In current times, a structured problem-solving perspective is advantageous as it reduces both cost and trial duration, and one such approach is response surface methodology. Response surface methodology is a robust optimization paradigm that involves the use of the coefficient of a polynomial to determine the best possible solution once it has been fitted to the data. Its prime objective is to propose optimal conditions to attain low surface drag and maximum heat transfer rate. An artificial neural network uses the Levenberg-Marquardt algorithm and the technique of feedforward backpropagation to learn to predict responses. The current examination is related to the Casson-Williamson nanofluid, as its modeling is essential for industrial and medical processes over a stretched surface replicating polymer sheets undergoing the extrusion process. In addition, the study is concentrated on the existence of inverse Darcy-Forchheimer, unsteady flow, and thermal buoyancy on a curved surface. This research explores the impacts of radiation, thermal sources, and homogeneous-heterogeneous chemical processes. The study also considers the Newtonian heating and Navier slip boundary conditions. The numerical methodology used is the Runge-Kutta Fehlberg method of 4th-5th orders. The findings reveal that the ideal value for the examined factors, the surface skin friction and Nusselt number, is 2. Negative sensitivity is observed for the thermal Buoyancy parameter, Hartman number, and inverse Darcy parameter on skin friction, with the magnitude increasing progressively from the lower to the higher levels of the magnetic parameter. Trained neural network modeling promises accurate results for skin friction and Nusselt numbers, with a minimum squared error of roughly 9.96e-20 and 1.0553e-13, respectively, and a regression coefficient nearly equal to 1, indicating flawlessly correlated data.

The extensive usage of blood-based hybrid-nanoparticles in blood clotting, tissue engineering and also biomedical scenarios has made them essential in the area of blood motion. Both silver (Ag) and molybdenum disulfide (MoS2) nanoparticles can be considered safe and stable for human application. The objective of this examination is to figure out the dual solutions and stability examination of blood-based hybrid nanofluid movement on the top of the unsteady permeable curved stretching/shrinking surface, using Ag and MoS_2 nanoparticles dispersed in blood. The analysis also considers the effects of porous medium, magnetic flux, Joule heating, exponential external heat source/sink and also viscous dissipation components remain intended in the study. The influence of a medium of porous is captured by applying Darcy–Forchheimer model. Using the transformation of similarity variable, governing nonlinear partial differential equations will be converted into system of nonlinear differential equations. The converted nonlinear differential equations are solved numerically by bvp4c problem solver in MATLAB software. The properties of the amount of hybrid nanoparticles, fluid suction or injection, curvature, unsteadiness and stretching/shrinking constraints on skin friction and local Nusselt number as well as the outlines of velocity and temperature are given in this paper. Since the fluid flow has two kinds of solution, stability examination is performed to discovery a stable point, that is, a physically realizable solution. The discovery reveals stable region for first solution and unstable region for second solution. Moreover, the solution of upper region has a positive eigenvalue and the solution of lower region has a negative one.

This study address the flow behaviour and irreversibility analysis of a tangent hyperbolic nanofluid within an upright microchannel. Parameter optimization is conducted using the ANOVA-Taguchi technique to identify the most influential factors. The Buongiorno model theory is utilized to describe nanoparticle transport mechanisms such as Brownian motion as well as thermophoresis. The formulation accounts for various effects, including non-linear thermal radiative heat flux, Hall current, buoyancy force, convective heating, and slip velocity at the channel walls. By introducing non-dimensional variables, the governing nonlinear equations are renovated into a non-dimensionl form and are simplified numerically by applying the Runge-Kutta-Fehlberg fourth-fifth order scheme conjugated with the shooting method. The integration of statistical optimization with numerical computation enhances efficiency and clarifies the interactions among physical parameters. The outcomes indicate that improving the magnetic parameter diminishes the primary velocity and temperature profile and also decreases irreversibility near the channel walls and enhances it in the core region, while simultaneously the secondary velocity increases. Similarly, the radiation parameter improves, resulting in enhanced entropy generation, whereas the Bejan number and temperaturdistribution decrease along the channel. The optimization study predicts a maximum thermal transmission rate of 0.074418. Overall, the proposed framework presents a practical approach for controlling thermo-fluidic performance in microscale systems, especially in magnetically controlled biomedical devices and advanced microchannel cooling applications.

