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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

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.

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).

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.

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]

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.

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 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]

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. [ABSTRACT FROM AUTHOR]

Understanding fluid flow, heat, and mass transfer over a stretching cylinder is essential in ascertaining the quality of wire coating and coper thinning. This study looks at the influence of the heat source, thermal radiation, chemical reaction, and natural convection of Carreau fluid flows over a vertical stretching cylinder immersed in a porous medium. Suitable similarity variables are applied to convert the partial governing equations arising in fluid flows, heat, and mass transfer into ordinary differential equations. The optimal homotopy analysis method is then utilized to solve the transformed highly nonlinear governing equations. The impacts of the relevant parameters such as the Weissenberg number, porosity, heat source parameter, radiative number, chemical reaction parameter, mixed convection parameter, and curvature parameter on the dimensionless velocity, temperature, and concentration distribution as well as for the skin friction, Nusselt number, and Sherwood number are discussed through graphs and tables. It is observed that the velocity shows an opposite behavior as compared to temperature, and concentration in shear-thinning, n

Due to their capacity to create better thermal conductivity than standard nanofluids, hybrid nano-fluids and modified nanofluids have notable applications in aerospace, energy materials, thermal sensors, antifouling, etc. This study aims to the modified and hybrid nanofluid flow with the Carreau fluid over a sloped shrinking sheet. The Cattaneo–Christov heat flux also takes into account. To determine the thermal efficiency of the heat, three different kinds of nanomaterials, copper oxide ( C u O) , copper (C u) , and alumina ( A l 2 O 3 ), are used. The similarity alteration commutes the insolubility of the model into ODEs. The conclusions are attained by program writing in MATLAB software and dealing with them through the bvp4c solver with the shooting method. The skin-friction amount decreases with the inclined sheet and local Weissenberg parameter for both modified and hybrid nanofluid. An upsurge thermal relaxation parameter declines the skin-friction coefficient for modified nanofluid flow and increases the skin-friction coefficient for hybrid nanofluid flow. The heat transfer rate is upsurged with modified and hybrid nanofluid for thermal relaxation parameter. Furthermore, the presentation includes the development of skin friction coefficient and Nusselt number values for specific parameters. Through benchmarking, numerical solutions are validated using certain limiting situations that were previously published findings, and typically solid correlation is shown. [ABSTRACT FROM AUTHOR]

The effect of a catheter on blood flow and heat transfer characteristics of a Carreau fluid model is investigated mathematically using the perturbation method involving appropriate small parameters. Physically, this study relates to the surgical technique when a catheter is injected eccentrically into an artery. The model describes fluid flowing in the gap between eccentric tubes where the inner tube is uniform and rigid representing the moving catheter while the other tube represents an artery with overlapping stenosis. A complete perturbation solution up to the first order is presented whereby a validation exercise carried out shows that previously published solutions involving Newtonian fluids may have some typographical errors. The present results obtained show that the axial velocity, wall shear stress, and temperature in the case of the eccentric catheter are higher than that of the concentric catheter. The average Nusselt number is enhanced with increasing catheter radius and the velocity of the catheter while it decreases with increased Weissenberg number, Prandtl number, and Eckert number. These results agree with physiological observations that the risks and complications associated with catheterization are alleviated when the eccentric position of the catheter is considered.

This study is aimed to explore the magneto-hydrodynamic Carreau fluid flow over a stretching/shrinking surface with a convectively heated boundary. Temperature-dependent variable thermophysical properties are utilized to formulate the problem. The flow governing equations are obtained with boundary layer approximation and constitutive relation of the Carreau fluid. The shooting method is utilized to obtain graphical and numeric outcomes. Additionally, initial guesses are generated with the help of Newton's method. The effect of Weissenberg number, Magnetization, stretching ratio, Prandtl number, suction/blowing parameter, and Lewis number is obtained on velocity, temperature and species continuity profile and analyzed. Shear stress rates and Nusselt number outcomes under body forces influences are present in tabulated data and discussed. It is observed that in absence of magnetization force, B = 0 and strong mass suction 5 ≤ S ≤ 7.5 effect high rates of Nusselt number is obtained. It is concluded that under the influence of power law index and non-linearity parameter maximum heat transfer and reduced shear stress rates are obtained. [ABSTRACT FROM AUTHOR]

