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Artificial neural networks have revolutionized machine learning by providing exceptional capabilities for modeling complicated mechanisms and solving various challenges. Backpropagation is an important training technique in the field of artificial neural networks. However, this technique must be optimized when working with complicated fluid dynamics. This study analyzes the three-dimensional radiative flow of a tangent hyperbolic fluid driven by the Cattaneo-Christov flux system across a porous stretching sheet using ANN backpropagation enhanced by Bayesian Regularization approach. Heat and mass transfer analysis includes thermal radiation, chemical reactions and Cattaneo-Christov flux model. Porosity, radiation, chemical reaction rate, and ion slip effect are among the important physical characteristics that are modified to see how they affect fluid dynamics. Using MATLAB's BVP4C solver, the velocity, temperature, and concentration profiles that result from these model equations provide the training dataset for ANNs. The dataset is divided into 80 % for training, 10 % for testing, and 10 % for validation. Performance plots, regression graphs, and error histograms are used to analyze the performance of the LMT-based ANN and demonstrate its high accuracy and efficiency. With an R2 value of 1, the ANN produced a mean squared error of around 10⁻11. Fluid mobility drops as the magnetic parameter grows, while the thermal profile exhibits an increasing trend. Similarly, decreasing fluid velocity is the outcome of raising the porosity parameter. The study's conclusions have great potential for use in sectors that need sophisticated cooling and heating equipment.

Hybrid nanomaterials significantly enhance thermal systems through improved thermal conductivity, efficient energy storage, and customized thermomechanical properties. Due to their superior thermophysical characteristics, binary/ternary hybrid nanofluids are crucial in fields such as industry, biomedicine, transportation, and pharmaceuticals. This study examines the hydromagnetic flows and heat transfer properties of binary (CuO+Fe3O4/sodium alginate) and ternary (CuO+Fe3O4+MoS2/sodium alginate) hybrid Maxwell nanofluids over a radially stretching rotating disk. The governing flow equations incorporate Cattaneo–Christov heat flux, mixed convection, velocity and thermal slips, nonlinear radiation, viscous dissipation, and Joule heating in a Darcy–Forchheimer porous medium. These axisymmetric partial differential equations are transformed into ordinary differential equations using similarity variables, and the spectral quasilinearization method (SQLM) is applied for numerical solutions. Results reveal the effects of relevant parameters on velocity, temperature, skin friction, and heat transfer rates. Novelty lies in the comparative analysis of flow behaviors and entropy production rates between binary and ternary hybrid Maxwell nanofluids. When the heat source is included, the Nusselt number increases by 12.9% for ternary nanofluids and 16.07% for binary nanofluids as the thermal relaxation number increases from 0.5 to 1.5. Furthermore, the ternary hybrid nanofluid shows greater resistance to Lorentz and porous medium forces and exhibits a higher temperature distribution and better thermal management capabilities compared to the binary hybrid nanofluid.

The researchers have shown significant interest in nanofluids due to their superior heat rate transport and enhanced thermal efficiency when compared to general fluids. The useful applications of nanofluids are electronic cooling, automotive engines, machinery, industrial equipment, lubrication processes, solar energy applications, targeted drug delivery, electronics cooling, solar collectors, automotive engines, etc. Therefore, this study focuses on examining the enhanced heat and mass transmission due to the nanofluid model along the curved stretched surface. For the nanofluid formation, the Fe3O4 nanoparticles are incorporated into the vacuum pump oil (VPO). The flow of nanofluid is typically characterized by the combined effects of magnetic influence and chemical reactivity. Further, the relevance of the heat source/sink, Brownian and thermophoretic diffusivity, and the Cattaneo-Christov heat flux are considered in the present analysis. The problem is expressed using PDEs, and these PDEs are renovated into nonlinear ODEs through the application of similarity transformations. An analytical procedure known as the HAM technique is used for the simulation of the resulting higher-order ODEs of the formulated problem. Furthermore, the validity of the present model's accuracy is confirmed through a comparison with the published findings. Some significant outcomes of the present problem are that the heat transference is larger for the curvature factor and thermal relaxation time factor. Nanoliquid velocity is lesser for the greater magnetic factor due to the Lorentz force mechanism. Further, it is examined that the nanoliquid solutal profile is lesser for the higher chemical reaction factor. Detailed discussions on numerous engineering and industrial applications are also extensively covered.

