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The purpose of this article is to study the effect of electromagnetic (EM) stimulation on the pipeline, which has an electrical and thermal effect in addition to the chemical reaction on the crude oil and makes a sinusoidal wave on the wall. Modeling the crude oil as Carreau fluid is done. EM stimulation is an effective and safe technology that can be used to improve fluid movement in a variety of industrial applications. The flow analysis by applying EM may avoid blocking the crude oil pipeline which leads to a loss of production and capital investment. The basic partial differential equations of momentum, temperature and concentration are reduced to a system of nonlinear partial differential equations, which is solved numerically by using the Rung–Kutta–Merson method with Newton iteration in a shooting and matching technique under the assumption of long wavelength and the effect of physical implanted parameters is represented through charts for velocity, temperature, and concentration and numerical application. [ABSTRACT FROM AUTHOR]

This article investigates the effect of activation energy on Carreau fluid flow in the presence of variable thermal conductivity, inclined magnetic field, melting heat transfer, and Soret and Dufour phenomena. A set of suitable similarity transformations is applied for transforming the governing partial differential equations (PDEs) considered for the physical phenomena into non-linear ordinary differential equations (ODEs). We obtained the results by solving the non-linear ODEs numerically which describe the behavior of the velocity, temperature, and concentration profiles of the fluid against the governing parameters and illustrated these graphically. The velocity distribution decreases for angle of inclination while it is increasing against higher melting parameter. The temperature distribution enhances with higher modified Dufour parameter and Prandtl number while it declines for thermal conductivity, activation energy, and melting parameters. The concentration distribution increases for melting parameter, Soret number, and activation energy parameter while decreasing for temperature difference parameter, Schmidt number, and reaction rate parameter. Skin friction, Nusselt and Sherwood numbers are evaluated numerically against the various governing parameters. Accuracy of the present work is illustrated and established through comparison carried out for skin friction versus magnetic parameter with existing results in the literature. [ABSTRACT FROM AUTHOR]

This paper presents the numerical study for heat generation and electro-osmotic effects of Carreau model of a fluid in an unsteady channel. This prepared model has many useful applications in microfluidic biological and mechanical research. Modified Darcy’s law, viscous dissolution and ciliated walls are also presented. Non-Newtonian fluids play a key role in biological fluids which include the motion of foods within digestive zone, blood circulation, including small bowls and migration of urine. The main findings for the research is to analyze the flow rate change and check the behavior of velocity and temperature by using different parameters. The modeled equations of Carreau fluid flow are depressed into a simple form by employing the long wavelength approximation and small Reynolds number, then the resulting equations are calculated numerically by using MATHEMATICA. Impact of physical factors are graphically figured for velocity, temperature field and streamline distribution. The results show that velocity modifies by applying different values of flow rate and Darcy’s number. Temperature profile increases for advanced values of Prandtl number, Eckert number and heat source parameter. For heat source parameter (γ=0.5, 0.6 and 0.7) and Eckert number (Ec=1.0, 1.5 and 2.0) the temperature profile expands at the region (y

The aim of this study is to perform analysis of nonlinear mixed convection flow of Carreau fluid due to curved stretched surface with slip condition. Furthermore, the effect of Darcy–Forchheimer and MHD is taken into consideration. The energy equation is modified by considering the variable thermal conductivity with Joule heating effect. The governing equations of the problem are highly nonlinear partial differential equations subjected to boundary conditions. Similarity transformations are used to transform nonlinear PDEs into ODEs. The numerical technique is used to solve this problem via bvp4c. The main outcomes of this research are that the impact of Forchheimer and porosity parameter tends to decrease the fluid velocity for shear thickening case. The velocity slip condition increases the fluid velocity only near the surface but the fluid velocity away from the surface does not get affected. The changing thermal conductivity raises the temperature of the fluid near the boundary. Furthermore, the impacts of various parameters on the velocity and temperature profiles are demonstrated. The values of local Nusselt number and skin friction for a different parameter are provided. [ABSTRACT FROM AUTHOR]

Many applications, including micro air vehicles, automotive, aerospace, refrigeration, mechanical–electromechanical systems, electronic device cooling, and micro heat exchanger systems, can be used to determine the heat flow in microchannels. Regarding engineering applications, heat flow optimization discusses the role of entropy production minimization. Therefore, this work explores new facets of entropy production in fully developed Carreau fluid heat transport in an inclined microchannel considering exponential space/temperature dependence, radiative heat flux, and Joule heating. The Carreau fluid model’s rheological properties are taken into account. Additionally, the influence of Hall slip velocity and convective boundary conditions is considered. Using appropriate transformation constraints, the governing equations are transformed into a system of ordinary differential equations, which are then numerically solved using the fourth- and fifth-order Runge–Kutta–Fehlberg method. Graphs illustrate a significant discussion of physical parameters on production of entropy, Bejan number, thermal field, and velocity. Our findings established that there is a dual impact of entropy generation for the exponential space/temperature-dependent, radiation parameter, Hall parameter, Weissenberg number, and velocity slip parameter. The Bejan number decreased with the Hall current and the Weissenberg number, and it enhanced with exponential space/temperature dependent. The convection constraint maximizes the entropy at the channel walls. The results are compared with exact solutions, which show excellent agreement.

Many scholars performed the analysis by using the non-Newtonian fluids based on the nano and hybrid nano particles in blood arteries to investigate the heat transport for cure in several diseases. These performances are presented to investigate the blood flow behaviour with extended form of the novel tetra hybrid Das and Tiwari nanofluid system attached by the Carreau fluid. The assessment of energy transport has been achieved based on the thermal radiation, heat source/sink, Joule heating, and viscous dissipation. The obtained partial differential equation from physical problem is transformed into ordinary differential equations (ODEs) by using the similarity variables. Furthermore, system of nonlinear ODEs attached with boundary conditions are transported into the system of first-order ODEs with initial conditions. For the numerical solution of obtained ODEs, the numerical solutions have been performed based on the RK method. The numerical results are plotted through figures, tables, and statistical graphs. Magnetic forces and inclined magnetic effects are caused to reduce velocity of blood. Temperature of blood within the arteries is increased by increasing the parameter of thermal radiation.

