Conical disk systems play a crucial role in biomedical engineering, rheometer, viscometry, and advanced thermal devices due to their ability to regulate flow and transport processes. In this study, we investigate the nonlinear behavior of tangent hyperbolic nanofluid flow across an inclined conical disk configuration, which uniquely combines a stretchable disk with a rotatable cone under the influence of magnetohydrodynamic (MHD) effects. By employing similarity transformations, the governing nonlinear partial differential equations are reduced to a system of dimensionless ordinary differential equations, which are then solved numerically using MATLAB’s boundary value solver bvp5c. To further improve computational efficiency and predictive capability, a feed-forward backpropagation neural network is trained on the numerical results for heat and mass transfer rates. The results demonstrate that increasing the disk inclination angle intensifies radial flow, while simultaneously reducing thermal and solutal transfer rates by 34.47

Magnetohydrodynamic porous duct flow systems are widely employed in engineering and bioengineering applications, such as thermal exchangers, exhaust ducts, and blood circulation devices. The current study explores the thermal and flow characteristics of a conducting fluid flow within a duct subjected to a spatially varying periodic magnetic field with a soft porous medium. The formulated partial differential equations with boundary conditions are transformed into a dimensionless form through appropriate scaling. These dimensionless equations are numerically resolved by employing the explicit central finite difference scheme in MATLAB R2024a. A graphical visualization and tabular data are provided to evaluate the effects of controlling parameters on velocity, induced magnetic field, volumetric flow rate, temperature, shear stress, and Nusselt number at the walls of the duct. The results highlight that a narrowing duct with a deforming fibrous medium diminishes the velocity magnitude, volumetric flow rate, induced magnetic field, and shear stress at the walls, while improving the rate of heat transfer by 80%. On the other hand, with the widening of the duct, the velocity magnitude, induced magnetic field, volumetric flow rate, and shear stress at the walls increase, whereas the Nusselt number is reduced by 43% in this case. Furthermore, a significant increment in the velocity and temperature maxima is observed with the enhancement of the periodically induced magnetic field within the flow system.

The investigation of ferrofluid flow through flexible channel holds significant importance in various fields, including biomedical engineering, microfluidic devices, and magnetohydrodynamic (MHD) pumping systems. This study examines the flow of a hydrocarbon-based ferrofluid flow within a flexible wavy channel under the combined effects of velocity slip and temperature jump conditions. The novelty of this work lies in the unified analysis of double-slip boundary conditions together with magnetic field interactions, oscillatory Reynolds number, wave amplitude, and wave steepness parameters. A similarity transformation is employed to convert the governing momentum and energy equations into a set of coupled, nonlinear ordinary differential equations as functions of leading dimensionless key parameters. The transformed system is numerically solved using MATLAB R2023a, to obtain the detailed graphical presentations for velocity and temperature fields. The results reveal that the axial velocity is enhanced by the magnetic interaction parameter and oscillatory Reynolds number, whereas, the tangential velocity exhibits an opposite trend with these parameters. Notably, lower wave amplitude induces stronger thermal variations, whereas, higher wave amplitude smoothens these fluctuations. At lower magnetic interactions, distinct dipole-like streamline patterns with strong recirculation zones are observed, which gradually diminish with stronger magnetic interactions. Moreover, increasing the velocity slip parameter reduces the skin-friction coefficient by nearly 33% and the Nusselt number by about 9%, while higher oscillatory Reynolds numbers strengthen convective transport, leading to a 47% increase in the Nusselt number at the lower-wall. This study contributes to the advancement of magnetically actuated pumping systems, improving the efficiency of various medical, industrial, and microfluidic equipment.

Elliptic partial differential equations (PDEs) are of vital importance in several physical and engineering phenomena, for example in heat conduction, electrostatic, fluid dynamics, and gravitation. The solution of elliptic PDEs using any traditional and modern numerical approach is consistently a difficult task, and constructing a numerical method that is accurate, efficient, simple, and reliable to handle these problems effectively is of great importance. The primary focus of this research is to develop a numerical scheme utilizing the knowledge of Vieta-Fibonacci wavelets and artificial neural networks to solve elliptic PDEs accurately. This hybrid combination of wavelet and neural networks has been designed in order to deal with the involved challenges while solving elliptic PDEs effectively. The detailed structure and algorithm of the Vieta-Fibonacci wavelet neural network (VFWNN) method for solving elliptic PDEs of the Poisson kind with Dirichlet boundary conditions have been illustrated. Several test examples are resolved using the VFWNN method which demonstrates the validity and computational strength of the proposed method. The comparison has been made with the other existing methods which ensure that the VFWNN method is an efficient PDE solver with the potential to solve scientific and engineering problems.

