The goal of this study is to analyze, predict, and optimize the thermal and frictional properties of the unsteady separated stagnation-point flow of a radiative magnetohydrodynamic (MHD) Williamson ternary hybrid nanofluid to minimize skin friction and maximize heat transfer. The nonlinear governing boundary-layer equations were derived to account for the combined effects of unsteady separated stagnation point flow of MHD Williamson ternary hybrid nanofluid over a stretching surface, incorporating thermal radiation and mass suction, and were solved numerically using MATLAB's bvp4c method. Using the data, a scaled conjugate gradient-based Artificial Neural Network (SCG-ANN) demonstrated exceptional predictive performance, with MSE ranging from 10#8315;(6) to 10#8315;(7) and correlation coefficient (R) exceeding 0.999, indicating high model accuracy and generalization potential. To determine the best operating conditions, the effect and interaction of unsteadiness ( $ \beta $ beta), nanoparticle volume fraction $ (\phi ) $ (phi), Weissenberg number ( $ We $ We), and radiation ( $ Rd $ Rd) on the skin friction coefficient and local Nusselt number were quantified using the Response Surface Methodology (RSM) with a Central Composite Design (CCD). The ANOVA findings showed significant models with R2 = 95.09% for skin friction and 95.46% for Nusselt number. With a composite desirability of 100%, the RSM optimization projected the minimal skin friction (0.4686) at $ \beta = 1.68179,\phi = - 1.61384 $ beta=1.68179,phi=-1.61384, and $ We = 1.68179 $ We=1.68179, and the highest Nusselt number (129.68) at $ \beta = - 1.68179, \phi = - 1.68179 $ beta=-1.68179,phi=-1.68179, and $ Rd = 1.68179 $ Rd=1.68179. Research shows that increasing the Weissenberg number and flow deceleration (negative $ \beta $ beta) reduce drag, while increasing nanoparticle volume fraction and radiation intensity enhance heat transfer. For designing advanced cooling, energy, and drag-reduction applications, the integrated SCG-ANN-RSM framework predicts and optimizes thermo-hydrodynamic transport in radiative non-Newtonian nanofluid systems using a computationally efficient and powerful hybrid modelling strategy.

In engineering and technology, the micropolar-Sutterby fluid serves as a favorable alternative to conventional non-Newtonian fluids due to its practical applicability, robustness, and computational efficiency. This analysis focuses on the thermophysical characteristics of the magnetized flow of a micropolar-Sutterby fluid bounded by an exponentially stretching surface, taking into account Cattaneo–Christov heat flux, thermal radiation, activation energy, and heat source, which have wide industrial applications. The surface is influenced by Darcy-Forchheimer effects and is situated within a porous medium. The impacts of mobile microorganisms with magnetic flux are also probed. Similarity transformations are used to convert the governing PDEs into coupled ODEs. The resulting ordinary differential equations are solved using the collocation-based MATLAB built-in solver bvp4c. Tables, graphs, and literature comparisons are used to demonstrate the impact of different parameters on the involved profiles. Quantitative results obtained from the numerical simulations reveal that increasing the thermal radiation parameter Rd from 0.5 to 0.9 enhances the local Nusselt number by 24.3

Oil and water-based immiscible nanofluids are being investigated for enhanced oil recovery in the petroleum industry. The removal of leftover oil from reservoir rocks can be enhanced by these nanofluids, which contain nanoparticles dissolved in either water or oil. The procedure of mixing several kinds of solid nanoparticles with different thermophysical properties such as bi/tetra-nanoparticles, into a nanoliquid based on kerosene and water, respectively is crucial. Two immiscible fluids’ flow characteristics in a horizontal semi-corrugated conduit are examined while both electric and magnetic fields are applied. Two regions make up the channel: the first is filled with a Casson bi-nanofluid that conducts electricity and contains two distinct kinds of nanoparticles, and the second is full of permeable matrix soaked in a conducting electrical Newtonian tetra-nanofluid having four distinct kinds of nanoparticles, which has been overlooked in prior published literature. Closed-form functions for the fluid velocity and temperature were identified by applying the perturbation method with appropriate conditions. For a range of problem parameter values, an analysis has been performed for velocity, temperature, shear stress, and heat transmission rate. The effects of pertinent fluid flow parameters on the Bejan and entropy generating numbers were evaluated in detail. The results show the behaviors of two immiscible nanofluids flowing in a channel are more affected by the electric fields, porous media, and the width ratio of the channel. Mathematical evaluation that is intended to be helpful in applications involving oil/water-based nanoflows is presented in the proposed study.