A theoretical study is examined to investigate the effects of enthalpy and radioactive heat transfer in generalized Newtonian fluid flow due to an inclined rotating disk. In order to examine the shear thickening phenomenon, Carreau fluid model is utilized. Here for the first time, Carreau fluid model under the enthalpy, thermophoretic effect, binary mixture and solar radiation effects are examined near an inclined rotating disk. In this study, the fluid flow does not give an analytical solution of the Navier–Stokes equations, except for the limit case, that is for high Reynolds number. The flow inside three-dimensional boundary layer flow is determined by the von Kármán transformations. The boundary layer equations are converted into system of ordinary differential equations by using von Kármán inequalities and solved numerically by employing shooting method. The effects of shear thinning and shear thickening fluids show a increase behaviour for base flow profiles. Thermal conductivity and mass diffusivity increases the temperature and concentration distributions. The azimuthal velocities increases by increasing values of $ \lambda _{T} $ λ T and $ \lambda _{C}\ $ λ C parameters. We present the numerical solutions for different physical parameters and discussed them for radial and azimuthal velocities and for heat and mass transfer profiles through graphs and tables. [ABSTRACT FROM AUTHOR]

This analysis aimed to explore the blood-based non-Newtonian hybrid nanofluid flow in elliptical stenosed artery with single- and multi-walled carbon nanotubes as nanoparticles. The Carreau fluid model is incorporated to assess the non-Newtonian rheology of blood-based nanofluid for mild stenosis. In particular, the carotid artery is responsible for delivering blood to the brain. If normal blood circulation is disrupted or in the case of severe stenosis, blockage of the carotid artery can lead to the development of brain disability or stroke, which in turn can lead to death. The idealized mathematical equation is transformed into a nondimensional form and solved analytically via the perturbation method through a novel polynomial technique. These analytical solutions are explored and explained graphically. The system’s disorder and variability are assessed by completing an entropy production analysis. The disruption in blood flow due to the presence of nanoparticles causes uncertainty in the flow nature. This uncertainty is dealt with by fuzzy analysis of temperature distribution by accounting for the nanoparticle volume fractions as triangular fuzzy numbers. It is noticed that stenosis shapes and height greatly impact the flow characteristics. The nanoparticles’ percentage in fluid affected the temperature profile. The non-Newtonian characteristics of blood are found to be more dominant along the minor axis, and an effectively higher disorder is produced in this direction. It is observed that the temperature of nanofluid emerged as a triangular fuzzy number of symmetric shape. [ABSTRACT FROM AUTHOR]

The melting phenomenon plays a critical role in optimizing the performance of power storage, electronic cooling, and semiconductor devices. The present study aims to analyze the melting effect on the flow of Carreau fluid over a stretchable cylinder, with special consideration given to the impact of quadratic thermal radiation. Similarity variables and the homotopy analysis method are used to simplify and determine the semi-analytical homotopic solutions of the governing equations. The present findings reveal that the melting parameter increases the heat transfer rate by more than 10% for both fluids, water (Pr = 0.71), and polymer (Pr = 10). However, as the temperature ratio due to quadratic radiation increases, the local Nusselt number for water has been reduced by 25%, and an even more substantial reduction is observed for the polymer. The present study offers valuable insights into achieving optimal efficiency in electronic devices.

In this article, the mixed convection transport of the Carreau fluid in the presence of a magnetic field and a heated circular cylinder within a trapezoidal cavity is examined. Finite element analysis is used via COMSOL Multiphysics software to simulate the two-dimensional flow. The heat equation is formulated by incorporating the effects of viscous dissipation and Joule heating. The upper and lower surfaces of the cavity are adiabatic, while the left and right sides are assumed to be cold. The temperature distribution and flow field within the cavity are visualized through the plotting of isotherms, streamlines, and total entropy generation. The graphical representations are performed to examine the impacts of physical parameters on temperature, flow field, and total entropy generation, including the Reynolds number , Weissenberg number [Formula: see text], Power law index [Formula: see text], Hartman number [Formula: see text], and Richardson number [Formula: see text] and [Formula: see text]. It is found that the Joule heating and Richardson number are the main reasons to increase the thermal performance within the cavity. The r[Formula: see text] ising values of power law index and Weissenberg numbers caused to reduce the velocity field, irreversible effects and circulation within cavity. While the entropy is augmented by elevating the Eckert and Hartmann numbers. Furthermore, heat transfer improved with the increased magnetic field and Eckert number.

The current work examines the significant impact of Carreau hybrid nanofluid, particularly in enhancing thermal conductivity and heat transfer efficiency, which is discussed through the integration of aluminium oxide (Al2O3) and titanium dioxide (TiO2) nanoparticles into a base fluid made of 50 % water and 50 % ethylene glycol solution. The study is concentrated on thermal radiation, chemical reactions, thermophoresis, Brownian motion, and viscous dissipation over an exponentially curved Riga surface. The governing partial differential equations are formulated in light of all of these factors. Subsequently, the original partial differential equations are transformed using boundary layer approximations and similarity variables to produce a dimensionless ordinary differential equation. The above equations are solved numerically using the shooting method and the MATLAB function bvp4c. The behaviour of heat and mass transfer in both shear thinning (pseudoplastic) fluids and shear thickening (dilatant) fluids is discussed in detail, and efforts are made to highlight some important parameters such as the Weissenberg number, the curvature radius, the modified Hartmann number, Thermophoresis, Brownian motion, and chemical reaction. The fundamental observation is that a rise in the Weissenberg number results in a reduction of skin friction and Nusselt number in shear thickening scenarios, whereas it leads to an increase in shear thinning scenarios. Increasing the modified Hartmann number results in elevated skin friction and a reduction in the Nusselt number for both shear thickening and shear thinning scenarios. These findings are advantageous for developing strategies applicable to heat control across many industries and biological sectors to enhance energy efficiency and system performance.

In this article, the effects of double-diffusive convection have been seen on the flow of Carreau fluid of variable density, maintained inside straight and converging (diverging) channels. Note that the channel walls exhibit variable porosity and undergo stretching or shrinking at non-uniform velocities. In the flow domain, we have taken into account the convective transport of both heat and species mass concentrations. The pseudo-similarity solutions of previous investigation for such type flows are not used here; however, a new set of similarity transformations has been formed, which reduced the governing system (PDEs) into an exact set of ordinary differential equations (ODEs), whereas the longstanding issues of semi-similarity solutions have been successfully resolved in this particular case and in general for all situations of Carreau fluid flow. In a nutshell, a unified mathematical approach has been devised in this paper. We strictly emphasized the simulation of flow of Carreau fluid and double diffusive convection in flow inside a channel with multiple dynamical behaviours and structures (both Jeffery Hammel and Poissuielle flow types) of walls. The boundary layer approximation and stream function assumptions have not been used in the study's execution. The shear thinning and shear thickening features of Carreau fluids with variable density cause them to exhibit lower axial velocity in the centre of a horizontal channel than Newtonian fluids due to variable wall configurations and complex flow conditions. By enhancing buoyancy driven convection, which promotes effective heat dispersion, increasing the solutal and thermal Grashof numbers (GrS=GrT>0) lowers the temperature field. It has been found that in channel with diverging walls (a1=2.0), increasing the modified Reynolds numbers (σ1,σ2>0) increases concentration profiles, i.e. ϕ(η), while decreasing them in channel with rectilinear walls (a1=0.0).

