Symbolic Regression (SR) offers an interpretable alternative to conventional Machine-Learning (ML) approaches, which are often criticized as "black boxes". In contrast to standard regression models that require a prescribed functional form, SR constructs expressions from a user-defined set of mathematical primitives, enabling the automated discovery of compact formulas that fit the data and reveal underlying physical relationships. In fluid mechanics, where understanding the underlying physics is as crucial as predictive accuracy, this study applies SR to model three-dimensional (3D) laminar flow in a rectangular channel, focusing on the axial velocity and pressure fields. Compact symbolic equations were derived from numerical simulation data, accurately reproducing the expected parabolic velocity profile and linear pressure drop, and showing excellent agreement with analytical solutions from the literature. To address the limitation that purely data-driven SR models may overlook domain-specific constraints, an innovative hybrid framework that integrates SR with Answer Set Programming (ASP) is also introduced. This integration combines the generative power of SR with the declarative reasoning capabilities of ASP, ensuring that derived equations remain both statistically accurate and physically plausible. The proposed SR/ASP methodology demonstrates the potential of combining data-driven and knowledge-representation approaches to enhance interpretability, reliability, and alignment with physical principles in fluid dynamics and related domains.

This study aims to investigate heat and mass transport characteristics in biomagnetic fluid flow, where blood is modeled as a biofluid containing spherical CoFe2O4 magnetic particles. A modified form of Tiwary and Das’s hypothesis is employed to incorporate the combined influence of magnetohydrodynamic (MHD) and ferrohydrodynamic (FHD) effects, an approach that has received little attention in previous works. Using a one-parameter group-theoretic technique, the governing partial differential equations describing momentum and energy transport are reduced to nonlinear ordinary differential equations with appropriate boundary conditions. Numerical solutions are obtained through MATLAB’s bvp4c solver to ensure computational accuracy and stability. The analysis demonstrates that Lorentz and Kelvin forces strongly affect velocity and temperature distributions, while particle radius and volume fraction significantly influence heat transfer rates and skin friction. Results are discussed for both co-moving and counter-moving plate scenarios. The outcomes of this study provide deeper physical insight into the simultaneous action of MHD and FHD mechanisms in biomagnetic fluids. Such understanding may support the development of advanced biomedical technologies, including targeted drug delivery systems, magnetic hyperthermia for cancer therapy, and improved magnetic resonance imaging (MRI)-based diagnostic procedures. The findings also establish a comprehensive framework for future theoretical and experimental investigations in complex biomagnetic transport phenomena under varying magnetic field strengths and physiological conditions.

This research explores the magnetohydrodynamic (MHD) stagnation-point flow (SPF) over a permeable surface that either stretches or shrinks, incorporated into a permeable surface while also accounting for internal heat generation and absorption effects. Through a similarity transformation, the complex nonlinear partial differential equations (PDEs) describing the flow as well as heat transfer are simplified into a system of nonlinear ordinary differential equations (ODEs). Consequently, these equations are numerically solved with MATLAB's boundary value problem solver, bvp4c. The obtained outcomes include detailed analyses into velocity as well as temperature profiles, along with evaluations of the skin friction coefficient (SFC), as well as the local dimensionless heat transfer rate, which quantify surface drag and the rate of heat transfer, respectively. Notably, for the shrinking sheet, the model results to dual solutions, whereas for the stretched sheet, only a single solution is obtained.The findings indicate that an improve in the permeability coefficient R leads to enhancements in both SFC and the heat transfer rate at the surface. Conversely, a rise in the heat generation or absorption parameter results in a decline in the surface heat transfer rate. Furthermore, a temporal stability analysis confirms that the primary (first) solution remains stable over time, while the secondary solution is found to be unstable.