Enhancing heat transfer efficiency and understanding fluid flow behavior in convergent–divergent channels remain significant challenges in advanced thermal engineering applications. This study presents a novel numerical investigation of magnetohydrodynamic (MHD) Jeffrey–Hamel flow through a porous medium in convergent and divergent channels by simultaneously incorporating Joule heating, thermal radiation, activation energy, and stretching/shrinking wall effects in the presence of hybrid nanofluids—an aspect not previously addressed in a unified framework. Three different hybrid nanofluids are considered to evaluate their comparative thermal performance. The governing equations are solved using the bvp5c method. The influences of key physical parameters, including the magnetic parameter, inertia parameter, porosity parameter, stretching/shrinking parameter, and channel angle, on the velocity distribution are examined. Additionally, the effects of the magnetic parameter, Eckert number, thermal radiation, and heat source strength on the temperature field are analyzed, along with the role of activation energy on the concentration profile. The results reveal that the combined effects of magnetic field and inertia significantly enhance the skin-friction coefficient. Heat transfer is intensified with increasing magnetic parameter and heat source strength, as reflected by higher Nusselt numbers, while mass transfer is enhanced with increasing magnetic parameter and decreasing activation energy. Among the hybrid nanofluids studied, the MoS2-Al2O3 water hybrid nanofluid exhibits the highest heat transfer efficiency, highlighting its potential for advanced MHD-based thermal systems.

This study numerically investigates magnetohydrodynamic Casson nanofluid flow over a curved stretching sheet under the influence of Joule heating, thermal radiation, activation energy, and chemical reaction. The Buongiorno model incorporating Brownian motion and thermophoresis describes nanoscale transport, while gyrotactic motile microorganisms stabilize the nanoparticle suspension through bioconvection. A systematic comparison between Newtonian and non-Newtonian (Casson) fluid models highlights the yield stress effects on transport characteristics. The governing equations of the Casson nanofluid flow are transformed into coupled ordinary differential equations by using similarity variables. The obtained system is solved by using the built-in bvp4c method of MATLAB. Key findings reveal that the Casson nanofluid exhibits lower velocity but higher temperature and concentration profiles than the Newtonian fluid due to enhanced viscous resistance and internal friction. Velocity decreases with increasing magnetic parameter (M) and buoyancy ratio (Nr), while it increases with mixed convection (λ). Temperature rises with higher thermal radiation (Rd), Brownian motion (Nb), thermophoresis (Nt), and thermal Biot number (β₁). Concentration enhances with increasing Brownian motion, mass Biot number (β₂), and activation energy (E). Microorganism density increases with motile Biot number (β₃) and curvature (A), but decreases with Peclet number (Pe) and bioconvection Lewis number (Lb). Skin friction rises with magnetic and Casson parameters, while the Nusselt, Sherwood, and motile density numbers show strong Biot number dependence. Results agree excellently with existing literature. This comparative analysis demonstrates that non-Newtonian behavior significantly alters thermal and transport characteristics, making the Casson model suitable for biomedical and industrial applications involving yield-stress fluids such as blood and polymer solutions.

