This work presents a sensitivity analysis of entropy generation in the peristaltically driven flow of a Newtonian fluid through a symmetric round conduit. The flow is governed by nonlinear coupled partial differential equations that describe the motion of the viscous fluid within the conduit. These are reduced to nonlinear equivalent ODEs by the long- wavelength and low Reynolds value assumptions. The solutions obtained using boundary value solver (bvp4c) routine yield several numerical findings. Empirical relation for entropy generation (S-G) and pressure rise per wavelength ( Delta P ) is developed using the RSM and ANN. The accuracies of the developed empirical relations are verified by calculating the R-2, along with an ANOVA. In this case, high R-2 = 98.99% and adjusted R-2 = 99.84% values clearly show a good fit of the model. The sensitivity analysis shows that (SG) and (Delta P) is maximum when the curvature parameter, (k), Brinkman number, B-r) Grashof number, (G(r)) increase. The artificial neural networks achieve minimal errors, with MSE ranging from 10 -(08) to 0, ensuring strong convergence and reliable predictions. Fitness analysis confirm accuracy and efficiency in all scenarios. The developed approach is particularly applicable in biomedical systems, such as artificial organs, micro-pumps, and drug delivery mechanisms, where efficient heat and fluid transport are critical.

Non-Newtonian biofluids are pumped through narrow channels under influence of external fields, and this pumping action is fundamental in biological systems such as mucociliary clearance in respiratory tract, ovum transport in fallopian tube, and CSF circulation. The present study deals with the magnetohydrodynamic (MHD) cilia-induced flow of a bilayer Prandtl fluid in a symmetric channel under the influence of simultaneous heat and mass transfer, viscous dissipation and Soret effects. Under conditions of both low Reynolds number and long wavelengths, two sets of nonlinear partial differential equations for the fluid flows are converted and numerically solved using a shooting-method-based NDSolve scheme in Mathematica, with interface and wall boundary conditions imposed across both fluid layers. Key results indicate that as the Prandtl number M is increased from 1.0 to 3.0, the velocity everywhere decreases, and the Nusselt number decreases, indicating a reduction in convective heat transfer at the wall. An increase in H from 1 to 3 helps accelerate the flow near the top ciliated wall and decelerate the flow near the bottom wall, while simultaneously reducing channel temperature by reducing viscous and Joule heating. Internal heat generation is greatly enhanced at higher Brinkman numbers, with Br = 3 yielding the highest temperature profiles throughout the channel. A rise in cilia amplitude ε increases high-velocity and high-temperature cores vertcally, enhancing cross-layer coupling and a uniform solute concentration distribution across the bilayer domain. Increasing the layer viscosities μ^(1) and μ^(2) always reduces flow, quenches convection, and hinders mass diffusion, whereas higher thermal conductivities κ^(1) and κ^(2) increase the boundary temperatures and improve heat transfer at the walls. These results are directly relevant to the rational design of nanoscale ciliary actuators and magnetically responsive nanocarriers, in which the coupling between non-Newtonian bilayer rheology and electromagnetic forcing can be harnessed to regulate the transport of nanoparticles, biomolecules, and nanostructured therapeutic agents within confined biological microenvironments. Such engineered bio-nano interfaces provide a promising route toward smart, stimuli-responsive nanodevices for precision nanomedicine, lab-on-a-chip diagnostics, and next-generation bionanotechnological pumping systems.

We present a robust computational algorithm for the simulation of incompressible two-phase fluid flows on a virtual cubic surface, i.e., the models of incorporating the conservative Allen-Cahn (CAC) equation into the Navier-Stokes (NS) equation. By using a phase-field approach, the proposed method effectively captures the evolution of complex interfaces among distinct fluid phases instead of explicit interface tracking. The projection method is combined with the finite difference method (FDM) to solve the governing equations in an efficient manner. In addition, a multigrid solver is adopted to handle the pressure Poisson equation, which improves computational accuracy and reduces computational cost. The virtual cubic surface is modeled as a two-dimensional unfolded domain to facilitate straightforward discretization while preserving geometric fidelity. Numerical experiments, including benchmark shear flow and vortex dynamics on the cubic surface, back up the efficacy of the method in handling two-phase flows. The computational results validate that the proposed scheme has significant potential to advance the simulation of multiphase incompressible flows on curved or complex surfaces. This approach provides an effective numerical method applicable to various scientific and engineering problems.