Unlike conventional Machine-Learning (ML) approaches, often criticized as "black boxes", Symbolic Regression (SR) stands out as a powerful tool for revealing interpretable mathematical relationships in complex physical systems, requiring no a priori assumptions about models' structures. Motivated by the recognition that, in fluid mechanics, an understanding of the underlying flow physics is as crucial as accurate prediction, this study applies SR to model a fundamental three-dimensional (3D) incompressible flow in a rectangular channel, focusing on the (axial) velocity and pressure fields under laminar conditions. By employing the PySR library, compact symbolic equations were derived directly from numerical simulation data, revealing key characteristics of the flow dynamics. These equations not only approximate the parabolic velocity profile and pressure drop observed in the studied fluid flow, but also perfectly coincide with analytical solutions from the literature. Furthermore, we propose an innovative approach that integrates SR with the knowledge-representation framework of Answer Set Programming (ASP), combining the generative power of SR with the declarative reasoning strengths of ASP. The proposed hybrid SR/ASP framework ensures that the SR-generated symbolic expressions are not only statistically accurate, but also physically plausible, adhering to domain-specific principles. Overall, the study highlights two key contributions: SR's ability to simplify complex flow behaviours into concise, interpretable equations, and the potential of knowledge-representation approaches to improve the reliability and alignment of data-driven SR models with domain principles. Insights from the examined 3D channel flow pave the way for integrating such hybrid approaches into efficient frameworks, [...] where explainable predictions and real-time data analysis are crucial.

This study explores free convective heat transfer in an electrically conducting nanofluid flow over a moving semi-infinite flat plate under the influence of an induced magnetic field and viscous dissipation. The velocity and magnetic field vectors are aligned at a distance from the plate. The Spectral Relaxation Method (SRM) is used to numerically solve the coupled nonlinear partial differential equations, analyzing the effects of the Eckert number on heat and mass transfer. Various nanofluids containing Cu, Ag, Al2O3, and TiO2 nanoparticles are examined to assess how external magnetic fields influence fluid behavior. Key parameters, including the nanoparticle volume fraction phi, magnetic parameter M, magnetic Prandtl number Prm, and Eckert number Ec, are evaluated for their impact on velocity, induced magnetic field, and heat transfer. Results indicate that increasing the magnetic parameter reduces velocity and magnetic field components in alumina-water nanofluids, while a higher nanoparticle volume fraction enhances the thermal boundary layer. Greater viscous dissipation (Ec) increases temperature, and Al2O3 nanofluids exhibit higher speeds than Cu, Ag, and TiO2 due to density differences. Silver-water nanofluids, with their higher move more The SRM results with those from the method's

The analysis of velocity slip and heat generation/absorption in fluid flow problems is crucial due to their significant impact on fluid behavior and heat transfer characteristics. The findings are vital for understanding and optimizing flow and heat transfer in industrial processes involving shrinking/stretching surfaces. Thus, this study aims to examine dual solutions of MHD stagnation-point flow over a stretching/shrinking sheet with suction/injection, velocity slip, and heat generation/absorption effects. The governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equations using a similarity transformation and solved numerically using the boundary value problem solver bvp4c, a built-in MATLAB software. Dual solutions are found for the shrinking case, while the stretching case yields a unique solution. Increasing suction and slip parameters broadens the range of dual solutions. Results show that suction enhances the skin friction coefficient and heat transfer, whereas velocity slip reduces skin friction but increases heat transfer. Heat generation lowers the local Nusselt number. It is observed that the first solution is stable, while the second is unstable.

In the present study, we concentrate on finding the dual solutions of biomagnetic fluid namely blood flow and heat transfer along with magnetic particles over a two dimensional shrinking cylinder in the presence of a magnetic dipole. To make the results physically realistic, stability analysis is also carried out in this study so that we realized which solution is stable and which is not. The governing partial equations are converted into ordinary differential equations by using similarity transformations and the numerical solution is calculated by applying bvp4c function technique in MATLAB software. The effects of different physical parameters are plotted graphically and discussed according to the outcomes of results. From the present study we observe that ferromagnetic interaction parameter had a great influenced on fluid velocity and temperature distributions. It is also found from the current analysis that the first and second solutions of shrinking cylinder obtained only when we applied particular ranges values of suction parameter. The most important characteristics part of study is to analyze the skin friction coefficient and rate of heat transfer which also covered in this analysis. It reveals that both skin friction coefficient and rate of heat transfer are reduced with rising values of ferromagnetic number. A comparison has also been made to make the solution feasible.