Electromagnetohydrodynamic (EMHD) mechanisms play a crucial role in controlling the momentum and energy transport in nanofluid-based systems, offering significant prospects for advanced thermal regulation systems. The current work investigates the coupled influence of externally imposed magnetic and electric fields on nanofluid flow across a rotating conical disk, incorporating Hall current and ion-slip effects to capture finite conductivity phenomena. Two novel scenarios are defined: (i) flow in the absence of an electric field, (ii) flow under an externally imposed electric field. Utilizing similarity transformations, the resulting nonlinear governing equations has been solved numerically using the bvp5c, and a comparative evaluation is done to authenticate the results. The influence of key parameters on velocity profiles (flow), temperature and heat transfer efficiency are illustrated through graphical and tabular results. The findings reveal that the externally applied electric field serves as a primary electromagnetic control mechanism, substantially enhancing the velocity magnitude and temperature distribution. In the absence of electric forcing, the flow exhibits outward radial motion along both the disk and conical surfaces and localized temperature gradients near the disk. The introduction of an electric field alters the flow structure by reversing the radial motion and suppressing axial inflow near the disk, strengthening axial outflow adjacent to the conical surface and redistributing thermal energy more uniformly across the flow domain. These results provide new physical insight into electromagnetic regulation of nanofluid transport and support the modeling and design of electromagnetically controlled systems.

The phenomenon of rotating flow across a disk holds significant importance in diverse fields, including advancements in aerodynamics, turbo-machinery design, blood flow analysis, oceanography, and meteorology. The current study aims to investigate the rotating family of flows, i.e., the Bodewadt, Ekman, and von Karman (BEK) ferrofluid flow past a vertically reciprocating and rotating disk. A similarity transformation reduces the governing momentum and energy equations to a set of coupled nonlinear ordinary differential equations, solved numerically using the bvp5c solver in MATLAB. Graphical visualizations of velocity components, streamlines, isotherms, and temperature fields are presented to illustrate the physical trends. The findings reveal that the radial velocity of the ferrofluid is a decreasing function of both the ferromagnetic interaction parameter and downward reciprocation, whereas the tangential velocity exhibits a direct relationship with these parameters. Notably, in Ekman flow, the radial velocity increases near the geostrophic region, where the Coriolis and pressure gradient forces reach equilibrium, underscoring the balance of rotational and pressure-driven effects. The analysis of thermal behavior indicates that temperature profiles escalate with thermal conductivity and upward reciprocation, while they decline with increasing Prandtl number, demonstrating the superior cooling efficiency of hydrocarbon-based ferrofluid (C1-20B). Hydrocarbon-based ferrofluid exhibits a higher surface heat transfer rate than water-based and fluorocarbon-based ferrofluids. These findings underline the potential of ferrofluids for electronic cooling, compact heat sink design, and improved drilling efficiency in oil and mineral extraction.