The flows of non-Newtonian fluids over deforming surfaces are common in polymer and coating process, as well as bioengineering applications. Inspired by these, the current paper considers the bioconvective magnetohydrodynamic flow of a Sisko fluid on an exponentially stretching surface with consideration to heat transfer, mass transfer, and chemical reaction effects. Similarity transformations are used to convert the governing nonlinear partial differential equations into a system of ordinary differential equations which are solved numerically using the MATLAB bvp4c solver. An artificial neural network (ANN) with the LevenbergMarquardt algorithm is created and trained on the numerical data to achieve a high predictive efficiency. The numerical observations indicate that the effect of increasing the Sisko fluid parameter is to increase the velocity field because of lower effective viscosity, and the effect of applying a magnetic field is to inhibit fluid motion as a result of the Lorentz force. The distribution of the thermal, concentration and motile microorganisms is determined to be highly dependant on the transport parameters governing the organism and this implies that heat transfer, mass diffusion, as well as bioconvective behavior responds differently. The ANN forecasts are in excellent agreement with the numeric solutions, with a low mean squared error and regression coefficients close to unity, and validates the robustness and accuracy of the proposed predictive model. The proposed hybrid numerical-ANN framework provides an accurate and computationally efficient tool for analyzing complex nonlinear transport phenomena and offers potential applications in advanced thermal management, chemical processing, and bioreactor design. Novelty of this work lies in the hybrid coupling of a bioconvective MHD Sisko fluid model with a Levenberg-Marquardtbased artificial neural network for an exponentially stretching surface, which has not been reported previously. Furthermore, the study demonstrates ANN-based high-fidelity prediction of bioconvective transport variables with mean squared errors of order 10-9, highlighting the distinct roles of shear-dependent viscosity, magnetic damping, and bioconvection parameters.

This study focused on the optimal analysis of thermodynamic irreversibilities in the peristaltic transport of a Bingham plastic fluid in an asymmetric porous channel. The governing differential equations were solved numerically using MATLAB's boundary value problem fourth-order method (bvp4c function) to estimate the pressure rise per wavelength () and entropy generation () under varying parameter conditions. Models for and were developed using Response Surface Methodology (RSM) and Artificial Neural Networks (ANNs) to provide comprehensive insights into the system. The coefficient of determination (R 2) value for the RSM model of was 99.98%, whereas that for was 99.76%. The ANN models demonstrated high precision, with error margins ranging from 10-3 to zero for and 10-4 to zero for . Sensitivity analysis revealed that was strongly influenced by the permeability parameter , whereas was more sensitive to low values of and intermediate to maximum values of Brinkman number . Model validation using residual plots, normal probability plots, and observation order comparisons confirmed the excellent agreement between the observed and predicted values. Both the RSM and ANN models achieved regression values near unity, demonstrating their robustness in modeling parameter interactions and system responses. This study establishes a reliable framework for analyzing peristaltic transport in complex fluid systems and provides valuable insights for optimization.