The unsteady flow of a Carreau fluid over a coated disk under the simultaneous effects of a thermal and concentration field with buoyancy forces is reported. The time-dependent diffusive stream of a Carreau fluid over a conducting coated disk is carried out with energy loss. The time-dependent partial differential equations are first converted into a scheme of ordinary differential equations by the appropriate transformations and are then solved by shooting method. Significant results for speed, hotness and concentration profiles are revealed and deliberated by the graphical outcomes. The numerical values of skin friction suggest that the viscoelastic parameter of the Carreau fluid causes a reduction in the skin friction coefficient due to the coated surface, but the Nusselt and Sherwood numbers increase with the rise of the viscoelastic parameter of the Carreau fluid because of the coated surface. The present model is useful in the field of mechanical engineering to design a tesla turbine for the flow of viscous fluid. [ABSTRACT FROM AUTHOR]

The 2-D steady flow of Carreau fluid by considering the effects of magneto-hydro-dynamic, viscous dissipation and activation energy on the surface of parabola is investigated numerically and graphically in this study. The motion, temperature and mass transfer of fluid is examined by considering the variable thermo-physical properties. The upper surface of a submarine, aircraft, bullet and car's bonnet are certain examples of the upper paraboloid surface. The motion of fluid over this surface is depending on the boundary layer which is made in the instant space on it. At the surface of object, the layers of fluid are presumed as stretchable. The independent variables of governing equations are reduced by suitable dimensionless physical variables. The numerical results of dimensionless determining equations are achieved by adopting the BVP4c method through MATLAB software. The graphs are conspired to understand the conduct of numerous parameters on velocity, temperature and concentration fields. It is analyzed that the velocity field is increased function of power law and viscosity coefficient. The decline in the temperature within the flow happens for Prandtl number whereas temperature rises for heat source parameter. The activation energy caused the increment in the concentration field.

We scrutinized the influence of nonlinear heat radiation on heat transmission evaluation of Carreau nanofluid and tangent hyperbolic nanofluid streams across a wedge with gyrotactic microorganisms by taking slip situations into consideration in this research article. The necessary nonlinear partial differential formulation is transmuted into non-linear ordinary differential equations by employing appropriate similarity variables, and these equations, including the boundary constraints are resolved in Matlab software utilizing Runge-Kutta fourth order via shooting tactic. A definite description of the framework is achieved by fluctuating the inputs of influential variables of the dependent functions and exhibited via graphs. The inhibiting flow velocity is portrayed by the intensifying inputs of buoyancy ratio, magnetic force, Rayleigh number, and eigenvalue. As a consequence of thermophoresis and Brownian motion of nano-particles, the temperature of the liquids initiates to ascend instantly. Because of differentiated viscous effects, the flow velocity for Carreau nanofluid is slower than that of tangent hyperbolic fluid and the temperature behavior is reversed. Further, the magnitude of skin friction factor for tangent hyperbolic nanofluid is almost half ofs that of Carreau nanofluid.

In this analysis, the authors reveal the effects of electro-osmosis on the multiphase flow of Carreau fluid in a microchannel in the presence of Hall currents and solid particles. Moreover, the compliant channel walls are subject to oscillation occurring at the surface. To investigate the problem quantitatively, mathematical models for fluid phase and particulate phase have been structured. A lubrication approach is adopted due to laminar flow and the small dimensions of the channel. To produce the data, a system of differential equations is produced with the help of a numerical process performed on Mathematica through a built-in NDSolve tool. The results are presented graphically to examine the effects of various physical factors on the flow quantities. From pictorial discussion, it is gathered that the Helmholtz–Smoluchowski velocity parameter and the presence of an increasing amount of solid particles increasing the heat exchange while producing electro-kinetic energy. It is also found that velocity is a direct function of solid particles and compliant walls, but an inverse link is seen in the presence of electro-kinetic energy. Such studies can be employed with microfluidic devices and \may also be productive in medical and mechanical research.