A theoretical study of velocity slippery, thermal jump, and mass slippery influences in three-dimensional (3D) hybrid nanofluid flowing via a biaxial extending surface is presented. In the present study, a hybrid nanofluid composed of nanoparticles of both magnesium oxide (MgO) and copper oxide (CuO) was formed homogeneously in methanol as a conventional fluid. The controlling boundary layer equations (continuity, impetus, heat, and concentration) are transmuted into nonlinear ordinary differential equations (ODEs) by employing suitable similarity transformations. The numerical solutions were done with the help of bvp4c code in the MATLAB software. Impact of emergent parameters is presented on velocity profile, temperature, and concentration. The physical interest of drag force, Nusselt, and Sherwood numbers are displayed with various parameters. Validation of computations has been performed with published results. Our results suggest that a linear biaxial stretching sheet has a greater significant impact on flow boundary. When the size of the nanoparticles boosts, the velocity outlines decreases, and the thermal and the concentration boundary layers increase. By the increment in the Brownian diffusion parameter, the heat transmission rate reduces while the mass transport rate upsurges. Incrementation thermophoresis parameters works to reduce the heat and mass transmission rate.

Magnetohydrodynamic boundary layer flow and heat transmission processes with a hybrid nanofluid film over a steady stretched sheet are taken into consideration. The impressions of an angled magnetic field, tangent hyperbolic flow, and viscous dissipation upon the momentum and thermal boundary layer are investigated. The leading equations are PDEs transfigured into nonlinear, ordinary ones that apply a non-dimensional transformation. Spectral relaxation methods are exploited for numerical solutions to non-dimensional governing equations with no-slip boundary conditions. This simulation was constructed with the cooperation of the application MATLAB. Present outcomes are matched with literature in the limiting cases and are an excellent agreement. To analyze the flow behavior, thermal physical characteristics, and the nature of the hybrid nanofluid particles’ transport properties, we look at various kinds of hybrid nanofluid particles with the base fluid ethylene-glycol ( EG ), which are Ferro–Copper, ( Fe_3O_4 –Cu) and Single walled carbon nanotubes–Copper Oxide, SWCNT-CuO . The consequences of emerging parameters such as Magnetic parameter, Prandtl number, Brinkman number, Power law index, Weissenberg number, and Angle of inclination are explored through graphs The local skin friction and Nusselt number are also graphically displayed with respect to the above parameters.

Theoretical and numerical investigation of an applied magnetic field on mixed convection flow of a biofluid through a vertical plate using contained heating or cooling is observed in this study. The mathematical formulation is that of the full Biomagnetic Fluid Dynamics (BFD) model which deals with on the ferrohydrodynamics (FHD) and magnetohydrodynamics (MHD)principle. In this work, the study is performed on a specific biofluid, viz. human blood. Assume that the magnetization very linearly with magnetic field strength, temperature dependency of dynamic viscosity and thermal conductivity is noticed. A system of non-linear equations with appropriate boundary condition is obtained by familiarizing suitable non-dimensional variables in the physical problem. For the numerical solution, we used finite difference method which is based on an efficient technique is applied in the problem. Computations for flow profiles, local skin friction coefficient and local heat transfer coefficient are performed with the magnetic parameter Mn, the viscosity/temperature parameter theta(r) and the thermal/conductivity parameter S-& lowast;. The effect of the localized heating or cooling is examined. The computational results presented graphically and have been validated in an appropriate manner. The study reveals that the impact of a magnetic field for blood flow in arteries is found significantly. The results presented bear the promise of valuable applications in physiology, medicine and bioengineering.

