The increasing presence of biological contaminants in wastewater poses serious challenges to safe water reuse and sustainable management. The effects of filtration on pollutant transport in a vertical porous channel are investigated mathematically and numerically in this work, taking into account nonlinear microbial growth controlled by generalized Haldane kinetics. Key characteristics, including viscosity, density, and diffusivity, are supposed to change nonlinearly with contaminant concentration, and the fluid is described as incompressible and dilatant. The Bivariate Spectral Quasi-Linearization Method (BSQLM) is used to solve the resulting system of nonlinear partial differential equations, and the Bivariate Spectral Chebyshev Collocation Method (BSCCM) is used for validation. The findings show that while higher inhibition and liquid-biofilm mass transfer coefficients successfully control pollutant concentration, porous filtration dramatically lowers flow velocity due to increased resistance and bio-clogging. With few residual errors, the numerical scheme exhibits great accuracy and quick convergence. Overall, the study establishes that coupling filtration mechanisms with generalized biokinetic models provides a robust framework for predicting contaminant behavior and enhancing the design of efficient wastewater treatment and reuse systems.

The analysis of heat transfer during the decomposition of nitrogen dioxide (NO2) is considered. The analysis is modelled in a cylindrical domain and an exothermic chemical reaction is assumed to initiate the decomposition reaction following a two-step reaction mechanism process. Heat loss to the environment is assumed to follow a convective means. The analysis of heat transfer is attained by studying the variation of selected kinetic parameters embedded in the governing partial differential equation. The non-linear differential equation is solved numerically by the semi-implicit finite difference method (FDM). Some selected parameters such as the rate of reaction indicate the acceleration of the exothermic chemical reaction to enhance the heat release, whilst others such as the activation energy show a retardation of the process to reduce the heat release.

This study is motivated by the need to understand thermal stability and entropy generation in reactive electroosmotic systems exhibiting temperature-sensitive fluid behavior. Such understanding is essential for advancing microscale technologies where chemical reactions and electrokinetic forces interact under confined conditions. In a couple-stress fluid going through a two-step exothermic reaction, the study intends to examine the effects of electroosmosis, temperature-dependent electrical conductivity and viscosity on temperature, velocity, entropy generation, and thermal stability. To describe electroosmotic effects at the channel wall, a nonlinear slip-dependent zeta potential is used. The spectral Chebyshev collocation method is used to solve the governing equations, and the Runge-Kutta shooting method is used to validate. Parametric analysis reveals that viscosity and couple-stress parameters reduce velocity and entropy generation, indicating their stabilizing effects on flow and irreversibility. In contrast, the two-step reaction parameter enhances velocity by 10 % and temperature by 192 %, with negligible impact on entropy generation, highlighting the role of exothermic kinetics in promoting thermal and momentum transport without significantly increasing thermodynamic losses. This work provides theoretical insights applicable to thermal management in microreactors, electrokinetically driven lab-on-chip devices, and chemical transport systems requiring precise thermal and mechanical stability.

Fluid flow along an inclined channel phenomenon is crucial in several geophysical, environmental, engineering, biological, and industrial processes, and in aerodynamics and hemodynamics. This present study examines the effect of a constant magnetic field on the entropy production rate in a steady flow of Casson fluid along an inclined heated channel. The governing equations for the flow of velocity, temperature, and entropy generation are formulated based on the Casson constitutive relations and thermodynamics’ first and second laws. The exact solutions are constructed for the dimensionless equations and validated with previous results in the literature. The effects of various fluid parameters on the flow, heat transfer, and entropy production rate are conducted and reported graphically with adequate discussion. The impact of the Hartmann number parameter reveals a decrease in both flow velocity and entropy generation rate, meanwhile it also enhances the fluid temperature distribution across the inclined channel. An opposite trend is, however, observed with the Casson fluid parameter.

This study presents a mathematical analysis of the collective effect of chemical reactions, variable fluid properties, and thermal stability of a hydromagnetic couple-stress fluid flowing through a microchannel driven by electro-osmosis and a pressure gradient. The viscosity of the biofluid is assumed to depend on the temperature, while the electrical conductivity is assumed to be a linear function of the drift velocity. The governing equations are derived non-dimensionalized, and numerical solutions are obtained using the spectral Chebyshev collocation method. The numerical solution is validated using the shooting Runge–Kutta method. The effects of varying the parameters on the thermal stability, temperature, velocity, and entropy profiles are discussed with adequate interpretations using tables and graphs. The results reveal that the chemical reactions and viscosity parameter increase the fluid temperature, while the Hartmann number decreases the temperature and increases the flow velocity and entropy generation. It was also observed that the chemical reactions and viscosity parameter increased the entropy at the channel walls, while the Hartmann number decreased the entropy at the core center of the channel. This study has tremendous empirical significance, including but not limited to biophysical applications of devices, engineering applications such as control systems, and thermo-fluidic transport.