The efficient heat transfer and control of heat flow is essential to electromagnetic propulsion and MHD power generation systems. The present study deals with the effect of magnetic field, thermal radiation, Brownian motion and thermophoresis on the unsteady flow and heat transfer characteristics of a second grade viscoelastic nanofluid flowing over a stretching surface. A physics-informed, closed-loop artificial intelligence framework is developed, integrating a feed-forward artificial neural network for real-time flow field prediction with response surface methodology for global sensitivity analysis and parameter optimization. Results reveal that the second-grade viscoelastic parameter enhances the velocity boundary layer, while increasing magnetic field intensity suppresses fluid motion through Lorentz force retardation. Thermophoretic effects augment thermal energy distribution, whereas increasing Schmidt number progressively diminishes nanoparticle concentration profiles. The model was statistically reliable as confirmed by ANOVA-RSM analysis at 95

This study investigates the steady, two-dimensional flow of Sisko fluid with a chemical reaction over an exponential sheet, employing suitable similarity variables to transform the governing partial differential equations into ordinary differential equations. The research is carried out in three phases: first, acquisition of the numerical data by the employment of the BVP4C solver; then, Response surface methodology (RSM) analysis serves to analyze the impact of the different parameters on the coefficient of skin friction, identifying which ones have the most influence. Finally, an artificial neural network (ANN) is employed to predict the numerically obtained skin friction coefficient and assess model accuracy. The effects of heat generation and chemical reaction are incorporated in the numerical formulation and reflected indirectly in the data used for ANN and RSM modeling. The response surface methodology (RSM) model, based on a central composite design (CCD), identified the porosity parameter (K-p) as the most influential factor on the skin friction coefficient (effect size = 0.72), followed by N (0.18) and A (0.10). Statistical analysis confirmed the model's reliability with a strong correlation (R-2 = 0.9944) and a non-significant lack-of-fit (p > 0.05). The ANN model achieved superior predictive accuracy, yielding a minimal mean squared error (MSE = 1.23 & times; 10(-9)) and high correlation (R-2 = 0.9960) during validation using a 70 %-15 %-15 % data split for training, validation, and testing. Comparative results show that both RSM and ANN effectively capture nonlinear parameter interactions, with ANN demonstrating slightly better generalization. These findings highlight the robustness and predictive capability of the proposed hybrid RSM-ANN framework for predicting skin friction behavior in non-Newtonian Sisko fluid flow.

Novelty: This study presents a pioneering integration of artificial neural networks (ANNs) with the LevenbergMarquardt algorithm to model bio-convective blood flow systems featuring tetra-hybrid nanoparticles in computational framework. The novelty lies in the synergistic coupling of quantum-inspired physics with classical Third grade non-Newtonian fluid mechanics under simultaneous electroosmotic, magnetic, laser heating, and thermophoretic influences. Unlike conventional approaches, this research bridges deep learning methodologies with complex multi-physics phenomena, establishing a transformative paradigm for analyzing nanoparticleassisted hemodynamics. The quantum-classical hybrid modeling framework represents a significant departure from traditional computational fluid dynamics approaches, offering unprecedented insights into entropy generation and bio-convective transport mechanisms. Applications: The research findings have direct translational potential across multiple biomedical frontiers: Enabling controlled nanoparticle navigation through vascular networks for site-specific therapeutic interventions. Facilitating early detection of pathological conditions through enhanced understanding of blood flow anomalies. Supporting development of adaptive sensing platforms for real-time monitoring of physiological parameters. Optimizing laser-activated nanoparticle systems for targeted cancer treatment. Providing predictive tools for understanding hemodynamic behavior in diseased arterial conditions. Methodology: The computational framework employs a multi-tier approach integrating analytical and machine learning techniques. Blood flow is mathematically modeled as a non-Newtonian Third grade fluid embedded with tetra-hybrid nanoparticles, subjected to electroosmotic forces, applied magnetic fields, laser-induced heating, and thermophoretic diffusion. The governing partial differential equations are solved using Mathematica 13.2's numerical solvers to generate high-fidelity datasets. An artificial neural network architecture trained via the Levenberg-Marquardt backpropagation algorithm serves as the predictive engine, learning complex nonlinear relationships between physical parameters and flow characteristics. Results: The ANN model achieves exceptional predictive performance with a mean squared error (MSE) of 4.2105 x 10- 7, demonstrating remarkable accuracy in capturing entropy generation patterns across Prandtl numbers ranging from 0.5 to 2.0 and Eckert numbers spanning 0.1 to 0.3. Comprehensive parametric studies reveal critical flow dynamics: increasing the Deborah number from 0.1 to 0.5 enhances peak velocity magnitude by 23 %, indicating significant viscoelastic contributions to momentum transport. Conversely, magnetic field intensification induces an 18 % reduction in velocity profiles due to electromagnetic Lorentz forces opposing flow motion. The entropy generation analysis identifies optimal operating regimes for minimizing thermodynamic irreversibility, essential for energy-efficient biomedical device design. Temperature distributions exhibit strong sensitivity to laser heating parameters and thermophoretic effects, with tetra-hybrid nanoparticle concentration profiles demonstrating enhanced dispersion characteristics. The validated ANN framework successfully predicts velocity, temperature, and concentration fields across diverse parameter spaces, confirming its utility as a rapid surrogate model for complex bio-convective systems. These findings establish computational foundations for designing next-generation nanoparticle-enhanced therapeutic and diagnostic technologies, advancing precision medicine capabilities through intelligent modeling of multiphysics biological flows.