In this paper, we scrutinize the impacts of multi-slip with multi-stratification on nanoparticles MHD Carreau fluid flow through a time-dependent stretching sheet with thermal radiation. We obtain the solution upon the application of the Keller-box scheme with an implicit finite-difference scheme. The influence of thermal stratification and concentration stratification is also being measured. The differentce parameters' influence on the fluid flow is illustrated in diagrams and the quantitative numerical findings are well-argued. For increasing heavier species, fluid index and the reaction rate parameter with Prandtl number Pr = 5. 0 , 10.0 and 15.0 will decrease the concentration of fluid within its boundary limits. Behavior of Weisenberg number on fluid velocity is studied. Moreover, fluid velocity declines as the unsteady parameter rises, while the temperature and concentration significantly decrease. Higher values of solutal and thermal stratification parameters result in decay of velocity and thermal boundary layer thickness while the fluid temperature reflects the opposite behavior. [ABSTRACT FROM AUTHOR]

AbstractBlood flow through a narrowed artery, such as those caused by plaque buildup, can lead to various cardiovascular diseases. Therefore, studying this phenomenon is essential for developing better treatments and understanding the underlying mechanisms. In this study, a mathematical model is developed to simulate blood flow through stenotic arteries with porous walls. The familiar Navier-Stokes equations are executed to simulate flow, whereas the Carreau fluid model is employed to quantify blood rheology. This model illustrates both Newtonian and non-Newtonian characteristics depending on the shear rate. Other than that, the blood vessel is assumed to have a moderate level of stenosis with porous saturated walls. The solution of the governed partial differential equations is carried out using a finite difference scheme. Our research addresses the impacts of permeability and rheological parameters, stenosis size, Brinkman number, and Prandtl number on blood velocity, flow rate, resistance to flow, wall shear stress, and temperature distribution. The findings demonstrate that while flow resistance reduces, the velocity, shear stress, and flow rate increase with high permeability values. Conversely, as the stenosis amplitude increases, the magnitude of wall shear stresses, velocity, and flow rates decrease. In addition, blood rheology also has the potential to transfigure the aforementioned factors. We also found that the temperature distribution increases with the Brinkman number, stenosis size, permeability, and power law exponent. It drives down, however, by raising the values of Weissenberg number and Prandtl number. Collectively, our findings contribute to a better understanding of the complicated phenomena of blood flow in stenotic vessels with porous saturated walls. The findings of this study may be valuable in establishing improved therapies for cardiovascular disorders and formulating more precise mathematical models to forecast blood flow in vivo. [ABSTRACT FROM AUTHOR]

Advancement in thermodynamical characteristics of non-Newtonian fluid is achieved by induction of gyrotactic microorganisms in order to attain efficient thermal and mechanical engineering setups, such as micromixers, microfluidic instruments, biosensors and microdevices. So, the primer motive to conduct this research is to examine the effectiveness of gyrotactic microorganisms in managing heat transport in Carreau nanofluid flow over a linearly stretched surface under the effectiveness of opposing and assisting buoyancy forces. Physical aspect of magnetic field along with slip boundary constraint also entertained by employing in vertical direction. The governing equations are modelled in the form of dimensional PDE's and latterly transformed into dimensionless ODEs via similarity transformations. Solution of obtained set of differential system is solved by implementing Runge-Kutta of order 4 and shooting procedure. Visualizations about influence of flow concerning parameters on associated distributions are analyzed in comparative manner for opposing and assisting buoyancy parameter. In addition, metrics for quantities of engineering interest, such as heat and mass fluxes for nano and gyrotactic particles along with skin friction coefficient are calculated in a comprehensive manner.

For treating and diagnosing cardiovascular diseases, the field of biomedical engineering is significant because it develops new ways and techniques. Stenosis is the narrowing of an artery, and it leads reduction in the flow rate of blood. This study investigates the blood flow mechanism in an artery using a mathematical model of Carreau nanofluid with four distinct nanoparticles. Tetra nanofluid model produces significant advancement in the simulation of blood flow through the stenosed arteries. The model is capable of predicting the pressure drop and velocity distribution for diagnosing and treating stenosis. The spectral relaxation approach is used to present the model's efficiency and effectiveness, which makes it a suitable method for solving the governing equations of this study. The findings of this study have important implications for the development of new treatments and diagnostic techniques for stenosis and other cardiovascular diseases.

The main source of thermal energy is the sun, and with the increase in solar technology, it is now being utilized in many devices such as sun-based panels, photovoltaic cells, batteries and lights, solar fabric, solar water pumping, etc. Nowadays, improvement in flight effectiveness of solar aircraft by utilizing solar energy and nanotechnology is being studied by many scientists. This article is also based on studying the effectiveness of solar aircraft based on solar energy and nanotechnology. For this purpose, few properties of heat transfer among symmetrical wings will be analysed such as porous surface, thermal radiation, convective condition and heat source/sink. It is considered that the ternary hybrid nanofluid moves through the internal side of parabolic trough solar collector. The current study examines the radiative flow of a Carreau tri-hybrid nanoliquid across a convectively heated stretching surface in porous media. Also entropy generation on Carreau fluid is analysed in this work. Energy equation is modelled through heat source/sink and thermal radiation. The well-established numeric technique BVP4c has been used to solve the system of differential equations in the form of concentration, energy and momentum. Several flow variables on fluid velocity, temperature, drag friction, the Nusselt, entropy generation and Bejan number are described in figures and tables. The main outcomes of the current investigation are that the velocity and temperature lowered with augmenting values of Weissenberg number We . Results will prove that THNF is larger in the case of HNF and NF. Further, the drag friction and thermal efficiency of Thnf ( MoS 2 + SiO 2 + Fe 3 O 4 /EG ), Hnf ( MoS 2 + SiO 2 /EG ) and Nf ( MoS 2 /EG ) are computed in percentage with numerous values. The second finding is the addition of entropy due to the increasing magnitude of radiative flow, Carreau fluid variable. When comparing the current results to the reported results, we get a close match. [ABSTRACT FROM AUTHOR]

The motivation for this comparative examination is to verify and account for the appropriate source of heat energy needed in production industries, and to expunge the contradictions between Fourier's law and the theory of relativity. In heat management such as in engineering systems, the required heat for effective melting, production rate, etc. dictates the appropriate heat generation/injection module to be deployed. However, the material medium of heat transmission can be a hindering factor for a successful convection process, to this, an investigation of the instantaneous heat propagation paradox is required. An assumption of induced flow due to stretching characteristics is made, and an appropriate similarity transformation is deployed for the governing systems. By means of the Galerkin-weighted residual method, numerical solutions to the system of equations are approximated and validated. In a limited case, the solution compares favourably with existing literature, while the novel comparative investigation predicts the dominance of the Temperature-Dependent Heat Source (TDHS) capable of weakening the fluid cohesive force and upsurging the energy rate in short time in contrast to the Space-Dependent Heat Source (SDHS) module. The material relaxation phenomenon indicated that more time will be needed for prosperous/equal heat energy transfer. Further heat transfer enhancement is called for higher discharge of radiation and boosting of thermal conductivity, while TDHS module is strongly recommended in highly required heat systems. [ABSTRACT FROM AUTHOR]