In spite of the enormous applications of heating combustible materials due to exothermic chemical reactions, scientists and engineers still face a problem with these materials’ tendency to self-ignition, which can result in significant property damage if serious precautions are not taken. Therefore, the thermal decomposition of combustible viscoelastic material in a cylindrical channel is investigated in this study. With a third-order constitutive model considered for viscoelastic flow behavior, momentum and energy-balanced governing equations are provided. The chemical reaction of the material is assumed exothermic and thus follows Arrhenius’s kinetics. A numerical solution is provided for the boundary value problem via the bivariate spectral collocation method, and the impact of kinetics parameters on the combustible viscoelastic material is investigated. Our findings show that self-ignition is delayed with higher values of activation energy parameter (ε\varepsilon ) and third-order parameter (γ\gamma ), as well as lower values of magnetic field parameter (MM), current density parameter (δ\delta ), and electrical conductivity exponent (mm).

This paper addresses the mixed convective flow and heat transfer in combustible third-grade fluids through a slant porous channel filled with permeable materials. The fluid layer in contact with the channel wall is exposed to asymmetrical slippage and isothermal conditions. We employ the spectral Chebyshev collocation method (SCCM) to the coupled nonlinear flow governing equations and validate using the Shooting–Runge–Kutta method (RK4). Fluid velocity and temperature profiles, local entropy generation, and irreversibility ratio are computed and analyzed quantitatively and qualitatively. The convergence of the numerical method was demonstrated. The flow and thermal effects results, entropy generation rate, and Bejan number revealed fascinating manifestations that have profound implications in the design of thermo-mechanical systems. In particular, the thermal analysis results are pertinent to optimal system designs that achieve efficient energy utilization.

Flow enhancement is one of the most significant challenges in microfluidics with extremely low permeability. Based on this, electrokinetics remediation for the double-layer flow of hybrid ternary nanofluid is proposed based on the transport of ions under constant pumping pressure. The entropy generation volumetric rate relation is also modeled. The nonlinear system of equations is formulated, solved, and validated numerically by the collocation method and shooting Runge-Kutta method. Results are shown graphically to explore the impacts of governing parameters, such as the Darcy parameter, viscosity index, electrokinetic, and Joule heating effects, on the velocity, temperature, entropy, and Bejan number profiles. The flow and heat transmission in microchannels may be significantly changed and controlled by the electric double layer. The findings show that as the electrokinetic parameter's magnitude increases, the flow is slowed down, and thermal dispersion is impeded by nanoparticle collisions, which lowers the velocity field and heat transfer profile.

This paper addresses the mixed convective flow and heat transfer in combustible third grade fluid through a slant channel that is filled with a permeable materials. The fluid layer in contact with the lower plate is exposed to convective heating with the ambient while at the upper plate, a constant heat flux is experienced. We employ the spectral quasilinearisation method (SQLM) to the coupled nonlinear flow governing equations. Fluid velocity and temperature profiles, local entropy generation and irreversibility ratio are computed and analysed quantitatively and qualitatively. A residual error analysis demonstrated high accuracy and convergence of the numerical method. The results on flow and thermal effects, entropy generation rate and Bejan number revealed fascinating manifestations that have profound implications in design of thermo-mechanical systems. In particular, the results of the thermal analysis are pertinent to optimal designs of systems that achieve efficient energy utilization.

In many geothermal engineering applications involving exothermic chemical reactions, gravity, and buoyancy forces significantly enhance yields, especially during the thermal and catalytic cracking involving high temperature and concentration differences of some heavy hydrocarbons. This investigation deals with the numerical examination of nonlinear double convective flow, heat, and mass transfer of reactive Sisko fluid based on chemical kinetics theory for fluid flowing through a nondeformable porous medium. The governing equations are formulated and made dimensionless, and numerical results are obtained by applying the Spectral Chebyshev Collocation Method and validated with the Shooting-RK4 method. The effects of several parameters on the flow, heat, mass transfer, and thermal stability of the combustible fluid are documented in graphical and tabular forms, with detailed explanations. The computation reveals that buoyancy forces and Frank-Kamenetskii parameters encourage a velocity and temperature distribution rise. This analysis gives an insight into friction reduction in many heat and mass transfer thermophysical systems such as automobiles, power generations, and lots more.