Homogeneous-heterogeneous chemical reactions are important in numerous thermal and industrial systems due to their control of the relationship between the heat transfer and chemical species transport of the bulk fluid and on reactive surfaces. Catalytic reactors, fuel cells and environmental purification systems are known to have the processes. This research is on the flow behavior of a tangent hyperbolic nanofluid under magnetohydrodynamic (MHD), species diffusion, and velocity slip conditions with a stretching surface. The governing nonlinear partial differential equations of momentum, energy and concentration are transformed into ordinary differential equations by similar transformations. The resulting system is numerically solved by the shooting method to investigate the effects of various physical parameters on the characteristics of flow. The results indicate that velocity profile decreases with an increase in magnetic parameter (M) and power-law index (n) due to an increase in the resistive forces. Streamline analysis indicates that the magnetic parameter plays a more important role in flow structure by comparison with the power-law index. Also, Artificial Neural Network (ANN) model is adopted to model the behavior of flow with results that are not only excellent compared with numerical results but also very accurate. The model may be applied in chemical processing, cooling technologies and biomedical systems.

The goal of this study is to investigate the magnetohydrodynamics boundary layer flow of non-Newtonian Casson hybrid nanofluid past a horizontal circular cylinder in a porous medium. The effects of viscous and ohmic dissipation are also taken into account. Blood is used as a base fluid, and aluminum oxide (Al2O3) and copper (Cu) nanoparticles are used to make it hybrid nanofluid. The local nonsimilarity method is employed to obtain dimensionless nonsimilar partial differential equations. A supervised learning approach based on artificial neural networks (ANNs) is used to compute approximate solutions. The accuracy and reassurance of the estimated solutions from ANNs are assessed through the correlation index and residual evaluation. The mean square error values is evaluated for different situations. The results indicate that nanoparticles significantly enhance both fluid flow and heat transfer primarily due to increased thermal conductivity and alterations in flow characteristics. The machine learning-based ANNs surpass traditional computational techniques due to their superior accuracy and efficiency and their ability to handle the complex data. Because neural networks can learn patterns and relationships from data without explicit programming, they are adaptable to a wide range of tasks.

Accurate forecasting of infectious diseases is essential for effective public health planning and pandemic preparedness. In this study, we present a novel forecasting framework based on Physics-Informed Neural Networks (PINNs) to model infectious disease dynamics. The proposed approach integrates epidemiological differential equations directly into the neural network training process, enabling the model to learn from both theoretical disease transmission dynamics and observed data. This hybrid structure helps reduce overfitting, a common limitation in purely data-driven deep learning models. The model is evaluated using state-level COVID-19 data from Pakistan. A Bayesian PINN framework is employed to estimate hidden epidemiological compartments, correct for potential underreporting, and quantify uncertainty in key transmission parameters. The model also infers time-varying quantities such as transmission rates, detection probability, isolation effectiveness, and the average infectious period. In addition, the influence of different underreporting scenarios on parameter estimates and epidemic forecasts is examined. Results indicate that the Bayesian PINN approach provides reliable predictions even with limited or noisy data. Model simulations show that forecasts of cases, deaths, and hospitalizations are consistent with established benchmark methods while maintaining a simpler and more flexible modeling structure. Overall, the findings demonstrate that PINN-based models offer a promising computational tool for improving infectious disease forecasting and supporting data-driven public health decision-making.