Present work discusses the effects of channel wall thickness on MHD flow employing permeability effect on porous wall liner. Couette steady Carreau fluid flow in a parallel inclined plate's channel at an inclination λ to the horizontal. We also examined how tilted magnetic fields affect channel flow with slope angle β. We employ slip boundary conditions. Convective boundary conditions affect heat transmission. An incompressible Carreau fluid's governing system equations are in Cartesian coordinates. Low Reynolds number and negligible inertial effects progress the problem formulation. Similarity of nondimensional quantities converts flow controlling equations into conventional differentiation equations. Series solutions for axial velocity are obtained analytically by regular perturbation technique method (RPTM) when Weissenberg number is small, while the energy equation is solved numerically by "Explicit Euler" method with size step of (0.1). Plotting graphs shows the effects of Hartmann number, Darcy number, channel and magnetic field inclination angles, and others on velocity and temperature distributions by using Mathematica program software. [ABSTRACT FROM AUTHOR]

This study aims to investigate the impact of suction/injection on a quadratic mixed convection magnetohydrodynamics Carreau fluid over a stretching cylinder. The effects of heat generation, Joule heating, convective boundary conditions, and variable thermal conductivity in the fluid region are considered. The developed partial governing equations are transformed into ordinary differential equations using the similarity transformation technique and then solved using the homotopy analysis method to determine the analytic approximation solutions. Due to the numerous pertinent parameters, the variations of the heat and mass distribution, as well as the physical quantities, such as the local skin friction and the Sherwood and Nusselt number, are analyzed through graphs and tables. Quadratic convection has enhanced the movement of the fluid, heat transport, and mass transfer in a situation without inflow either at the surface of the cylinder or flat plate. The mass transition phenomenon has reduced the impact of quadratic convection. Furthermore, in the suction situation, the concentration and velocity of the fluid field are increased but an opposite tendency in the temperature distribution is observed. The present analysis is essential in controlling the quality of wire coating, pipe coating, or heat exchangers, which depends on the fluid characteristics, as well as its concentration and thermal diffusion. [ABSTRACT FROM AUTHOR]

This paper reports the hydromagnetic mixed convective flow of Carreau nanofluid in the vertical microchannel with slip and convective mechanisms at the boundaries. The main objective of this work is to analyze the conduct of Carreau nanofluid under Buongiorno model as the nano-dimensions of the particle consequently encounters random motion. The novel impression is to retain the outcomes of the flow in regard of Brownian dispersion and thermophoresis. Parameters such as buoyancy ratio, Brownian motion and thermophoresis along with the effect of Weissenberg number are discussed on distribution of flow, temperature and concentration by using graphical demonstration. Surface drag coefficients, transport rates of mass and heat are portrayed by numeric esteems. Boundary layer approximations are made use to tackle nonlinear ordinary differential equations and solved by Runge–Kutta–Fehlberg 4-5th-order method. Results so obtained elaborate that velocity depletes with increased values of buoyancy ratio and viscosity ratio parameter along with power law index. Also, distinct behavior is noticed in the profile of velocity for varying Weissenberg number by boosting and diminishing depending on the power law index. On the other hand, temperature rises by augmentation of thermophoresis and Brownian motion parameter. Elevation in Brownian motion parameter accelerate the concentration panel of the fluid. Skin friction and Nusselt number are in direct proportion with buoyancy ratio parameter and Brownian motion parameter, respectively. [ABSTRACT FROM AUTHOR]

The influence of radiation, convective heating, second-order velocity slip condition, and activating energy on the magnetohydrodynamic (MHD) Carreau nanofluid and heat/mass transfer are depicted in this study. A mathematical model of the Carreau nanofluid is carried out with the effects of thermophoresis and Brownian motion. The governing equations are transformed into a system of nonlinear coupled differential equations. Solutions of the transformed equations are derived by the Matlab BVP4c solver. Efficiency of the solver is verified by the comparison with the previous literature. Moreover, the impacts of the dimensionless velocity, temperature, and nanoparticle concentration profile are elucidated graphically. [ABSTRACT FROM AUTHOR]

The primary objective of this article is on examining the impacts of mixed convection flow of Carreau fluid past a permeable stretched surface with Soret and Dufour effects occurring. For the flow analysis, the Darcy–Forchheimer porosity medium is taken into account. In order to explore the transference analysis of heat and mass rate, the impressions of thermal radiation, viscous dissipation, heat generation, Arrhenius activation energy, convective heat, and mass conditions impacts are taken into consideration. The flow equations initially exist as PDEs, and we then employ appropriate similarity transformations to change them into ODEs. The R–K 4th order strategy based on the shooting approach is used to numerically solve these ODEs. Through various emergent variables, the numerical results for the velocity, temperature, and concentration fields are shown. For a variety of distinct variables, the physical quantities such as friction factor, local Nusselt number, and local Sherwood numbers are shown. It is noted that the velocity field is reduced by the magnetic field, but the Weissenberg number buoyancy ratio parameter exhibits the opposite tendency. Further, it is noticed that temperature and concentration distributions have an improving tendency for the Soret and Dufour parameter. [ABSTRACT FROM AUTHOR]

The main focus of the current work is to explore the thermal energy and mass transfer process in conducting non-Newtonian fluid flows through a horizontal microchannel along with irreversibility analysis and to investigate the upshot of nonlinear radiation, convective boundary conditions, and magnetic field. Slip regime and temperature jump conditions are used at the boundaries of the microchannel. The rheological characteristics of the Carreau fluid model are also considered. The Buongiorno model was used in this study, which emphasizes the light on Brownian motion and thermophoresis phenomena that occur during the fluid flow. Runge-Kutta Fehlberg's fourth-fifth order was used to solve governing equations numerically. The significant influence of effective parameters on entropy generation, Bejan number, velocity, temperature and concentration profile have been discussed with the aid of graphs. Entropy generation rises by 40% at the left wall of the channel and is depleted by 36% at the right wall when the thermophoresis parameter augments by 200%. Amplified concentration profile with an escalation of Brownian parameter. The temperature field diminishes with inflation of the radiation parameter. [ABSTRACT FROM AUTHOR]

