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Aim: This paper presents a numerical simulation of the mixed convective boundary layer (BL) motion of a bio-rheological liquid over an inclined plate under viscous dissipation and Joule heating effects. This is significant because of the various applications of electro-osmotic force, inclined plates, and viscoelastic fluids in the biochemical engineering and industrial domains. Furthermore, the BL flow is controlled by electromagnetic force (EMF). In this study, a non-Newtonian liquid model, called the Jeffrey fluid model, was employed. Method: The rheological equations of the current study are nonlinear partial differential equations (PDEs). By applying a set of similarity transformations, these PDEs become coupled ordinary differential equations (ODEs), which are then solved numerically using the NDSolve method under realistic boundary constraints. Outcomes: Numerical solutions for the velocity profile (f′(ξ)), temperature distribution (Θ(ξ)), skin friction (shearing stress), and Nusselt number (heat transfer rate) were obtained subject to convective boundary constraints. These numerical outcomes are dependent on 12 embedded parameters: Hartmann number (Ha), viscoelastic time relaxation parameter (γ), inclination of the inclined plate with a horizontal line (ω), mixed convection parameter (Ω), electro-osmotic parameter (k), electro-osmotic velocity parameter (Uhs), Prandtl number Pr, Brinkman number (Br), suction/blowing parameter (s), velocity slip parameter (ϱ), Joule heating parameter (Γ), and thermal slip parameter (δ). The authors discussed how these embedded variables affect rheological features through graphs and tables. The numerical solutions of viscous liquids are also discussed, and these outcomes are compared with the numerical solutions of a viscoelastic liquid. The enhancements of the Θ(ξ) and f′(ξ) are largely dependent on the Joule heating parameter, Brinkman number, and Hartmann number. As the Prandtl number increases, diminishing behavior is observed in the Θ(ξ) and f′(ξ). Increasing the magnetic and viscoelastic parameters increases the magnitudes of the skin friction coefficient and local Nusselt number. The validation of the numerical procedure is discussed by recovering the outcomes of research works from the available literature. Significances and applications: This mathematical study has diverse applications in the electromagnetic multiphase processes, magnetic power generators, chemical engineering phenomena, polymer industry systems, and the thermal enhancement of mechanical and industrial processes.

The current thermal slip and Magnetohydrodynamic analysis plays prominent importance in heat insulation materials, polishing of artificial heart valves, heat exchangers, magnetic resonance imaging and nanoburning processes. The main objective of the existing article is to deliberate the impact of thermal slip, thermal radiation and viscous dissipation on magnetized cone embedded in a porous medium under reduced gravitational pressure. Convective heating characteristics are used to increase the rate of heating throughout the porous cone. For viscous flow along a heated and magnetized cone, the conclusions are drawn. The simulated nonlinear partial differential equations are transformed into a dimensionless state by means of suitable non-dimensional variables. The technique of finite differences is implemented to solve the given model with Gaussian elimination approach. The FORTRAN language is used to make uniform algorithm for asymptotic results according to the boundary conditions. The influence of controlling parameters, such as thermal radiation parameter Rd, Prandtl number Pr, porosity parameter Ω, viscous dissipation parameter Ec, δ thermal slip parameter, Rg reduced gravity parameter and mixed convection parameter λ is applied. Graphical representations were created to show the consequences of various parameters on velocity, temperature and magnetic field profiles along with fluctuating skin friction, fluctuating heat and oscillatory current density. It is found that velocity and temperature profile enhances as radiation parameter enhances. It is noted that the amplitude and oscillations in heat transfer and electromagnetic waves enhances as magnetic Prandtl factor increases.

As a result of the tremendous development in all aspects of life, the increase in population day after day, and the scarcity of freshwater resources sufficient to meet the needs of living organisms on the surface of the earth, all these reasons called for an attempt to treat wastewater supported by solar energy to raise the temperature and facilitate the operation of wastewater treatment systems. Health in areas rich in solar resources. This approach reduces reliance on traditional energy sources in the wastewater treatment process. This theoretical study aims to create a mathematical model for the problem of water turbidity due to impurities (wastewater). Non-Newtonian fluid flow equations are established in the case of using renewable energy in various forms of nano-photocatalysts, as well as magnetic force which has a clear effect on the process of removing impurities from the aqueous fluid. Chebyshev's method was used to obtain numerical solutions to the problem, which were represented by a set of drawings and tables, that highlighted the effective role of both the magnetic field and sunlight in eliminating impurities. Increasing the Lorentz force and thermal radiation increases the strength of hydrogen bonding of the aqueous liquid mixed with particles of tri-hybrid nanomaterials larger than the aqueous liquid, which in turn works to raise the thickness of the thermal layer in the blade shape higher than the rest of the shapes. This study gives future insights for further research endeavors.

Present problem concerns may be found in a wide variety of various industries and can takes a shape of round toroid's, cones, cylinders, domes (Axisymmetric shape), and other shapes as well. When it comes to actual applications, they are represented by aerosol cans, submarine pressure hulls, cooling towers, offshore drilling rigs, radomes, nuclear reactors, and other similar objects. The aim of this research is to improve the performance of MHD flow using Darcy-Forchheimer, viscous dissipation, porous and a heat source. The viscous convective permeable flow flowing over a bullet-shaped item is examined. Using similarity transforms, the structure of nonlinear differential equations are converted into dimensionless ODEs. Employing bvp4c shooting technique to decrypt the findings. Influences of physical entities on velocity and temperature were sketched and briefly described. Physical behaviors of skin friction and heat transfer rates are explored. The fluid velocity drops for Eckert number Ec as ε 1.0, The fluid velocity improves with Darcy-Forchhemier parameter Fr when stretching ratio ε 1.0. The enhancement of fluid temperature is observed with enhanced Ec. A enhance in the Darcy-Forchhemier parameter Fr associated with improvement in skin-friction. In the absence of viscous dissipation, hall current, porous and Dacy-Forchhemier influences, the outcomes are in splendid covenant with those of previous studies.

