probability density functions, we prove that the Bell-CHSH inequality is violated in a Bell test. To test the validity of the spin-entanglement model, we propose a Bell test experiment with time delay. Such an experiment starts with a typical Bell test that confirms the violation of the Bell-CHSH inequality but adds an extra step that Bob’s measurement is delayed with a period of time after Alice’s measurement. As the time delay increases, the test results are expected to change from violating the Bell-CHSH inequality to not violating the inequality.

Influence of MHD Flow on Shrinking Sheet with Partial Slip and Heat Generation at Stagnation Point

Fluid dynamics encompasses the fundamental principles of continuity, momentum, and energy conservation, which are applied through mathematical models like the Navier-Stokes equations. These equations are essential for describing how fluid properties like velocity, pressure, and density change in response to forces and environmental conditions. Thus, this study attempted to explore the characteristics of flow and heat transfer of a shrinking sheet in magnetohydrodynamics (MHD), along with the effect of partial slip and heat generation on the system. We employ a similarity transformation technique for turning the governing partial differential equations into ordinary differential equations. These equations are solved numerically through shooting method in Maple, and the results are compared to the previous research. The analysis shows that the suction parameter and velocity slip parameter have an increasing effect on both the skin friction rate and the heat transfer rate. In the meantime, the heat transfer rate decreases as the parameter increases for the heat generation, magnetic parameter, Eckert number and thermal slip parameter. The bvp4c solver in MATLAB is implemented to conduct a stability analysis and determine the physically feasible solution. According to our research, the stability of the solution occurs only in the first solution.

A reflective overview: European vocational education and training reform: The Copenhagen process 2002 to 2024

The European Union’s (EU) Copenhagen Process on cooperation in vocational education and training (VET), initiated in 2002, is a voluntary method that coordinates VET reform in Europe. In terms of this process, EU Member States, the European Commission and the social partners agree, at ministerial level, on common VET-related objectives, priorities and statistical targets to be met over a five- to ten-year period. Progress is monitored and political momentum maintained through regular reporting, exchanges of experience and periodic revisions of reform objectives, priorities and statistical targets. This article outlines the origins of the Copenhagen Process and discusses its evolution and influence on European VET policy since 2002. The writer argues that the process has proved to be an effective working method. It has strengthened European cooperation in VET, provided the basis for common European instruments and principles, influenced national reforms and raised the profile of VET in other policy areas. The writer also argues that, while the Copenhagen Process has operated at a multinational, trans-European level, its principles of partnership, resources and momentum can be used to create effective partnerships and networks at national, sectoral and local levels in order to bring about VET reform.

Optimization of thermal performance in hybrid nanofluids for parabolic trough solar collectors using Adams–Bashforth–Moulton method

Solar energy is an environment-friendly renewable energy source that fulfills the energy consumption requirements of the world economy. Concentrated solar power is the most advanced technology that efficiently absorbs concentrated solar energy. This article investigates the solar energy storage and entropy generation in concentrated parabolic trough solar collectors for hybrid nanofluid flow due to a spinning tube inserted at the receiver's center. The effect of copper nanoparticles and multi-walled carbon nanotube mixture is studied using water-based fluid. Also, the influence of an external magnetic field is investigated with thermophoresis and Brownian motion of nanoparticles. The governing equations for the nanofluid flow are derived using the conservation principles of mass, momentum, energy, and concentration with boundary layer assumptions and no-slip boundary conditions. The governing equations are numerically solved using Adam Bashforth and Adam Moulten's fourth-order predictor-corrector numerical scheme along with the shooting approach. The numerical outcomes for the fluid velocity, temperature, Nusselt number, and entropy formation against various influential physical parameters have been discussed using graphs. It is observed that the augmenting Eckert number, thermophoretic diffusion parameter, and Brownian motion parameter escalates the thermal profiles. Also, an escalation of the radiation parameter and Lewis number enhances the Nusselt number. Thermal energy storage in solar collectors utilizing different terminologies is crucial for improving the efficiency and performance of solar thermal collectors.