The characteristics of dual solutions of the flow of Casson or Carreau fluids mixed with microorganisms over a permeable shrinking surface are investigated considering the effects of the magnetic field, heat dissipation, Dufour, and thermophoretic diffusivity. By using similarity transformations, the governing equations are converted into a set of ordinary differential equations. These reduced equations are then solved by the well-known shooting technique along with the fourthorder Runge-Kutta method. A comparison is provided and the present solutions hold a good agreement with available published results. The local skin friction coefficient, local Nusselt number, local Sherwood number, and local density number of the microorganisms are found to increase with the increase of non-Newtonian Casson parameter, Dufour, and thermophoretic diffusivity number, however, it decreases with Weissenberg number. The remarkable finding which is not revealed yet is that dual solutions exist in the flow of Casson or Carreau fluids with microorganisms over a permeable shrinking sheet subject to a certain combination of parameters. Moreover, the domain of the occurrence of dual solutions broadens on account of higher values of all physical parameters of the problem except the Eckert number. [ABSTRACT FROM AUTHOR]

In this study, we investigate blood flow in a small artery with a constriction using gold nanoparticles (Au) in the presence of microorganisms, mass, and heat transfer. The non-Newtonian behavior of blood fluid in slight arteries is quantitatively inspected by simulating blood flow using the Carreau fluid model. Momentum equations incorporating magnetohydrodynamics (MHD) and Darcy–Forchheimer porous media are used to model the fluid flow. Heat transfer properties, including thermal radiation, joule dissipation, and bio-convective microorganisms, are investigated. Blood serves as the base fluid for the nanofluid, which contains gold nanoparticles. The system's nonlinear partial differential equations are transformed into nonlinear ODEs through suitable transformations. To obtain numerical solutions for these ODEs, the homotopy analysis method is used. The physical implications of flow restrictions are compared with fictitious fluid flow using physical interpretations. Additionally, investigations into the interpretations of blood flow based on drag force and heat transfer are being conducted. ANOVA, or analysis of variance, is a dependable statistical tool used to evaluate regression models and a variety of statistical tests. These investigations include error assessments, total error evaluations, F-values, p-values, and model fit assessments. These statistical investigations were applied to the dataset at hand, with the goal of achieving a robust 95% level of confidence. We investigate the effects of minute adjustments in parameters on both the heat transfer rate and the friction factor rate using these analyses. The study intends to dive deeper into the potential effects of minor changes in one or more factors on the overall effectiveness of surface friction rate and the larger domain of thermal energy transfer. This will be performed by employing sensitivity analysis approaches. This strategy allows us to obtain a better understanding of how minor changes to specific parameters might affect the speed of thermal energy conveyance and fluid flow management. Furthermore, it lays the framework for future studies aimed at optimising system designs. Article highlights: We examined blood based MHD Au-nanofluid flow in the presence of microorganisms applying Carreau fluid model. To model the nanofluid flow we used, Darcy-Forchheimer porous media and heat transfer properties. Analysis of variance is a dependable statistical tool is used for the finding of regression models and a variety of statistical tests. [ABSTRACT FROM AUTHOR]

The two-dimensional boundary layer of steady mixed convective magnetohydrodynamic Carreau dusty fluid flow across a stretched sheet was investigated in this study. The primary focus of this study is to examine the impact of Carreau nanofluid on dusty flow dynamics in the presence of the Lorentz force. The partial differential equations governing the fluid flow, temperature, and concentration fields for both the fluid and dust phases are transformed into ordinary differential equations using an appropriate set of similarity transformations. Numerical outcomes are acquired employing the Runge–Kutta technique combined with the shooting method, implemented on the MATLAB platform. The numerical outcomes are compared to previous research and confirmed to be in very good accord. Finally, the impacts of relevant factors of physical and technical importance on flow and thermal transport characteristics are visually and tabulated. The results for the linear density and the quadratic density profile, it is noted that the linear density decreases compared to the quadratic density for velocity, but the opposite behavior is observed for temperature profile. The advancement of dusty nanofluids has resulted in significant enhancements in the heat transfer mechanism, which finds application in manufacturing, industry, and nanotechnology research. Research on dusty flows benefits on is beneficial in air pollution studies, dust entrainment in a cloud formed during a nuclear, vehicle emissions of smoke and other pollutants, crude oil purification, petroleum production, combustion, industrial effluent emissions, capillary blood flow, paint sparing, and, more.

Blood flow analysis plays an essential role in understanding cardiovascular health and disease. An accurate mathematical modeling is necessary in context of representation of arterial domains and blood flow to learn about the intricate blood dynamics. In this paper we concentrate on the mathematical representation of the geometry of the blood artery and find numerical solutions with mild to severe degree of stenosis. Blood dynamics has steady state, in-compressible and laminar nature. The shear thinning non-viscous effects of blood are assumed to obey Carreau model. Fluid flow is based on coupled Navier-Stokes equation in three dimensions and is simulated with finite volume approach over hexahedral elements. Application of a constant mass flow rate at the inlet of the artery results in a number of observations to analyze. It is found that with increasing the severity of stenosis, blood velocity becomes higher in stenotic region, pressure drop along the arterial domain is increased and shear stresses exerted by blood on walls of artery are increased.

In this review, a quantitative research survey has been carried out on the peristaltic motion of Carreau fluid (CF), one of the most important generalized Newtonian fluids. This work primarily focuses on the fluid's diverse behavior as a result of the application of various methods and the broad overview of peristaltic Carreau fluids in a range of impacts such as Heat and Mass Transfer, Magnetohydrodynamic (MHD), stagnation point, stretching and shrinking sheet, porous medium and boundary layer. The various approaches that have been explored to depict this exquisite physiological transport system, which is frequently seen in nature, are summarized in this paper. The extensive study of biofluid peristaltic pumping pertaining to its importance in various fields, perspectives, and mathematical models is also discussed in depth. This analysis compares the major components and research results over the last 50 years and offers recommendations for future advancement. Researchers can use this analysis as a platform for building peristaltic transport for various applications.