Understanding combustion efficiency and reduction of environmental pollution are one of the main challenges encountered in porous medium combustion in many industrial and chemical engineering applications in recent times. In this regard, we present a numerical investigation of the thermal stability of the process and inherent irreversibility analysis for combustible third-grade fluid through a non-Darcian porous medium. The heated fluid flow is assumed to flow steadily through parallel Riga plates with asymmetrical convective cooling. The equations governing the flow are formulated and converted to a set of nonlinear Boundary-value problems by dimensional analysis. The dimensionless equations are solved numerically by spectral Chebyshev collocation method and validated with Shooting-Runge-Kutta method. Graphical and tabular illustrations are provided with sufficient explanations for the velocity, temperature, entropy generation, and irreversibility ratio profiles.

Oxygen concentration plays crucial roles in the combustion of polymeric materials, such as reducing the emission of poisonous gases (nitrogen oxide and carbon monoxide) and enhancing heat transfer. However, in the combustion process, a continuous increase in reaction rate without considering heat loss to the environment may result to the accumulation of heat in the stockpile and possible self-ignition. As such, this work investigates the thermal decomposition of an electrically combustible visco-elastic material in a cylindrical stockpile, considering O-2 consumption. The nonlinear partial differential equations governing the problem were solved numerically via the bivariate spectral collocation approach (BSCA). The impact of kinetic parameters (for example, heat generation, oxygen consumption rate, and activation energy.) on the flow, heat and mass transfer, and thermal stability of the system were studied and qualitatively discussed.

Flow instability is a major challenge experienced in medical, engineering and industrial settings globally. For instance, flow instability linked with irregular cardiac output of the heart leads to organ malfunctioning in the medical field, it also encourages mechanical vibrations in the case of fluctuating flow rate, and several other applications. In this study, linear stability analysis is conducted to monitor the behavior of a small disturbance that is imposed on hydromagnetic Casson fluid that flows steadily through a saturated porous medium. A new variant of the Orr-Sommerfield equation is obtained and solved numerically by using spectral point collocation weighted residual approach with eigenfunction expansion of the Chebyshev polynomial as the admissible trial function. Based on the QZ algorithm, numerical results are obtained for wave and Reynold’s numbers, wave velocity as functions of Magnetic field intensity and porosity shape parameters. Results are validated against previously released data. The biophysics of the heart, particularly in cardiac rhythm analysis, as well as several other medicinal and technical applications, is among the areas where the current work has applicability.

A mathematical investigation of a thermodynamical system linked with energy management and its impact on the environment, especially climate change, is presented in this study. In this regard, a numerical investigation of the flow and heat transfer of hydromagnetic third-grade liquid through a porous medium. The permeability of the medium and electrical conductivity of the fluid are assumed to be temperature functions. The appropriate mathematical formulations for momentum, energy, and entropy equations are presented in both dimensional and dimensionless forms. We obtained the numerical solutions using the spectral version of the Chebyshev collocation method and compared the result with the shooting Runge–Kutta method. Numerical results for velocity, temperature, entropy, and Bejan profiles are communicated through tables and graphs with adequate physical interpretation. The thermal stability of the thermo-fluid system that guarantees the prevention of spontaneous fluid heating that fuels climate change is also included in the analysis.

One of the significant water-related health challenges globally is due to pollutant fate. Contaminants endanger the lives of humans, animals, and even plants. The present mathematical analysis explains reactive wastewater sludge ejected into a drinking water source from wastewater treatment plants. The assumption that wastewater sludge follows a power-law constitutive relation leads to nonlinear momentum and concentration equations. The contaminants are assumed to follow a nonlinear irreversible first-order sorption model. The numerical solution of the coupled problem is solved using the Bivariate Spectral Local Linearization Method and validated with the spectral Chebyshev weighted residual method. Profiles are presented for dimensionless flow velocity and concentration. Comprehensive explanations for the obtained results are provided with relevant applications.