A robust 3D finite element framework for monolithically coupled thermo-hydro-mechanical analysis of fracture growth with frictional contact in porous media

This paper presents the formulation of a robust integrated framework for the coupled multiphysics and multiple fracture growth analysis in porous media. The finite element-based thermo-hydro-mechanical method for fracture growth with frictional contact (THMf-g) simultaneously solves monolithically coupled equations, incorporating contact and frictional constraints from fracture sliding. It also implements an adaptive process to update fields and geometries during fracture growth, effectively modeling emerging new surfaces. The thermo-hydro-mechanical fields are derived from fundamental principles of mass, momentum, and energy conservation and discretized numerically using the finite element method. Fractures are represented as sub-dimensional surfaces embedded within the volume of the porous medium. The growth of multiple fracture is modeled based on stress intensity factors at fracture tips, with fracture aperture and permeability emerging as dynamic properties of the system. The main novelty of this work lies in extending the implicitly solved monolithic coupling to include frictional and growth modeling for multiple non-planar fractures of emerging geometry in three dimensions. This includes the direct incorporation of cubic terms in the fracture flow equations and convection terms in the heat transfer equations, adopting an incremental method to solve these coupled, nonlinear equations. To ensure energy conservation, the heat equations are resolved using an implicit scheme, establishing a velocity dependency on pressure fields and introducing quadratic terms into the heat equation. Furthermore, the heat transfer equation has been revised to account for the work done on the fluid, enhancing the accuracy of thermal modeling. A contact mechanics leader–follower strategy tracks a conformal mesh split at each fracture, accounting explicitly for permeability changes during deformation and growth, effectively reducing computational complexity and cost. The iterative process for applying these constraints and the solution of the coupled THMf-g with friction and growth is described. The method does not require calibrated material properties or artificial length scale parameters, and relies on laboratory-measured properties with direct physical interpretation. A detailed algorithm is presented, including the updating of apertures and handling of new surfaces during the simulation, accounting for the variability of fracture permeability and its interplay with contact and friction during the non-linear solution. Several validating numerical experiments are benchmarked against analytical solutions, demonstrating the accuracy and reliability of the proposed framework in capturing complex fracture behavior and multiphysics interactions in porous media.

Oil pipeline multiple leakage detection and localization based on sensor fusion

Oil pipeline simultaneous leak localization solely using pressure and flow rate measurements at its boundaries is inherently challenging. Localizing simultaneous leaks in a long-distance oil pipeline solely using pressure and flow rate measurements at its boundaries presents significant challenges. This difficulty is attributed to the presence of noisy measurements and the interdependence of two equations that incorporate variables associated with leaks, leading to inherent uncertainty. Additionally, existing leak detection methodologies predominantly utilize a single sensor, which contributes to a high incidence of false alarms, particularly in complex operational environments. Developing nonlinear equations based on momentum and continuity principles is proposed as a comprehensive solution for estimating multiple leak locations in oil pipelines. In this study, multiple leaks in long-distance oil pipelines are estimated using a two-stage decision-making scheme. Arrays of sensors are proposed for improving the reliability and accuracy of simultaneous leak estimations, replacing the traditional reliance on single sensors along pipelines. The Fisher fusion technique combines inaccurate measurements to improve estimation accuracy. This method integrates sensor measurements even when noise is present, demonstrating its effectiveness. Comprehensive simulationsare conductedusing OLGA multiphase flow simulation software to evaluate the robustness and practicality of the proposed approach under various real-time conditions found in long-distance oil pipelines. The system’s state convergence and precision in simultaneous leak estimations highlight the effectiveness of the proposed strategy in oil pipeline control. Across various scenarios, the leak detector showed a 28 % improvement for a 50 Km oil pipeline.