Abstract In this study, we investigate blood flow in a small artery with a constriction using gold nanoparticles (Au) in the presence of microorganisms, mass, and heat transfer. The non-Newtonian behavior of blood fluid in slight arteries is quantitatively inspected by simulating blood flow using the Carreau fluid model. Momentum equations incorporating magnetohydrodynamics (MHD) and Darcy–Forchheimer porous media are used to model the fluid flow. Heat transfer properties, including thermal radiation, joule dissipation, and bio-convective microorganisms, are investigated. Blood serves as the base fluid for the nanofluid, which contains gold nanoparticles. The system's nonlinear partial differential equations are transformed into nonlinear ODEs through suitable transformations. To obtain numerical solutions for these ODEs, the homotopy analysis method is used. The physical implications of flow restrictions are compared with fictitious fluid flow using physical interpretations. Additionally, investigations into the interpretations of blood flow based on drag force and heat transfer are being conducted. ANOVA, or analysis of variance, is a dependable statistical tool used to evaluate regression models and a variety of statistical tests. These investigations include error assessments, total error evaluations, F-values, p-values, and model fit assessments. These statistical investigations were applied to the dataset at hand, with the goal of achieving a robust 95% level of confidence. We investigate the effects of minute adjustments in parameters on both the heat transfer rate and the friction factor rate using these analyses. The study intends to dive deeper into the potential effects of minor changes in one or more factors on the overall effectiveness of surface friction rate and the larger domain of thermal energy transfer. This will be performed by employing sensitivity analysis approaches. This strategy allows us to obtain a better understanding of how minor changes to specific parameters might affect the speed of thermal energy conveyance and fluid flow management. Furthermore, it lays the framework for future studies aimed at optimising system designs. Article highlights We examined blood based MHD Au-nanofluid flow in the presence of microorganisms applying Carreau fluid model. To model the nanofluid flow we used, Darcy-Forchheimer porous media and heat transfer properties. Analysis of variance is a dependable statistical tool is used for the finding of regression models and a variety of statistical tests.

This research presents a numerical investigation for flow around a stagnation point on a vertical stretching cylinder placed in a pseudo-plastic fluid. An important Carreau fluid model is adopted to account for pseudo-plastic effects in the flow field. Buoyancy force term arising due to the vertical boundary is formulated under the well-known Oberbeck-Boussinesq approximation. Conservation equations simplified under boundary layer assumptions are solved for the similarity solutions for full range of parameter A, giving the ratio of free stream velocity to the cylinder surface velocity. The shape of velocity profile is dependent on the choice of the parameter A. Contributions of buoyancy force term and shear-thinning effect in both assisting and opposing flow situations are scrutinized. A striking influence of rheology is such that resisting wall shear declines and cooling rate of the surface drops whenever shear-thinning effect is present. However, cooling rate amplifies as the strength of buoyancy force is enhanced. Special cases of the model including the case of flat surface and Newtonian fluid are presented separately.

The current article communicates a numerical investigation on laminar flow of dissipative generalized Newtonian Carreau nanofluid flowing through vertical conduit with converging and diverging plane walls. Thermal and concentration characteristics due to enthalpy change, activation energy, and non-linear thermal radiation have been examined in the presence of buoyancy forces. The channel walls for both temperature and volumetric fraction are assumed to be isothermal. The instability mechanism of nanofluids is reported using a two-phase nanofluid model, which works reasonably well for nanoparticle concentrations below a certain threshold. A Jeffery-Hamel (J-H) flow model is developed by assuming an incompressible purely radial flow of Carreau nanofluids with heat and mass transportation. Using the suitable non-dimensional variables, the resulting nonlinear partial differential equations are turned into a system of ordinary differential equations. The modified governing equations are then numerically solved using the built-in boundary value problem solver bvp4c, on the template form of commercial software MATLAB. The impacts of material, geometrical and thermophysical parameters governing the J-H problem are discussed and illustrated. Results indicate that higher buoyance forces incline the velocity profiles in converging enclosure, while a slight reduction is perceived in opposing forces. A significant decrease of wall heat transmission is reflected for larger values of activation energy and radiation parameter. For endorsing this communication, a comparison analysis is established with existing research and noticed a remarkable agreement. Practically, the flow inside converging and diverging channels are deployed in nuclear reactors that use plate-type nuclear energies, high heat-flux condensed heat exchangers, high-performance micro-electronic cooling systems, jets, rockets nozzles, and jet propulsion inlet.

The authors are attracted to conduct this study because it has a major dependence on scientific, medical, and engineering works, including heat exchangers, geothermal systems, reactive flow in chemical reactors, blood flow in arteries, groundwater flow, enhanced oil recovery, cooling systems in aircraft, and nanofluidic devices. In this study, the problem deals with the two-dimensional incompressible and boundary layer natural convective flow of Carreau fluid flowing on a surface with variable thickness. The thermophysical properties like viscosity, thermal conductivity, and thermal diffusivity are chosen to be variable. For the determination of solutal and thermal analysis, we have considered the enthalpy change along with activation energy, heat generation, Soret, and Dufour effects. The model equations occur in PDEs form and then converted into ODEs via similarity transformations. The numerical outcomes of ODEs are obtained by using a multistep method named the Adams-Bashforth method with the Predictor and Corrector technique in MATLAB software. The velocity of fluid is a decreasing function of the Weissenberg number, the viscosity coefficient, and both the Grashof numbers of thermal and solutal transport. The temperature field is a growing function of the Damkholer number, thermal conduction coefficient, and heat generation parameter, whereas declining behavior is noted due to the Dufour number and thermal difference coefficient. The concentration field rises with the increasing inputs of the activation coefficient and Damkholer number, but a decline in concentration is noted due to the Soret number. The numerical determination of skin friction, the Nusselt number, and the Sherwood number are also discussed here, and these numerical outcomes are compared with Bvp4c and Bvp5c built-in commands on Matlab software.

The formulation of heat transfer in non-Newtonian fluid models remains a topic of great interest for researchers. The ultimate flow differential equations in this direction are non-linear and hence difficult to solve analytically. Therefore, we offer state of the art determination of film effectiveness (heat transfer coefficient) by using artificial intelligence (AI). To be more specific, the Carreau fluid model is formulated at a flat surface along with thermal slip, heat generation, velocity slip, chemical reaction, and magnetohydrodynamics (MHD) effects. The developed equations are reduced by the application of the Lie symmetry approach and solved by the shooting method. The artificial neural networking model is built by using 132 sample values of the Nusselt number (NN) as an output. Porosity parameter, Prandtl number, magnetic field parameter, and heat production parameter are all inputs. 92 (70 %) is designated for training, whereas 20 (15 %) is designated for validation and testing. The Levenberg-Marquardt algorithm is used to train the neural network. The constructed artificial neural networking (ANN) is best to predict the NN at a flat surface. It is observed that for large Prandtl numbers, the magnitude of the Nusselt number is greater for porous surface, but the converse is true for magnetic field parameter.