The study of electromagnetohydrodynamic (EMHD) fluid flow through a rectangular duct is significant in heating, ventilation, and air conditioning (HVAC) systems, EMHD pumps, heat exchangers, biosensors, and liquid metal cooling systems. Understanding the interplay between electric and magnetic fields, which is vital for several industrial applications, serves as the driving force behind this study. The effects of nonlinear convection and Hall current are incorporated for the first time, contributing novel insights to the understanding of these phenomena. The analysis of the Hall effect is essential because of the applied magnetic and electric fields, while the induced magnetic field is ignored due to the high magnetic diffusivity condition. The impact of the nonlinear Boussinesq approximation makes it particularly suitable for high-temperature conditions and ideal for modeling heating and cooling devices. In addition, geometric effects are analyzed by considering narrow, square, and wide-shaped ducts. The formulated underlying equations are computed through the explicit central finite difference scheme (ECFDS) in Matlab R2023A. Computations have been performed for influencing parameters like as the Hartmann number, Brinkman number, nonlinear convection parameter, Grashof number, and Hall parameter. The study highlights the substantial influence of the nonlinear convection parameter, Hall parameter, and duct shape on flow behavior and thermal transport. Results indicate that the augmentation in the Hartmann number, Grashof number, nonlinear convection parameter, and Brinkman number augments the flow, whereas the flow rate declines with an elevation in the Hall parameter through ducts. An augmentation in the Hartmann number boosts the volumetric flow rate by 35% and about 95% rise in the Nusselt number at the cold wall of the duct. Additionally, the transition in the shape of the duct from narrow to wide elevates the volumetric flow rate and thermal transfer in the left wall of the duct. This study may play a vital role in optimizing and controlling the performance of heat exchanger devices.

Electromagnetohydrodynamic driven nanofluid flows in vertical ducts are of considerable interest due to their applications in biomedical devices, nuclear reactors, and solar thermal systems. This study investigates the response of a spatially periodic magnetic field and an externally applied electric field on the motion of an electrically conducting nanofluid within a vertical duct, accounting for Hall current and ion-slip effects. Two configurations are considered: flow under the influence of an applied electric field and a flow free from an electric field. The underlying set of partial differential formulations are nondimensionalized and computed numerically via the explicit central finite difference scheme implemented in MATLAB. The influence of key physical parameters on velocity, temperature, and engineering parameters is elaborated through graphical and tabular illustrations. The findings reveal that a higher Hartmann number reduces velocity in the absence of an electric field but enhances it when the electric field is applied. Hall current and ion-slip parameters decline the fluid temperature for the inclusion of an electric field, and negligible impact otherwise. The results indicate that the presence of an electric field enhances the volumetric flow rate by approximately 42%, whereas the absence of an electric field leads to a reduction of about 52%. Moreover, the Nusselt number at the left wall of the duct increases by 80% in the presence of an electric field, while a negligible reduction of around 0.002% is observed in its absence. The outcomes of the study are crucial in developing effective electromagnetic control strategies for the thermal-management devices.

This study presents two numerical methods for solving tenth-order differential equations: the Vieta-Fibonacci wavelet method (VFWM) and the reproducing kernel Hilbert space method (RKHSM). The VFWM approximates the unknown function using Vieta-Fibonacci wavelets, transforming differential equations into algebraic ones and solving them via the collocation method. A key contribution is the derivation of the operational matrix of derivatives for Vieta-Fibonacci wavelets, enhancing computational efficiency without sacrificing accuracy. The RKHSM generates approximate and analytical solutions in series form, effectively addressing nonlinear problems. Both methods are evaluated for convergence, accuracy, and computational efficiency. Applications to three test problems demonstrate that VFWM excels in handling higher-order derivatives and boundary conditions, while RKHSM offers flexibility for a range of nonlinear issues. These methods are reliable, precise, and efficient, with potential applications in fields such as fluid dynamics and astrophysics. The study concludes with suggestions for future extensions, including fractional-order differential equations and advanced models. (c) 2026 L&H Scientific Publishing, LLC. All rights reserved.

Leptospirosis is an emerging zoonotic disease that poses significant threat to public health and economy. The main objective of this work is to analyze the dynamics of leptospirosis disease considering temperature dependent growth of bacteria with optimal control. The existence and stability conditions of the equilibrium states are analyzed in relation to the basic reproduction number. To assess how well the model represents reality, we calibrated it using actual leptospirosis case data from Kerala, spanning from January 2018 to December 2023. Through sensitivity analysis, we identified the most influential model parameters of the disease control and found that the temperature-dependent bacterial growth rate is among the most critical. Further, study considers three types of control measures: a prevention strategy targeting susceptible humans, medical treatment for those infected and proper sanitation of the environment. Pontryagin’s Maximum Principle has been employed to determine the most effective control strategy for reducing the leptospirosis infections. The numerical results show that integrating treatment, prevention and environmental sanitation as control measures proves to be the most effective strategy for minimizing the impact of the disease within the community. The combined control measures yields approximately a reduction of 36.4