This research introduces novel theoretical assumptions through the use of a non-Newtonian Sutterby model with variable properties, demonstrating significant advancements in thermal conductivity and diffusivity, which enhance heat and mass transfer in fluids, particularly under magnetic field exposure. The study is notable for its comprehensive examination of the effects of thermal radiation and viscous dissipation within a porous medium. Additionally, it explores the role of suction velocity to provide a deeper understanding of transport phenomena. Following the Darcy hypothesis, the fluid flow is modeled as resulting from the linear stretching of an elastic sheet in a saturated porous medium. The core physical model, comprising equations of mass, motion, concentration, and energy, is transformed into ordinary differential equations using appropriate similarity transformations. Employing the shooting technique, the numerical solution to the problem is obtained. The research uncovers and quantitatively analyzes intriguing physical parameters influencing velocity, concentration, and temperature fields. These parameters are further investigated both numerically and graphically, providing valuable insights into their effects. Quantitative outcomes include enhanced thermal and concentration fields by magnetic field parameter, the porous parameter and the viscosity parameter and improved the rate of mass transfer by both suction and thermophoretic parameters, highlighting the model’s efficacy in optimizing fluid dynamics especially under magnetic fields. The obtained outcomes were juxtaposed with previous studies, revealing a significant level of agreement.

This manuscript reveals the mathematical analysis of dual simulations in tangent hyperbolic rheology with heat transfer and mass diffusion mechanisms on the needle (expanding and shrinking). Further, Darcy's Forchhiermer law is employed with the occurrence of the magnetic field. The models of hybrid nanofluid utilized are Xue- and YO-hybrid nano-fluid models. Such models apply to optical fibers, solar systems, surgical implants, electronics and biological applications. Transfer of thermal energy and diffusion related to mass species occurs employing a non-Fourier approach. The transformed system of non-linear Odes is numerically tackled with the finite element method. Furthermore, because fluid and needle movement have opposite directions, the current model has dual solutions. It visualizes that such a complicated model is not solved numerically by FEM. The applications of Xue- and YO-hybrid nano-fluid model with non-Fourier's law on a needle are implemented for the first time. It concludes that adding YO-hybrid nanofluid with base fluid increases the entropy profile, temperature profile and Bejan profile rather than the Xue hybrid nanofluid model. Fields of concentration, velocity and heat energy for the case of upper solutions are higher than concentration, velocity and heat energy profiles for lower solutions.

The study of heat transfer in fluid flows across permeable media is critical in many engineering applications, including energy systems, cooling technologies, and chemical processes. This study aims to explore the impact of joule heating and heat transfer flow of an MHD dissipative ternary fluid through a channel embedded in a Darcy-Forchheimer permeable medium. Additionally, the utilization of ternary hybrid nanofluids, composed of a base fluid and three nanoparticles Al2O3, MoS2, and Cu has emerged as a promising avenue for augmenting thermal conductivity and heat transfer rates. The governing equations are transformed into a set of coupled ordinary differential equations by employing similarity variables and simplified by bvp4c and artificial neural network (ANN) model approaches. Results reveal that significant enhancement in the velocity field at the lower channel wall and reductions at the upper wall, while fluid temperature decreases with increasing Prandtl number. Further, the heat transfer rate increases with an increase in the Prandtl number and magnetic field whereas the skin friction decays with an increase in the magnetic field. Meanwhile, the comparison was carried out for the temperature field by using bvp4c & ANN and the results are strongly correlated.

The growing fascination with nanofluid flow is motivated by its potential applications in a variety of industries. Therefore, the objective of this research article is to conduct a numerical simulation of the Darcy porous medium flow of Newtonian nanofluids over a vertically permeable stretched surface, considering magnetohydrodynamic mixed convection. Various attributes, such as the impacts of slip, thermal radiation, viscous dissipation, and nonuniform heat sources, are integrated to explore the behavior of the flow. The utilization of the boundary layer theory helps to describe the physical problem as a system of partial differential equations (PDEs). These derived PDEs are then converted to a system of ordinary differential equations (ODEs) through the application of suitable conversions. The outcomes are obtained using the finite difference method, and the effects of parameters on nanofluid flow are compared and visualized through both tabular and graphical representations. The outcomes have been computed and subjected to a comparative analysis with previously published research, revealing a remarkable degree of agreement and consistency. Consequently, these innovative discoveries in heat transfer could prove beneficial in addressing energy storage challenges within the contemporary technological landscape. The noteworthy main findings indicate that when the porous parameter, magnetic number, velocity slip parameter, viscosity parameter, and Brownian motion parameter are assigned higher values, there is an observable expansion in the temperature field. Due to these discoveries, we can enhance the management of temperature in diverse settings by effectively modulating the heat flow.

The analysis focuses on investigating the phenomenon of double-diffusive convection using the Sisko nanofluid model. It particularly highlights the impact of induced magnetic flux, viscous dissipation, and heat radiation within an asymmetric geometry having multiple slip conditions. To ascertain the salient of the Brownian diffusion coefficient and thermophoresis, we have incorporated viscous dissipation, heat radiation, and the Buongiorno model. The Soret and Dufour parameters describe the convective double diffusion phenomenon. The mathematical formulation is constructed through equations governing magnetic force function, concentration, temperature, momentum, and continuity. These formulations yield nonlinear partial differential equations to explain the designated flow. To simplify the nonlinear partial differential equations, the lubrication paradigm of mathematical simulations is employed. The subsequent system of coupled nonlinear differential equations is calculated numerically through the NDSolve function, which is a built-in program of Mathematica. Numerical results and graphs give evidence that supports the significance of different flow quantities in physiological contexts. The findings from this investigation are anticipated to contribute to the development of intelligent magneto-peristaltic pumps, particularly in thermal and drug administration applications. The current investigation suggests that the distribution of temperature reduces as the coefficient of radiation increases due to a system’s high heat emission and consequent effects of cooling. Furthermore, the increased influence of heat radiation raises the concentration profile. It is also highlighted that heat radiation has the potential to raise a fluid’s temperature, which raises the volume fraction of nanoparticles.

The remarkable thermophysical characteristics of hybrid nanofluids have considerable potential for the enhancement of heat transport. In recent years, there has been a significant surge in research on hybrid nanofluids, with findings indicating that these fluids are favorable for transferring heat in engineering contexts. Therefore, a theoretical investigation is conducted by adopting the Tiwari and Das model, for viscous dissipative flow and thermal transport characteristics of hybrid nanoparticles in an exponentially permeable porous shrinking surface. The convective thermal transport features are studied with the influence of Ohmic heating and thermal generation/absorption effects. To solve the governing flow problem of hybrid nanofluid, a standard conversion and numerical approach are used. The numerical findings indicated the existence of dual solutions within a certain range of shrinking and mixed convective parameters. The results reveal that high suction strength higher the skin friction coefficient and heat transfer rate. It also reveals that in opposing flow regions, the addition of nanoparticles resulted in a lower heat transport rate and a higher skin friction coefficient. Furthermore, as thermal radiation increases, the rate of heat transfer increases, but the upward Eckert number exhibits the opposite behavior. In addition, the velocity profile displays opposite behavior for upper and lower behavior when the mixed convection parameter increases. Besides, the thermal distribution grew when the radiation, Eckert number, and Biot number were raised.