The main purpose of this study is to present the mathematical modeling of steady three dimensional boundary layer flow of incompressible non-Newtonian Cross nanofluid over a bidirectional stretching surface. The Buongiorno nanofluid model and Cattaneo-Christov heat flux model are also assumed in present work. It is considered that bidirectional stretching sheet is embedded in Darcy-Forchheimer porous media. Additionally, the motion of fluid flow is induced due to the bidirectional stretching surface. Boundary layer theory is invoked to model the basic partial differential equations of current study. The modeled partial differential equations are reduced to ordinary differential equations with the assistance of appropriate transformation and then solved numerically through Runge Kutta Fehlberg scheme along shooting method. The ranges of involved physical parameters in present study can be explained as 0≤βT≤0.08,0≤βC≤0.25,0≤Fr≤6.0,0≤β≤4.0,0≤We₁≤1.0,0≤We₂≤1.5,0≤Nt≤0.4,0.1≤Nb≤1.2,0≤n≤2,0≤δ≤1.2,0.7≤Pr≤6.2,1≤Sc≤7. It is engrossing to reveal that surface mass transfer rate is an aggrandizing function of concentration relaxation parameter but reverse behavior is noticed for enrich thermal relaxation parameter. Additionally, fluid velocities f′(η) and g′(η) reduce due to the improvement of Forchheimer number and porosity parameter. Furthermore, magnitudes of the surface drag forces along x− and y− directions grow for larger approximation of Weissenberg numbers.

The Darcy-Frochheimer flow scheme is important in industries where a higher stream rate impact is a prominent occurrence, for illustration in, petroleum engineering. In this article, we examine the influence of Cattaneo-Christov heat fluxing on the Darcy-Frochheimer flowing of Sutterby nanofluid via stretchable surface with chemical reactive and heat radiative impacts. The highest-order nonlinear partial differential equation is addressed by utilizing the similarity function and a new set of the ordinary differential equation is developed. The obtained equations are then numerically examined by using the bvp4c technique in MATLAB. The graphical behavior for numerous aspects, such as velocity, solutal, thermal, drag coefficient, and heat transport are illustrated for particular non-dimensional variables. Skin fraction and velocity curve decline with the higher valuation of the Frochheimer number. An augmentation in the thermal curve due to the heat source/sink parameter, but a reverse pattern is observed for the Nusselt number. The reverse pattern is observed for the thermal curve and solutal curve via a higher Brownian motion parameter. Compared with the existing research, the outcomes demonstrate excellent congruence.

The flow of fluids containing nanoparticles is important in industrial applications, particularly in nuclear reactors and nuclear system cooling to enhance energy efficiency. In connection to this, the convective boundary layer flow of ternary hybrid nanofluid (water-based graphene-CNT-Silver) flow over a curved stretching sheet with activation energy is investigated in this article. In addition, the non-Fourier heat flux model is taken into account. The use of similarity variables transforms the existing partial differential equations into an ordinary differential equation, which is then numerically solved using the Runge-Kutta-Fehlberg fourth and fifth order (RKF-45) method combined with a shooting approach. The set of graphical results for the significant parameters on thermal, concentration, and velocity profiles is explored. Results reveal that the heat transport in ternary hybrid nanoliquid rises as the thermophoresis and Brownian motion parameters rise. The Biot number influences the thermal profile positively, whereas the increasing Schmidt number and Stefan blowing parameter values reduce mass transport. The curvature parameter has positive impact on skin friction and mass transport rate but negative impact on heat transport rate. The concentration profile rises with increased activation energy parameter, but declines with increased chemical reaction rate parameter.