This study demonstrates a rigorous numerical simulation that maps out the parametric sensitivity analysis, mesh grid independence and structural convergence verification of magnetohydrodynamic (MHD) Casson Hybrid Nanofluid (TiO2+Ag-NPs/WEG) flow and heat transfer over a rotating porous stretching surface. The fluid model consists of a 50:50 water and ethylene glycol solution with the addition of titanium dioxide and silver nanoparticles that are widely used for biosensing, medical diagnostics, and photocatalytic devices. The mathematical model is capable of accounting for all the above effects in a comprehensive manner, such as Joule heating, Brownian motion, thermophoresis, space-dependent heat source and exponential activation energy of Arrhenius. The transformed boundary layer ODE's are solved by the adaptive three-step Lobatto IIIa finite-difference formula, which is embedded in the bvp4c collocation solver in MATLAB. Precise grid independence tests demonstrate that the maximum local residual error is always within a convergence tolerance of 10-6 and precisely bounded by the relative error of 10-7 and absolute error of 10-9 when the number of nodes in the system is increased from N = 40 to N = 200\ nodes. The quantitative sensitivity analysis indices computed using response derivatives identify precisely the percentage margins of influence of the competing operational parameters for surface shear stress and wall heat flux. In addition, two-dimensional internal flow streamlines are plotted to trace the trajectory bending of the internal flow which shows the zones of fluid deceleration and cross flow splitting’s of flow trajectories which are rotational. The parametric evaluations show that the influence of the different porosities of the surfaces and magnetic fields restrict the primary flow velocity. The velocity fields enhance with increasing Casson parameters and sheet rotation quantities, but decrease with the inclined magnetic field. In terms of percentage, the local heat transfer rate reduces by up to 12.80% when the magnetic inclination is changed from 20° to 80° while it increases by up to 70.25% and 114.67% upon the increase of the thermal Biot number from 2.0 to 8.0 and the heat source parameter from 0.4 to 1.2, respectively.

The present paper mathematically models and thermodynamically optimizes bioconvective polymeric nanofluids for advanced material coatings and smart surface processing on curvilinear permeable materials. It is a transport system which couples the motile gyrotactic microorganisms with complex interfacial phenomena such as second-order dual-velocity slip conditions, Arrhenius activation energy, cross-diffusion, and an irregular, space-dependent internal heat source/sink. The governing non-linear partial differential equations for the transport properties are converted to a set of coupled ordinary differential equations, using curvilinear similarity transformations, and then solved using an adaptive finite difference scheme, implemented in the MATLAB bvp4c framework. The sensitivity study and maximum absolute error residual evaluation are found to be rigorous and independent of the grid with a high numerical stability and accuracy up to 10-8 of the computations. Geometrical trajectory mapping shows that the streamlines of the fluid flow parallel to the curved surface, has a dual-slip condition and causes a decrease in shear stress near the boundary. Using energy pathway tracking through heatline distributions, it is shown that the internal heat generation and Dufour cross-diffusion have a considerable effect on extending the thermal penetration length compared with the effect of standard isotherms, thus providing a practical way for tuning solidification rates in material manufacturing. Finally, entropy optimization profiles demonstrate that the Bejan number is always within physical limits (0 ≤ Be ≤ 1). The Brinkman number is then found to increase the system enters into the friction-dominated irreversibility state, while first and second order slip parameters can effectively reduce the viscous dissipation and retain the thermal dominance in the polymer boundary layer.

