We investigate solute dispersion in a two-phase system comprising a Casson fluid flowing in a tube and its surrounding wall phase that allows interphase solute exchange to mimic solute transport in blood and tissue phases. A pulsatile pressure gradient is imposed, and Gill’s classical methodology is extended to two-phase flows to analyse solute transport. The key parameters are the diffusivity ratio between wall and fluid phases ( $\lambda$ ), the partition coefficient ( $\beta _p$ ), the Womersley number ( $\alpha$ ), the yield stress ( $\tau _y$ ), the wall thickness ( $\delta _h$ ) and the initial dimensionless radius of the solute source ( $a$ ). In the long-time limit, increasing $\lambda$ , $\beta _p$ and $\delta _h$ reduces the phase-averaged convection ( $K_1$ ) and dispersion ( $K_2$ ) coefficients, owing to solute accumulation in the wall where convective and shear-induced transport are absent. Short-time behaviour is dictated by the rate of solute transfer to the wall. Larger $\alpha$ enhances both $K_1$ and $K_2$ , while larger $\tau _y$ suppresses them. The presence of a wall phase permits $K_2$ to reach $O(10^{0})$ , compared with $K_2 \sim O(10^{-3})$ without a wall, and can delay the onset of steady state to dimensionless time $t \sim O(10^{2})$ . Strong solute exchange and increasing wall thickness diminish downstream solute penetration, while non-Newtonian effects promote interphase transfer. These results provide mechanistic insight into solute exchange across fluid–wall interfaces, relevant to solute transport in blood flow and engineered permeable systems.

The non-Newtonian fluid plays a crucial part in a wide range of uses, likely optimizing cooling technologies in electronic devices, improving drug delivery systems, and enhancing thermal management in reactors. However, the flow over the conducting Riga plate is used in controlling thermal and concentration boundary layers, which offers potential improvements in energy efficiency. The primary objective of the proposed analysis is to examine the radiative time-dependent buoyancy-driven Casson fluid across an electrically conducting slanted Riga plate, considering the impacts of a heat source and variable suction. For the thermal and solute transport phenomena, the effect of heat dissipation, thermal radiation, and chemical reaction is considered. The utilization of suitable dimensionless quantities plays a role in converting the governing equations into a set of partial differential equations in non-dimensional form. Further, these are numerically solved by utilizing the Finite Element Method in the Galerkin Weighted Residual Approach (FEM-GWRA) with linear interpolation functions. The variation of several characterizing parameters is presented through graphs. The significant outcomes include the Casson parameter and titled angle controls momentum boundary layer, while thermal diffusion and mass diffusion contribute to the mass and energy boundary layer thickness accordingly. However, the important findings are that the non-Newtonian Casson parameter manifests in two ways on the velocity distribution, and it favors a significant hike in the speed of the fluid close to the surface, but when the domain grows, the profile attenuates significantly. More precisely, the heavier species, along with the reacting parameter, act as a controlling parameter in restricting the fluid concentration. The validation of the current outcomes is also disclosed in good correlation in comparison with earlier investigations.

Stability analysis of viscous–Casson fluid interface with heat and mass transport

Tribodynamic behavior of two-phase Casson fluid in a rectangular channel with thermal radiation, Hall current, and ion slip effects

Intelligent computing of mixed convection Casson fluid flow with microrotation effects using neural network backpropagation