This study numerically investigates the steady two-dimensional flow of a Carreau nanofluid with activation energy and motile microorganisms over a plate, wedge and stagnation point. The effects of the magnetic field, the Brownian motion, the viscous dissipation and the thermophoresis examined for both shear-thinning and the shear-thickening fluids. The similarity transformations are implemented to convert the governing equations into a non-dimensional form for easier analysis. The Runge–Kutta Method and the shooting technique are employed for finding numerical solutions using MATLAB plot form. The obtained numerical results were analyzed for a broad spectrum of dimensionless parameters and are discussed through graphs. These encompass 0.1≤M≤0.4,0.2≤We≤0.8,0.1≤S≤0.4,0.1≤Ec≤0.4,5≤Sc≤20,0.01≤R≤0.04,1≤Nb≤4,0.1≤Nt≤0.4,0.1≤Pe≤0.4,0.01≤Pr≤2and 1≤Ec≤2.5. These ranges were explored concerning velocity, temperature, concentration, diffusion, wall frictional factor and heat transfer rate, both through numerical computations and graphical representations. A decreasing trend in profiles is observed except for velocity, as the magnetic field parameter increases. The flow over a plate exhibits lower skin friction, heat transfer, mass transfer and gyrotactic microorganism compared to other geometries. The Brownian motion leads to a decreased nanoparticle concentration and motile microorganism density, while increasing thermophoresis has the opposite effect. The suction/injection parameter increases fluid velocity but decreases the temperature, the concentration and the motile microorganism density. The shear-thinning nanofluids demonstrate higher rates of the heat transfer, the mass transfer and the motile microorganism compared to shear-thickening fluids. Furthermore, the present analysis demonstrates that as the Peclet number and bioconvective Schmidt number increase, there is a corresponding decrease in microorganism concentration. Additionally, the higher activation energy E is found to enhance the concentration field due to the reduced chemical reaction rates.

The use of heat transfers in heat exchangers, nuclear cooling, solar collectors, and electrical devices is crucial nowadays. Hybrid nanofluid can be used in these applications to get the best results because it encourages efficient heat transmission compared to conventional fluid. The effect of radiation and an unsteady Carreau hybrid fluid in addition to non-Fourier heat flux over a shrinking sheet is discussed in this paper. Using similarity transformations, the newly developed system of partial differential equations (PDEs) is converted into a set of ordinary differential equations (ODEs), that are next numerically addressed by utilizing shooting approach and the Runge–Kutta (RK) method. Tables and graphs are used to present utilizing the computational findings for nondimensional temperature, velocity, heat transfer rate, and friction between fluid and hybrid nanoparticles. Additionally, for certain physical factors, the physical quantities in numerical values were also presented (such as the friction factor and local Nusselt number). We made a restricted case comparison between the current findings and the body of prior research. Our finding confirms that the temperature profile is strengthened by the heat generation parameter and the effect of radiation. The porosity parameter decelerates the momentum boundary layer thickness near the plate. [ABSTRACT FROM AUTHOR]

In this review, a quantitative research survey has been carried out on the peristaltic motion of Carreau fluid (CF), one of the most important generalized Newtonian fluids. This work primarily focuses on the fluid's diverse behavior as a result of the application of various methods and the broad overview of peristaltic Carreau fluids in a range of impacts such as Heat and Mass Transfer, Magnetohydrodynamic (MHD), stagnation point, stretching and shrinking sheet, porous medium and boundary layer. The various approaches that have been explored to depict this exquisite physiological transport system, which is frequently seen in nature, are summarized in this paper. The extensive study of biofluid peristaltic pumping pertaining to its importance in various fields, perspectives, and mathematical models is also discussed in depth. This analysis compares the major components and research results over the last 50 years and offers recommendations for future advancement. Researchers can use this analysis as a platform for building peristaltic transport for various applications. [ABSTRACT FROM AUTHOR]

The use of nanofluid technology as a working fluid in specific channels has significantly improved heat exchange rate in thermal engineering applications. In fact, nonuniform channels with converging or diverging nature offer special characteristics that encourage the most favorable usage, such as flow via nozzles, diffusers, and reducers in polymer processing and flow through canals and rivers. Therefore, we present a theoretical and mathematical insight into nanofluid heat transfer and flow dynamics in nonuniform channels utilizing non-Newtonian nanofluids. The impacts of heat absorption/generation and Joule heating in a magnetohydrodynamics (MHD) flow of Carreau nanofluids with viscous dissipation are addressed in this research. Brownian and thermophoresis diffusion are deliberated to investigate heat and mass transport mechanisms. The modeled flow equations are numerically handled using a built-in MATLAB code based on the finite difference approach (bvp4c) for velocity, temperature, and concentration fields. The consequences of the variable magnetic field, heat absorption/generation, and numerous physical parameters on the temperature and concentration field are surveyed. The results indicate that the increasing values of heat generation/absorption parameter and Eckert number significantly enhances the temperature profiles in convergent channel. In addition, a similar behavior is observed for higher Prandtl number. An assessment of solutions achieved in this article is made with existing data in the literature. [ABSTRACT FROM AUTHOR]

Recently, the scientists have been giving attention to the mutual pattern of heat and mass transportation such that both phenomena endorsed the novelty in engineering and industrial equipment's approximating polymer progressions, fridge-freezers, plastics extrusion, fissile vessels, irrigation of arithmetical kit and power generation, heat exchangers. The Fourier and Fick laws are exploited to describe exactly how the heat and mass transportation via a space with influence of temperature and concentration will change, respectively. Furthermore, the worth of thermo-solutal transport has increased extremely in numerous industrial and scientific progresses, for instance, nutrition processing, utensils power aerials, crop mutilation, thermal transmission in fleshy tissue and numerous others. The theme of this effort is to explore the theory of thermo-solutal relaxation times (Cattaneo–Christov double diffusion) and thermo-solutal stratifications in Carreau fluid with the properties of magnetic field. Additionally, the thermal phenomenon of variable conductivity and mechanism of solutal diffusivity had been studied. The bvp4c algorithm has been exploited for the solution process. It is remarkable to note that both velocity fields decay for magnetic factors. The higher estimations of variable conductivity and thermal stratification factors have contradicting performance on temperature field. Additionally, mass stratification depreciates the concentration of Carreau fluid. The coefficient of skin frictions is enhanced for magnetic factor. The comparative tables compared with past studies with worthy settlement have also been shown. [ABSTRACT FROM AUTHOR]

Non‐Newtonian fluids are significantly better on the heat transfer phenomenon than from Newtonian fluids, so they are helpful in air conditioner and heating and cooling devices. Blood, toothpaste, ketchup, paint, and so on are common non‐Newtonian fluids in our daily lives. Among other non‐Newtonian fluid models, Carreau fluid has gained much acceptance because of its unique features of shear thinning and shear thickening for various power‐law index ranges. Owing to the significance of Carreau fluid, a theoretical analysis has been performed here to investigate the steady three‐dimensional flow of Carreau fluid between two coaxially rotatory and stretching disks. The fluid is electrically conducting, and a uniform magnetic field is applied perpendicularly to the circulating disks. The heat transfer phenomenon is also analyzed in the presence of nonlinear thermal radiation, viscous dissipation, joule heating, and nonuniform heat source/sink subject to convective boundary conditions. With specific similarity transformation, the leading partial differential equations are diminished to ordinary differential equations and then tackled numerically through the Keller–Box scheme. It is concluded that with the increase in the disks rotation rate, the fluid velocities enhances whereas plummets for the magnetic parameter and predicts two diverse behaviors for Weissenberg number. The temperature escalates for the magnetic parameter, radiation parameter, temperature ratio parameter, Eckert number, heat source/sink parameters, and lower disk Biot number, but the temperature decays for rotation parameter, Prandtl number, and upper disk Biot number. [ABSTRACT FROM AUTHOR]

