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Background: Non-newtonian fluids, especially shear thinning fluids, have several applications in the polymer industry, food industry, and even everyday life. The viscosity of shear thinning fluids is decreased by two or three orders of magnitude due to the alignment of the molecules in order when the shear rate is increased, and it cannot be ignored in the case of polymer processing and lubrication problems. Objective: So, the effects of viscosities at the low and high shear rates on the heat and mass boundary layer flow of shear thinning fluid over moving belts are investigated in this study. For this purpose the generalized Carreau model of viscosity relate to shear rate is used in the momentum equation. The Carreau model contains the five parameters: low shear rate viscosity, high shear rate viscosity, viscosity curvature, consistency index, and flow behavior index. For the heat flow, the expression of the thermal conductivity model similar to the viscosity equation due to the non-Newtonian nature of the fluid is used in the energy equation. Methods: On the mathematical model of the problem, boundary layer approximations are applied and then simplified by applying the similarity transformations to get the solution. The solution of the simplified equations is obtained by numerical technique RK-shooting method. The results are compared with existing results for limited cases and found good agreement. Results: The results in the form of velocity and temperature profiles under the impact of all the viscosity’s parameters are obtained and displayed in graphical form. Moreover, the boundary layer parameters such as the thickness of the regions, momentum thickness, and displacement thickness are calculated to understand the structure of the boundary layer flow of fluid. Conclusion: The velocity and temperature of the fluid are decreased and increased respectively by all viscosity’s parameters of the model. So, the results of the boundary layer fluid flow under rheological parameters will not only help engineers to design superior chemical equipment but also help improve the economy and efficiency of the overall process.

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.

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.

This theoretical analysis considers a biomechanical model in which the Carreau fluid model characterizes the viscoelastic nature of growing human embryo and secreted fluid. This model incorporates transport mechanisms that involve the swaying motions of ciliary cells, peristaltic contractions of smooth muscle cells and pressure gradient at the ampullar region entrance. Series form solutions of the resulting partial differential equations are obtained using the regular perturbation method. A theoretical estimate of effects of the condition of pressure gradient, geometric parameters and fluid model parameters on the flow variables that have relevance to the problem of growing embryo transport in the human fallopian tube is presented through the discussion of graphs. Furthermore, an analogy between the linearly viscous fluid, and the shear thinning and shear thickening characteristics of the Carreau fluid model is also presented. The pertinence of the obtained results with growing embryo transport in the human fallopian tube revealed that when shear thickening characteristics of the Carreau fluid model are considered then complete mitotic divisions take place properly with an estimated appropriate residue time about 3–4 days. Smaller size trapped boluses of the secreted fluid make the smooth forwarding of the growing embryo in the human fallopian tube when shear thinning characteristics of the Carreau fluid model are taken into account. Key modulators: progesterone ( $$P_{4})$$ and estradiol ( $$E_{2}$$ ), prostaglandin $$E_{2}$$ ( $$PGE_{2}$$ ) and prostaglandin $$F_{2\alpha }$$ ( $$PGF_{2\alpha }$$ ) constraint the growing embryo transport.

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 thermal entrance problem (also known as the classical Graetz problem) is studied for the complex rheological Carreau fluid model. The solution of two-dimensional energy equation in the form of an infinite series is obtained by employing the separation of variables method. The ensuing eigenvalue problem (S–L problem) is solved for eigenvalues and corresponding eigenfunctions through MATLAB routine bvp5c. Numerical integration via Simpson’s rule is carried out to compute the coefficient of series solution. Current problem is also tackled by an alternative approach where numerical solution of eigenvalue problem is evaluated via the Runge–Kutta fourth order method. This problem is solved for both flat and circular confinements with two types of boundary conditions: (i) constant wall temperature and (ii) prescribed wall heat flux. The obtained results of both local and mean Nusselt numbers, fully developed temperature profile and average temperature are discussed for different values of Weissenberg number and power-law index through graphs and tables. This study is valid for typical range of Weissenberg number W e ≤ 1 $\left(We\le 1\right)$ and power-law index n < 1 $\left(n{< }1\right)$ for shear-thinning trend while n > 1 $\left(n{ >}1\right)$ for shear-thickening behaviour. The scope of the present study is broad in the context that the solution of the said problem is achieved by using two different approaches namely, the traditional Graetz approach and the solution procedure documented in M. D. Mikhailov and M. N. Ozisik, Unified Analysis and Solutions of Heat and Mass Diffusion, New York, Dover, 1994.