Enhancing the thermal performance of photovoltaic (PV) systems is critical for improving their electrical efficiency, particularly in high-irradiance environments where panel overheating leads to significant energy losses. This study presents a novel mathematical framework for optimizing heat transfer in PV-integrated cooling systems using a non-Newtonian carbon nanotube (CNT)-based nanofluid. A thin-film Casson nanofluid flow and heat transfer characteristics are studied over a vertical stretching surface under combined convective and radiative boundary conditions. The model incorporates linear, quadratic, and nonlinear Rosseland approximations to capture varying intensities of radiative heat transfer, and the Casson fluid formulation captures shear-thinning behavior. The governing equations are transformed using similarity variables and solved numerically via MATLAB’s BVP4C solver, with validation against benchmark studies. The analysis shows that increasing the Casson parameter suppresses fluid velocity but enhances thermal energy retention, with up to 12% improvement in heat transfer performance. Nanoparticle loading improves buoyancy-driven flow but introduces a trade-off with thermal boundary layer thickness. Stronger radiation coupling improves heat transfer near the boundary. The conduction–radiation coupling boosts the Nusselt number by 5%–11%. The nonlinear radiation modeling results in the most significant gains in thermal performance. The velocity rises approximately 14%–16% and temperature surges 62% under high irradiance. These insights provide actionable strategies for designing next-generation systems, bridging the gap between theoretical fluid dynamics and sustainable energy engineering.

The present study examines the magnetohydrodynamic flow of a Casson fluid over a stretching sheet while accounting for homogeneous and heterogeneous chemical reactions. The model further incorporates Joule heating, viscous dissipation, nonlinear thermal convection, and radiative heat transfer relevant to moderately high temperatures. The resulting system of nonlinear ordinary differential equations is tackled numerically using the RKF-45 method combined with a shooting technique. The influence of key physical parameters on the velocity, temperature, and concentration fields is thoroughly analyzed through graphical and tabulated results. The findings indicate that increases in homogeneous and heterogeneous reaction parameters reduce species concentration and shrink the associated boundary layers, while stronger thermal radiation and higher temperature-ratio parameters elevate the fluid temperature.

Abstract The current model explores the role of dual simulations of Casson fluid and Sisko fluid in mass diffusion and heat energy on expanding and shrinking disk, considering tri- and hybrid nano-fluid, which investigates the performance between fluid flow and heat energy for the cases of the lower branch and the upper branch. The correlations of tri, mono and di nanofluid are called Hamilton Crosser and Yamada Ota models. The desired form of PDEs (partial differential equations) is obtained using transformations. The numerical simulations of ODEs (ordinary differential equations) are attained using a finite element approach. Soret and Dufour influences are considered with Joule heating and thermal conductivity (variable). Such a model is applicable in heat exchangers, thermal insulation, and engine cooling systems. The Taguchi approach is used for heat transfer rate. The results reveal the simulations of motion, concentration and heat energy in view tables, graphs and contour graphs. Skin friction and temperature gradient are discussed for UBS (upper branch solution) and LBS (lower branch solution) with various parameters. The opposite trend in momentum boundary layer thickness occurs when N and B a are enhanced. The mass diffusion can be reduced when Sc and Kc are enhanced.

ABSTRACT This paper presents a comprehensive numerical analysis of magnetohydrodynamic (MHD) Casson nanofluid movement over a permeable, linearly stretching sheet, integrating the contributions of non‐uniform heat generation or absorption and chemical interaction. The mathematical model integrates the non‐Newtonian attribute of Casson fluids and enhanced thermal conductivity due to nanoparticles. A uniform transverse magnetic field is imposed, and wall permeability is considered to simulate suction or injection. Through similarity variables, the primary nonlinear partial differential equations are reduced to a system of ordinary differential equations, which are then solved numerically. The influences of various physical parameters including the Casson fluid parameter, magnetic field strength, heat source/sink coefficient, and chemical reaction rate on velocity, microrotation, temperature, and concentration profiles are thoroughly investigated. The study reveals that magnetic damping, viscous resistance, and stretching elevate Casson nanofluid motion and thermal transport, with Brownian motion intensifying internal energy and thermal boundary thickness. Increasing chemical consumption actively removes flow species. Using response surface methodology (RSM), model accuracy and significance are assessed through analysis of variance (ANOVA) for different parameters. Using RSM and ANOVA, model accuracy is validated through regression metrics.