This study investigates the influence of thermal radiation and chemical reaction on magnetohydrodynamic (MHD) mixed convection flow of a Casson fluid over a nonlinear stretching sheet embedded in a porous medium. The non-Newtonian behavior of the Casson fluid, combined with the effects of buoyancy forces, magnetic field, and porous medium resistance, is analyzed under nonlinear stretching conditions. The governing partial differential equations (PDEs) are transformed into a system of nonlinear ordinary differential equations (ODEs) using suitable similarity transformations. The resulting equations are solved numerically using the Runge-Kutta-Fehlberg (RKF) method with shooting technique. The impacts of key parameters such as the Casson fluid parameter, magnetic field strength, thermal radiation, chemical reaction rate, porosity, and nonlinear stretching exponent on velocity, temperature, and concentration profiles are discussed in detail. Additionally, the skin friction coefficient, Nusselt number, and Sherwood number are evaluated to assess the flow dynamics, heat transfer, and mass transfer characteristics. The results reveal that thermal radiation enhances temperature distribution while chemical reaction significantly alters concentration profiles. The findings provide valuable insights for industrial and engineering applications involving non-Newtonian fluid flows, thermal processing, and mass transfer control.

ABSTRACTThe significance of this study is to understand the complex interplay between fluid flow and surface roughness. Modeling surface roughness adds a new dimension for examining fluid dynamics, which is essential for understanding phenomena like drag force, heat transfer, and mass transfer. In this context, the aim of the present work focuses on modeling the magnetohydrodynamic peristaltic slip flow of Casson nanofluid and analyzing the role of multiple slip effects over a non‐uniform rough channel. A novel rough non‐uniform model is effectively governed by a set of nonlinear coupled governing partial differential equations, which are simplified under long wavelength and creeping flow approximations. The resulting simplified equations are solved numerically using Mathematica's built‐in ND‐Solve tool. The study primarily examines the velocity, temperature, and concentration profiles graphically for various pertinent physiological parameters. Additionally, engineering interests like skin friction coefficients, Nusselt numbers, and Sherwood numbers are reported in tabular form, revealing intrinsic flow oscillations. The results are further explored by analyzing pressure drop, friction force, and bolus shapes created by the sinusoidal motion of the fluid. Such insights are vital for comprehending internal fluctuations during peristaltic transport. In summary, skin friction and Nusselt numbers are typically higher for rough versus smooth surfaces. Also, roughness induces stresses, conductive‐convective heat transfer, and viscous effects. Further, magnetically activated rough surfaces and nanoparticle interactions create flux balances. Magnetic effects reduce bolus size due to resistive forces. The findings of this study have important applications in biomedical engineering, aerospace engineering, heat transfer enhancement, and environmental remediation.

The flow dynamics of an obliquely impinging nanofluid toward a rotating disk holds significance in many industrial processes like mixing tanks, stir-welding, turbo-machinery, and combustion chambers of jet engines. The presence of a phase-changing material emanates a melting heat phenomenon around the solid surfaces in these applications. Motivated by this, the current study investigates the flow characteristics of an obliquely stagnating nanofluid flow over a disk in rotating motion. The disk is stretchable in nature and is endowed with melting heat transport. The flow field is controlled by an external magnetic field acting in the transverse direction, while thermal radiation effects are also considered. The governing mathematical model is reduced to its self-similar form with the usage of appropriate similarity transformations and then solved numerically using the in-built bvp5c numerical solver in MATLAB R2023a. The plot variations reveal substantial impacts of the magnetic field on the flow field. The temperature is increased with the radiation parameter.