This study aims to determine the optimal circumstances for heat and momentum transport in a TiO2/ethylene glycol nanofluid in an unsteady stagnation-point flow over an EMHD Riga plate, with a focus on nanoparticle aggregation. The originality of this study is in the concurrent integration of (i) aggregation and non-aggregation nanofluid models, (ii) Electromagnetic hydrodynamic (EMHD) actuation using a Riga plate under transient decelerating flow conditions, and (iii) a hybrid data-driven optimization framework integrating numerical modelling, artificial neural network (ANN), and response surface methodology (RSM) for sensitivity and thermal optimization. The controlling partial differential equations are transformed into a system of ordinary differential equations by the application of appropriate similarity transformations. The MATLAB bvp4c solver solves the nonlinear ordinary differential system produced by the unstable boundary-layer equations. High-fidelity numerical data are used to train a Levenberg-Marquardt backpropagation artificial neural network (LMB-ANN) for precise prediction of velocity and temperature fields. RSM with a face-centered central composite design evaluates parameter sensitivity and optimizes the Nusselt number for both aggregation and non-aggregation cases, considering the unsteadiness parameter (- 0.1 <= beta <= - 1.0), the nanoparticle volume fraction (0.01 <= phi <= 0.04), and the thermal radiation parameter (0.2 <= Rd <= 1.0). Results show that increasing the negative unsteadiness parameter (stronger deceleration) drastically affects the flow structure, resulting in dual velocity profiles and a constant thermal boundary-layer thickness decrease. Nanoparticle aggregation increases effective viscosity, reducing velocities while improving thermal regulation. RSM study indicates that unsteadiness is the predominant component affecting heat transport, followed by thermal radiation and nanoparticle volume percentage. The quadratic regression models show high accuracy, with R & sup2; values over 99.9% for both aggregation phases. Aggregation improves thermal stability by making heat transport less susceptible to parameter changes, according to sensitivity studies. Advanced thermal management systems, micro-electromechanical devices, and energy conversion technologies can benefit from the integrated numerical-ANN-RSM framework's robust and efficient prediction, optimization, and control of EMHD nanofluid flows with aggregation effects.

This research aimed to analyze the effects of both without aggregation and with aggregation of nanoparticles (i.e., titania‐ethylene glycol ) on the velocity and temperature profiles over a permeable MHD stretching/shrinking sheet with permeability parameter and thermal radiation . For the purpose of studying nanoparticle aggregation, the improved Maxwell‐Bruggeman and Krieger‐Dougarty models are applied. By applying the similarity transformation, the simple partial differential equations that arise from mathematical modeling are transformed into nonlinear ordinary differential equations. The calculated nonlinear equation is then numerically solved using the Runge‐Kutta‐Fehlberg 4th‐5th (RKF45) order method with shooting technique and analytically via the Adomian decomposition method (ADM). For validation, the outcomes of this inquiry are linked with those outcomes that are available in the literature. In addition, the acquired analytical ADM data are compared to numerical RKF45, homotopy analysis method (HAM)‐package values, and those given in the literature. It is found that the skin friction coefficient is also lower in the presence of aggregation effects than in the absence of such effects. Furthermore, when the sheet is shrinking, the heat transport (HT) coefficient decreases and increases, respectively, with stretching. The aggregation of nanoparticles reduces the HT coefficient.

Aims and objective: This study investigates the impact of varying electrical conductivity and viscosity on flow using a unique design of intelligent Bayesian regularization neural networks (IBRNN). The study focuses on several significant physical aspects of a tetra hybrid nanomaterial model consisting of silver ( Ag ) , magnesium oxide ( MgO ), titanium dioxide ( TiO 2 ) and zirconium oxide (ZrO2) with water as the base fluid. Investigating the effects of mass suction, variable electrical conductivity, variable viscosity, Magnetohydrodynamic, Joule heating, thermal radiation, Smoluchowski temperature, and Maxwell Velocity slip conditions on a combination of tetra hybrid nanoparticles is the primary goal of this work. Design/methodology: The study approach entails converting fundamental equations into a dimensionless form by the use of a similarity transformation technique and implementing the bvp4c numerical computing method in MATLAB. We train and test the ANN-IBR approach exploitation reference datasets derived from numerical calculations to estimate flow solutions under diverse physical parameter situations. The ANN-IBR model effectively moves water and tetra hybrid nanoparticles around a cylinder, according to the research. Mean square error analysis, transition state analysis, histogram analysis, and regression analysis show its accuracy compared to reference data. Findings: The temperature profile exhibits dual behavior in response to the magnetic parameter and the changing electrical conductivity parameter. As the radiation parameter rises, the temperature profile and the Nusselt number both grow in magnitude. Mass suction enhances the velocity profile, whereas the viscosity parameter diminishes the velocity profile of tetra hybrid nanofluid. There is a 0.6% improvement in heat transfer rate for tetra hybrid nanofluid with Rd = 0.8 and a 0.82% improvement for S = 2.0 when compared to nanofluid. This work influences the development of thermal protection systems for aerospace applications, where materials reach high temperatures and thermal load management depends on their conductive properties.