Dynamics modeling and simulation for satellites with optoelectronic tracking system

Abstract The optoelectronic tracking system has a strong coupling between the tracking platform and the satellite platform when rapidly capturing and precisely tracking targets. The high-precision coordinated pointing control between the two platforms is a key technology in the fields of space laser communication, debris monitoring, and astronomical observation. Firstly, a multi-level coupled dynamics model of spacecraft with two-dimensional (2D) motion platforms is derived based on the principles of momentum and angular momentum. Secondly, the coupling effects between the motion platform and the satellite platform are theoretically analyzed, highlighting the impact of platform orientation, configuration, mass properties, and motion characteristics on the coupling. Finally, a multi-level platform cooperative control strategy is proposed, and the model and control strategy are validated through mathematical simulations. The results show that the model is consistent with simulation results from ADAMS software, and the satellite attitude accuracy under the cooperative control strategy can be improved by 80%.

Study on Aerodynamic Characteristics and Stability of a Vehicle with Inverted Dihedral and Momentum Lift Augmentation

Abstract Inspired by the wave-rider idea and momentum principle, the vehicle with inverted dihedral and momentum lift augmentation is a new aerodynamic configuration of high-speed gliding vehicle in the near-space, which has achieved a high lift-to-drag ratio and long-distance sliding. Numerical simulation of aerodynamic characteristics and stability of the aircraft are carried out in this paper. The lift-to-drag characteristics, longitudinal-directional stability and lateral-directional stability are evaluated based on the National Numerical Wind tunnel’s high-speed simulation software, named NNW-HyFLOW. An unstructured/hybrid grid is used in the calculation at the typical ballistic points of altitude of 10-75km and Mach number of 3-25. The results shows that the lift-to-drag ratio reaches a peak value of 4.11 at the altitude of 30km and attack angle of 8°. This value is decreased when the altitude raises. The usable lift-to-drag ratio is over 3 in the glide phase range from 30 to 50 kilometres. This vehicle shows better longitudinal-directional stability at large angles of attack than at small in the reentry phase and glide phase, which can be optimized by adjusting the center of mass or pitching rudder. It has a weak instability in the lateral direction at small angle of attack in the glide phase. Therefore, it is suggested to avoid to work at the high altitude with a small angle of attack. Or, the lateral-directional stability can be strengthened at this altitude by improving the V-tail.

Research on Rock Energy Constitutive Model Based on Functional Principle

The essence of rock fracture can be broadly categorized into four processes: energy input, energy accumulation, energy dissipation, and energy release. From the perspective of energy consumption, the failure of rock materials must be accompanied by energy dissipation. Dissipated energy serves as the internal driving force behind rock damage and progressive failure. Given that the process of rock loading and deformation involves energy accumulation and dissipation, the rock constitutive model theory is expanded by incorporating energy principles. By introducing the dynamic energy correction coefficient, according to the law of the conservation of energy, the total energy exerted by external loads on rocks is equal to the energy dissipated through the dynamic energy inside the rocks. A new type of energy constitutive model is established through the functional principle and momentum principle. To validate the model’s accuracy, a triaxial compression test was conducted on sandstone to examine the stress–strain behavior of the rock during the failure process. A sensitivity analysis of the parameters introduced into the model was conducted by comparing the model results, which helped to clarify the innate laws of significance of these parameters. The results indicated that the energy model more accurately captures the non-linear mechanical behavior of sandstone under high-stress loading conditions. The model curve fits the test data to a high degree. The fitting curve was basically consistent with the changing trend of the test curve, and the correlation coefficients were all above 0.90. Compared with other models, the model based on the energy principle not only accurately reflects the rock’s stress–strain curve, but also reflects the energy change law of rock. This has reference value for the safety analysis of rock mass engineering under loading conditions and aids in the development of anchoring and support schemes. The research results can fill in the blanks that exist in the energy method in terms of rock deformation and failure and provide a theoretical basis for deep rock engineering. Moreover, this research can further improve and extend the rock mechanics research system based on energy.