The ferrofluid flow between torsionally oscillating disks spans a spectrum of applications including enhanced damping, controlled energy dissipation, magnetic actuators, cooling systems, and advanced aerospace engineering. The current study focuses on exploring ferrofluid flow between torsionally oscillating disks under two selected cases: Case I, where the lower disk torsionally oscillates, while the upper disk remains stationary, and Case II, where both the upper and lower disks torsionally oscillate, with the upper disk 180° out of phase relative to the lower disk. The novel influence of magnetic field-dependent (MFD) viscosity and variable thermal conductivity are incorporated into the dealing equations of the ferrofluid flow to identify the real character of ferrofluids. The parametric form of these equations is obtained after utilizing the suitable similarity function approach. The patterns of the velocity field, streamlines, and isotherms along with velocity and temperature variations are portrayed pictorially after a series of experimental simulations conducted in the MATLAB environment by using Bvp5c. The variations in the coefficients of skin friction and Nusselt numbers at the respective upper and lower disks are displayed through graphs and tables with the emerging parameters. It is identified from the comparative analysis of above-mentioned two cases that the ferrofluid flow is a decreasing function of the MFD parameter due to the reduced magnitude of velocity, whereas a direct relationship between the ferrofluid temperature and the MFD parameter is noted. It is also observed that ferrofluid heat is enhanced by the variable thermal conductivity owing to the increased value of the ferrofluid temperature. This study may be helpful in understanding magnetically damped oscillations.

As humans age, they experience deformity and a decrease in their bone strength, such brittleness in the bones ultimately lead to bone fracture. Magnetic field exposure combined with physical exercise may be useful in mitigating age-related bone loss by improving the canalicular fluid motion within the bone's lacuno-canalicular system (LCS). Nevertheless, an adequate amount of fluid induced shear stress is necessary for the bone mechano-transduction and solute transport in the case of brittle bone diseases. The underlying mechanisms of how magnetic fields, in combination to mechanical loading, affects the canalicular fluid motion still need to be explored. Accordingly, this study aims to develop a computer model to investigate the role of magnetic fields on loading-induced canalicular fluid flow in a curvy lacunar canalicular space with irregular osteocyte cell processes and walls. Moreover, this study considers canalicular fluid as non-Newtonian fluid, i.e., Jeffery fluid. In addition, a machine learning model was further employed for the estimation of parameters which significantly influence the canalicular fluid flow in response to loading and magnetic field. The results show that static magnetic field modulates the loading-induced canalicular fluid flow. Additionally, present study accelerates the fluid induced wall shear stress in case of osteoporosis.

This study explores the unsteady squeezed flow of ferrofluid through a fibrous porous medium confined between two circular disks. The lower disk remains stationary, while the upper disk undergoes sinusoidal oscillations in the axial direction, inducing time-periodic compression of the porous matrix. The variation in the inter-disk gap drives transient deformation, altering porosity and permeability over time. A normalized Kozeny-Carman relation is employed to model the deformation-dependent permeability, effectively capturing the relationship between porosity evolution and flow resistance. The combined influence of oscillatory squeezing motion, fibrous microstructure, and magnetic effects significantly alters flow resistance, mixing behavior, and heat transfer characteristics. A similarity transformation reduces the governing momentum and energy equations to a set of coupled nonlinear ordinary differential equations, which are solved numerically using the MATLAB built-in boundary value problem solver (bvp5c). The results reveal that increasing the squeezing parameter intensifies axial flow and enhances thermal transport, while stronger ferromagnetic interactions redirect streamlines radially and suppress vertical compression, contributing to flow stabilization. The overall velocity magnitude increases under stronger compression of the porous medium, enhancing ferrofluid displacement and axial pumping effects. The heat transfer of ferrofluid is found to be damped by augmented ferromagnetic interactions, whereas it is enhanced with increasing oscillatory compression. The skin friction coefficient and Nusselt numbers on both disks exhibit distinct oscillatory patterns, with sharp peaks occurring during early compression phases. These insights provide a foundation for engineering adaptive thermal and lubrication systems, where oscillatory compression and magnetic field are harnessed to regulate transport and stability.