Artificial neural networks have greatly enhanced computational modelling, providing substantial improvements in accuracy, versatility, and predictive performance across a multitude of scientific disciplines. Discharging effluent is a critical concern for industrial plants and environmental treatment systems, as it may pollute water bodies, impacting both ecosystem health and compliance with environmental standards. The efficient monitoring and control of water pollution is crucial for preserving water resources and complying with environmental standards, both of which are vital for safeguarding ecosystems and providing sustainable industrial practices. Herein, we utilize an artificial neural network model to examine the heat transfer and irreversibility in axial bioconvective Williamson nanofluid flow inside a horizontal microchannel under variable thermal conductivity and external pollutant sources, as well as non-uniform heat generation/absorption with no-slip and wall boundary constraints. The analysis is further extended, based on the Buongiorno model, to capture transport mechanisms of nanoparticles that include both thermophoresis as well as Brownian motion. The governing nonlinear equations are converted into dimensionless ordinary differential equations, and we obtain a numerical solution using the shooting technique with Runge–Kutta–Fehlberg. A feedforward backpropagation artificial neural network is then trained on the resulting data. Results reveal that the nanofluid temperature rises in higher values of non-uniform heat generation/absorption and variable thermal conductivity parameters. Similarly, the concentration of motile microorganisms improves for both the bio-convection Lewis number as well as the Peclet number due to stronger thermal gradients. In general, the developed ANN model exhibits great predictability and authenticity toward characterizing complex nanofluid systems. The mean squared error values for all the cases with respect to the ANN model are in the range of e−10 to e−5, which is suggesting good model performance. This scenario is where gradient descent optimization converges near e−8. The correlation between training, testing, and validation data indicates a very strong linear relationship with an average absolute error of only R = 1, substantiating the accuracy and consistency of the proposed ANN-based model.

Optimization of the transport mechanisms in non-Newtonian fluids is fundamental to enhance the thermal efficiency of a wide range of engineering systems. In this study, Soret-Dufour effects on the flow of Walter-B fluid in a stretchable porous channel are investigated. The main focus is to optimize the surface drag by using response surface methodology and sensitivity analysis. The model integrates a magnetic field, buoyancy, linear radiation, a porous medium, heat generation, chemical reaction, and second-order slip with suction/injection. The resulting nonlinear governing equations are solved by utilizing the Runge-Kutta-Fehlberg approach employed in a shooting technique, while RSM-ANOVA is used to identify dominant parameters. These outcomes show that the skin friction varies from-7.5407 to-7.8544, with minimum drag magnitude occurring at Re = 1, M = 0.5, and Kp = 1, corresponding to variation in the surface approximately around 4%. Enhancing porosity reduces the surface drag, and also magnetic parameters suppress the local shear. The radiation and cross-diffusion effects cause a rise in thermal and concentration components. ANOVA consequences reveal that porosity K-p is the most influential parameter (66.8%), followed by Re(29.0%) and M (1.6%), with p < 0.001 indicating a substantial model significance. Markedly, controlling porosity is identified as the most effective strategy for reducing surface drag and improving a system's performance. These conclusions are applicable to several industrial processes, such as polymer extrusion, surface coating operations, and heat dissipation management in porous channel configurations.

