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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.

The present study deals with an investigation of steady two dimensional flow of Carreau fluid covering both the motile microorganism and nanoparticles through a nonlinear stretching surface. The idea of microorganisms is implied just to stabilize suspended nanoparticles by convection. The governing PDEs (partial differentiable equations) are transform set of nonlinear ODEs (ordinary differentiable equations) using suitable similarity transformation. The higher order nonlinear differential equation is solved by using Runge-Kutta-Felhberg method with shooting technique. The performance of fluid microorganism density, concentration, temperature, velocity in different values of physical parameters are scrutinized. Here the numerical results are presented through graphs and tables. Numerical values of skin friction, local Nusselt, Sherwood and local density numbers are analyzed. This study is relevant to navel microorganisms using few technologies combining with the Nano fluid with Bio-convection.

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.

In this paper, a mathematical model of the blade coating of the viscoelastic Carreau fluid which passes through the gap between the fixed blade and the moving substrate is constructed. The governing equations in the light of LAT (Lubrication Approximation Theory) are nondimensionalized and the solutions for the velocity, volumetric flow rate and the pressure gradient are calculated using Adomian decomposition method and numerical method. Numerical results for the velocity distribution using shooting method have also been studied by taking the full lubrication equation without taking the binomial expansion. Some important quantities from the industrial point of view like pressure, shear stress, Nusselt number and skin friction coefficient are also calculated. The important results are graphically shown at the end. With exponential blade coater, we have calculated velocity profiles, pressure, coating thickness and blade loading. Some of these results are shown graphically while other are presented in the tabulated form.

Carreau fluid flows past a stretching cylinder is elucidated in the present study. The transformed self-similarity and dimensionless boundary layer equations are solved by using the Homotopy analysis method. A convergence study of the method is illustrated explicitly. Series solutions of the highly nonlinear differential equations are computed and it is very efficient in demonstrating the characteristic of the Carreau fluid. Validation of the series solutions is achieved via comparing with earlier published results. Those results are obtained by using the Keller-Box method. The effects of the Weissenberg number and curvature parameter on the velocity profiles are discussed by graphs and tabular. The velocity curves have shown different behavior in and for an increase of the Weissenberg number. Further, the curvature parameter K does increase the velocity profiles.

This investigation aims to analyze the effect of boron nitride nanotubes (BNNTs) suspension on the heat transfer performance of ethylene glycol (EG) based nanofluids flow curved microchannel driven by two pumping mechanisms i.e., peristaltic pumping and electroosmotic pumping. The shear-thinning aspect of BNNTs-EG nanofluid is characterized by employing the Carreau fluid model. A uniform magnetic field is also imposed along the curved conduit, and the effect of Hall currents and ion slip generated by this magnetic field is also considered. The analysis has been performed in the presence of the Joule heating phenomenon. The no-slip conditions for velocity and convective boundary conditions for heat and mass transfer are enforced along channel walls. The Poisson-Boltzmann equation in curvilinear coordinates is employed to find the electric potential distribution in the fluid medium subject to the approximation of lower zeta potential. Further, the system of equations is executed numerically by Maple 17 subject to the lubrication linearization principle. The impression of important physical parameters on the flow characteristics is displayed graphically. It is found that more cooling effect is generated by considering a channel with relatively less curvature and fluid momentum is assisted by the larger curvature parameter i.e., in a straighter channel. It is further revealed that the increasing volume fraction of BN nanotubes reduces the shear-thinning characteristics of BNNTs-EG nanofluid, however, it enhances the thermal conductivity of the nanofluid and removes the heat from the system more efficiently.

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.

In this study, a two-dimensional mixed convective MHD stagnation point flow of Carreau fluid past an infinite plate in a porous medium is studied. Effects of cross-diffusion, thermal radiation, hea...

The aim of this paper is to study the combined effects of the concentration and the thermo-diffusion on the unsteady oscillation flow of an incompressible Carreau fluid through an inclined porous channel. The temperature is assumed to affect exponentially the fluid's viscosity. We studied fluid flow in an inclined channel under the non-slip condition at the wall. We used the perturbation series method to solve the nonlinear partial differential equations. Numerical results were obtained for velocity distribution, and through the graphs, it was found that the velocity of fluid has a direct relation with Soret number, Peclet number, and Grashof number, while it has a reverse variation with chemical reaction, Schmidt number, frequency of oscillation, and Froude number.

This research article addresses the physical characteristics of the Carreau fluid flow through a horizontal microchannel subjected to Newton boundary conditions, Hall current, and heat generation. ...