This study encapsulated the research methodology utilized in the flow behaviors of Williamson nanofluid and analyzed the associated mass heat transfer. The study concentrated on examining the magnetohydrodynamic behavior of nanofluids in the presence of heat generation effects and the inclusion of dissipative energy on a vertical nonlinear stretching surface submerged within a Darcy porous medium. The rationale for including variable viscosity and variable conductivity in this research was to precisely evaluate the mechanisms of heat and mass transfer, particularly with regard to the fluctuations in fluid properties. The objective was to enhance the understanding of how these varying properties impact the overall heat and mass transfer processes. The initial formulation of the phenomenon, initially presented as partial differential equations, was transformed into ordinary differential equations by employing appropriate dimensionless variables. The ultimate streamlined version of the model was then numerically solved utilizing the shooting method. By employing the numerical shooting method, we portrayed nanofluid patterns in velocity, temperature, and concentration fields, alongside essential parameters such as skin friction coefficient, Sherwood number, and Nusselt number. The significant key findings highlighted that both the porous parameter and the magnetic number increasingly affected temperature and concentration distributions. Additionally, increasing the thermophoresis parameter resulted in higher concentration and corresponding temperature levels. Graphical presentation and physical explanations were used for analysis, and the study's outcomes were compared to existing literature, affirming a strong agreement that validated the solutions.

This work conducts a numerical examination into the influence of a magnetic field and viscosity dissipation on the movement of a micropolar fluid over the surface of a vertical, hollow circular cylinder via conjugate mixed convection. In this investigation, we obtained a numerical solution for a non-linear differential equations-based modeling system by employing MATLAB and the bvp4c solver, which operates on a two-equation model. We show graphically how micropolar materials, conjugate heat transfer, viscous energy dissipation, buoyancy factors and magnetic field affect the temperature at the interface, local skin friction and heat transfer. By contrasting the acquired results with those found in the published research, which exhibit a high degree of concordance, the validity of the methodology is proven.

A metallic alloy is a combination of two or more elements, often called a compound or a solution. Steel is largely composed of carbon, a nonmetal, but alloys are often made up of metals. In this article, the authors have explored the electrically conducting water-based viscous nanofluids flow past a rotating disk surface. The nanofluids flow is composed of titanium and aluminum alloys where water is used as a base fluid. Two important cases, namely the stretching case and the shrinking case, were investigated to analyze the flow behaviors due to the different embedding factors. The impacts of viscous Joule heating, thermophoresis, Brownian motion, activation energy, nonlinear thermal radiation, and chemical reaction are investigated here. By employing an appropriate set of variables for shifting the leading equations to dimension-free form. The mathematical model is solved numerically by incorporating the bvp4c MATLAB scheme. Current work is validated with previous studies. The outcomes showed that the radial velocity increases when the disk surface stretches and reduces when the disk surface shrinks. On the other hand, the Azimuthal velocity increases when the disk surface shrinks and reduces when disk surface stretches. Both the radial and Azimuthal velocities are the diminishing functions of the magnetic factor, whereas temperature is the growing function of magnetic factor. In addition, the temperature is more influenced by the magnetic factor in the case of nonlinear radiation. The higher magnetic factor increases skin friction. In addition, the stretching case experiences more surface drag than the shrinking case. It is found that nanofluid flow containing titanium alloy has perceived the greater impacts of the embedded factors compared to the nanofluid flow containing aluminum alloy.

Nanofluids have improved thermophysical properties compared to conventional fluids, which makes them promising successors in fluid technology. The use of nanofluids enables optimal thermal efficiency to be achieved by introducing a minimal concentration of nanoparticles that are stably suspended in conventional fluids. The use of nanofluids in technology and industry is steadily increasing due to their effective implementation. The improved thermophysical properties of nanofluids have a significant impact on their effectiveness in convection phenomena. The technology is not yet complete at this point; binary and ternary nanofluids are currently being used to improve the performance of conventional fluids. Therefore, this work aims to theoretically investigate the ternary nanofluid flow of a couple stress fluid in a vertical channel. A homogeneous suspension of alumina, cuprous oxide, and titania nanoparticles is formed by dispersing trihybridized nanoparticles in a base fluid (water). The effects of pressure gradient and viscous dissipation are also considered in the analysis. The classical ternary nanofluid model with couple stress was generalized using the fractal–fractional derivative (FFD) operator. The Crank–Nicolson technique helped to discretize the generalized model, which was then solved using computer tools. To investigate the properties of the fluid flow and the distribution of thermal energy in the fluid, numerical methods were used to calculate the solution, which was then plotted as a function of various physical factors. The graphical results show that at a volume fraction of 0.04 (corresponding to 4% of the base fluid), the heat transfer rate of the ternary nanofluid flow increases significantly compared to the binary and unary nanofluid flows.

This research offers an analysis of the mixed convective transient boundary film Casson fluid stream and thermal distribution through a vertical sheet. Viscous dissipation and the Soret effect are introduced to support the flow in stimulating heat capacity. Unlike typical investigations, the present flow-formulated model is done to capture an induced magnetic field. The Boussinesq approximation is used to describe the nonlinear formulated partial derivatives governing the heat transfer fluid that is non-dimensionalized using suitable dimensionless quantities. The transformed partial derivative model is numerically solved via the spectral Chebyshev technique and the results of shear stress, current density, Nusselt number, and mass gradient are tabulated. The role of numerous terms on dimensionless flow rate, induced velocity, and heat transfer with species distribution is discussed in a very effective way. Velocity and temperature of liquid decline as boosting the material parameter. Enhancing values G r , G r m favor the flow momentum and oppose the temperature profile. [ABSTRACT FROM AUTHOR]