The present work purposes to examine the consequence of varying thermal conductivity as well as viscosity on an unsteady flow of Williamson nanofluid within the microchannel. The variable characteristics change in line with the fluid temperature. We consider the mixed convective flow in the existence of linear radiation, no-slip, and convective mass along those heat boundary constraints. The optimal conditions for heat transmission rate and surface drag are obtained by means of sensitivity analysis applied with the response surface technique. Moreover, the Levenberg-Marquardt approach is used in the construction and training of artificial neural network models. Results show that the velocity profile maximises at the right wall of the channel and depletes at the left wall for an increasing Weissenberg number. Higher values of variable thermal conductivity cause depletion in the thermal profile. Skin-friction coefficient declines on enhancing variable viscosity parameter and magnifies when Weissenberg number upsurges. The heat transport rate is enriched by higher values of variable viscosity as well as the thermal conductivity parameter. At a low level of thermal Grashof number, positive sensitivity is exhibited by the variable viscosity parameter and negative sensitivity by the Weissenberg number. The positive sensitivity of the radiation parameter has a higher impact on the Nusselt number. The constructed neural network framework shows excellent predictive capability for both skin-friction coefficient and the Nusselt number, with regression coefficients 0.999 and 0.99554. The very small mean squared errors, 5.4833e−12 and 1.3021e−9, are extremely low, confirming the strong reliability of the model outcomes.

The presents study investigates the novel thermal analysis of porous wavy fin made of copper and aluminium under dehumidification conditions, examining temperature distribution, efficiency variations and moisture interaction. Dehumidification facilitates latent heat release during condensation, introducing simultaneous heat and mass transfer governed by the interaction between the fin surface temperature and the dew point temperature. Moisture flow is modelled using Darcy's equation and the governing equations are solved numerically using the Runge-Kutta-Fehlberg 4th-5th (RKF 45) order method, with results validated against existing literature. Due to copper superior thermal conductivity, it exhibits higher thermal efficiency and temperature distribution with temperature increasing by 127.45% in copper and 193.59% in aluminium, reflecting the impact of material properties on heat retention. As relative humidity (RH) increases by 66.66%, temperature distribution decreases by 85.5% in aluminium and 51.46% in copper, leading to reduced efficiency. Similarly, a 200% increase in the fin profile aspect ratio (aRL) results in a temperature reduction of 24.61% in aluminium and 16.35% in copper, emphasizing the balance between surface area and convective heat transfer. These findings underscore the importance of optimizing fin geometry and material properties for applications in HVAC systems, refrigeration, and heat exchangers, where efficient heat dissipation under dehumidification conditions is crucial.

Optimizing the heat transfer rate for the flow of fluid is fruitful for the industries as well as in the biomedical field. The current study is focused on statistical analysis of heat transmission of the Carreau nanofluid flow by the inclusion of the responses in terms of Nusselt number through response surface methodology. Nonlinear mixed convection is considered to study the natural as well as forced convection. Cattaneo-Christov heat transmission model is employed along with heat generation. Also, entropy generation is considered to analyze the amount of heat disorder in the flow system. After modelling the problem through mathematical expressions, graphs of solutions have been obtained. Results demonstrated that the lower skin friction for higher unsteadiness parameter while keeping the mixed convection factor at its lowest value. Larger rate of heat transmission is obtained for higher value of the thermal relaxation parameter when the unsteadiness parameter is kept low. The Nusselt number decreases by 8-10 % for increasing unsteadiness parameter. Heat dissipation parameter show negative sensitivity at low, medium and high level of Eckert number and positive sensitivity is exhibited by thermal relaxation parameter. For the experimental setup by response surface methodology, the better correlation coefficient is 100 % attained.

The Jeffrey fluid model, known for describing viscoelastic non-Newtonian fluids, is widely applied in polymer processing due to its significance, biological systems, and manufacturing processes. However, most existing studies focus on simplified geometries, neglecting complex interactions and advanced physical effects. Despite extensive studies on Jeffrey fluid flow, the combined effects of inverse Darcy resistance, Cattaneo-Christov double diffusion, magnetic field, Joule heating, chemical reaction, and Newtonian heating over a curved stretching sheet remain unexplored. This study aims to analyse the significance of entropy generation in such a flow system and optimize heat transfer efficiency. To achieve this, the fourth-fifth order Runge-Kutta-Fehlberg method is used for numerical solution, while ANOVA-Taguchi optimization technique is employed to determine optimal conditions for enhancing heat transfer performance. Here, the study reveals that an increase in the Deborah number enhances the velocity profile, while higher values of the inverse Darcy number and inertial co-efficient suppress it. A rise in the thermal relaxation parameter reduces the temperature profile, whereas Newtonian heating increases it. The concentration profile decreased as the concentration relaxation parameter increased, while it increased as the chemical reaction parameter increased. Furthermore, a greater inverse Darcy number and inertial co-efficient result in increased entropy generation, while the Bejan number initially rises near the boundary before gradually decreasing. The ANOVA-Taguchi analysis reveals that the Hartmann number has the least contribution to minimize entropy generation by only 1.66%, while the Eckert number has the most dominant effect at 78.92%.