This paper is concerned with the investigation of steady, two-dimensional, and laminar boundary layer flow of a biomagnetic fluid over a continuously moving sheet in the presence of a magnetic dipole. The magnetic field resulting from the dipole is contemplated to be strong enough to saturate the biofluid. The magnetization of the fluid is regarded to be a linear function of temperature. The solution procedure involves the reduction of a nonlinear system of coupled PDEs into ODEs that comprise five parameters. The transformed ODEs along with the boundary conditions are then solved numerically by introducing an efficient numerical technique based on the finite difference algorithm. The velocity, as well as temperature profiles within the boundary layer, are illustrated at specified values of free stream velocity (U-infinity), wall velocity (U-w) and ferrohydrodynamic interaction parameter(beta) .The demonstration of the Nusselt number and friction factor are achieved for various governing parameters. Considering a specified Prandtl number (Pr = 7) and normalized velocity difference|U-w-U-infinity|, higher values of the friction factor are obtained for increasing beta and U-w > U-infinity than that of U-infinity > U-w. Whereas for the Nusselt number, we attain higher values for decreasing beta and U-w > U-infinity . Moreover, an increase in the velocity ratio U-infinity /U-w results in a decrease in both heat transfer rate as well as friction factor. Further-more, as beta increases, the heat transfer rate decreases yet we get higher values for higher Pr ,U-infinity and U-w. In the case of friction factor, it increases with increasing beta and gives higher values for lower Pr, U-infinity and higher U-w . We have also depicted the stream lines for the 2-D boundary layer flow of the biomagnetic fluid for different beta. The graphical results manifest that the flow field is greatly impacted by the ferrohydrodynamic field, which could be of interest in medical as well as bioengineering implementations, like, magnetic drug delivery in blood cells, separating RBCs as well as controlling the flow of blood during surgical procedures.

The significance of non-Newtonian fluids is preferred for nanomaterials due to industrial demands. Consequently, this research examines the influence of magnetic fields and power law variation on nanofluids with heat transfer and heat source effects on a movable flat stretching surface assessing the nanoparticles effectiveness. Using numerical simulations via Runge–Kutta method, local similarity is used to convert partial differential equations into ordinary differential equations transformations to solve numerically and engage the local non-similarity method (LNS). The Mathematica package is used to calculate different parameters obtained in the solutions. The velocity and temperature are affected by different nanoparticles and parameters, and these effects are depicted graphically. The results obtained are compared to previous studies to confirm the inference. The output of this analysis shows that increasing the volume fraction of copper nanoparticles has a significant effect on thermal and momentum dispersion, while silver nanoparticles exhibited the maximum thermal conductivity, which is a substantial and obvious discovery regarding the increase in heat source. Furthermore, an increase in the volume fraction leads to a decrease in both the skin friction coefficient and Nusselt number. These findings have important implications for creating more efficient cooling systems for electronics and other industrial applications.

This work explores the interaction of mixed bio convection and non-Newtonian fluid flow around a vertical cylinder. It considers different slip effects and the impact of suction/injection boundary conditions. The inquiry is driven by the substantial implications of comprehending the interconnected dynamics of living organisms and non-Newtonian fluids, with wide-ranging applications in biotechnology, medicine, and environmental science. The study incorporates the intricacies of shear-thinning or shear-thickening fluids by utilizing the generalized power-law model to capture non-Newtonian rheological phenomena. The vertical cylinder, selected as the archetype geometry, functions as a fundamental structure encountered in several engineering applications. The slip effects, which can vary from no-slip to full slip, are included in the model to represent the interactions between the fluid and solid. Additionally, the suction/injection boundary conditions are used to simulate external forces that are provided to govern the motion of the fluid. The study utilizes similarity transformations to convert the governing equations and employs the MATLAB BVP4c scheme to solve the resulting ordinary differential equations. It investigates a parameter space that encompasses non-Newtonian parameters, slip coefficients, bio convection parameters, and suction/injection parameters. The results demonstrate intricate relationships between bio convection, non-Newtonian rheology, slip effects, and suction/injection. These findings state the suction parameter (s>1) and dilatant fluid (n>1) have great influence on heat, mass and motile microorganism rate and also slip parameters are responsible for reducing flow profiles. The study's findings enhance our comprehension of intricate fluid dynamics when biological activity and non-Newtonian behaviour are present. This provides valuable insights for the efficient design and optimization of processes involving vertical cylinders in fields such as biotechnology, medicine, and environmental engineering.