This study looks at the Darcy–Brinkman flow across a stretched sheet of porous dissipation and frictional heating. The geometry of a steady flow of dust particles fluid through a porous material in the existence of slip effect and porous dissipation is the subject of this study. The equations that govern the system are shown and summarized as boundary layer assumptions, and then modified into framework of first-order DEs using the similarity approach. By using similarity transformation, a two-dimensional nonlinear partial differential equation is decreased to a sequence of nonlinear ordinary differential equations (ODEs). Then, by employing numerical techniques such as Maple packages, the solution of system of nonlinear equations is represented using the RK4 method. The numerical findings are derived under specific unique situations. The Nusselt number and coefficient of skin-friction are also given numerically. The increase in Brinkman number γ raises the temperature profile for both the dusty and the fluid phases. The results also demonstrate that rise in the suction number S falls the temperature distribution within the boundary layer for the dusty phase and fluid phase. For a variety of flow quantities of attention, the variation of parameters is studied, and the outcomes are reported in the shape of graphs and tables. Several industrial processes make advantage of boundary layer flow and heat transfer over such a stretched surface in porous materials. [ABSTRACT FROM AUTHOR]

AbstractThe purpose of this work is to provide a comprehensive analysis of the bioconvective Williamson nanofluid stream while considering the effects of radiation flux and viscous dissipation. The two-phase nanofluids model is applied where the impacts of the Brownian motion and thermophores are significant. This study starts with introducing the governing system in case of the irregular plane then it transforms to the regular plane. The solution methodology is based on the fully implicit finite difference method. The major results disclosed that maximizing the amplitude parameter causes a diminishing in the velocity and skin-friction coefficients. Also, lower behaviors are noted for the velocity in the case of Williamson nanofluids compared to a Newtonian nanofluid case (We = 0). Furthermore, the considered range of Biot number (1 ≤ Bi ≤ 5)causes an improvement in the surface temperature up to 71.59%. [ABSTRACT FROM AUTHOR]

In this paper, we investigate time-dependent viscous fluid flow along a stretching horizontal plate with a magnetic field, viscous dissipation and chemical reaction are taken into account. A system of partial differential equations (PDEs) governs the flow problem. Using dimensionless variables, system of governing PDEs is transformed to dimensionless form. Finite difference method is the numerical method adopted for solving the system of PDEs with initial and boundary conditions. The impact of potential variables is examined and displayed graphically. The results suggest that greater Hartmann number reduces velocity and temperature, whereas raising chemical reaction parameter decreases concentration profile. Moreover, skin friction increases with Hartmann number values and Nusselt number, showing increase for growing Eckert and Prandtl numbers. [ABSTRACT FROM AUTHOR]

The purpose of the present research is to examine the effect of porosity in the presence of radiation and viscous dissipation on two-dimensional unsteady MHD mixed convection heat and mass transport at the stagnation point. The governing time-dependent nonlinear PDEs are converted into nonlinear ODEs by utilizing similarity transformations. The mathematical formulation is produced as ordinary differential equations. The achieved coupled system has been analyzed using the Keller–Box technique. Newton's system dictates that the coefficients must be accurate and refined. To assess the influence of flow-controlling factors on relevant profiles, quantitative, and qualitative analysis are conducted. With the use of graphs and tables, beneficial talks were conducted. Under the restricting conditions, prior findings were compared and determined to be satisfactory. The conclusions of this study are widely accepted by the scientific community. [ABSTRACT FROM AUTHOR]

In this paper, the significance of slip situations in the appearance of a porous medium and frictional heating on unsteady fluid flow through a porous medium has been explored. The numerical solutions of the differential equation representing fluid flow through a porous material, including slip effects, are presented. We have obtained a nonlinear ordinary differential equation using a similarity transformation. Under velocity and thermal slip conditions, the Maple packages were used to numerically solve the resulting set of nonlinear problems. Both the velocity and temperature increase when the Brinkman viscosity ratio parameter γ increases. The effects of the nondimensional parameters on the governing flow velocity and temperature are examined with graphical profiles. The implications of relevant parameters on dimensionless temperature, velocity, local Nusselt number and skin friction coefficient are shown and explained. For various flow quantities of interest, the fluctuation of parameters is investigated, and the results are given in the system of graphs and tables. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of the flow and heat transfer performance of ternary nanofluid flowing past a stretching sheet under the influence of quadratic convection. The ternary nanofluid is formed by suspending three different nanoparticles, namely, titanium dioxide (TiO2), silver (Ag) and graphene (Gr) in the base fluid water (H2O). Thus, the ternary nanofluid obtained is Gr–Ag–TiO2/H2O where the hybrid nanofluid Ag–TiO2/H2O forms the base fluid for the resulting ternary nanofluid. The addition of TiO2 nanoparticles enhances the photocatalytic nature of the base fluid and makes it useful in various applications concerning the medicinal field. The presence of Gr helps in intensifying the thermal conductivity of water while the suspension of silver nanoparticles ensures chemical stability. Meanwhile, the thermophysical properties of the ternary nanofluid are mathematically defined and the system of equations that describe the flow of a ternary nanofluid past a stretching sheet is framed using differential equations. The outcomes of this study are interpreted through graphs for velocity and temperature profiles of the ternary nanofluid. It was mainly observed that the thermal conductance of ternary nanofluid was higher than the monophase and hybrid nanofluid. Also, the presence of quadratic convection had a prominent impact on the ternary nanofluid flow. The Nusselt number was found to be greater for spherical nanoparticles and it was found to be least for blade-shaped nanoparticles. [ABSTRACT FROM AUTHOR]

This study is important because it helps us understand how different factors affect the flow of a special liquid with nanoparticles, giving us useful insights for real-world applications in engineering and technology. This research endeavours to explore and contrast the distinctive characteristics of steady, 2-D magnetohydrodynamic (MHD) and stagnation point (Hiemenz) flow in Carreau hybrid nanofluid {Fe3O4-Cu/kerosene-engine Oil} over a nonlinear stretching-shrinking surface with its counterpart, Carreau nanofluid. Employing the laws of conservation encompassing mass, momentum, energy, and concentration, our methodology probes into the multifaceted effects induced by factors like the suction-injection, viscous dissipation, Newtonian heating, Hiemenz flow, thermophoresis, and Brownian motion. This exploration involves the transformation of governing partial differential equations (PDEs) into a system of ordinary differential equations (ODEs) through similarity transformations. Subsequently, the intricate non-linear ODEs are meticulously solved using the bvp4c technique, an adept MATLAB built-in function, facilitating a comprehensive understanding of the intricate dynamics within the system. In addition, the study considers the Newtonian heating effect and a numerical comparison is performed to verify the results. Using two different types of nanoparticles dispersed in kerosene-engine oil, it was demonstrated that increasing Weissenberg parameter values resulted in significant improvements in thermal conductivity and thermal velocity. A comparison of Carreau nanofluid thermal and concentration velocities with Carreau hybrid nanofluid showed that Carreau nanofluid values are lower (Weissenberg parameter). Higher Biot numbers are associated with reduced temperature velocity and elevated concentration velocity. [ABSTRACT FROM AUTHOR]