In the field of artificial intelligence and machine learning, physics-informed neural networks (PINNs) have received considerable attention because of their extensive applications in flow problems. PINN is a highly effective tool for discovering the intrinsic physics behind transport phenomena by incorporating governing equations into the training procedure of the neural network. The system of nonlinear partial differential equations is developed using non-Newtonian Casson fluid over a cylinder under the effect of a magnetic field in a porous medium. TensorFlow was employed to create and train the models, and the predicted results were compared with the reference solutions using the bvp4c method. This study compared the numerical and predicted solutions for parameter variation. The desired solutions were obtained by extending the parameter values, which required more neurons and hidden layers. To examine the prediction with PINNs, we used four number of hidden layer and three two number of neurons in the PINN design. In addition, the infinite boundary condition requires a suitable number of layers and neurons to be accounted for when the faraway boundary is set at a larger distance from the origin. The variations of various parameters are analyzed on flow output, i.e. velocity and temperature profiles. The interesting of Lorentz force is examined on fluid velocity and heat transfer analysis. It is noted Lorentz force have opposing effects on velocity. It is also noted that with the growing value of thermal radiation results in the increment of heat transfer.

This study explores the development of empirical relationships for the critical transport performance parameters, pressure rise and entropy generation in peristaltic flow of Bingham fluid through curved channel. The aim is to study structural fluid dynamics to evaluate the interaction between fluid transport and channel geometry by considering effects of curved channel structure on pressure distribution, and heat losses and hence improving efficiency. To do so a consistent correlations of input parameters like curvature, Bingham number, and Brinkman number and the corresponding output responses is developed using a combination of Response Surface Methodology (RSM) and Artificial Neural Networks (ANNs). Numerical solutions of the governing equations are obtained using MATLAB's bvp4c solver, ensuring precise modeling of the flow dynamics. Optimiality is ensured by parameter sensitivity analysis and residual assessments reveal that the Bingham number has a significant impact on pressure rise, while the curvature parameter plays a pivotal role in entropy generation. Although the Brinkman number has minimal effect on pressure rise, its influence on entropy generation exhibits a complex, parameter-dependent behavior. The developed models are rigorously validated, showing strong predictive accuracy with low error margins and high correlation coefficients across training, testing, and validation phases. The findings of this research offer critical insights into optimizing peristaltic flow in practical, non-Newtonian fluid systems, contributing to the advancement of fluid management technologies and systems efficiency.

Gyrotactic microorganisms, including unicellular algae and bacteria, exhibit directed movements in fluid flow. They form three-dimensional coherent structures in complex flows from mixing by commercial mixers. This study uses gyrotactic microorganisms as a new approach to chemical reaction systems, flow visualization and fluid mechanical measurements. The resulting planktonic fluid motion promotes mixing, and gyrotactic organism trajectories are used to visualize perpendicular flow components produced by different mixer configurations. The motions of sunset-colored bodies indicate complex fluid motion. Bioconvection is a phenomenon where microorganisms swim in response to gravity or light, affecting the boundary layer flow of stretching surfaces. In this work, a computational machine learning method is used to investigate the thermal performance of an EMHD non-Newtonian fluid on a porous stretching surface with Dufour-Soret effect of a hybrid nanofluid. MATLAB's built-in function bvp4c is used to create and transform the governing flow equations into appropriate ODEs. The Levenberg-Marquardt scheme with a Backpropagation Neural Network is the basis for supervised machine learning, which is used to calculate over a porous stretching surface of an approximated solution of a hybrid nanofluid of a non-Newtonian fluid. This study uses the MSE, correlation index, error histogram, and linear regression to validate the supervised machine learning approach. An optimal solution is obtained at epochs 344, 1000, 92, 195, 51, 56, and 126, with MSEs is 1.62 , 1.97 , 1.26 , 8.28 , 7.84 , 7.84 , and 7.97 , respectively. This investigation shows an increment in thermal, concentration, and microorganism boundary layer thickness against the raising value of pertained parameters, that is, thermal radiation, Lewis number and bio-convection.