We examine the time-independent boundary layer flow (BLF) and heat transmission of Sisko fluid subject to convective boundary conditions. The fluid flow is impacted through a nonlinear stretchable sheet in the existence of a magnetic field and porous medium. Thermal conductivity effect assumed on heat transfer. We also consider the impact of Joule heating and radiation in the present study. The model's leading partial differential equations (PDEs) have reduced similarity transformation into nonlinear ordinary differential equations (ODEs). The resultant nonlinear coupled differential equations are tackled in MATLAB built-in solver (bvp4c). The repercussions of relevant parameters like material parameter (A), the combined effect of the magnetic and porous medium parameter (C), Sisko fluid power law index (n), stretching parameter (r), Prandtl number (Pr), Joule heating parameter (J), temperature parameter (?) and radiation parameter (NR) on the temperature, velocity, skin friction coefficient, and Nusselt number are scrutinized via graphs. The results show that higher inputs of material parameter have a growing impact on velocity and a decreasing effect on temperature distribution. Increasing Sisko fluid has a reducing impact on velocity distribution. As witnessed from sketches, the shear stress distribution is decreased to increase the impression of the Sisko fluid, material, stretching, and combined parameters. Our computed outcomes are linked to existing outcomes and get the finest contract. The results benefit from comprehending the flow characteristics, flow behavior, and how to foresee it for individuals involved in designing high-temperature machinery in the industry.

In this paper, new symmetry reductions and similarity solutions for Burgers equation with moving boundary are obtained by means of Lie’s method of infinitesimal transformation groups, for a linearly moving boundary as well as a parabolically moving boundary. By using discrete symmetries, new analytical solutions for the problem under consideration are presented, for two cases of the moving boundary: one moving with constant velocity and another one rapidly oscillating.

Cryosurgery is demarcated as cell annihilation by freezing temperature yielded through a cryogenic probe. Thermal distributions nearby the adjacent blood vessels can impede the iceball evolution and surgery success. Hence it is indispensable to predict the dynamic temperature and iceball propagations in biological tissues and its implications earlier through simulation methods. In this study, a transient two-phase flow and heat transfer model is presented to envisage the thermal extents inside and outside cryoprobes. The bioheat transfer model is adopted to inspect the temperature evolutions during multiprobe cryosurgery of hepatic tissues embedded without or with blood vessels. During computer simulation, a 3D geometry model is introduced as the living tissue vicinities embedded with two simplified hepatic arteries when multiple cryogenic probes are injected into the tissues with uniform insertion depth system. The tissues are pondered as non-ideal materials in which phase transition occurs over a temperature range, and the effects of blood perfusion and metabolic heat generation in the tissues are also considered. Computational analyses are then carried out to scrutiny the impact of the blood vessels on the temperature developments of tissues. The results show that when blood vessels are treated as a heat source during treatment, oscillatory thermal outlines are found along the anticipated pathways during multiprobe cryosurgery, and the relative maximum temperature increased upto ∼98 % at the midpoint of the route BB′ for the ablation time of 100 s. The irregular ice fronts are also obtained with the manifestation of blood vessels compared with no blood vessel case, and the iceball volume enclosed by the − 40°C isothermal front decreased significantly from 22.94 cm3 (without blood vessels) to 6.92 cm3 (with blood vessels) for the ablation time of 300 s. The simulation outcomes indicate that multiprobe cryosurgery is more effective to destroy the tumor tissue properly when the target tissue is located faraway from blood vessels. The simulation platform is anticipated to be valuable in optimizing pre-treatment plans of cryosurgery schemes while determining the number and position of cryoprobes in the neighborhood of blood vessels.

The flow and heat transfer of a steady, viscous biomagnetic fluid containing magnetic particles caused by the swirling and stretching motion of a three-dimensional cylinder has been investigated numerically in this study. Because fluid and particle rotation are different, a magnetic field is applied in both radial and tangential directions to counteract the effects of rotational viscosity in the flow domain. Partial differential equations are used to represent the governing three-dimensional modeled equations. With the aid of customary similarity transformations, this system of partial differential equations is transformed into a set of ordinary differential equations. They are then numerically resolved utilizing a common finite differences technique that includes iterative processing and the manipulation of tridiagonal matrices. Graphs are used to depict the physical effects of imperative parameters on the swirling velocity, temperature distributions, skin friction coefficient, and the rate of heat transfer. For higher values of the ferromagnetic interaction parameter, it is discovered that the axial velocity increases, whereas temperature and tangential velocity drop. With rising levels of the ferromagnetic interaction parameter, the size of the axial skin friction coefficient and the rate of heat transfer are both accelerated. In some limited circumstances, a comparison with previously published work is also handled and found to be acceptably accurate.