AbstractUnderstanding and controlling the shape factors of nanoparticles in fluid flow problems is crucial for optimizing heat transfer, fluid dynamics, and various applications such as nanofluids, drug delivery systems, catalysis, and nanocomposites. By tailoring the shape of nanoparticles, researchers can manipulate their behavior, dispersion, and interaction with the fluid, thereby optimizing the desired outcomes in various fluid flow problems. The meticulous investigation uncovers the integration of entropy analysis in the investigation of the flow of convective magnetohydrodynamic hybrid nanofluid via a permeable flat surface, by considering the effects of Dufour, viscous dissipation, Soret, and thermal radiation. The governing equations are converted into a set of nonlinear ordinary differential equations using appropriate similarity transformations and then solved using the bvp4c solver, a MATLAB in-built function. We display the outcomes for three different shape factors: platelet, cylinder, and brick. The skin friction coefficient increases at a rate of 1.5089 for the brick shape, 1.5163 for the cylindrical shape, and 1.5212 for the platelet shape when the mixed convection parameter lies between 1 and 5. In this regard, it is important to point out that there is a direct connection between the increase in temperature transmission rate and a decline in the Dufour number (Du). It has been found that when 1≤Du≤2.4, the Nusselt number decreases by 0.4786, 0.4618, and 0.4512 for brick, cylinder, and platelet shapes, respectively. A decline in the Sherwood number is noticed with an upsurge in the Soret number (Sr). There is a noticeable increase of 0.0442 in the Sherwood number for brick shape, 0.0438 for cylinder form, and 0.0435 for platelet form when 1≤Du≤2.4. Furthermore, we have detected that an increase in the radiation parameter amplifies both the Bejan number and the profiles of entropy formation. [ABSTRACT FROM AUTHOR]

The study of nanoliquid characteristics and their heat performance have attracted the interest of engineers. These engineered fluids have high thermal conductivity due to which such liquids are reliable for different engineering applications including heating/cooling of buildings, thermal and mechanical engineering, etc. Therefore, the current research design provides a new ternary nanoliquid model for the heat transport process under induced magnetic field effects, mixed convection, heating source and thermal radiations. The modeling has been done by implementing the ternary fluid characteristics and supportive transformations and then for results simulation; bvp4c is coded successfully. It is scrutinized that a higher inductive magnetic field (0.1–0.4) and nanoparticles amount (0.01–0.07) are better to resist the movement while the wedge parameter (λ1) promotes it. By promoting the heating source, Eckert and Rd, the heat transfer process is observed rapidly while the mixed convective number α controls it. Further, the particular used ternary nanoliquid is examined and found to be good for cooling purposes. [ABSTRACT FROM AUTHOR]

The researchers have reported numerous numerical and analytical efforts in recent years to understand technological and industrial processes. Microelectronics, heat exchangers, solar systems, energy generators are just a few numbers of recent applications of heat and mass transfer flow. Two dimensional steady incompressible MHD flow of micropolar fluid over an inclined permeable surface with natural convection is investigated in this research work, with the contribution of thermal radiation under thermophoretic effects as a heating source. As a result of this infestation, mathematical model of the problem equations based on energy, momentum, angular momentum, mass, and concentration are developed. To convert the current problem into dimensionless ordinary differential equations, non-dimensional variables have been assigned. The evolved mathematical model is numerically solved aside utilizing Shooting technique along with 4th order R-K method solver in MATHEMATICA. The outcomes are displayed and analyzed through figures and tables. Finally, skin friction, Nusselt and Sherwood numbers are tabulated for distinct parameter-factors. To validate the accuracy of numerical method used in this problem, we compare the numerical results with available findings, and it is evident that the outcomes of current work are in good agreement with those reported in the literature. Improving the values of thermophoresis, radiation factors, and Schmidt number, declines the velocity. Higher values of radiation parameter, thermophoresis parameter, the microrotation increase near plane-surface and gradually diminishes far away from plane-surface. Profiles of temperature enhances with increasing the viscous dissipation parameter. Profiles of the concentration decreases by increasing the thermophoresis parameter and Schmidt number. Profiles of Skin friction and mass transfer rate decreases for magnetic field, thermal radiation and Schmidt number values. [ABSTRACT FROM AUTHOR]

This study encapsulated the research methodology utilized in the flow behaviors of Williamson nanofluid and analyzed the associated mass heat transfer. The study concentrated on examining the magnetohydrodynamic behavior of nanofluids in the presence of heat generation effects and the inclusion of dissipative energy on a vertical nonlinear stretching surface submerged within a Darcy porous medium. The rationale for including variable viscosity and variable conductivity in this research was to precisely evaluate the mechanisms of heat and mass transfer, particularly with regard to the fluctuations in fluid properties. The objective was to enhance the understanding of how these varying properties impact the overall heat and mass transfer processes. The initial formulation of the phenomenon, initially presented as partial differential equations, was transformed into ordinary differential equations by employing appropriate dimensionless variables. The ultimate streamlined version of the model was then numerically solved utilizing the shooting method. By employing the numerical shooting method, we portrayed nanofluid patterns in velocity, temperature, and concentration fields, alongside essential parameters such as skin friction coefficient, Sherwood number, and Nusselt number. The significant key findings highlighted that both the porous parameter and the magnetic number increasingly affected temperature and concentration distributions. Additionally, increasing the thermophoresis parameter resulted in higher concentration and corresponding temperature levels. Graphical presentation and physical explanations were used for analysis, and the study's outcomes were compared to existing literature, affirming a strong agreement that validated the solutions. [ABSTRACT FROM AUTHOR]

Vertical channels between two flat plates have a wide range of engineering applications, such as heat exchangers, chemical processing equipment, electronic component heat dissipation, and so on. Therefore, this analysis examines the significance of variable properties on the dynamics of Casson fluid in a vertical channel due to mixed convection when thermal radiation and Joule heating are significant. The combined effects of temperature-dependent plastic dynamics viscosity and temperature-dependent thermal conductivity are analyzed. Moreover, the evaluation of heat transfer is further supported by viscous dissipation. The nondimensional governing equations are elucidated analytically by employing the perturbation technique and numerically by Runge–Kutta method with the shooting technique. The effects of numerous dimensionless parameters on fluid flow are featured through graphs. In addition, the heat transfer rate and the skin friction coefficient are computed and scrutinized. It is reported that velocity depicts enhancing behavior for higher values of the Casson fluid and slip parameters. It has also been observed that the fluid temperature decreases with an increase in the thermal conductivity parameter. [ABSTRACT FROM AUTHOR]