Artificial intelligence and machine learning revolutionizing the domain of fluid mechanic due to their precise modeling, optimization, and understanding the complex and nonlinearity more efficient. The author uses the AI-based Levenberg–Marquardt Scheme with a Backpropagation Neural Network (LMS-BPNN) to investigate the flow stability of MHD boundary layer flow of Casson Hybrid Nanofluid (CHNF) over a porous shrinking sheet. The partial differential equations (PDEs) that describe Casson hybrid nanofluid are transformed into a system of ordinary differential equations (ODEs) with efficient similarity variables. The initial/reference solution is generated using bvp4c function (an embedded MATLAB function designed to solve systems of ODEs) for various input parameters as demonstrated in scenarios 1–5. There are three options to divide numerical data: 80% for training, 10% for testing, and an additional 10% for validation. The LMS-BPNN is used to obtain the approximate solution for scenarios 1–5. The effectiveness and reliability of the proposed LMS-BPNN are validated through fitness curves based on correlation index ( R ), error, and regression analysis. It is noted that velocity and temperature profiles satisfy boundary conditions asymptotically for Senario1-5 with LMS-BPNN. Intelligent algorithms are used to calculate the dual solution for evaluating flow performance results. The perturbation scheme is applied to an unsteady boundary layer problem to obtain the eigenvalues problem. An unsteady solution f(η , τ ) converges to steady solution f_o(η ) for τ→∞ when γ≥ 0 . However, an unsteady solution f(η , τ ) diverges to a steady solution f_o(η ) for τ→∞ when γ <0 . It is found that the boundary layer thickness for the second (lower branch) solution is higher than the first (upper branch) solution. This investigation is the evidence that the first (upper branch) solution is stable and reliable. The analysis of errors demonstrates the consistency and reliability of the intelligent algorithm.

The main purpose of the Levenberg–Marquardt Scheme to analyze flow behvaiour of on non-Newtonian fluid models influenced by gyrotactic microorganisms. The research addresses the complexities of fluid behavior in the presence of biological entities and thermal effects. The study begins by establishing the governing equations for fluid flow as partial differential equations, which are transformed into ordinary differential equations. The numerical solution are obtained with MATLAB ODEs solver bbvp4c. The Levenberg–Marquardt Scheme (LMS) is integrated with a Backpropagation Neural Network (BPNN) to enhance the accuracy of predictions. The efficacy of the proposed LMS-BPNN model is assessed using various statistical metrics, i.e. correlation index, linear regression, and mean squared error. These metrics confirm that LMS-BPNN model provides reliable predictions for fluid dynamics in this context. The study graphically illustrates the effects of various parameters on the momentum boundary layer, thermal boundary layer, species concentration, and motile microorganism behavior. The results show a strong correlation between numerical and predicted results, validating the effectiveness of the LMS-BPNN approach for modeling complex fluid behaviors involving non-Newtonian fluids and gyrotactic microorganisms. The thermal boundary layer grow as there is an increment in the values hybrid nanoparticles ( ϕ_1=ϕ_2 ). The study contributes valuable insights into how various parameters affect momentum, thermal dynamics, and concentration distributions in fluid mechanics, paving the way for future research in this domain.