In the current analysis, the energy transport through a hybrid nanofluid (HNF) across a rotating as well as a stretching surface is evaluated. The HNF has been prepared by the dispersion of aluminum alloys (AA7072-AA7075) also known as aerospace aluminum or aircraft aluminum in water. Engineering systems use aluminum alloys with a broad range of characteristics. Due to the lighter weight of aluminum alloys, they are also used in vehicle engines, especially in crankcases and cylinder blocks. The flow of hybrid nanofluid is studied under the significance of thermal radiation, activation energy, heat source, and magnetic field. The flow phenomena are mathematically expressed in the form of a system of nonlinear PDEs. That system of PDEs is first transformed into the dimensionless system of ordinary differential equations (ODEs), by employing similarity variables and then solved through the MATLAB built-in package bvp4c. The results are derived for velocity, concentration, and energy curves versus diverse physical parameters. It has been observed that the temperature profile of hybrid nanoliquid rises with the mounting values of Eckert number, heat source, and thermal radiation. Furthermore, it can be noticed that the energy transference rate rises up to 9.9261% by the addition of AA7072-NPs in water, while in case of hybrid nanofluid, the energy transference rate rises up to12.8972%. It has been also noticed that the hybrid nanoliquid (AA7072-AA7075/water) has superior thermal characteristics than the simple nanofluid.

Enhancing the heat transfer rate in hybrid nanofluids is significantly aided by a porous medium. The nanofluid's heat transfer to the surrounding medium is more efficient due to the presence of a porous medium. The porous space also provides additional surface contact points for heat exchange and an intricate network of interconnected pores. This article elaborates on the incorporation of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) into water to perform hybrid nanofluids. The flow is considered over a Riga plate surrounded by a variable porous space. Various applications, including thermal management systems, microelectronics cooling, and energy conversion devices, benefit greatly from the study of hybrid nanofluid flow on a Riga plate. The similarity equations of the flow problem are easily tackled with the homotopy analysis method (HAM) built on fundamental homotopy mapping. Furthermore, with the increments in paramount parameters, the skin friction coefficient and heat transfer rate are remarkably meliorated under a higher modified Hartmann number. All reverberations are illustrated in graphs and Tables. From the obtained results it is observed that flow control capabilities are enabled by the variations in permeability in the case of hybrid nanofluids. It is also observed that hybrid nanofluids are more capable to enhance the thermal capabilities of the traditional fluids as compared to the mano nanofluids.

The entropy generation analysis has great significance in the industrial sectors with heat transmission and fluid flows, for evaluating the irreversibility aspect of a system in heat transfer operations. The numerical simulation of entropy generation and the nanofluid flow across a permeable curved surface subject to cross-diffusion and irregular heat source/sink is reported in the current investigation. The thermodynamics 2nd law is used to simulate the entropy optimization. The flow phenomena are mathematically described by partial differential equations (PDEs), which are derived in a curvilinear coordinate system. The system of ODEs (ordinary differential equations) is derived by using the similarity conversion, which is further numerically calculated through the parametric continuation method (PCM) using MATLAB software. The results reveal that the velocity slip and curvature parameters improve the velocity profile whereas the inverse effect is observed against the surface permeability parameter. It can also be noticed that the entropy optimization enhances with the variation in Brinkman number and temperature ratio parameter. The impact of Schmidt number and chemical reaction decline the mass transmission ratio.