In this article, a mathematical model is envisaged to scrutinize the aspect of swimming motile microorganisms for a steady 3D convectional flow of Carreau nanofluid towards a bidirectional stretching surface. Various researchers in the past years are collaborating in the nanotechnology field due to their improvement in heat capacity, chemotherapy for cancer, microelectronics, cooling of energy storage devices, cooling of nuclear system, air conditioning, and nanochips, etc. The novel concept of activation energy and double stratification effects is considered to analyze the flow problem. Thermal relaxation time and concentration relaxation time properties are both determined by implementing Cattaneo–Christov heat and mass flux in the energy and mass equation. Suitable similarity approximations are utilized to convert governing PDEs into dimensionless ODEs. Numerical solutions are determined to utilize collocation finite difference technique and 3-stages Lobatto-IIIa formula. The obtained ODE’s are resolved numerically by engaged built-in function bvp4c solver in computational software MATLAB. The behavior of various emerging parameters and local skin friction coefficients, local Nusselt numbers, local Sherwood number, motile microorganisms’ number, velocity profiles, temperature profile, concentration profile and microorganism concentration is elaborated with the help of graph and tables. Carreau fluid model is a particular category of established non-Newtonian fluid that exemplifies shear thinning $$ n1 $$ at high shearing rates. It is examined that the influence of the mixed convection parameter against temperature distribution both the temperature field of the shear-thinning/thickening liquids is reduced.

We study the propagation of a plane‐strain hydraulic fracture driven by a shear thinning fluid following a Carreau rheology. We restrict to the impermeable medium case and quantify in details the impact on fracture growth of the shear‐thinning properties of the fluid between the low and high shear‐rates Newtonian limits. We derive several dimensionless numbers governing the evolution of the solution. The propagation notably depends on the ratio between the two limiting viscosities, the fluid shear‐thinning index, a dimensionless fracture toughness and a characteristic time‐scale capturing the instant at which the fluid inside the fracture reaches the low‐shear rate Newtonian plateau. We solve the problem numerically using Gauss‐Chebyshev methods for the spatial discretization of the coupled hydro‐mechanical problem and a fully implicit time integration scheme. The solution evolves from an early time self‐similar solution equals to the Newtonian one for the large‐shear rate viscosity to a late time self‐similar solution equals to the low‐shear rate Newtonian solution. The transition period (corresponding to the shear thinning part of the rheology) exhibits features similar to the power law rheology, albeit quantitatively different. Comparisons of hydraulic fracture growth predictions obtained with a power‐law model confirm its inadequacy for realistic fluids used in practice compared to the more physical Carreau rheology: the Newtonian plateau at high and low shear rates cannot be neglected.

Many non-Newtonian fluids, including polymers, exhibit both shear-thinning and viscoelastic rheological properties. A lattice Boltzmann (LB) model is developed for simulation of the flow of thinning–elastic fluids through porous media. This model applies the Oldroyd-B constitutive equation and the Carreau model, respectively, to account for the viscoelastic and shear-thinning behaviors of the thinning–elastic fluid in porous media. Both rheological features are captured well by this model and are verified against analytical solutions. The thinning-then-thickening viscosity curve of the thinning–elastic fluid observed in experiments is reproduced by the present pore-scale simulations. In addition to the traditional extensional theory, we propose other important mechanisms for the increase in apparent viscosity of viscoelastic fluids at higher shear rates. The mechanisms proposed include the reduction in conductivity due to stagnant fluid, the compressed effective flow region, and larger energy dissipations caused by the viscoelastic instability. We find that the viscoelastic thickening effect is more prominent in porous geometries with a large pore–throat ratio.

Mucus transport mediated by motile cilia in the airway is an important defense mechanism for prevention of respiratory infections. Cilia motility can be affected by temperature difference and magnetic field. In this research, we investigate the combined effects of magnetic field and buoyancy force due to temperature difference. In the present study, mixed convective flow of a Carreau fluid model through a ciliated channel is modeled and analyzed by a symplectic metachoronal wave. The momentum and energy equations for the Carreau fluid are modeled and simplified by the stream function and small Reynold’s number approximation. The influence of magnetic parameter, Carreau fluid parameter, Brinkmann number and Weissenberg number on velocity, temperature and pressure gradient are presented via graphs. It is observed that Hartmann number helps to decelerate the flow, whereas Weissenberg number, Grashof number and Carreau fluid parameter are responsible for the accelerated flow. The heat transfer rate rises by increasing the values of Hartmann number, Weissenberg number, Carreau fluid parameter and Brinkmann number.