Importance of heat and mass transmission is intensified remarkably in many engineering and technological process such as air conditioning, equipment power collectors, damage of crops, food processing, heat exchangers, refrigeration and many more. Main focus of this exploration is to disclose a mathematical modeling of chemically reactive Maxwell nanofluid for axisymmetric flow case by considering Cattaneo-Christov heat flux model and revised nanofluid model. Additionally, Arrhenius activation energy and magnetic field aspects are invoked. Furthermore, heat source/sink and nonlinear radiation repercussions are accounted. The essential partial differential equations of this exploration are modeled with the assistance of boundary layer estimation and then reconstructed into nonlinear ordinary differential equations by invoking proper transformation. Runge-Kutta-Fehlberg scheme is deployed to establish numerical solutions of this inspection. It is interesting to reveal that fluid temperature and associated boundary layer thickness grow due to enhancement of temperature ratio parameter and heat source parameter. Additionally, higher estimation of magnetic parameter and radiation parameter reflect in the depreciation of wall heat transfer. Furthermore, nanoparticle concentration is denigrated for exalting values of reaction rate parameter. For the endorsement of this communication, a comparison assessment is established with existing investigation and disseminated in excellent resemblance.

In the disciplines of engineering and research, the most demanding uses of nanofluids are of great interest. The use of nanotechnology in current science encouraged scholars to investigate nanofluid models from several perspectives. The main idea of the existing research is to describe the impacts of Darcy-Forchheimer flow of hybrid nanofluid and single nanofluid fluid through a slippery nonlinear, non-uniform starching surface. These nanofluids are developed by submerging different nanoparticles Zirconium dioxide and Aluminum in Kerosene Oil base fluid. The effects of Cattaneo-Christov (C-C) heat flow and radiative flux are examined in this work. Furthermore, radiative, heat source/sink, and dissipation of viscosity impacts are taken into account. This research use mathematical modelling to create and renovate a system of PDEs for fluid flow based on a few standard ones. The produced nonlinear framework is numerically calculated within the shooting approach using the bvp4c solver provided in the computational MATLAB software. When compared to typical profiles, the nature of a range of criteria is visually presented and analyzed in depth.

This study aims to scrutinize the mathematical formulations of hybrid nanoliquid filled with thermal radiation, Cattaneo-Christov heat theory and melting phenomenon in a nozzle of rocket engine that includes hybrid nanoparticles (CoFe2O4,CdTe,TiO2,Cu&CuO)and kerosene (C12H26−C15H32) utilized as a base fluid. The main governing equations are changed into a system of nonlinear ODEs using the similarity transformations. The outcomes of the transformed equations have been achieved by using the bvp4c method in MATLAB software. Physical flow characteristics are shown against velocity, temperature, and entropy generation profiles. The velocity is declined for the greater values of volume fraction of nanoparticles and melting parameter. The thermal field is declined for the variations of thermal relaxation and melting parameters while boomed for the higher values of Biot number. The entropy generation profile is increased for the greater estimations of Reynolds number and Brinkman parameter. To the best of the researcher's information, no one has previously attempted to characterize melting phenomena and Cattaneo-Chrsitov heat theory applications by studying the flow through a nozzle of rocket engine using (CoFe2O4,CdTe,TiO2,Cu&CuO) hybrid nanoparticles and kerosene (C12H26−C15H32) as a base fluid.

In the current article, the Von Kármán swirling flow problem is extended for Navier’s slip condition on the stretching rotating disk surface with the Cattaneo-Christov heat flux. A semi-analytical study of magnetohydrodynamic Casson fluid flows over the rotating disk is carried out. The impact of the Joule heating and convective heating condition is considered. The nonlinear partial governing equations are transformed into the ordinary governing equations by using the similarity transformation technique and then solved by the optimal homotopy analysis method. The consequence of the pertinent parameters such as the Biot number, magnetic field, stretching parameter, Eckert number, and slip velocity parameter on the velocity and temperature distribution are graphically displayed and discussed. The study reveals that the thickness of the momentum boundary layer is decreased by the slip velocity parameter and the stretching rate of the disk. The thermal relaxation time has enhanced the rate of heat transfer and reduced the temperature profile in the fluid.