Researchers are searching state-of-the-art materials continuously for solutions to our most pressing queries regarding regulating heat transfer and energy preservation. One viable strategy is to add small solid metal particles to common fluids to increase their thermal efficiency. Examining the properties of a steady-state magneto-Casson squeezing flow that takes heat and mass transfer into consideration was the aim of this work. The right way to solve the flow problem was to treat it as a time-independent system with a medium that contains permeability between the plates. The authors also examined the effects of thermophoresis, Brownian motion, and magnetic fields on the flow. To conduct the investigation, a similarity transformation was applied to the momentum, mass, and energy equations governing the flow of the system. This change resulted in a fifth-order nonlinear ordinary differential equation (NODE), which described the velocity profile. A second-order NODE that was coupled to this NODE managed the temperature and concentration distribution. The authors used the homotopy analysis method to obtain analytical results that were nearly correct. Following the acquisition of the solutions, more investigation was carried out to ascertain the impact of diverse elements on the temperature, concentration, and velocity profiles. Among these were skin friction, the Nusselt number, porosity, thermal radiation, thermophores, Prandtl, Levis, and Eckert numbers. Effective heat transfer is essential to many industries, including the automotive, microelectronics, defense, and industrial sectors. Cooling various devices and equipment effectively remains a major challenge, though. By applying this theoretical method to improve the heat transfer ratio, the authors hope to meet the demands of the engineering and industrial sectors.

There is great potential for improving the efficiency and sustainability of many industrial processes, including advanced cooling systems and aerospace applications, by understanding the dynamics of shape factor in magnetohydrodynamic (MHD) Al2O3-Cu-TiO2/H2O nanofluid on a stretching disk as a function of Joule heating and thermal stratification. Given the aforementioned applications, the primary aim of the present study is to examine the entropy analysis pertaining to the hydromagnetic radiative stagnation point flow of a ternary nanofluid. This flow is facilitated by a stretching disk that possesses various characteristics, including shape factor, variable viscosity, thermal stratification, viscous dissipation, mass suction, and joule heating. We used suitable transformations to turn a set of linked partial differential equations (PDEs) into a system of ordinary differential equations to simplify the computations. The bvp4c solver solved these altered equations numerically. Our results were almost identical to prior findings in particular circumstances, with a relative error of 0%. Visualizations showed velocity, temperature, entropy production, Bejan number, skin friction, and the Nusselt number. Velocity profile decreases and temperature profile increases for variable viscosity parameter. Higher Eckert number and radiation parameter values boost thermal profiles, according to our research. Temperature distribution decreased as magnetic and suction parameters increased. Both the Eckert number and the thermal stratification parameter have a negative correlation with the Nusselt number. Compared to conventional fluids, nanofluids enhance the Nusselt number of nanofluids by roughly 10.2%, hybrid nanofluids by around 19.8%, and ternary hybrid nanofluids by about 38.05% when the parameter S is set from 2.0 to 2.4 with nanoparticles of 0.01. Nanoparticle volume fraction, Eckert number, and Brinkmann all contribute to an increase in entropy production and a drop in the Bejan number. Radiation also helps with Bejan and entropy.

The main objective of the current endeavor is to monitor hypothetical processes utilizing a Sisko tri-hybrid fluid over a rotating disk with entropy generation suspended in Darcy-Forchheimer porous medium. Electro Magneto Hydro Dynamics (EMHD), non-linear thermal radiation and exponential and thermal- space dependent heat source/sink coefficients are considered with the intent of conceiving an Runge-Kutta-Fehlberg method with shooting procedures integrated with a combination of an Adaptive Neuro-Fuzzy Inference System (ANFIS) and Reptile Search Algorithm (RSA). Then, ANFIS-RSA, is used to predict the Nusselt number, skin friction co-efficient in radial and tangential velocities. Reliable self-similarity variables have reduced a non-linear partial differential set of equations into an ordinary differential equation. According to the empirical evidence, Sisko fluid parameter rises the radial velocity whereas for magnetic field and Darcy-Forchheimer the azimuthal and axial velocities visualizations decreasing trend, respectively. The entropy generation and Bejan number rises for electric and radiation effects. Also, ANFIS-RSA indicates that the model attained a high level of precision in terms of radial velocity (98.13%), tangential velocity (98.18%) and Nusselt number (98.91%). Thus, the longer rendering of the nanoparticles used here might, makes them potentially helpful for regulating the therapeutic impact in the management and treatment of cancer.