An electrically conducting three-dimensional non-Newtonian Carreau fluid induced by bidirectional movement of sheet is demonstrated in this work. The aspects of viscous heating, radiation and first-order chemical reactions are executed in energy transport and mass species expressions. The new variables are defined to restructure the mathematical problem into single independent variable equations (ordinary differential equations, ODE’s). The re-framed ODE’s are exploited numerically through the utilization of MATLAB bvp4c package. The derived results of velocities, temperature and species distributions are executed graphically and interpreted physically for both shear-thinning and shear-thickening cases. Also, the numerical results for coefficients of skin friction, rate of mass transfer in terms of Sherwood number and rate of heat transfer in terms of Nusselt number are presented in tables. The influence of local Weissenberg numbers on Sherwood and Nusselt numbers for shear-thinning fluids is reversed to shear-thickening fluids. The thermal radiation parameter enhances the fluid temperature, and chemical reaction parameter decelerates the fluid concentration.

This article deals with heat transfer analysis on the Electro-magnetohydrodynamic Carreau fluid flow through a pair of rectangular plates. A Darcy–Brinkman–Forchheimer medium is considered for physical modeling. The flow is induced due to the Lorentz force, which occurs owing to the presence of an extrinsic imposed magnetic and electric field. The solutions are obtained with the help of numerical and semi-analytical/numerical schemes. A differential transform method (DTM) is employed to resolve the nonlinear coupled differential equations. The obtained solutions are discussed and plotted against all the physical parameters, and the Nusselt number is also addressed with the help of the table. The present outcomes are also plotted for the Newtonian fluid model as a particular case. The comparison of DTM is presented with a numerical shooting method for the Nusselt number. It is concluded from the analogy that the DTM method is very adaptive and stable to solve the nonlinear differential equations.

This paper aims to study variable viscosity effects of mass transfer on the MHD oscillatory flow of the Carreau fluid through a porous channel. It is assumed that fluid viscosity varies with variou...

The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Benard convection for non-Newtonian inelastic fluids. The rheological model considered here is the Darcy–Carreau model, which is an extension to porous media of Carreau rheological model usually used in clear fluid media. The linear stability approach showed that the critical Rayleigh number and wave number corresponding to the onset of convection are the same as for Newtonian fluids. By employing weakly nonlinear theory, we derived a cubic Landau equation that describes the temporal evolution of the amplitude of convection rolls in the unstable regime. It is found that the bifurcation from the conduction state to convection rolls is always supercritical for dilatant fluids. For pseudoplastic fluids, however, the interplay between the macroscale properties of the porous media and the rheological characteristics of the fluid determines the supercritical or subcritical nature of the bifurcation. In the parameter range where the bifurcation is supercritical, we determined and discussed the combined effects of the fluid properties and the porous medium characteristics on the amplitude of convection rolls and the corresponding average heat transfer for both pseudoplastic and dilatant fluids. Remarkably, we found that the curves describing these effects collapse onto the universal curve for Newtonian fluids, provided the average apparent viscosity is used to define Rayleigh number.

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.

This paper investigates the linear stability of the flow in the two-dimensional boundary-layer flow of the Carreau fluid over a wedge. The corresponding rheology is analysed using the non-Newtonian Carreau fluid. Both mainstream and wedge velocities are approximated in terms of the power of distance from the leading edge of the boundary layer. These forms exhibit a class of similarity flows for the Carreau fluid. The governing equations are derived from the theory of a non-Newtonian fluid which are converted into an ordinary differential equation. We use the Chebyshev collocation and shooting techniques for the solution of governing equations. Numerical results show that the viscosity modification due to Carreau fluid makes the boundary layer thickness thinner. Numerical results predict an additional solution for the same set of parameters. Thus, a further aim was to assess the stability of dual solutions as to which of the solutions can be realized. This leads to an eigenvalue problem in which the positive eigenvalues are important and intriguing. The results from eigenvalues form tongue-like structures which are rather new. The presence of the tongue means that flow becomes unstable beyond the critical value when the velocity ratio is increased from the first solution.

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.

This paper aims to study a mathematical model showing the effects of mass transfer on MHD oscillatory flow for Carreau fluid through an inclined porous channel under the influence of temperature and concentration at a slant angle on the centre of the flow with the effect of gravity. We discussed the effects of several parameters that are effective on fluid movement by analyzing the graphs obtained after we reached the momentum equation solution using the perturbation series method and the MATHEMATICA program to find the numerical results and illustrations. We observed an increased fluid movement by increasing radiation and heat generation while fluid movement decreased by increasing the chemical reaction parameter and Froude number.