Thermohaline convection in MHD Casson fluid over an exponentially stretching sheet

Abstract This study investigates the thermohaline convection in MHD Casson fluid over an exponentially stretching sheet. This study has practical significance in industrial processes, materials processing, energy systems, and environmental applications. The governing equations describing the conservation for an electrically conducting fluid flow, thermal and concentration transports are considered based on the principles of mass, momentum, energy and concentration equations. Our first step involves transforming the governing nonlinear partial differential equations into a coupled nonlinear ordinary differential equations with the help of suitable similarity transformations. Second step, infinite domain [0, ∞) of the problem to a finite domain [0, 1] through a coordinate transformations. This specific choice is motivated by the wavelet's significance in the finite domain of [0, 1]. Third step, we effectively solve the resulting coupled nonlinear ordinary differential equations using the numerical Hermite wavelet method (HWM). This approach proves to be a valuable technique for obtaining significant results and insights in our study. Finally, the effect of known physical parameters on velocity, temperature and concentration are analysed through tables and graphs.

Effects of cushion gas pressure and operating parameters on the capacity of hydrogen storage in lined rock caverns (LRC)

Hydrogen storage in lined rock cavern (LRC) provides versatile site selection, safety, and stability, making it a crucial option. In this study, a thermodynamic mathematical model was established to describe hydrogen storage in LRC. This model is based on the principles of mass, momentum, and energy conservation while accounting for the authentic compressibility of hydrogen. The thermodynamic fluctuations throughout the seasonal cycle in LRC were computed using the COMSOL Multiphysics software. The results indicate that when the hydrogen storage reaches the operating pressure, the lower the cushion gas pressure, the greater the hydrogen injection amount, and the higher the utilization rate of storage capacity. With a cushion gas pressure of 3 MPa, the storage capacity utilization rate exceeds that at 15 MPa by 41 %. The injection rate of 0.5 kg/s results in a temperature increase of 12 °C compared to 0.27 kg/s, indicating twice the temperature rise at the 0.27 kg/s injection rate. An increase in injection temperature corresponds to higher temperature within the hydrogen storage cavern upon completion of gas injection, thereby reducing the time needed for gas injection to reach maximum operating pressure. A lower initial cavern temperature results in a longer time for hydrogen storage to reach maximum operating pressure.

Lid driven flow and heat transfer due to various positions of slit in a square cavity: FEM approach

A detailed analysis of flow and heat transfer due to moving slit within the various positions of the square cavity is discussed. The slit is strategically positioned and examined at three distinct locations within the square cavity i.e. left, middle and right. Constraints of the square cavity are defined for horizontal and vertical walls. The top horizontal wall is considered to be adiabatic while the other three walls of the cavity are cold. The upper surface of slit is heated and movable towards the left side of the square cavity while the other three walls are immovable and cold. The fluid flow dynamics and heat transfer within the enclosed cavity are governed by fundamental principles of mass, momentum and energy conservation. These governing principles manifest as intricate mathematical equations, specifically nonlinear partial differential equations accompanied with boundary conditions. The key parameters under consideration include the Reynolds number (250≤Re≤1500) and Richardson number (0.01≤Ri≤5) at a fixed Prandtl number (Pr=6.2). These parameters are analysed through variations in velocities, temperature profiles, isotherms and streamlines. In the computational framework, the momentum and temperature equations are addressed through quadratic interpolation, while linear interpolation is applied to handle the pressure equation. The numerical solution, utilizing Newton's technique and the PARADISO solver, is rigorously validated against existing research methodologies, establishing its methodological reliability. Higher Reynolds numbers shift the primary vortex towards the middle and create corner recirculation zones, more uniform heat distribution, while higher Richardson numbers increase buoyancy forces, reducing isotherm contours and affecting heat distribution. The local Nusselt number increases with the increase in Reynolds numbers but shows small changes with the increase in Richardson number. The value of Nusselt number decreases as the position of slit changes from left to right.

On the comparison between phenomenological and kinetic theories of gas mixtures with applications to flocking

We study the comparison between the phenomenological and kinetic models for a mixture of gases from the viewpoint of collective dynamics. In the case in which constituents are Eulerian gases, balance equations for mass, momentum, and energy are the same in the main differential part, but production terms due to the interchanges between constituents are different. They coincide only when the thermal and mechanical diffusion are sufficiently small. In this paper, we first verify that both models satisfy the universal requirements of conservation laws of total mass, momentum, and energy, Galilean invariance and entropy principle. Following the work of Ha and Ruggeri (ARMA 2017), we consider spatially homogeneous models which correspond to the generalizations of the Cucker Smale model with thermal effect. In these circumstances, we provide analytical results for the comparison between two resulting models and also present several numerical simulations to complement analytical results.