This study systematically investigates the impact of key physical parameters on entropy generation and thermal behaviour in a micropolar nanofluid flowing through a horizontal microchannel using the ANOVA–Taguchi method. The Buongiorno model is employed to represent nanoparticle transport mechanisms accurately, including Brownian motion and thermophoresis. The analysis also incorporates the effects of magnetic field, fluid suction/injection, and wall boundary conditions. The presence of a porous medium is modelled using the Darcy–Forchheimer theory, while micropolar fluid theory accounts for microstructural effects through microrotation and microinertia. The resulting nonlinear governing equations are resolved numerically using the fourth–fifth order Runge–Kutta–Fehlberg method and validated via the differential transform method to ensure accuracy. The findings indicate that the material parameter increases microrotation in the upper region of the channel but reduces in the lower region, whereas the microinertia parameter exhibits the opposite trend. Higher values of the material parameter also lead to reduced entropy generation, indicating improved thermodynamic performance. Optimization analysis identifies a maximum thermal transfer rate of 1.09233 for the system. According to the ANOVA results, the Prandtl number is the most influential parameter, contributing 79.73% to the total effect on entropy generation, while the Darcy number has a minimal influence of 0.07%. These results highlight the significant role of fluid thermal properties and microstructural parameters in controlling entropy generation and heat transfer in micropolar nanofluid flows through porous microchannels.

The Cattaneo-Christov heat flux model with nonlinear radiation, exponential heat generation, Joule heating, and homo-heterogenic reactions in association with the melting heat peripheral condition has been envisioned as a mathematical representation of the unsteady flow of Prandtl nanofluid carried over a curvy sheet that allows stretching and shrinking geometry enabled by a magnetic dipole. Solution graphs were generated using the Runge-Kutta-Fehlberg 4–5th order tool. Other parameters are simultaneously set to their default values while showing the solution graphs for all flow defining profiles with the appropriate parameters. Each produced graph has been the subject of a thorough discussion. This study investigates the effects of various parameters on velocity, thermal, and concentration distributions in stretching and shrinking sheets. The velocity curves for the magnetic and stretching/shrinking parameters exhibit rising or decreasing trends during sheet stretching, reversing when the sheet shrinks. Significant differences in thermal distribution are observed between stretching and shrinking sheets for each parameter. The velocity distribution decreases with increasing unsteadiness parameter values due to the time factor, while thermal distribution increases and concentration decreases with rising unsteadiness. The melting heat parameter enhances temperature distribution in both stretching and contracting cases. Additionally, an increase in the homogeneous reaction parameter decreases concentration, while the heterogeneous reaction parameter reduces the mass transfer profile in both cases. The thermal relaxation parameter negatively impacts thermal panels when the sheet is stretched and positively when contracting, with both the Nusselt and Eckert numbers increasing regardless of the sheet’s motion. Streamlines and isotherms are provided to further illustrate the flow and heat patterns.