A hybrid nanofluid phenomenon is considered involving nanoparticles since such particles are potential medication transportation devices in biomedical applications. Moreover, the expansion in heat transfer can be accomplished by refining either flow mechanism with geometry annoyance or thermal conductivity of the liquid itself. This motivation encourages the authors to discuss the three-dimensional study to analyze the peristaltic transport of Carreau nanofluid in a cross section of rectangular channel. This investigation may be helpful in physiology, chemical industries and biomedical apparatus, especially for the dismissal of cancerous cells where the heat convection would be followed by nanoparticles through three-dimensional tube/duct. In the current analysis, the constitutive equations are managed under the assumptions of long wave length and low Reynolds number assumptions. After making use of some suitable dimensionless quantities, we gathered the partial differential equations in nonlinear coupled form which are then handled by the homotopy perturbation method. The variational occurrence of all emerging parameters affecting the flow is analyzed through graphical treatment. Velocity distribution is also plotted for three dimensions. Trapping scheme is also presented through streamlines which implies the flow phenomenon through circulating empty bolus traveling toward the flow. A comparative analysis is also made to differentiate between the behavior of Newtonian and Carreau fluid model. On the other hand, the pumping rate of 3D rectangular channel is also contrasted with a 3D square duct. It is analyzed that peristaltic pumping rate is enhanced with the growing values of aspect ratio, power law index and local nanoparticles Grashof number; however, the inverse results are faced with Brownian motion factor and Weissenberg number. It is also visualized that pumping rate in 3D rectangular enclosure is higher than a square duct geometry.

The existing analysis deals with heat transfer occurrence on peristaltic transport of a Carreau fluid in a rectangular duct. Flow is scrutinized in a wave frame of reference moving with velocity c away from a fixed frame. A peristaltic wave propagating on the horizontal side walls of a rectangular duct is discussed under lubrication approximation. In order to carry out the analytical solution of velocity, temperature, and pressure gradient, the homotopy perturbation method is employed. Graphical results are displayed to see the impact of various emerging parameters of the Carreau fluid and power law index. Trapping effects of peristaltic transport is also discussed and observed that number of trapping bolus decreases with an increase in aspect ratio β.

The heat flow in microchannels can be established in numerous applications such as micro air vehicles, mechanical–electromechanical systems, cooling of electronic devices and micro heat exchanger systems. Heat flow optimization deliberates the function of entropy generation minimization (EGM) in engineering applications. Hence, this paper investigates the heat transport of non-Newtonian magneto-Carreau fluid in a microchannel with EGM. Mathematical modeling incorporates the Carreau fluid model. Further, viscous heating, Joule heating and convective heating aspects are also analyzed. The physical features of entropy production in the flow of non-Newtonian Carreau fluid in a microchannel are the major focus of this model. Dimensionless variables are executed for the simplicity of basic equations. The subsequent system is treated by using finite element method. Behaviors of effective parameters on velocity, Bejan number, entropy generation rate and temperature are interpreted. It is established that EGM is occurred for larger values of Weissenberg number. The Carreau fluid exponent is positively related to Bejan number, whereas it is negatively related to EG, temperature and velocity fields.

Conservation laws and variable conductance (viscosity, thermal conductivity and mass diffusion coefficient) models are used to develop mathematical problems describing transport mechanism in Carreau fluid over a non-uniformly moving surface. Boundary conditions are developed by no-slip theory. Mathematical models are transformed into suitable residual integrals which are approximated by Galerkin approximations. The obtained residuals are used for solving problems numerically using finite element method. Numerical investigation of variable viscosity, thermal conductivity and mass diffusion coefficients is carried out to examine the impact of parameters. The heat dissipated as a result of friction among the particles of the fluid of constant viscosity is greater than the heat dissipated due to friction force in the fluid of temperature-dependent viscosity. It is also found that ohmic phenomenon in the fluid of variable viscosity is prominent than that in the fluid of constant viscosity. Therefore, this fact must be in mind while using the fluids of variable viscosity in engineering applications. The transport of mass in fluid of constant viscosity is greater than that in fluid of variable viscosity. The Lorentz force is observed to oppose the flow. Hence, flow is decelerated by an increase in the intensity of magnetic field. The rate of transportation of mass has shown increasing trend when mass diffusion coefficient increases due to temperature.

This article addresses the time-dependent flow of magnetized rheological Carreau nanoliquid conveying microorganisms over a moving wedge with velocity slip and thermal radiation features. Carreau fluid is auspicious to depict several types of physical issues because this fluid model has the capability of revealing the rheology of multiple specific fluids such as fluids with brief-chain suspension particles, fluid crystals, detergents, and blood in animals and humans. The mathematical formulation is developed by combining the impact of infinite shear rate viscosity. The physical aspects for both static and moving are discussed in detail. At first, relevant similarity transformations are employed to obtain dimensionless form of equations, and then renovated equations have been solved numerically by employing bvp4c via MATLAB based on shooting technique. Both the numerical and graphical results against physical quantities, such as velocity temperature, nanoparticles concentration and density of gyrotactic microorganism, are observed under the influence of physical parameters.