A machine learning-based surrogate model for bubble dynamics and cavitation behavior under sinusoidal waves has broad applications in science and engineering. In biomedical ultrasound, it can predict the nonlinear oscillations of microbubbles for targeted drug delivery, suboperation, and high-intensity focused ultrasound (HIFU) therapy, thereby enhancing treatment safety and efficiency. In underwater acoustics and naval engineering, such models help analyze cavitation around ship propellers and sonar devices, reducing noise, vibration, and material erosion. Similarly, in energy systems like hydropower turbines, compressors, and pumps, surrogate models accelerate the prediction of cavitation-related performance losses and blade damage, supporting more reliable design and operation. In order to find anomalies, cancers, and other medical issues, it is also used to evaluate organs like the liver, kidneys, heart, and blood arteries. Fluid dynamics difficulties have been greatly impacted by the development of contemporary simulation tools such as Artificial Intelligence (AI) and Machine Learning (ML). This study employs an Artificial Neural Network (ANN)-based machine learning technique to analyze the behavior of spherical gas bubbles in Carreau fluids when exposed to an ambient sonic field. The developed dynamical system is trained using an ANN application, specifically the Levenberg–Marquardt Scheme (LMS), a highly nonlinear and complex network. ANN approximation utilizes the Adam optimizer function to evaluate the numerical solution of the problem. The effectiveness of the proposed scheme is tested through the computation of MSE and the comparison of the optimal curve function against the iterative approach.

The main objective of the conducted research is to present an application of computational fluid dynamics simulation technique to investigate the numerical solution of hybrid nanofluid flow in branched channel carotid arteries configurations with blood as a base fluid. This study examines blood flow across three different branched channel arteries configurations: lateral bifurcation, T-bifurcation, and Y-bifurcation. The numerical treatment of momentum and thermal boundary layer is examined using computational fluid dynamics. The significance of bifurcation shape and hybrid nanoparticles’ interaction on flow behavior and thermal performance are elaborated. The primary objective of computational fluid dynamics simulation with COMSOL is to demonstrate the variations in pressure and velocity profile due to blood flow in branched channel carotid arteries configurations. The artificial neural network algorithm is applied to obtain the predicted solution velocity and flow patterns. The comparison of the predicted solution with the artificial neural network algorithm and the numerical solution with COMSOL is provided. The graphical results demonstrate that T-bifurcation exhibits the maximum pressure and velocity, however, Y-bifurcation maintains the most efficient and consistent flow. This paper combines, in a unique way, ANNs with CFD simulations to obtain high accuracy of predictions in complex geometry velocity and patterns of flow has been validated against the results by using COMSOL. This bridge between machine learning and fluid dynamics, it provides valuable insights into optimizing channel shapes for enhanced flow efficiency. Future research would pursue turbulence modeling, pulsatile flow, and variations of Silver-Gold hybrid nanofluids, leading to higher heat transfer and lower resistance. Such discoveries possess great potential in biomedical applications. These range from cardiovascular drug delivery to surgical planning and diagnostic tools.

This work aims to evaluate the implications of pressure rise per wavelength and frictional forces on peristaltic circulation in a fourth-grade fluid using sensitivity analysis. To accomplish this objective, the frictional forces and pressure increase per wavelength are modelled empirically, linking them to functions that vary with the governing parameters of the issue. First, we use artificial neural networks (ANN) and response surface methodology (RSM) to build an empirical model between responses and governing parameters. This allows us to observe the sensitivity of the frictional forces, rise in pressure per wavelength, and transport parameters. We use the analysis of variance (ANOVA) table to determine the coefficient of determination, which indicates how well the empirical model fits the data. The empirical model shows a 100% coefficient of determination for both frictional forces and pressure increase, indicating a great goodness of fit. After a thorough sensitivity analysis, it is clear that changes to the thermal buoyance parameter have a significant effect on the pressure rise and frictional forces at all levels (-1, 0, and +1). This demonstrates that the Deborah number has the least influence on these responses, and that the thermal buoyancy parameter is the most crucial component in determining how these forces behave in the system.