Due to recent advancements in technologies and nanoscience, research focused on nanofluid because of its extensive use in many industrial as well as manufacturing processes. Nanotechnology is widely utilized in many photovoltaic devices, mini-computers, bio-medical apparatus, electronic instruments, cooling and heating of metallic plates, etc. In view of numerous applications, the basic aim of this work is to explore the impact of ferromagnetic nanomaterial on 2-D Eyring–Powell fluid flow. Characteristics of viscous dissipation and Brownian movement are under consideration. Some suitable similarity transformations are exploited to achieve the nonlinear ODEs from PDEs. Bvp4c scheme is implemented to solve the resultant system of equation. Impacts of ferromagnetic interaction parameter, thermophoresis parameter, viscous dissipation, Curie temperature and Eckert number are perceived for velocity, temperature as well as concentration fields. Moreover, velocity, thermal and mass transfer gradients are deliberated and scrutinized graphically. It is found that upsurge in the value of magnetic parameter reduces the velocity profile while it deteriorates against Eyring–Powell fluid parameter. Additionally, greater thermophoresis parameter has augmented the temperature profile while it shows opposite behavior for concentration. [ABSTRACT FROM AUTHOR]

AbstractThe main objective of this framework is to formulate a model and carry out simulations on the thermally radiative biological fluid of Sisko nanofluid in the non-uniform channel under the combined effects of induced magnetic field and double diffusion convection. Additionally, the effect of viscous dissipation is also considered. The formulation and simplification of the governing equations for a non-Newtonian fluid with nanoparticles and double diffusion convection is done under the assumption of a long wavelength and low Reynolds number. The numerical solution of the proposed problem is calculated by utilizing built in commands in Matlab and Mathematica software’s. The equation solving is done in Mathematica using the integrated program NDSolve. The framework provides expressions for temperature, nanoparticle fraction, velocity, pressure rise, stream functions, pressure gradient, magnetic force function, and concentration. The impact of relevant parameters on various physical quantities is then analyzed through graphical representations, considering both non-Newtonian and viscous fluid behavior. The present analysis suggests that an increase in heat radiation may cause a drop in the temperature distribution strengthening the cooling effects. It is also found that as Brinkman number rise, buoyancy-driven flows have a greater impact on fluid motion. The concentration of nanoparticles may fluctuate and maybe decrease in specific locations in accordance with the dispersion and advection phenomena. Furthermore, this finding has important ramifications for the field of biomechanics too. Examples include comprehending chyme movements in the digestive system and the prospective use in operations to regulate blood flow by adjusting the magnetic field’s strength. [ABSTRACT FROM AUTHOR]

The axisymmetric stagnation point flow of brick and blade-shaped Silver and Copper nanoparticles immersed in an ethylene glycol base fluid under the influence of an induced magnetic field over an unsteady radial stretching surface is investigated in this study. The unsteady phenomenon is considered because most flow issues in practice are unsteady. The fundamental laws of mass, momentum, and energy conservation are used to present the physical model. Heat transmission is also examined under the effects of magnetohydrodynamics, Joule heating, viscous dissipation, and convective boundary conditions to give a realistic physical investigation. Scaling analysis transforms the flow-governing issue into a collection of higher-order nonlinear ODEs. These are, then, solved numerically using the fourth-order Runge–Kutta and shooting techniques. Moreover, the numerical technique is validated by calculating residual error. It is concluded that, compared to the Ag–EG nanofluid, the Cu–EG nanofluid had the highest IMF, lowest temperature, minimum surface drag, and maximum heat flux, making it the ideal choice for creating a radial module. [ABSTRACT FROM AUTHOR]

AbstractNumerically the hybrid nanofluid (HNF) flow comprised of aluminum alloys (AA7072 and AA7075) across to coaxial cylinders is reported in the current analysis. The fluid flow has been examined under the significances of viscous dissipation, thermal radiation, and exponential heat source/sink. The HNF is produced by the addition of AA7072 and AA7075 nanoparticles (NPs) in water. Aluminum alloys are mostly used for the electric module wrapping, electronic machinery, automotive body, solar and wind energy controlling, and energy generation. The flow has been modeled in the form of momentum, continuity, and energy equations. Numerical approach parametric continuation method (PCM) is used to solve the modeled equations. The consequences of flow and energy parameters on the temperature, velocity, skin friction, and Nusselt number have been revealed through figures. It has been noticed that the fluid velocity drops, whereas the energy curve develops with the escalating influence of magnetic field. Furthermore, the rising numbers of AA7072 and AA7075 NPs in the water diminish the fluid velocity and temperature profile. Energy curve accelerates with the growing values of Brinkman number. It can be observed that the energy transmission rate upsurges up to 13.17% by the addition of AA7072 and AA7075 NPs in the water. [ABSTRACT FROM AUTHOR]

In this paper, a non-isothermal study of the calendering processes is presented using Carreau–Yasuda model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. The final equations are solved numerically using "bvp4c" to find stream function and velocity profiles, while the hybrid numerical method which is the combination of shooting and finite difference methods is used to solve the energy equation. Graphs show the impact of the concerned material parameters on various quantities of interest. The pressure distribution decreases with the increasing values of the slip parameter and Weissenberg number. The mechanical variables show an increasing trend with the increasing values of the slip parameter and Weissenberg number. The temperature distribution increases with an increase in the Brinkman number, while temperature shows declining trend near the roll surface with the increasing values of the slip parameter. The force separating the two rollers, total power input into both rolls, increase with the increasing values of the Weissenberg number and slip parameters. The results show that the Newtonian model predicts higher pressure in the nip zone than the Carreau–Yasuda model. It is interesting to note that for the case of shear thinning, the Carreau–Yasuda model predicts 30% less pressure in the nip region when compared to the Newtonian model. [ABSTRACT FROM AUTHOR]