This work aims to analyze the impacts of the magnetic field, activation of energy, thermal radiation, thermophoresis, and Brownian effects on the hybrid nanofluid (HNF) (Ag++silicon oil) flow past a porous spinning disk. The pressure loss due to porosity is constituted by the Darcy–Forchheimer relation. The modified Buongiorno model is considered for simulating the flow field into a mathematical form. The modeled problem is further simplified with the new group of dimensionless variables and further transformed into a first-order system of equations. The reduced system is further analyzed with the Levenberg–Marquardt algorithm using a trained artificial neural network (ANN) with a tolerance, step size of 0.001, and 1,000 epochs. The state variables under the impacts of the pertinent parameters are assessed with graphs and tables. It has been observed that when the magnetic parameter increases, the velocity gradient of mono and hybrid nanofluids (NFs) decreases. As the input of the Darcy–Forchheimer parameter increases, the velocity profiles decrease. The result shows that as the thermophoresis parameter increases, temperature and concentration increase as well. When the activation energy parameter increases, the concentration profile becomes higher. For a deep insight into the analysis of the problem, a statistical approach for data fitting in the form of regression lines and error histograms for NF and HNF is presented. The regression lines show that 100% of the data is used in curve fitting, while the error histograms depict the minimal zero error −7.1e6 for the increasing values of Nt. Furthermore, the mean square error and performance validation for each varying parameter are presented. For validation, the present results are compared with the available literature in the form of a table, where the current results show great agreement with the existing one.

Non-Fourier's and non-Fick's models enhance predictions of transient heat transfer and mass transport, especially in situations where local equilibrium assumptions break down. In such fluids, the deformation rate and shear stress display a natural parabolic structure. In this study, we explore the flow of a Jeffrey fluid with dual diffusions, incorporating both non-Fourier's and non-Fick's assumptions, through the gap of disk-cone devices. Four scenarios have been investigated for fluid flow and heat transformation in this work comprising of (i) both disc and cone are rotating in opposite directions (ii) both disc and cone are rotating in same direction, (iii) cone is rotating while disc remains stationary, (iv) disc is rotating while cone remains stationary. After deriving the basic equations, a set of appropriate variables is used to convert them into dimension from form. RK-4 (Runge-Kutta 4th order) technique has been employed for the solution of nonlinear equations. As the magnetic and Maxwell factors experience a notable increase, whether the disk and cone move in tandem, in opposite directions, or involve a static cone with a gyrating disk, or a static disk with a gyrating cone, the transverse velocity panels consistently exhibit retardation in all scenarios. Additionally, a noteworthy observation is made when evaluating changes in both the Nusselt number and the Sherwood number, with significantly more pronounced variations detected at the surface of the disk compared to that of the cone. There has been a 35% increase observed in the thermal flow rate with variations in the thermophoresis factor.

Oldroyd-B nanofluids have potential applications in enhanced heat transfer systems, drug delivery, and advanced material processing due to their unique rheological properties. Bioconvection finds applications in biomedical research, environmental monitoring, and optimizing fluid dynamics in biotechnological and pharmaceutical processes. Due to these applications current study investigates bioconvection in the flow of Oldroyd-B nanofluid subjected to Joule heating with convected boundaries. For uniqueness of problem the effects of variable thermal connectivity and activation energy are taken into account. In addation the influence of heat source and thermal radiation are part of current model. The governing PDEs of mathematical model incorporates the Oldroyd-B rheological framework to capture the viscoelastic nature of the fluid are transformed into ODEs via similarity solution. Matlab platform via shooting algorithm is involved to solve transformed system. Through numerical simulations the stimulus of involving parameters on velocity f '(eta) temperature theta(eta), concentration phi(eta), and microbes profile X(eta) are displayed in graphical and tabulated form. For strength of problem the streamlines and graphs of physical quantities are also designed. It is noted that velocity curve decreased for increasing value of magnetics parameter and opposite trend is noted in velocity distribution for increasing value of mixed convection parameter lambda. Moreover, thermal layer theta is diminished for growing value of Pr, a reverse relation is noted in concentration profile for value of activation energy.