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 (Ra) values and decreases with increasing variable viscosity (β1) and material parameter (We) values.

Here, our aim is to investigate incompressible Carreau fluid with variable properties and nanoparticles are considered to improve the heat transfer. This communication also highlights the thermal significances of Joule heating and incorporation of activation energy. The mathematical modelling is constructed for physical phenomena by assuming boundary layer problems. The well-known numerical treatment Bvp4c is utilised to solve the nonlinear problem and transform the governing equations. The effects of different physical parameters are assessed graphically and numerical data of skin friction and Nusselt number are explored. It is noted that activation energy rate improves and it decreases with the exponential fitted rate on concentration profile. The simulations for the current model reveal that the Nusselt number decreases with increasing exponential fitted rate and elastic deformation while for activation it increases. [ABSTRACT FROM AUTHOR]

Understanding fluid flow, heat, and mass transfer over a stretching cylinder is essential in ascertaining the quality of wire coating and coper thinning. This study looks at the influence of the heat source, thermal radiation, chemical reaction, and natural convection of Carreau fluid flows over a vertical stretching cylinder immersed in a porous medium. Suitable similarity variables are applied to convert the partial governing equations arising in fluid flows, heat, and mass transfer into ordinary differential equations. The optimal homotopy analysis method is then utilized to solve the transformed highly nonlinear governing equations. The impacts of the relevant parameters such as the Weissenberg number, porosity, heat source parameter, radiative number, chemical reaction parameter, mixed convection parameter, and curvature parameter on the dimensionless velocity, temperature, and concentration distribution as well as for the skin friction, Nusselt number, and Sherwood number are discussed through graphs and tables. It is observed that the velocity shows an opposite behavior as compared to temperature, and concentration in shear-thinning, n < 1 , and shear-thickening, n ≥ 1 , fluid for an increment of the Weissenberg number. Additionally, the thickness of the momentum, thermal, and concentration boundary layer is enhanced by the curvature of the cylinder. [ABSTRACT FROM AUTHOR]

This study investigates novel aspects of entropy generation in fully developed heat transport of Carreau fluid in a porous vertical microchannel. Consequences of thermal radiation and viscous heating are included in the thermal energy equation. The no-slip velocity and convective heating temperature boundary conditions are also considered. Darcy–Forchheimer model model is also considered. The system is solved using the bvp4c approach to solve a dimensional less two-point boundary value problem derived from the governing equations. The effect of effective factors on the Bejan number, entropy generation rate, temperature, and velocity are depicted in graphs. According to our analysis, increasing values of the Weissenberg number reduce entropy generation on the left and right sides of the channel, whereas the Bejan number improves on both sides of the channel and is highest in the middle. The fluid transport in a microchannel is enhanced by the pressure gradient. Because of the boundary conditions of convection heating at the microchannel walls, entropy production is at its maximum. Furthermore, the magnitude of the Bejan number is lessened due to a stronger viscous heating mechanism, whereas entropy production gets larger. The results were compared to those previously reported in the literature as a limiting case of the problem, and they were found to be in excellent agreement.

Correlations for the Nusselt number for the fully-developed laminar flow of Carreau fluids through circular pipe subject to a uniform heat flux have been sought. Based on the mathematical expression, the Nusselt number could be obtained by numerical integration. To evaluate the Nusselt number for many conditions, an efficient integration method has been proposed. Using the obtained Nusselt number for different material constants and flow conditions, an improved correlation method has been proposed. The proposed correlation could reduce the maximum error from 3% to 0.9%. [ABSTRACT FROM AUTHOR]