The main objective of this work is to investigate viscous dissipation with thermal radiation impact on hybrid nanofluid flow past an exponentially stretching/shrinking surface along with magnetic field, and heat source/sink, slip boundary conditions. Water ( H 2 O ) is taken into account as a host fluid with copper (Cu) and alumina ( Al 2 O 3 ) as nanoparticles to form a hybrid nano liquid (Cu- H 2 O - Al 2 O 3 ). The coupled nonlinear partial differential equations are transmuted into ordinary differential equations using similarity transformations with boundary conditions and these ODEs are simplified using numerical solver, bvp4c in MATLAB. The behavior of momentum profiles, thermal profiles, coefficient of heat transfer ( Nux 1 ) and skin friction ( C f ) are explored using the pertinent parameters such as Eckert Number ( Ec 1 ), Suction Parameter ( S 1 ), Magnetic Parameter ( M 1 ), Radiation Parameter ( Nr 1 ), Stretching/Shrinking Parameter (λ ), Velocity and thermal slip Parameters ( A 1 and B 1 ), Heat source/sink Parameter ( β 1 ), variable thermal Parameter (ϵ ), thermal conductivity (K) are depicted through tables and graphs. The present simulation is more stable and convergence when compared with the existing literature, which is portrayed in the tables. The presence of dual solution is noticed for Ec 1 = 0.1, 0.11,0.12 when the critical value of λ (= λ c ) are λ c 1 = - 1.60355799, λ c 2 = - 1.60322884 and λ c 3 = - 1.6031348, respectively. The existence of the dual solution is reported due to the presence of shrinking surface and suction, further, the first solution is found to be more stable. The novelty of the current simulation includes MHD hybrid nano liquid flow stability with the impact of viscous dissipation and thermal radiation past stretching/shrinking permeable plane to fill the research gap in the existing literature. The applications of transmission of heat by stretching/shrinking surface in boundary layers is used in various fields such as polymer and material processing, biomedical engineering, heat exchangers, etc. [ABSTRACT FROM AUTHOR]

The aim of this analysis is to examine theoretically the mixed convection flow of an incompressible and electrically conducting viscous fluid via an infinite vertical porous plate. This research is unique in that it examines the effects of a magnetic field, Soret, heat source, chemical reaction, and Joule heating on heat and mass transmission. The mathematical model regulating the flow has been developed using partial differential equations and then converted via proper similarity transformations to a system of ordinary differential equations containing the momentum, energy, and concentration equations. Though several hypotheses have been advanced to explain the idea of boundary layer flow, the present analysis's use of the bvp4c scheme suggests excellent agreement with the results of a previously published data in the limiting sense. Graphs and tables illustrate the numerical results of the solutions for flow field, temperature, and species concentration, furthermore the coefficient of friction factor, heat, and mass transfer characteristics. The range of parameters selected is as follows: Gr = Gm [0.1 − 3], M [1–2], χ = [ 0.1 - 0.5 ] , Ec [0.01–0.1], N [1 = 3], Pr [0.71–3], R [0.2–1], δ [0.5–3], Sc [0.22–0.61], and Sr [0.1–0.5]. The novel result shows among the major finding that velocity and concentration are decreasing while increasing values of Soret number. [ABSTRACT FROM AUTHOR]

This research communication scrutinizes the time-dependent radiated stagnation point flow by a movable Riga surface in the presence of porous medium, mixed convection, viscous dissipation and radiative heat flux. Both electro-magneto effects are accounted for. The considered flow problem comprises Copper (Cu) as a nanoparticle in a continuous phase fluid water (H2O). The mathematical modeling depends on two main expressions, i.e., momentum equation of Navier-Stokes equation under certain conditions and temperature expression. The coupled system is further simplified via similarity transformations. Computational outcomes are computed by implementing bvp4c technique with the shooting concept. The obtained results are displayed graphically. It is noticed from the computations that both Nusselt number (NN) and skin friction coefficient (SFC) declined subject to increasing estimations of the unsteady variable. The final outcomes and main key points are listed in the conclusion section. [ABSTRACT FROM AUTHOR]

Online viscosity information on processing lines can reflect the material flow resistance and offer valuable guidance for manufacturing across various industries. Considering the accuracy, devices, and processes involved in injection molding, characterizing the melt's flow state during material processing poses a significant challenge. To reduce investment in viscometers, avoid influencing the components' surface aesthetics due to the installation of sensors, and make the flow state detect online in mold, this study designs a rheometric mold with cylindrical runners for identifying the in situ viscosity of molten resin during injection molding. The detection conditions of injection speed and cavity pressure variations, the entrance effect, and the viscous dissipation for Polycarbonate are analyzed under various conditions. The in situ viscosity is identified and compared with the standard cross‐WLF model. The result shows that the melt velocity and cavity pressure variations during the filling process create a stable environment for in situ rheological characterization and the detected viscosity is related to the shear rate, melt temperature, and channel dimension in injection molding. The designed mold with cylindrical runners for determining the in situ thermal‐rheological behavior of polymer is distinguished successfully and exhibits prospects for the development of low‐cost, nondestructive, and inner‐mold measurement in manufacturing applications. [ABSTRACT FROM AUTHOR]

The issue of Jeffrey nanofluid peristaltic flow in an asymmetric channel being affected by an induced magnetic field was studied. In addition, mixed convection and viscous dissipation were considered. Under the supposition of a long wave length and a low Reynolds number, the problem was made simpler. The system and corresponding boundary conditions were solved numerically by using the built-in package NDSolve in Mathematica software. This software ensures that the boundary value problem solution is accurate when the step size is set appropriately. It computes internally using the shooting method. Axial velocity, temperature distribution, nanoparticle concentration, axial induced magnetic field, and density distribution were all calculated numerically. An analysis was conducted using graphics to show how different factors affect the flow quantities of interest. The results showed that when the Jeffrey fluid parameter is increased, the magnitude of axial velocity increases at the upper wall of the channel, while it decreases close to the lower walls. Increasing the Hartmann number lads to increases in the axial velocity near the channel walls and in the concentration of nanoparticles. Additionally, as the Brownian motion parameter is increased, both temperature and nanoparticle concentration grow. [ABSTRACT FROM AUTHOR]