The rate of change of thermal and flow transfer inside an enclosed box under the effect of thermal radiation and convective flow driven by buoyancy force with discrete heating and cooling is analyzed here. The “Cattaneo-Christov heat flux’ (CCHF) pattern is implemented in thermal energy model. The study is performed by varying the partial thermal sources namely the heater and cooler along the sidewalls of the region in three different positions: bottom, center and top. The inactive parts of the box, namely vertical sidewalls and the horizontal flat walls are maintained adiabatic. Numerical computations to examine the distinctiveness of liquid flow & heat transport are calculated by method of finite volume. The solutions are displayed as graphs. Thermal radiation effects inside the cavity illustrate a drift from linear to nonlinear increase in rate of change of heat. This is observed when the Grashof value is altered from 103 to 106. The results obtained show that middle-middle location of the partial heater/cooler in the cavity produce enhanced heat transfer rate and top-bottom position showed low rate of heat transfer.

In this effort, solid hybrid nanofluid flowing and thermal transport characteristics over a slippery, nonlinear, uniform stretching surface are proposed. The influence of nanosolid particle shapes, permeability material, viscous dissipative flow, Cattaneo-Christov heat flux and radiate flux are studied. The predominant flow equations are systemized in form of partial-differential equations (PDEs). Keller-box's computational method is the employed method to identify the self-similar resolution for transformed principles into the ordinary-differential equations (ODEs) by appropriate transmutations. Williamson hybrid nanofluidcontaining of dual varied types of nanoparticles, named Copper (Cu) and Zirconium dioxide (ZrO2) in the rich viscid; based fluid of kind EO- Engine Oil is utilized in this research. The remarkable consequence of this analysis is reached by comparison of thermal transmission level of such type of fluid (ZrO2−Cu/EO), which has increasingly more gains to traditional nanofluids (Cu−EO). The lamina-figured elements effect the utmost major thermal conductivity in the boundary-layer, whilst the lowermost thermal conductivity is detected in sphere geometric nanoparticle.

Purpose: The current mathematical model is developed to scrutinize the consequence of bioconvective cross diffusion flow of magnetized viscous nanofluid past multiple geometries (cone, wedge and plate) with convective boundary conditions. Together the nanoparticles and motile microorganism are incorporated into the dimensionless nonlinear differential expressions. The behavior of Cattaneo-Christov heat and mass flux is accounted for energy and concentration expressions. The influence of activation energy and thermal radiation are considered. The mathematical model is reduced into an ordinary one by using adequate similarity transformation. Buongiorno model is utilized for nanofluid (nanoliquids) analysis. Methodology/approach: The renovated dimensionless self-similarity systems are then solved numerically by utilizing shooting technique built-in function bvp4c solver with the help of commercial software Matlab. The obtained results are verified and an outstanding agreement has been found. Engineering quantities of interest are observed physically. Findings: The features of various emerging parameters against velocity distribution, thermal distribution, and solutal field of species, microorganism concentration as well as skin friction coefficient, gradient of temperature, local Sherwood number and density number of motile microorganisms are interpreted and deliberated in tabulated and graphical form. Results: The results indicate that velocity field is raises via larger Grashof number. The resultant velocity is decline via larger magnetic parameter. Larger estimation of thermal Biot number increases the heat transfer. Larger thermal relaxation parameter reduces the temperature of fluid. The concentration of nanoparticles is declines via concentration relaxation parameter. The microorganism's field is declines by varying the variations of Peclet number.

The current article scrutinizes the magneto-bioconvective of nanofluid flow through a rotating cone using Cattaneo-Christov heat flux model. Additionally, we analyzed about cross diffusion (thermo-diffusion, diffusion-thermo), porous medium, Stefan blowing and Navier slip effects respectively. The governing equations are converting into system of self-similar ordinary differential equations by using suitable transformation. A parametric study focusing the impact of involved parameters on different fields such as motile microorganisms density, nanoparticle concentration, temperature and (swirl, tangential) velocities are plotted via graphs along with the motile microorganisms density numbers, Sherwood number and local Nusselt number are analyzed through table. As a matter of fact is temperature field declined for increasing thermal relaxation parameter γ. An increase in magnetic parameter M resembles to diminish in energy field. The suction of wall is noted for different number of when flow of swirl is reduce meanwhile injection blowing reveals conversely. The swirl velocity field diminishes to free stream from wall (cone surface) with every numbers of Nr including for stationary cone (Nr=0). Further, Dufour (Du) and Soret (Sr) number have entirely reverse behavior on temperature and nanoparticle concentration fields.