The development of ground-breaking solar power technologies is considered to be one of the critical clarifications towards satisfying an increasing demand of energy worldwide. The use of solar energy had a remarkable benefit in current span. Furthermore, in numerous utilizations, there is a sympathetic requirement for augmenting the poor thermal conductivity of standard materials so as to develop effective heat transfer fluids. Nanofluids are fluid disturbances of nanoparticles and extensive analysis has been presented on nanofluid application in heat transfer developments. The crucial objective of this article is to scrutinize numerically the structure of unsteady 3D radiative flow of magneto-Carreau nanofluid. The Buongiorno’s relation is utilized to visualize the effects of thermophoresis and Brownian motion on the Carreau nanoliquid. The convective heated surface is ponder for heat transfer mechanisms. Additionally, zero nanoparticle mass flux condition on the surface of the boundary is accounted. An appropriate conversion is used to alter the PDEs into ODEs and then elucidated numerically via bvp4c tactic. The impact of distinct parameters for shear thinning/thickening liquids on velocity component, nanoliquid temperature and concentration of Carreau fluid are reported graphically. It is deceptive that the temperature of the Carreau nanoliquid declines as the Prandtl number Pr augment. Additionally, it is reported from these plots that the concentration of Carreau nanoliquid decay as the Brownian motion parameter boost up and opposite performance is being acknowledged for thermophoresis parameter.

This article reports a revised relation of nanofluid considering the properties of activation energy in mixed convection Carreau nanofluid, The Brownian and thermophrotic diffusions are incorporated. The aspects of convective heat transport, viscous dissipation, Joule heating and chemical reaction are also reported. The numerically bvp4c scheme has been established for the solutions of nonlinear ODEs. The local Weissenberg number and magnetic parameter decline the velocity field. The radiation and thermophoretic parameters enhance the temperature of Carreau fluid. Furthermore, on both Bejan number and entropy generation, the impact of radiation and magnetic parameter have been studied.

Here, a moving permeable surface for the mixed convective flow of Carreau nanofluid with thermal radiation is considered in detail along with the variable conductivity and thermophoresis diffusion aspects. Mathematical modeling of the existing physical model is formulated for a system of PDEs to simplify it in ODEs. Before applying Bvp4c technique, first the nonlinear higher order ODEs are converted into first-order ODEs and then employed for solution development. Velocity, temperature, and concentration are conducted for Carreau nanofluid. The role of rheological parameters on velocity, temperature, and concentration is examined. Numerically, it is observed that in a moving permeable surface the skin friction and Nusselt number significantly affect the skin-heat-transport mechanisms. Moreover, the velocity of Carreau nanofluid is boosted for the ratio of rate constants. The values of suction parameter increase corresponding to less velocity, while the opposite behavior occurs for the ratio of rate constant. An augmentation in Brownian motion parameter corresponds to a lower concentration of cross-nanofluid, while the inverse situation is detected for greater thermophoresis parameter.

Cilia beating is a naturally occurring phenomenon that can be utilized in fluid transport in designing several biomechanical devices. Inspired by the ubiquity of bio-fluids (which are non-Newtonian), we report the characteristics of shear-rate-dependent viscosities on fluid flow generated by the wavy propulsion of magnetic cilia. We assume that the metachronal waves of these cilia form a two-dimensional wavy channel, which is filled with generalized Newtonian Carreau liquid. Galilean transformation is employed to relate fixed and moving frames. The constitutive equations are reduced under the classical lubrication assumption. The resulting fourth-order nonlinear differential equations are solved via a perturbation approach using the stream function. The effects of four dominant fluid parameters (shear thinning/thickening, power-law index, and zero- and infinite-shear-rate viscosity), magnetic parameter (Hartmann number), and metachronal wave parameters on fluid velocity, pressure rise per wavelength, and trapping phenomenon are shown in graphical results and explained thoroughly. This study could play an advisory role in designing a magnetic micro-bot useful in the biomedical industry.

In addition to the non-Newtonian fluids (NNFs), thermophysical analysis of NNF–NNF is very useful for developing unsteady flow measurements. This work aims to provide a two dimensional (2-D) optimization problem consisting of thermal radiation, viscous dissipation, and inclined magnetic force based on the Buongiorno mathematical model. In this way, the unsteady 2-D flow is simulated through a permeable shrinking wall, and the governing partial differential equations are reduced to a set of ordinary differential equations which can be easily solved by the robust homotopic approach (RHA). It is shown that the present RHA agrees very well with those numerical and analytical findings available in the open literature. In fact, it can be concluded that employing a desirable solution methodology is essential for nonlinear boundary value problems combined with the thermophysical properties.