Viscous dissipation effects on time-dependent MHD Casson nanofluid over stretching surface: A hybrid nanofluid study

The study of MHD Couple stress Casson flow of a hybrid nanofluid is paramount due to its potential applications in advanced engineering systems, particularly in enhancing thermal management and efficiency in various industrial processes. In this report, the time-dependent MHD Couple stress Casson flow of a hybrid nanofluid, considering the impact of viscous dissipation over a stretching surface is presented. The main objective of this work is to enhance the transformation heat ratio to meet the requirements of the engineering and manufacturing industries. After conducting a continuity check, the issue is analyzed by applying the principles of momentum and energy conservation. This modeling process generates nonlinear partial differential equations (PDEs), which are subsequently converted into nonlinear ordinary differential equations (ODEs) using similarity transformation and thermophysical characteristics. To solve these ODEs, the Approximate Analytical Method HAM is employed, which is specifically designed for nonlinear problems. Velocity decreases with increasing values and higher nanoparticle volume fraction further slows the velocity. Increasing the couple-stress parameter reduces velocity; increasing the Casson parameter slows the velocity profile. Increasing magnetic parameter enhances temperature profile; profile escalates with Eckert number magnitude. These findings contribute significantly to understanding the structure, interactions, and dynamics of such liquids, aligning with the focus on exploring the fundamental properties of simple, molecular, ionic, and complex liquids.

Multi-parameter modeling for prediction of gas–water production in tight sandstone reservoirs

Tight sandstone reservoirs are significant sources of natural gas reserves. As traditional reserves become increasingly scarce and costly, optimizing the development of these reservoirs becomes crucial. This study introduces a novel two-phase gas–water flow model for single wells, incorporating both Darcy and non-Darcy flow equations. These equations are derived from mass conservation and momentum principles for both gas and water phases. Using data from a real tight gas well, our model, which includes stress-sensitive phases for gas and water, outperforms traditional Darcy flow models. Specifically, the average relative deviations in daily production rates were 0.1815% for gas and − 0.2677% for water, which are significantly smaller compared to traditional Darcy flow models. Further application of the non-Darcy flow model reveals strategies to enhance well performance. For example, mitigating liquid lock damage within a 2 m radius near the well could restore the permeability from 0.045 to 0.143 mD, thereby tripling the daily gas production. This non-Darcy flow model is easy to implement and shows significant potential in forecasting production yields in tight sandstone reservoirs, highlighting its importance in the petroleum and natural gas industry.

An approach for multiscale two-phase flow simulation in the direct simulation Monte Carlo framework

To simulate multiscale gas flow with solid particles, Burt's model, based on the Direct Simulation Monte Carlo (DSMC) framework, is widely used to predict gas–solid interactions under the assumption of a negligibly small solid particle diameter compared to the local gas mean free path. However, Burt's model could become inaccurate when the solid particle is large relative to the local gas mean free path. This study introduces the Gas–Solid Synchronous (GSS) model, which predicts gas–solid interactions in continuum gas regions without assuming the local gas flow regime around a solid particle. Similar to Burt's model, the GSS model includes gas-to-solid and solid-to-gas interaction models to consider bidirectional interaction between two phases. The GSS gas-to-solid model is established by selecting accurate semi-empirical force and heat transfer models in comparison with DSMC simulation results. The GSS solid-to-gas model is developed based on the principles of momentum and energy conservation and validated against Burt's solid-to-gas model. The results show that Burt's model could overestimate the interphase force and heat transfer rates when its assumption on solid particle diameter does not hold, but it can reproduce non-equilibrium characteristics of two-phase flows where gas velocity distribution functions do not follow the Maxwell–Boltzmann distribution. By contrast, the GSS model can accurately predict gas–solid interaction in continuum gas flows, while it cannot capture the non-equilibrium nature of two-phase flows. The characteristics and limitations of the two models indicate that using a valid model for each gas–solid interaction could be crucial for accurate simulation of multiscale two-phase flows.