This research concentrates on the 2-D, steady, laminar, viscous, incompressible boundary layer flow of a biomagnetic fluid containing two different magnetic particles (CoFe2O4andFe3O4) over a continuously moving horizontal plate in the presence of a magnetic field generated by a magnetic dipole. For the mathematical formulation the comprehensive concept of Biomagnetic Fluid Dynamics (BFD) is adopted incorporating the principles of FerroHydroDynamics (FHD) and MagnetoHydroDynamics (MHD). The physical problem which is constituted by a coupled system of Partial Differential Equations (PDEs) along with corresponding boundary conditions, is transformed into a coupled system of nonlinear Ordinary Differential Equations (ODEs) subject to analogous boundary conditions by establishing newly simplified similarity transformations. The transformed ODEs along with the boundary conditions are then solved numerically by introducing an efficient numerical technique based on a finite difference algorithm. Verification of this work has been also done by comparing the obtained results with previously published results and found in quite good agreement. The significant effects caused by the variation of the governing parameters such as the skin friction, heat transfer rate and wall pressure are presented more intricately. It has been contemplated that including magnetic particles with pure blood enhances the impact of the magnetic field on the flow, temperature and pressure profiles which could be of interest engineering implementations, like, magnetic drug delivering in blood cells, separating RBCs (Red Blood Cells), controlling the flow of blood during surgeries, treating cancer by producing magnetic hyperthermia etc.

This article, motivated by hybrid magnetic coating manufacturing developments, utilizes a neural network-based computational program to study the dynamics of hybrid magnetic nanofluids with entropy generation. A new physico-chemo-mathematical model has been presented to simulate the hybrid magnetic nano-coating flow along a stretching surface to a porous medium with viscous heating. A Rosseland flux model is used for radiation heat transfer and Darcy's model for the isotropic porous medium. The stretching sheet is porous, and wall suction or injection are possible. A robust neural network has been deployed to optimize the physical parameters controlling the transport characteristics of hybrid nanofluids. Specifically, two hybrid nanoparticle combinations are addressed, namely graphite oxide (GO)-molybdenum disulfide (MoS2) and copper (Cu)-silicon dioxide (SiO2), both with engine oil as the base fluid. The dimensional boundary layer model is transformed via suitable scaling variables from a partial differential system into a dimensionless non-linear coupled ordinary differential system. The transformed boundary value problem is solved numerically with the BVP4C subroutine in the symbolic software MATLAB, which achieves exceptional accuracy. Validation with previous simpler studies is conducted and a good correlation is obtained. The neural network optimization analysis incorporates Bayesian regularization as the training algorithm. The Bejan entropy generation minimization (EGM) analysis shows that with increasing radiation parameter R-d, both entropy generation rate and Bejan number are increased. Furthermore, an elevation in Brinkman number Br leads to an upsurge in entropy generation rate and a downtrend in the Bejan number. The numerical solution of the boundary value problem reveals that with an increment in nanoparticle solid volume fraction phi(2), magnetic parameter M, inverse permeability parameter epsilon, surface injection parameter (s<0), Eckert number Ec and radiation parameter R-d and with a decrement in suction parameter (s>0) and Prandtl number Pr, there is a strong enhancement in temperature magnitude and thermal boundary layer thickness. With greater nanoparticle solid volume fraction phi(2), magnetic parameter M, inverse permeability parameter epsilon, suction parameter s and a reduction in thermal buoyancy parameter lambda, strong flow deceleration is induced, and momentum boundary layer thickness is increased. The skin friction coefficient is substantially boosted with lower values of magnetic parameter M, inverse permeability parameter epsilon, suction parameter s and higher values of thermal buoyancy parameter lambda. There is a significant decrement also computed in Nusselt number with a greater radiation parameter R-d. The simulations provide a good benchmark for future extensions that may consider non-Newtonian behavior.