Microscale heat transfer is essential in advanced thermal systems such as microchannel heat sinks, micro-heat pipes, and bio-microfluidic devices, where passive pumping mechanisms drive biological fluid transport. This study presents a novel bio-thermal pumping model for Casson fluid flow through a rough-walled vertical microchannel under the influence of a magnetic field. The governing equations are derived from mass, momentum, and energy conservation principles. Using dimensionless analysis and lubrication theory, analytical solutions are obtained and validated numerically via the Collocation method (implemented in MATLAB using bvp5c). The model examines the effects of surface roughness, thermal gradients, and buoyancy forces on velocity profiles, pressure distribution, volumetric flow rate, and streamline patterns. Additionally, entropy generation and the Bejan number are evaluated to assess thermodynamic efficiency. Quantitative results show that increasing wall roughness reduces the volumetric flow rate by 16.90%, whereas thermal buoyancy enhances it by 51.02%. The heat source parameter and Casson rheology further augment the pumping capacity by 31.20% and 68.49%, respectively. In contrast, magnetic field effects strongly suppress fluid transport, decreasing the flow rate by 70.56%. Key findings reveal that wall roughness dampens near-wall velocity while enhancing central flow, and that the Brinkmann number critically governs entropy production. These insights contribute to the optimization of microscale thermal-fluidic systems, particularly in biomedical applications where membrane-based heat transfer is pivotal.

Double diffusive bio-peristaltic propulsion of hydromagnetic Casson fluid through a conduit under the influence of Hall current and thermal radiation

Mathematical modelling of periodic MHD Casson fluid flow for sinusoidal boundary conditions in terms of chemical responses and thermal radiation

The phenomenon of stretching sheets is vital in various industrial and engineering applications, significantly affecting the efficiency and quality of processes such as the manufacturing of electronic components, aerodynamic designs, among many others. This study explores the interaction between magnetohydrodynamics and a reactive viscous Casson-Jeffery nanofluid. The physical system is transformed into a mathematical framework by applying conservation laws and translating observable phenomena into precise, quantifiable equations. Non-similar analysis is used to transform the system into a dimensionless form. The system of differential equations is simplified into a linear form through the local linearization technique, after which a bivariate spectral-based method is implemented to approximate the solutions of the differential equations. The convergence analysis of the method confirms the reliability of the spectral-based technique. It was found that the Casson fluid has greater thermal sensitivity and higher fluid flow than the Jeffrey fluid. However, when the fluid parameters are varied, the Jeffrey fluid exhibits higher concentration peaks under identical conditions than the Casson fluid. The Jeffery fluid has higher species sensitivity than the Casson fluid. The findings of this study contribute to the understanding of complex fluid dynamics, which can be used to validate or refine theoretical models and simulations.

This article investigates the influence of wall permeability on channel flows and addresses the lack of studies that quantify entropy generation in magnetized Casson fluid models using wavelet-based numerical schemes. We introduce a Fibonacci Wavelet Collocation Method (FWCM) to efficiently solve the transformed nonlinear ordinary differential equations and demonstrate its applicability to the coupled momentum and energy equations. The analysis includes detailed graphical and numerical evaluations of entropy generation, temperature, and velocity fields, along with the Bejan number, Nusselt number, and skin-friction variations. The results reveal that entropy generation increases by approximately 18–22% with a higher Biot number and by nearly 15% with increasing Grashof number, while it decreases by about 12% for higher Eckert numbers. Magnetic field strength exhibits a dual effect, producing both suppressing and enhancing behaviors depending on parameter ranges. The FWCM solutions show strong agreement with previously published data, confirming both accuracy and robustness.