The conical-disk apparatus consists of a cone in contact with a disk at its apex and is widely utilized in medical devices, viscometers, rheometers, and conical diffusers, among other applications. The present investigation considers the flow of hybrid nanofluid (Ag+ Al2O3/H2O) across a conical-Riga disk surface, where the disk is assumed to possess Riga properties. The three-dimensional flow is maintained by the rotation of a Riga disk and a conical surface with different angular velocities. The governing equations are mapped to the dimensionless form with the appropriate use of similarity functions, which are then simulated through the bvp5c algorithm of MATLAB R2023a. The sensitivity analysis of local Nusselt numbers of Riga disk and conical surface is obtained by using a statistical approach known as Response Surface Methodology (RSM). In this process, the emerging parameters are randomly selected, and the corresponding responses are noted. These responses are further utilized to construct a response surface using regression analysis. The Nusselt numbers of Riga disk and conical surface are positively sensitive with the modified Hartmann number, while the Nusselt number of conical surface is negatively sensitive with the rotation of Riga disk. For nanofluid (Al2O3 +H2O), The Riga disk serves as an efficient heat exchanger due to the predominance of convective thermal transport under the influence of modified Hartmann number, width parameter, (and Prandtl number. However, for : rotation parameters of Riga disk and conical surface, the conical surface becomes more fficient in heat transfer than the Riga disk. The emodified Hartmann number exhibit a positive influence on the radial velocity profile of the hybrid nanofluid, while an adverse effect is observed for the tangential velocity and temperature profiles. These results can play a critical role of Riga disk in optimizing hybrid nanofluid performance for the desired cooling of conical-disk configurations.

Numerous scientific fields have made extensive use of nanofluids for a wide variety of purposes. In recent years, nanofluids have significantly improved thermal conductivity, opening up a variety of new applications in industry and commerce. The role of porous media in the heat transport of nanoparticles is the primary focus of our investigation. The behavior of the shape factor with the hybrid nanofluid is considered. The equations that govern the given problem were solved using suitable transformations. The graphs are utilized to examine the flow model for various pertinent parameters. The obtained graphical results and table values show a keen relationship between the temperature and the velocity against the variations of the radiation parameter, porous medium, shape factor, Reynolds number, Grashof number, and other variables. The P & eacute;clet number and thermal radiation parameter are found to have a negative impact on the fluid's temperature and velocity, as indicated by the visual representation of the solutions. Viscous forces slow the flow of the fluid, while buoyant forces predominate. In contrast to spherical-shaped nanoparticles, blade-shaped nanoparticles exhibit the greatest increase in temperature and velocity. The heat is transferred from the fluid to the plate. In contrast to spherical-shaped nanoparticles, blade-shaped nanoparticles exhibit the greatest increase in velocity by 1.61%. An optimum rate of heat transfers of up to 189.805% and 86.96% is obtained at the plates y = 0 and y = 1, respectively. In the absence of a porous medium, the hybrid nanofluid maximizes its heat and volume flow rate by approximately 1.97% and 1.986%, thereby aiding in controlled flow and thermal management within the channel.

The disk-cone geometry has significant applications in bioengineering and medical technologies, including Rheometers, conical diffusers, viscometers, and biomedical devices. This study investigates the influence of nanoparticle shapes on the flow dynamics and thermal performance of a hybrid nanofluid composed of iron oxide (Fe3O4) and titanium oxide (TiO2) nanoparticles suspended in ethylene glycol (C2H6O2) as the base fluid. The analysis is conducted within a conical gap formed by a Riga disk-cone geometry (RDCG) under varying rotational and contra-rotational conditions. The Riga disk's electromagnetic properties are incorporated to examine their impact on fluid behavior. Five distinct nanoparticle shapes namely, spheres, bricks, blades, cylinders, and platelets are considered to evaluate their effects on velocity and heat transfer profiles. The governing equations of the system are transformed into a self-similar form using appropriate similarity mappings and numerically solved with the bvp5c method in MATLAB R2023a. The solution datasets are subsequently utilized to train a Levenberg-Marquardt backpropagation neural network (LMB-NN) to model and validate the system's behavior. The model demonstrates high reliability, with minimal absolute errors for critical parameters, confirmed through regression analysis, error histograms, and mean squared error (MSE) evaluations. Results reveal that increased rotational speeds of the Riga disk and cone significantly enhance the convective heat transfer, optimizing the cooling efficiency. Moreover, the contra-rotational motion maximizes velocity profiles for brick-shaped nanoparticles, whereas blade-shaped nanoparticles yield the highest peaks in heat transfer rates. These results highlight the critical role of nanoparticle shapes in optimizing hybrid nanofluid performance for advanced bioengineering applications.