The dynamics of blood conveying gold nanoparticles (GNPs) are helpful to the health workers while air conveying dust particles over rockets is helpful to space scientists during the testing phase. However, little is known on the significance of thermal diffusivity in these aforementioned cases. In this report, the partial differential equation suitable to unravel the implication of increasing partial slip and viscous dissipation on the dynamics of the mixture of (i) blood and nano size of GNPs (ii) air and dust particles on an object with an increasing diameter (uhspr) is investigated. The density, zero shear rate viscosity, heat capacity, and thermal conductivity treated in this study vary with volume fraction nanoparticles. In the second case, the interaction between the solid particles and air is incorporated into the momentum equation using the Stokes drag. Transformation and parametrization of the two‐dimensional nonlinear partial differential equations were obtained with the aid of suitable similarity variables. Thereafter, the numerical solutions of the corresponding boundary valued problems were obtained using the classical Runge–Kutta integration scheme together with shooting techniques and Matlab bvp5c package. Enhancement in the rate of viscous dissipation is a major factor suitable to increase the velocities of both fluids, boost temperature distribution across both fluids, and local skin friction coefficients. There exist a significant difference between the effect of partial slip on the dynamics of blood‐gold nanofluid and dusty fluid. [ABSTRACT FROM AUTHOR]

This investigation explores the convective MHD stagnation point flow of Casson nanofluid over a stretching sheet in a porous medium with higher-order chemical reactions and multiple slips. The examination of heat transmission is carried out in the presence of radiation, Joule heating, viscous dissipation, and heat source. The system of governing equations was simplified by using appropriate transformations. The transformed equations are tackled numerically by a Runge-Kutta-based shooting technique with the bvp5c MATLAB package. The graphical and numerical results for different parametric values are discussed and presented through figures and tables. It is observed that an increase in the magnetic field, porosity, and Casson fluid parameter reduces the velocity whereas the opposite trend is seen in the case of the thermal Grashof number and the solutal Grashof number. Increasing values of radiation and Joule heating parameters, and the Eckert number, lead to an increase in the temperature. The chemical reaction, suction, and solutal slip were found to reduce the concentration. The friction factor was reduced due to the higher values of the thermal Grashof number, solutal Grashof number, and the velocity ratio parameter. Also, an increase in the Brownian motion and thermal slip parameters decreases the heat transfer rate. The computational results of the Nusselt number have been validated with published results in the literature and found good agreement. The results obtained in this investigation have applications to biomedical, engineering, and industrial sectors such as food processing, polymer manufacturing, glass, and fiber production, improving oil recovery, and material processing. [ABSTRACT FROM AUTHOR]

This study intends to investigate the hybrid nanofluids with viscous dissipation and thermal radiation effects of the magnetic field. The consequences of viscous dissipation and thermal radiation on the transport of Cu - Ti02/water and CUO - Ti02/water were investigated. Magnetic fields create a hybrid nanofluid with varying viscosity. By including suitable symmetry variables, the fluid model is converted into ordinary differential equations. For the numerical simulation, we employ the bvp4c package in MATLAB. The evaluation's findings are laid out graphically in tables and charts. Various values of crucial parameters such as variable viscosity, magnetic field, viscous dissipation. and nanoparticle volume fraction are investigated better to grasp the nature of the problem's energy flow and energy transfer characteristics. It was revealed that raising the magnetic field and the suction/injection parameters reduces velocity. While the fluid velocity remains constant, the temperature distribution changes as the viscous dissipation, magnetic field, and thermal radiation parameters increase. [ABSTRACT FROM AUTHOR]

The remarkable thermophysical characteristics of hybrid nanofluids have considerable potential for the enhancement of heat transport. In recent years, there has been a significant surge in research on hybrid nanofluids, with findings indicating that these fluids are favorable for transferring heat in engineering contexts. Therefore, a theoretical investigation is conducted by adopting the Tiwari and Das model, for viscous dissipative flow and thermal transport characteristics of hybrid nanoparticles in an exponentially permeable porous shrinking surface. The convective thermal transport features are studied with the influence of Ohmic heating and thermal generation/absorption effects. To solve the governing flow problem of hybrid nanofluid, a standard conversion and numerical approach are used. The numerical findings indicated the existence of dual solutions within a certain range of shrinking and mixed convective parameters. The results reveal that high suction strength higher the skin friction coefficient and heat transfer rate. It also reveals that in opposing flow regions, the addition of nanoparticles resulted in a lower heat transport rate and a higher skin friction coefficient. Furthermore, as thermal radiation increases, the rate of heat transfer increases, but the upward Eckert number exhibits the opposite behavior. In addition, the velocity profile displays opposite behavior for upper and lower behavior when the mixed convection parameter increases. Besides, the thermal distribution grew when the radiation, Eckert number, and Biot number were raised. [ABSTRACT FROM AUTHOR]

This article presents the numerical investigation of magnetohydrodynamic (MHD) heat transfer flow of viscous dissipation in vertical channel in the presence of Newtonian heating. The coupled nonlinear dimensional partial differential equations governing the flow are converted into non-dimensional form by using the appropriate dimensionless quantities and parameters Implicit Finite Difference Method (IFMD) was employed in finding the solutions of dimensionless partial differential equations. The expressions of velocity, temperature, skin friction and Nusselt number were obtained numerically and discussed using line graphs. From the outcome of the results, it was concluded that rising the Newtonian heating parameter (γ), Brinkman number (Br), lead to a substantial growth of velocity and temperature. Similarly, as the, Grashof number (Gr) increases, the velocity of the fluid also gets increased. The skin friction gets enlarged with the increase of Grashof number (Gr), Newtonian heating parameter (γ), Brinkman number (Br) and time at y = 0 and y = 1. The Nusselt number gets reduced with the increase in Newtonian heating parameter (γ), Brinkman number, and time parameter at y = 0 and increases at y = 1. [ABSTRACT FROM AUTHOR]

In the field of heat and mass transfer applications, non-Newtonian fluids are potentially considered to play a very important role. This study examines the magnetohydrodynamic (MHD) bioconvective Eyring–Powell fluid flow on a permeable cone and plate, considering the viscous dissipation (0.3 ≤ Ec ≤0.7), the uniform heat source/sink (−0.1 ≤ Q0 ≤ 0.1), and the activation energy (−1 ≤ E1 ≤ 1). The primary focus of this study is to examine how MHD and porosity impact heat and mass transfer in a fluid with microorganisms. A similarity transformation (ST) changes the nonlinear partial differential equations (PDEs) into ordinary differential equations (ODEs). The Keller Box (KB) finite difference method solves these equations. Our findings demonstrate that adding MHD (0.5 ≤ M ≤ 0.9) and porosity (0.3 ≤ Γ ≤ 0.7) effects improves microbial diffusion, boosting the rates of mass and heat transfer. Our comparison of our findings to prior studies shows that they are reliable. [ABSTRACT FROM AUTHOR]