The current study intends to predict the optimised condition to attain the objective of acquiring highest heat transfer rate to develop an efficient model. The transient flow of Carreau nanofluid within a microchannel when channel walls are susceptible to radiation is contemplated. Buongiorno model is employed, which emphasizes the repercussions of Brownian motion and thermophoresis phenomena; also, mixed-convective flow is accounted. The modelled problem gives rise to partial differential equations, which are non-dimensionalized employing non-dimensional quantities. The resultant equations are solved numerically using the finite difference method. Results of analysis demonstrate that the Weissenberg number for n<1 depicts shear thinning nature, and for n>1, depicts shear thickening nature, decreasing velocity. The skin friction coefficient increases when solutal Grashof number rises for the high range of the Reynolds number. The Sherwood number increases when Schmidt number is less for increased value of Reynolds number. Optimization method reveals the highest heat transfer rate of 7.3687 for the considered model. ANOVA results show that the manipulation of Reynolds number is crucial with 57.29% impact and the manipulation of Prandtl number has minor impact of 1.41%on Nusselt number. Shear thinning nature of Carreau fluid finds its application in extrudability, printability and injectability and shear thickening nature is extensively used in industrial polishing, explosion resistance.

Response surface methodology plays a crucial role in optimising system performance by analysing the effects of key variables, such as channel dimensions and fluid flow conditions, to enhance heat transfer efficiency while minimizing pressure drops. This study focuses on the parametric optimization of Williamson fluid flow through a vertical, porous microchannel using response surface methodology, sensitivity analysis, and numerical simulations. The investigation incorporates the effects of the Hall current, non-linear thermal radiation, buoyancy forces, heat sources, convective heat transfer, and slip boundary conditions. The governing equations are solved numerically using the Runge-Kutta-Fehlberg method in conjunction with the shooting technique. The results reveal that increasing the magnetic parameter reduces entropy generation near the channel walls while increasing it within the flow region. The primary velocity diminishes, while the secondary velocity and thermal profile exhibit significant enhancements. Furthermore, an increase in the radiative parameter leads to higher entropy generation and Bejan number values, though the thermal profile declines with this parameter. Sensitivity analysis demonstrates that the magnetic parameter and Prandtl number exhibit positive sensitivity, while the temperature difference has a negative sensitivity. The squared coefficient is calculated to be 100 %, indicating excellent agreement between the predicted and observed values. These findings provide valuable insights into optimising the thermal and fluid characteristics of Williamson fluid flow in microchannel applications, with potential implications for advanced engineering systems.

The purpose of this analysis is to be acquainted with the repercussions of dust particles in the trihybrid nanofluid flow. Solid particles present in the manner of dust, soot, or ash immersed in the liquid influence the flow characteristics. We consider modelling a viscous fluid in the presence of radiation. The objective of the examination is to reduce the surface drag. In view of this, the Taguchi statistical approach is implemented to predict the optimized conditions to obtain the least surface drag. The computational solutions are attained by a finite difference scheme. Results of the examination reveal that an increase of particle concentration and particle mass parameter has depleted the fluid velocity. On escalating dusty fluid parameter, thermal distribution depletes. Shape of the nanoparticle that has the highest temperature is lamina-shaped, and the least temperature is produced when spherical-shaped particles are considered. The fluid phase with nanoparticles has a higher velocity and temperature than the dusty phase. The Taguchi analysis has prompted the optimized condition at the first level of nanoparticle volume fraction and the third level of pressure gradient parameter, radiation parameter, temperature relaxation time parameter and particle mass parameter to produce the highest signal-to-noise ratio corresponding to the least skin friction coefficient. Analysis of variance shows that the regulation of nanoparticle volume fraction on response variate was critical, subsequently by the temperature relaxation time parameter.

Artificial neural network due to its versatile applications is used in various domains. It helps in analysing large datasets which might be difficult to accomplish by conventional models. They help in modelling and analysing complex fluid flow problems and when properly trained they help in predicting the flow structures. Thus, this study focuses on constructing an artificial neural network design to solve mathematical problem of Casson fluid flow in the presence of non-linear radiation and a magnetic field. The study focuses on the flow that changes with time in a microchannel, resulting in partial differential equations that are computed with the help of finite difference approach. The occurrence of irreversibility in the medium is analysed in relation to the flow, and a neural network model is developed. The numerical results indicate that the irreversibility produced in the medium increases as the radiation parameter and temperature difference parameter increase. The mean squared error values achieved for all the scenarios fall within the range of e−12 to e−8, indicating the successful interpretation of the neural network model constructed in tight correlation with the target data. Gradient descent was performed within the range of e−8, and the error histograms have the lowest values within the range of e−8 to e−6. The regression analysis and plotfit demonstrate a high degree of concordance between the data points for training, testing, and validation, with an approximate correlation coefficient ≈1. An investigation of absolute error conducted for various parameters reveals that the errors fall within the range of 10−4 to 10−5.