In this article, we consider the effects of double diffusion on magnetohydrodynamics (MHD) Carreau fluid flow through a porous medium along a stretching sheet. Variable thermal conductivity and suction/injection parameter effects are also taken into the consideration. Similarity transformations are utilized to transform the equations governing the Carreau fluid flow model to dimensionless non-linear ordinary differential equations. Maple software is utilized for the numerical solution. These solutions are then presented through graphs. The velocity, concentration, temperature profile, skin friction coefficient, and the Nusselt and Sherwood numbers under the impact of different parameters are studied. The fluid flow is analyzed for both suction and injection cases. From the analysis carried out, it is observed that the velocity profile reduces by increasing the porosity parameter while it enhances both the temperature and concentration profile. The temperature field enhances with increasing the variable thermal conductivity and the Nusselt number exhibits opposite behavior. [ABSTRACT FROM AUTHOR]

The effect of a catheter on blood flow and heat transfer characteristics of a Carreau fluid model is investigated mathematically using the perturbation method involving appropriate small parameters. Physically, this study relates to the surgical technique when a catheter is injected eccentrically into an artery. The model describes fluid flowing in the gap between eccentric tubes where the inner tube is uniform and rigid representing the moving catheter while the other tube represents an artery with overlapping stenosis. A complete perturbation solution up to the first order is presented whereby a validation exercise carried out shows that previously published solutions involving Newtonian fluids may have some typographical errors. The present results obtained show that the axial velocity, wall shear stress, and temperature in the case of the eccentric catheter are higher than that of the concentric catheter. The average Nusselt number is enhanced with increasing catheter radius and the velocity of the catheter while it decreases with increased Weissenberg number, Prandtl number, and Eckert number. These results agree with physiological observations that the risks and complications associated with catheterization are alleviated when the eccentric position of the catheter is considered. [ABSTRACT FROM AUTHOR]

Numerical investigations are carried out to study incompressible hydromagnetic flow of Carreau liquid subjected to the stretching sheet, which saturates the porous medium space. Energy equation incorporated the influence of Joule heating and heat source/sink while mass equation is influenced by chemical reaction term. Similarity transformations are used to acquire non-linear Ode's system. Qausilinearisation technique with finite difference method is implemented for the numerical results. The results of emerging parameters are presented in graphical and tabular forms. It is observed that porosity parameter and local inertia coefficient parameter enhance the velocity profiles. Chemical reaction parameter decays the concentration field. [ABSTRACT FROM AUTHOR]

This study presents the computational analysis of three dimensional Casson and Carreau nanofluid flow concerning the convective conditions. To do so, the flow equations are modified to nonlinear system of ODEs after using appropriate self-similarity functions. The solution for the modified system is evaluated by numerical techniques. The results show the impacts of involving variables on flow characteristics and the outcomes of the friction factors are evaluated as well. In this study, the outcomes to local Nusselt number and Sherwood numbers are evaluated. Favourable comparison is performed with previously available outcomes. The achieved results are similar to solutions obtained by other researchers. The results are presented for flow characteristics in the case of Casson and Carreau fluids. Velocities are reduced for the growing values of permeability and velocity slip parameters in case of Casson and Carreau nanofluids. Temperature field enhances with the hike in the estimations of thermophoresis parameter and the thermal Biot number in case of Casson and Carreau nanofluids. Enhancing values of velocity slip parameter results in decrease in the skin friction coefficients and the rate of heat transfer, and rise in the rate of mass transfer in case of Casson and Carreau nanofluids.

In the present study, Carreau fluid model over a stretching cylinder is considered. The boundary layer approximation is incorporated in order to solve the mathematical model. The formulation has been constructed and then by employing similarity transformations, the boundary layer equations are rehabilitated into self-similar form. The simplified nonlinear boundary value problem is solved numerically with Keller box and shooting method. The interesting findings for essential parameters are examined through graphs and tables. In each graph and table comparison has been made through Keller box and shooting method. It was found that both solutions are in excellent agreement. Computations are performed for different values of Weissenberg number We, curvature parameter K and power law index n. The results show that velocity profile increases for large values of Weissenberg number We (n>1) and curvature parameter K while opposite behavior is noticed in the case of skin friction coefficient.