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

Unsteady MHD Oscillatory Slip Flow of Casson Fluid with Heat Source/Sink Through a Porous Medium

This article introduces a newly developed Orthogonal Bessel Polynomial and employs the operational matrix of integration approach for solving problems with finite and infinite boundary conditions. In this method, we utilise the Bessel polynomial operational matrix collocation technique to convert the differential equations into a system of nonlinear algebraic equations at various collocating points. These resulting algebraic equations are then solved to obtain the solution to the differential equations. To validate the method, we tested it on multiple problems and observed that the results demonstrated excellent convergence and agreement. For solving differential equations with infinite boundary conditions, we considered the flow of MHD Casson fluid over a permeable stretching sheet, solved the governing equations, and analysed the resulting graphs, providing insights into the behaviour of the system. It could be observed from the calculations below that orthogonal Bessel polynomials provide better results when compared to the non-orthogonal Bessel polynomials.

This study extends prior work on magnetohydrodynamic (MHD) mixed convective heat transfer in Casson fluids by investigating unsteady flow past a sphere, focusing on the rarely explored phenomenon of flow reversal. The Casson model, which captures a blood-like non-Newtonian behavior, makes the study relevant to biomedical and engineering applications. Minimizing flow reversal is crucial for improving flow stability, reducing energy loss, and enhancing transport efficiency. The governing equations are solved using the Keller box method, implemented via a MATLAB algorithm. Flow reversal is analyzed through graphical representations of velocity and temperature profiles, with particular attention to the effects of magnetic field strength, mixed convection, and the Casson parameter. This study is the first to report the effect of the magnetic field on flow reversal in unsteady Casson fluid flow past a rigid sphere. Results show that the magnetic field parameter reduces the momentum boundary layer thickness, effectively minimizing flow reversal. This novel finding has practical implications in applications such as improving drug delivery by regulating blood flow in arteries and optimizing thermal management in electronic cooling systems around spherical components in aerospace and industrial designs.

The current study investigates the MHD Couette flow of Casson fluid between two similar plates filled through a deformable permeable medium. The correlated phenomenon of the liquid flow and solid deformable permeable medium is considered. The effect of suitable parameters on the liquid velocity and solid displacement is examined in detail. The influences found for the existing flow characteristic indicated numerous exciting behaviours that warrant additional analysis of the deformable permeable media. The importance of drag, magnetic parameter, Casson parameter, upper plate velocity, slip parameter and skin friction on the fluid flow is discussed with the support of graphs. Furthermore, we noticed that skin friction increases with enhancement in the fluid fraction in the porous layer. Alexion and Kapellos1 repeat a similar analysis in the case of plane Couette-Poiseuille flows further into a homogeneous poroelastic layer. Major Findings: Magnetic fields can influence blood vessels in living organisms, potentially reducing inflammation in affected areas, thereby aiding in pain relief and tissue repair. Motivated by these effects, we have conducted a study on fluid flow through deformable porous layers. Furthermore, magnetic fields can impact cellular permeability, enhancing the absorption of nutrients and facilitating the removal of waste products.

This research explores the steady, two-dimensional motion of an incompressible bioconvective Casson fluid over a permeable, linearly stretching surface. The mathematical formulation accounts for velocity slip, convective surface heating, and nonlinear drag effects through the Forchheimer law. The impact of several physical phenomena, magnetic field application, suction, internal heat generation, chemical reactions and viscous dissipation on the thermal and solutal behavior of the fluid is comprehensively analyzed. The fluid’s viscosity and diffusivity are modeled as functions of temperature and concentration, respectively. The heat generation and viscous dissipation substantially intensify the thermal boundary layer, leading to elevated heat transfer rates. Through appropriate similarity transformations, the governing nonlinear partial differential equations are reduced to a set of ordinary differential equations. These equations are then solved numerically using the shooting method coupled with a suitable iteration scheme. Moreover, parametric investigations show how key dimensionless quantities affect the velocity field, thermal distribution, concentration and microorganism density. The graphical representation of results provides a deeper understanding of the influence of controlling parameters on flow and transport phenomena. The following results are generated by fixing values as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. These findings have practical relevance in the design of bio-inspired cooling systems, polymer processing and biomedical flows involving magnetic regulation. The presented results contribute to engineering applications where control of thermal and mass transport in complex fluids is essential.