This work theoretically investigates the interfacial transport of immiscible fluid layers in an inclined fluidic channel in the presence of magnetofluidic actuation. Immiscible binary system consists of both non-Newtonian fluid (top layer) and Newtonian fluid (bottom layer), while the Carreau fluid model is used to describe the rheology of non-Newtonian layer. We develop a theoretical framework consistent with the homotopy analysis method (HAM) to obtain the approximate analytical solutions for the underlying thermofluidic transport features. By depicting the auxiliary parameter curve (⁠ℏ -curve) of flow velocity and temperature distribution, we ascertain the effective reliability of the theoretical method developed here. We demonstrate both velocity and temperature variations in the channel for a set of involving parameters pertinent to this analysis. Albeit the flow configuration considered in this analysis is not complex, yet, the method developed here seems to be efficient in capturing underling transport features, retaining the simultaneously acted implications of fluid rheology and magnetohydrodynamics. Form the consistency observed in predicting the flow velocity for any values of shear-thinning parameter, including nonintegers, our semi-analytical method is deemed pertinent to predict the thermohydrodynamics of immiscible multilayer system even by accurately capturing the intervening effects of fluid rheology and applied fields. [ABSTRACT FROM AUTHOR]

The flow of Carreau nanofluid with generation/absorption and magnetic field can be valuable for modifying solar energy production. In this work, we extended the study of the MHD boundary layer flow of Carreau nanofluid with heat generation/absorption close to a stagnation point over the radially extending plate. Likewise, the features of radiation and magnetic field with convective boundary conditions are considered. Further, keeping in view the importance of chemical reactions, their effect is also incorporated during the modelling process. For motivation, the impact of thermophoresis and Brownian motion has been taken into account. Using appropriate similarity transformation, we converted nonlinear governing partial differential equations of Carreau nanofluid into a couple of nonlinear ODEs. Using a recognized shooting technique and the MATLAB bvp4c solver and Mathematica ND-solve built-in command, we were able to get numerical results for these modeled ODEs. Through graphs and tables, the effects of different physical parameters like magnetic, Weissenberg number, Brownian motion, thermal radiation, thermophoresis, Prandtl number, chemical reaction, and rate of heat generation/absorption on non-dimensional velocity, temperature, and concentration profile are discussed.

The purpose of this study is to investigate the effect of an elastic wall on the peristaltic flow of Carreau fluid between two concentric cylinders, where the inner tube is cylindrical with an inelastic wall and the outer wall is a regular elastic sine wave. For this problem, cylindrical coordinates were used, and a short wavelength “relative to channel width for its length,” as well as the governing equations of Carreau fluid in the Navier–Stokes equations. Then, the analytical solution has been investigated by using the regular perturbation technique. The solutions obtained by this perturbation are up to the fourth-order in dimensionless Weissenberg number (We{W}_{{\rm{e}}}). The performed computations of various parameter values such as velocity, shear stress, and wave frame streamlines are discussed in detail for different values of the Weissenberg number (We{W}_{{\rm{e}}}). The obtained results demonstrate that the fluid velocity increases with the increase in the value of We{W}_{{\rm{e}}} and some features of the wall, while the opposite behavior is observed with the increase in other features of the wall. Hence, the presented numerical analysis reveals many aspects of the flow by considering a non-Newtonian Carreau fluid model, and the presented model can be equally applicable to other bio-mathematical studies. The results were evaluated using the Mathematica software program. The Mathematica program was used by entering various data for the parameters, where the program showed the graphs, then the effect of these parameters became clear and the results were mentioned in the conclusion.

Nanomaterial flow has fascinated the concern of scientists across the globe due to its innovative applications in various manufacturing, industrial, and engineering domains. Bearing aforementioned uses in mind, the focal point of this study is to examine the Carreau nanofluid flow configured by the Riga surface with Arrhenius catalysts. Microorganisms are also suspended in nanofluid to strengthen the density of the regular fluid. Time-dependent coupled partial differential equations that represent the flow dynamics are modified into dimensionless patterns via appropriate non-dimensional variables, and handled through an explicit finite difference approach with stability appraisal. The performances of multiple flow variables are examined graphically and numerically. Representation of 3D surface and contour plots for heat transportation and entropy generation are also epitomized. The findings express that the modified Hartmann number strengthens the motion of nanomaterial. Reverse outcomes for heat transport rate and entropy are seen for the radiation variable. Concentration diminishes for chemical reaction variable. Activation energy enhances the concentration of nanomaterial, whereas reduction happens in the movement of microbes for bio-Lewis number. Greater Brinkman variable heightens the entropy.

This paper mainly analyzes the heat transfer of Carreau fluid over an infinite rotating disk, and two models with variable thermal conductivity are considered. Firstly, the governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by similarity transformation. Then, the boundary value problem is solved by improved bvp4c method, which reduced sensitivity to the initial values. For shear-thinning and shear-thickening fluids, the effects of Carreau fluid index n and Prandtl number Pr on velocity and temperature fields are shown and analyzed when 0.5 ≤ n ≤ 1.5 , 1 ≤ P r ≤ 10. Furthermore, the thermal conductivity is computed under two cases. [ABSTRACT FROM AUTHOR]

Despite the wide application of combustion in reactive materials, one of the challenges faced globally is the auto-ignition of such materials, resulting in fire and explosion hazards. To avoid this unfortunate occurrence, a mathematical model describing the thermal decomposition of combustible polymer material in a rectangular stockpile is formulated. A nonlinear momentum equation is provided with the assumption that the combustible polymer follows a Carreau constitutive relation. The chemical reaction of the polymer material is assumed to be exothermic; therefore, Arrhenius's kinetic theory is considered in the energy balance equation. The bivariate spectral local linearization scheme (BSLLS) is utilized to provide a numerical solution for the dimensionless equations governing the problem. The obtained results are validated by the collocation weighted residual method (CWRM), and a good agreement is achieved. Dimensionless velocity, temperature, and thermal stability results are presented and explained comprehensively with suitable applications. Some of the obtained results show that thermal criticality increases with increasing power law index (n) and radiation (R a) values and decreases with increasing variable viscosity ( β 1 ) and material parameter ( W e ) values. [ABSTRACT FROM AUTHOR]