In this paper, the MHD Pouisille flow of a third-grade fluid in a non-Darcian porous plate with a slip effect has been studied. The flow flux and the heat transfer analysis were carried out. The governing non-linear ordinary differential equations (ODE) were non-dimensionalize and the resulting ODE is solved by perturbation techniques. The result obtained was illustrated and presented in form of graphs. The effect of various governing flow parameters on velocity and temperature profile was taken into account. It is found that the effect of cross Renold’s number, the porosity of the medium, and the Magnetic parameter are to slow down the velocity profile of the fluid while it increases with an increase in Darcy’s number and slip parameter. Also, it is noticed that the temperature profile of the fluid increases with the increase in the value of the Prandtl number.

Energy management and heat control whenever a reactive viscous fluid is the working medium has been one of the greatest challenges encountered by many in the field of chemical and industrial engineering. A mathematical approach to thedetermination of critical points beyond which the working environment becomes hazardous is presented in the present investigation together with the entropy generation analysis that guarantees the efficient management of expensive energy resources. In this regard, the nonlinear mixed convective flow behavior of a combustible third-grade fluid through a vertical channel with wall cooling by convection is investigated. The mathematical formulation captures the nonlinearities arising from second-order Boussinesq approximation and exponential dependence of internal heat generation, viscosity, and thermal conductivity on temperature. The resulting nonlinear boundary value problems were solved based on the spectral Chebyshev collocation method (SCCM) and validated with the shooting-Runge–Kutta method (RK4). The nonlinear effects on the flow velocity, temperature distribution, entropy generation, and Bejan heat irreversibility ratio are significant. Further analyses include the thermal stability of the fluid. Findings from the study revealed that flow, temperature, and entropy generation are enhanced byincreasing values of the Grashof number, the quadratic component of buoyancy, and the Frank-Kameneskii parameter, but are reducedbyincreasing the third-grade material parameter. Moreover, it was shown that increasing values of the third-grade parameter encourages the thermal stability of the flow, while increasing values of the linear and nonlinear buoyancy parameter destabilizes the flow. The present result is applicable to thick combustible polymers with increased molecular weight.

The present model concentrates on entropy generation on a steady incompressible flow of a Casson liquid past a permeable stretching curve surface through chemical reaction and magnetic field effects. The exponential space‐dependent heat source cum heat and mass convective boundary conditions are accounted for. The resulting nonlinear boundary layer model is simplified by the transformation of similarity. Chebyshev spectral technique is involved for obtaining numerical results of the converted system of the mathematical models. Behavior of the determining thermo‐physical parameters on the profiles of velocity, temperature, concentration, skin friction, heat, mass transfer rate, rate of entropy generation, and finally the Bejan number are presented. The major point of the present investigation show that the curvature term weakens the mass transfer profile as the fluid temperature reduces all over the diffusion regime. A decrease in heat generation strengthens the species molecular bond, which prevents free Casson particle diffusion. Furthermore, the mass transfer field diminishes in suction and injection flow medium.

This article analyzed carbon dioxide (CO2) emission from the combustion of reactive materials modeled in a cylindrical domain. Reactive materials in this case involve carbon-containing substances that react spontaneously with the oxygen of the surrounding environment under the influence of an exothermic chemical reaction. In this analysis, the reactant (oxygen) consumption was neglected. The nonlinear differential equation governing the problem was solved numerically using the Finite Difference Method embedded within the Maple software. It was found that there are kinetic parameters that enhance the emission of CO2, like the rate of reaction, and others, like the heat loss parameter, retard the CO2 emission during the exothermic chemical reaction.

Accurate determination of optimum flow and heat transfer condition is one of the major challenges faced in the application of magnetic fluid in the field of medicine and engineering, especially when applied as ferrofluids for targeted drug deliveries, treatment of hyperthermia, sealants in computer hard drives, lubricants in car shafts. In view of these important applications, a mathematical investigation of the flow and heat transfer behavior of reactive magnetic fluids containing nanostructures is presented based on a couple of stress constitutive models. The reactive fluid is assumed to flow through inclined magnetized solid boundaries for energy conversion. The formulation leads to nonlinear coupled equations. The dimensionless equations are numerically solved using the spectral Chebyshev assumed solution for the weighted residual technique, and the correctness of the solution is confirmed using the shooting Runge–Kutta method. The effects of various fluid parameters on velocity, temperature, skin friction, and heat transfer rates are described in tabular and graphical form, along with suitable physical explanations. Thermal analysis computations are also presented. According to the findings, an enhanced couple of stress fluid and variable viscosity parameters reduced the skin drag and heat transfer rate at the bottom wall. Furthermore, the thermal stability of the flow can be achieved with increasing values modified Hartman number while increasing couple stress parameter encourages thermal instability in the flow domain.