A material energy–momentum flux-driven phase field fracture mechanics model

This work introduces a novel phase-field solution for simulating and predicting fractures in elastic solids. In this phase-field fracture model, the crack growth is driven by the material energy–momentum flux, or the configurational force, inspired by a null divergence conservation law and its variational theory, as opposed to employing the commonly-used strain energy degradation-based crack-driving force. By doing so, the crack growth or material configuration change is solely determined by the physics-based Eshelby energy–momentum tensor density or the Rice J-integral flux, which aligns with the energy release criterion of the celebrated Griffith–Eshelby–Rice (GER) theory. The proposed method not only preserves the integrity of the balance of linear momentum principle at the crack tip region but also provides a correct stress asymptotic field in front of the crack tip, while accurately capturing the crack growth.The key idea of this approach is to use the monotonically increasing and irreversible phase field as the marker of the configurational change and form a phase field variational principle that couples the strain energy density with the configurational strain energy density. By doing so, we only damage the displacement field as a form of diffused crack, as opposed to damaging the stress field in the existing phase-field method. Moreover, from numerical simulations, we discovered two different forms of energy release rates: one drives crack growth by enforcing the crack-tip traction loading condition, and one drives crack growth by enforcing the crack-tip deformation condition, demonstrating the ability of the proposed method to naturally capture two distinct modes of fractures due to two different near-crack tip loading conditions. These advancements and contributions reveal previously unknown insights into fracture mechanics.

Theoretical investigation of MHD peristalsis of non‐Newtonian nanofluid flow under the impacts of temperature‐dependent thermal conductivity: Application to biomedical engineering

AbstractNanofluids have the tendency to improve thermal characteristics in a broader context, such as in medical processes, industrial cooling applications, the transportation industry, hybrid engines, gas temperature reduction, microelectromechanical systems, refrigerators, nuclear reactors, vehicle temperature control, pharmaceutical processes, thermal management of vehicles, microelectronics, and chillers, etc. Therefore, present study investigates the effects of a two‐dimensional magnetohydrodynamics (MHD) peristalsis of the Carreau–Yasuda nanofluid through conservation principles of energy, concentration, mass, and momentum, which is motivated by recent advances in biological engineering. The entire system is coupled through viscous dissipation, Joule heating, a heat sink/source, Brownian diffusion, a radially magnetic field, thermophoresis motion, and slip conditions in a curved geometry. Lubrication theory is employed in order to simplify governing equations. Resulting systems are nonlinear coupled differential equations whose exact solutions are difficult to obtain, therefore, numerical solutions are obtained through NDSolve in Mathematica. The key features of peristaltic movement and rates of heat transfer are disseminated in detail via graphical representations for variation of various physical parameters. According to the findings, for greater values of the Hartman number, temperature increases, and the velocity decreases. The validity of these results has been verified using the existing published articles of Hayat et al. (2018).

Double diffusive on powell eyring fluid flow by mixed convection from an exponential stretching surface with variable viscosity/thermal conductivity

This study delves into heat and mass transference in fluids with variable thermo-physical features, focusing on the Powell-Eyring fluid, a non-Newtonian substance with unique characteristics. Our aim is to understand the behaviour of this shear-thinning fluid when interacting with an exponentially stretchable surface, considering factors like mixed convection and thermal radiation. Our research endeavours to uncover novel findings in this intricate domain, with the primary objective of comprehending how this shear-thinning fluid behaves when it encounters an exponentially stretchable surface. To address the complexities, we employ the BVP4C technique in MATLAB, transforming governing equations into nonlinear partial differential equations (PDEs) based on mass, linear momentum, and energy conservation principles. The Homotopy method aids in navigating these equations, and numerical solutions are calculated using the powerful BVP4C technique for ordinary differential equations (odes). The research stands out for its comprehensive exploration of the interplay among diverse factors, including radiation, variable viscosity, mixed convection, and activation energy constraints. Findings are presented through tables and graphs, offering valuable insights into the intricate physical phenomena within this multifaceted field.