In the present analysis, the thin film flow of Carreau fluid under the consequences of uniform magnetic field over an extending substrate has been addressed. Marangoni convection and couple stress has been also considered to find the fine point of the flow. The intention behind this work is to enhance the energy and mass transmission rate over a stretching sheet. The phenomena have been modeled in the form of system of PDEs, which are reduced to dimensionless system of ODEs through the similarity approach. The results are obtained via analytical framework homotopy analysis method (HAM). The behavior of velocity, energy and mass transmission profile has been disclosed through Figures and Tables. It has been perceived that the fluid velocity declines with the improvement of magnetic field, Marangoni parameter, unsteadiness term, thin film parameter, Weissenberg number We2and couple stress parameter K. The thermal energy transmission rate enhances with the consequences of thermophoresis term and Brownian motion factor, while reduces with the effect of Prandtl number.

The main objective of this manuscript is to glance into the assets of nanoparticles in the stream of generalized micropolar fluid and Carreau fluid against an intensified elongated surface. In order to assess, the heat and mass diffusion occurrences, the Cattaneo-Christov implications are also experienced in the temperature and concentration computations. In contrast to prior studies, rheological attributes on non-Newtonian fluids are depicted by engaging micropolar fluid and Carreau liquid that reflect a clear difference of transport phenomena. By minimizing the number of independent factors, the regulating equations are transmuted into non-dimensional types, that are then tackled numerically using the RK-4 algorithm along with the shooting strategy. For velocity, micro-rotation, temperature and concentration distributions, a visualization evaluation of the entangled flow parameters is executed. It has been revealed that expanding magnetic and micropolar constraints enhance micro-rotation velocity. The unsteadiness parameter enhances all three physical quantities, surface drag force, Nusselt number, and Sherwood number.

Due to their capacity to create better thermal conductivity than standard nanofluids, hybrid nano-fluids and modified nanofluids have notable applications in aerospace, energy materials, thermal sensors, antifouling, etc. This study aims to the modified and hybrid nanofluid flow with the Carreau fluid over a sloped shrinking sheet. The Cattaneo–Christov heat flux also takes into account. To determine the thermal efficiency of the heat, three different kinds of nanomaterials, copper oxide (CuO), copper (Cu), and alumina (Al2O3), are used. The similarity alteration commutes the insolubility of the model into ODEs. The conclusions are attained by program writing in MATLAB software and dealing with them through the bvp4c solver with the shooting method. The skin-friction amount decreases with the inclined sheet and local Weissenberg parameter for both modified and hybrid nanofluid. An upsurge thermal relaxation parameter declines the skin-friction coefficient for modified nanofluid flow and increases the skin-friction coefficient for hybrid nanofluid flow. The heat transfer rate is upsurged with modified and hybrid nanofluid for thermal relaxation parameter. Furthermore, the presentation includes the development of skin friction coefficient and Nusselt number values for specific parameters. Through benchmarking, numerical solutions are validated using certain limiting situations that were previously published findings, and typically solid correlation is shown.

This study is aimed to explore the magneto-hydrodynamic Carreau fluid flow over a stretching/shrinking surface with a convectively heated boundary. Temperature-dependent variable thermophysical properties are utilized to formulate the problem. The flow governing equations are obtained with boundary layer approximation and constitutive relation of the Carreau fluid. The shooting method is utilized to obtain graphical and numeric outcomes. Additionally, initial guesses are generated with the help of Newton’s method. The effect of Weissenberg number, Magnetization, stretching ratio, Prandtl number, suction/blowing parameter, and Lewis number is obtained on velocity, temperature and species continuity profile and analyzed. Shear stress rates and Nusselt number outcomes under body forces influences are present in tabulated data and discussed. It is observed that in absence of magnetization force, B = 0 and strong mass suction 5≤S≤7.5 effect high rates of Nusselt number is obtained. It is concluded that under the influence of power law index and non-linearity parameter maximum heat transfer and reduced shear stress rates are obtained.