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. Published version The authors extend their appreciation to the deanship of scientific research at the King Khalid University, Saudi Arabia, for funding research groups under the research grant number R. G. P. 1/162/42.

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 We 2 and 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 Carreau fluid flow in an infinite pipe, caused by a time-dependent pressure gradient, is studied theoretically. The problem is axisymmetric in space and reduces to a nonlinear parabolic equation for the axial velocity. This equation has no analytical solution in the general case of different Carreau numbers. In the present paper, it is proved that the velocity and its gradient, as well as their differences from the similar Newtonian solution, are bounded by explicitly found constants, which depend on the Carreau number and on the other parameters of the viscosity rheological model. These estimates are also discussed for the special case of oscillatory pressure gradient using some numerical examples at different Carreau numbers.

Flow of a shear-thinning fluid over curved and moving geometries demands special attention owing to its industrial and biological relevance. The present study reports a numerical simulation of the motion of a Carreau fluid over a transversely oscillating cylinder for a range of oscillation amplitudes and frequencies at Reynolds number ). This difference in the flow structures is explained through the behaviour of the diffusion terms in the vorticity transport equation. Further, the effect of the Carreau fluid properties on the flow structures and vorticity dynamics are also discussed.

The present study is based on the flow of two dimensional Carreau fluid with heat generation and activation energy over the surface of paraboloid. The thermophysical properties such as viscosity, thermal conductivity and thermal diffusivity are chosen to be temperature dependent. The stretching in the fluid layers is presumed at the surface of object. This problem is modified by using law of conversation of momentum, mass, heat and concentration. The governing PDEs are transformed into dimensionless form by using appropriate similarity transformations. The numerical and graphical results are obtained via solving the ordinary differential equations by “RK4” technique with shooting method through MATLAB software. The impacts of key parameters on velocity, temperature and concentration fields are solved numerically and graphically. The power law index and variable viscosity parameter shows increment in velocity field. The temperature of fluid rises due to high values of heat source parameter while temperature shows reduction with Prandtl number. The cause of increment in concentration of fluid is due to activation energy and thermal diffusivity parameter.

A sophisticated theoretical and mathematical model is proposed. It is verified that this model can estimate and monitor the detailed behavior for the steady Carreau fluid flow past a nonlinear stretching surface and the predicted phenomena due to the presence of heat flux, thermal radiation, and viscous dissipation. Despite the fact that some properties of the fluid do not depend on the temperature, the fluid thermal conductivity is assumed to depend on the temperature. Based on accelerating the fluid elements, some of the kinetic energy for the fluid can be turned to the internal heating energy in the form of viscous dissipation phenomena. The contribution in this study is that a similar solution is obtained, in spite of the high nonlinearity of the Carreau model, especially, with the heat flux, variable conductivity, and viscous dissipation phenomena. Some of the major significant findings of this study can be observed from the reduction in the fluid velocity with enhancing the Weissenberg number. Likewise, the increase in the sheet temperature is noted with increasing the Eckert number while the reverse behavior is observed for increasing both the radiation parameter and the conductivity parameter. Finally, the accuracy and trust in the proposed numerical method are validated after benchmarking for our data onto the earlier results.

In this paper Carreau fluid is taken into account to study its peristaltic flow with Hall and ion-slip effects. The study is carried out in a wave frame of reference for both asymmetric and symmetric channel. Analysis of heat transfer is accomplished by taking the effects of viscous dissipation and ohmic heating into our consideration. Modeling of fundamental equations is followed by the construction of closed form solutions for pressure gradient, stream function and temperature while assuming Reynold׳s number is very low and wavelength very long. The closed form solutions are generated with the help of perturbation technique considering Weissenberg number to be very small. The effects of emerging parameters on pumping, trapping, axial pressure gradient, pressure rise, velocity profile and temperature are analyzed through the graphical representation. Keywords: Hall effect, Ion slip, Ohmic heating, Perturbation method

The following paper describes a numerical simulation of a complete bypass of a stenosed human artery. The considered geometry consists of the narrowed host tube and the bypass graft with a 45-degree angle of connection. Different diameters of the narrowing are tested. Blood is the fluid with shear rate–dependent viscosity; therefore, various rheology mathematical models for generalized Newtonian fluids are considered, namely Cross model, modified Cross model, Carreau model, and Carreau-Yasuda model. The fundamental system of equations is based on the system of generalized Navier-Stokes equations. Generalized Newtonian fluids flow in a bypass tube is numerically simulated by using a SIMPLE algorithm included in the open-source CFD tool, OpenFOAM. The aim of this work is to compare the numerical results for the different mathematical models of the viscosity with the changing diameter of the narrowed channel.