Artificial intelligence has proliferated across numerous fields of study as a result of its rapid and accurate response times. Its application in fluid dynamics for optimization and computational training of numerical outcomes has been remarkably significant. In the current investigation, a neural network model is constructed to analyze the propagation of Prandtl fluid across a curved, stretched surface that is subjected to radiation and comprises porous media. The interaction between a homogeneous and heterogeneous chemical reactions, magnetic field, thermal stratification, and second-order slip at the surface boundary are all factors under consideration. The numerical solutions of resulting equations are trained and validated using a neural network fitting tool and the Runge Kutta Fehlberg 4 th to 5 th order method. A total of 15% of data is designated for testing and 15% for validation, whereas 70% of it is utilized for training. Levenberg Marquardt training algorithm is employed to assess the functioning of network through regression analysis and mean squared error. In addition, mean square error is estimated to be 10 -8 to10 -6 . Additionally, the elastic parameter has decreased velocity while the Prandtl fluid has increased velocity. The thermal relaxation parameter has reduced the temperature, and the non-linear stretching index has enlarged the solute profile. As a result, this research shows that artificial neural networks can serve as a substitute for long-term computation prediction. However, the model’s fluid flow structure can serve as a guide for creating an optimal industrial design.

This article focuses on the effect of buoyancy on the bioconvective micropolar fluid flow over a curved stretching surface under the slip boundary condition. Mass and heat transfer analysis has been analysed using double stratification, linear heat source, chemical reaction and Joule dissipation. The calculation of entropy formation in a fluid flow is also illustrated. Using similarity transformations, the system of partial differential equations is transformed into a set of ordinary differential equations through reduction in the flow equations and then solved using the numerical methodology of the Runge-Kutta-Fehlberg 4th-5th order method. The mathematical data obtained for velocity, temperature, concentration, and bio-convection profiles with different parameters are presented on graphs. The velocity profile exhibits consistent behaviour near the buoyancy parameter, with a decrease at the boundary followed by an increase. Notably, the velocity profile experiences a decrease under first-order slip conditions but grows when subjected to second-order slip. Micro-rotation intensifies as the material parameter increases. Conversely, temperature and concentration has declined as the stratification parameter rises. Additionally, the concentration of motile microorganisms reduces with increasing Lewis number and Peclet number. As the material parameter rises, entropy decreases while the Bejan number increases.

The current study explains the parametric entropy optimization by using response surface methodology and sensitivity analysis. It also reveals the effects of thermal radiation and exponential heating on Casson-Williamson nanofluid flow over an exponentially stretching curved sheet. The Darcy-Forchheimer model is used and stratification, Navier slip, and suction-injection are used as the boundary constraints. The 4-5th ordered Runge-Kutta Fehlberg approach is enacted to exhibit the modelled flow. Response surface methodology is used to build the statistical design for the Weissenberg number, magnetic parameter, and Brinkmann number. The thermal stratification parameter, according to the findings, lowers temperature. Entropy grows as Brinkmann number and magnetic parameter increase. When the thermal buoyancy factor assumes the minimum value, little shear stress is seen for high Forchheimer number. The squared co-efficient is confirmed to be 99.99%. The Pareto chart confirms that point 2.2 is the important point. Entropy surface diagrams in three dimensions demonstrate that with high magnetic parameter, high Brinkman number levels and low Weissenberg number levels, high entropy is obtained. For low and medium levels of magnetic parameter, Weissenberg number exhibits positive sensitivity, whereas for high and medium levels of magnetic parameter, it exhibits negative sensitivity.