The flow induced by a sphere rotating inside an incompressible, non-Newtonian, Carreau fluid has been investigated numerically. The rotating sphere is enclosed in a concentric cubic box with solid ...

This notion reports the properties of nanofluid on Carreau fluid for dual nature study. Nanofluids are stable colloidal suspensions of metals or non-metals element in a disreputable liquid, which intensify the heat transfer of the solution and exaggerate the storage aptitude. The forthright intention of this communication is to scrutinize the impact of binary chemical reaction and activation energy on dual nature study for unsteady flow of Carreau magnetite nanofluid owing to porous shrinking/stretching sheet. Additionally, the aspects of convective heat transport with nonlinear thermal radiation and heat sink/source have been established for heat transport phenomenon. With the assistance of compatible conversions the partial differential equations (PDEs) are condensed into ordinary differential equations (ODEs) which are then elucidated numerically via bvp4c scheme and the impacts of intricate parameters are exposed explicitly. It is noted that the intensifying values of activation energy parameter and fitted rate constant causes to an enhancement in the nanofluid concentration.

Effects of a controllable axial flow on the stability of rotational flow of pseudoplastic fluids in the gap between concentric cylinders are studied. It is assumed that the outer cylinder is fixed while the inner one has simultaneous and independent rotational and translational motions. The fluid follows the Carreau model and mixed boundary conditions are imposed. The four-dimensional low-order dynamical system resulting from Galerkin projection of the conservation of mass and momentum equations includes highly non-linear terms in the velocity components originating from the shear-dependent viscosity. In the absence of the axial flow as the pseudoplasticity effect increases, the critical Taylor number at which the rotational flow loses its stability to the vortex structure decreases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a Newtonian fluid in rotation. The existence of an axial flow, induced by the translational motion of the inner cylinder, appears to further advance the emergence of the vortex flow. Complete flow analysis together with viscosity maps are given for the entire flow field.

During this article, we have a tendency to show the peristaltic activity of magnetohydrodynamics flow of carreau fluid with heat transfer influence in an inclined tapered asymmetric channel through porous medium by exploitation the influence of non-slip boundary conditions. The tapered asymmetric channel is often created because of the intrauterine fluid flow induced by myometrial contraction and it had been simulated by asymmetric peristaltic fluid flow in an exceedingly two dimensional infinite non uniform channel, this fluid is known as hereby carreau fluid, conjointly we are able to say that one amongst carreau's applications is that the blood flow within the body of human. Industrial field, silicon oil is an example of carreau fluid. By exploitation, the perturbation technique for little values of weissenberg number, the nonlinear governing equations in the two-dimensional Cartesian coordinate system is resolved under the assumptions of long wavelength and low Reynolds number. The expressions of stream function, temperature distribution, the coefficient of heat transfer, frictional forces at the walls of the channel, pressure gradient are calculated. The effectiveness of interesting parameters on the inflow has been colluded and studied.

This study numerically investigates the heat transfer characteristics, including the fluid flow, of a Taylor vortex flow system with shear-thinning fluids. Governing equations were solved using OpenFOAM® 4.0 code. The Carreau model was utilized as the rheological model. The parameter (n) in the Carreau model, which describes the slope of a decrease in viscosity with an increase in shear-rate, was varied from 1 to 0.3. The local Nusselt number (NuL) decreased with the increase in shear-thinning property. Furthermore, a correlation equation between the effective Reynolds number (Reeff) and a global Nusselt number (NuG) was proposed. Using this equation, NuG was evaluated within a ± 10% error in the case of n = 1, 0.7, 0.5, and within a ± 20% error in the case of n = 0.3. The size of the Taylor vortices became axially larger with an increase in the shear-thinning property. Furthermore, the thickness of the boundary layer of velocity and temperature increased with an increase in shear-thinning property. The ratio of the thickness of velocity boundary layer to the temperature boundary layer monotonically decreased with an increase in Reeff.

New results are reported on the simulation of the rise of a deforming gas bubble surrounded by a liquid exhibiting a blend of shear-thinning and viscoelastic properties. A new hybrid model has been developed which combines the Carreau model for emulating the shear-thinning property of the liquid with the Chilcott–Rallison (FENE-CR) model for treating the viscoelastic property of the liquid. The new hybrid model allows one to independently prescribe the shear-thinning and viscoelastic properties (parameters) to be used in a simulation. Computations are implemented using the coupled level-set and volume-of-fluid (CLSVOF) method for tracking the deforming gas-liquid interface. The interfacial jump conditions are enforced with the sharp interface approach. In this study, four parameters, the Eotvos number, Carreau model parameter for slope of decreasing viscosity, Deborah number, and viscoelastic polymer dumbbell length parameter, are varied in order to gain understanding for gas bubble morphology in liquids exhibiting both shear-thinning and viscoelastic properties. It is found that the extent of bubble deformation and the liquid stress field are strongly dependent on these 4 parameters with minimal correlation between the shear-thinning and viscoelastic parameters. The ability to vary shear-thinning and viscoelastic parameters independently of one another in the new hybrid Carreau and FENE-CR model enables one to conveniently predict gas bubble behavior in complex liquid polymers by parameterizing the new model from basic measurements of the polymers.

In this study, a fluid-structure interaction (FSI) simulation of the blood flow in the femoral artery with a small occlusion is presented. For a more accurate simulation of the real conditions, computerized tomography (CT) scan was used to obtain a 3-D model of leg blood vessels, while the vessel was modeled as an isotropic elastic wall. By assuming a heartbeat period of 0.5 s, the inlet condition was considered as a time-dependent pulse using a non-Newtonian flow model. Blood flow was assumed nonlinear and incompressible, and Carreau model was used for blood rheological model. By considering unstable blood flow at the inlet, the involved hemodynamic parameters are velocity profile, vortices shapes, pressure drop, and streamlines. Furthermore, to determine the relationship between flow geometry and the vascular wall, wall shear stress (WSS) was calculated. By taking the real geometry of the vessel and fluidity of blood into account, comparison of computational results indicated a significant difference in velocity distribution and shear stress depending on whether the fluid-structure interaction is considered Newtonian or non-Newtonian. The results showed that employing Newtonian models for the blood flow does not lead to promising results at occluded areas and beyond them.

The 3D flow of non-Newtonian nanoliquid flows past a bidirectional stretching sheet with heat transfer is investigated in the present study. It is assumed that viscosity of the liquid varies with temperature. Carreau non-Newtonain model, Tiwari and Das nanofluid model are used to formulate the problem. The impacts of Joule heating, nonlinear radiation and non-uniform (space and temperature dependent) heat source/sink are accounted. Al-Cu-CH3OH and Cu-CH3OH are considered as nanoliquids for the present study. The solution of the problem is attained by the application of shooting and R.K. numerical procedures. Graphical and tabular illustrations are incorporated with a view of understanding the influence of various physical parameters on the flow field. We eyed that using of Al-Cu alloy nanoparticles in the carrier liquid leads to superior heat transfer ability instead of using only Aluminum nanoparticles. Weissenberg number and viscosity parameter have inclination to exalt the thermal field.

The study depicts the variations in the hydromagnetics flow of a Careau fluid in a semi permeable curved channel with convective boundary condition. Furthermore, Rosseland approximation is also considered to analyze the non-linear thermal radiation effects. Curvilinear co-ordinates system has been adopted for the mathematical modeling of the flow equations. The attained set of governing equation are then converted into non-linear dimensionless differential equations, by making use of similarity variables which are later treated by shooting method. In addition, the Newton?s Raphson method is also incepted to improve the accuracy of the obtained numerical result. The velocity field and temperature distributions are affected by various involved parameter which are presented in graphs and in table form. It is noticed that the velocity profiles are influenced by the change in the Weissenberg number.

The risk factor increases at the carotid bifurcation as a result of the changes in the flow of blood and phenomena such as flow separation, the rotational flow, and the effects of the shear stress induced by the walls, increasing the risk of injury. In this regard, using Fluent 16.2, the present study numerically simulates the pulsatile flow of the blood in a patient specific elastic carotid artery with physiological pulses and non-Newtonian and turbulent models. The result of wall shear stress showed that the being of rigid or elastic walls is much more important than that of the Newtonian or non-Newtonian blood, which in turn is more important than laminar or turbulent flow. Moreover by investigating Oscillatory Shear Index (OSI), it was found that the region prone to disease becomes larger as a Newtonian blood is assumed and that the k-ω model shows a larger area prone to disease compared with the k-e model. In addition, the pressure contours showed that both pressure drop and critical pressure are higher in the k-e and turbulent flow models in comparison with the k-ω and laminar flow models, respectively. By comparing various non-Newtonian models with the Newtonian one, it was found that the critical pressure is higher in the Carreau model compared with the Newtonian model where, in turn, it is higher than that in the Power-law model.