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 study, viscoelastic properties of xhantan gum (XG) and sodium alginate (SA) hydrogelling system were investigated at a definite range of volumetric mixing ratio of XG-SA in detail. The Carreau model gives reasonable fitting (R2 > 0.994) to the XG-SA system given by steady shear rheological measurements. In addition, the four element Maxwell model is used to capture well linear viscoelasticity of this system resulting from oscillatory shear rheological measurements. These measurements show that the viscoelastic relaxation times are longer for the higher ratio of XG in the hydrogel system. Physical gel network formations were also confirmed using scanning electron microscopy (SEM) of XG-SA.

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.

We propose a micromixer for obtaining better efficiency of vortex induced electroosmotic mixing of non-Newtonian bio-fluids at a relatively higher flow rate, which finds relevance in many biomedical and biological applications. To represent the rheology of non-Newtonian fluid, we consider the Carreau model in this study, while the applied electric field drives the constituent components in the micromixer. We show that the spatial variation of the applied field, triggered by the topological change of the bounding surfaces, upon interacting with the non-uniform surface potential gives rise to efficient mixing as realized by the formation of vortices in the proposed micromixer. Also, we show that the phase-lag between surface potential leads to the formation of asymmetric vortices. This behavior offers better mixing performance following the appearance of undulation on the flow pattern. Finally, we establish that the assumption of a point charge in the paradigm of electroosmotic mixing, which is not realistic as well, under-predicts the mixing efficiency at higher amplitude of the non-uniform zeta potential. The inferences of the present analysis may guide as a design tool for micromixer where rheological properties of the fluid and flow actuation parameters can be simultaneously tuned to obtain phenomenal enhancement in mixing efficiency.

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...

We study the hydrodynamics of the boundary layer flow of Carreau fluid over a moving wedge embedded in a porous medium in the presence of the applied magnetic field. The velocity of the wedge and mainstream is approximated by the power of distance from the leading boundary layer edge. Governing equations that model a non-Newtonian fluid in the boundary layer are reduced to an ordinary differential equation using the appropriate similarity transformations. The Chebyshev collocation and shooting algorithms based results show that there are non-unique solutions in the boundary-layer for the same system parameters. When the velocity ratio parameter is reduced the wall shear stress on the surface starts to increase till a critical value beyond which no solution exists. Thus, linear stability based on eigenvalue analysis helps to determine which of these non-unique solutions is physically realizable. When the magnetic field and permeability effects on the boundary-layer flow are increased the system shows unique solutions which are always stable. A detailed mechanism behind these results is discussed.

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.

The technology for producing products from composite materials with a thermoplastic binder differs from the traditional method, which uses thermosetting resins as a binder, and requires new design and technological solutions. This article proposes a two-stage method for manufacturing a long-length carbon strand of high-voltage cables: in the first stage, a thermoplastic layer of the polyetheretherketone (PEEK) melt is applied to the carbon filament; in the second stage, a strand is formed from the filaments and crimped by hydrodynamic drawing. Using the finite-element method, a numerical model of the process of applying a thermoplastic layer to a carbon filament in an axisymmetric die is constructed. To describe the rheological properties of the PEEK melt, a generalized Carreau model was selected and the material constants included in it are determined. The velocity, pressure, and temperature fields are calculated, and the effects of the applied pressure and strand drawing speed on the thickness of the applied layer are estimated. For the stage of formation and impregnation of the strand composed of carbon filaments covered with a thermoplastic layer, a dimensionless equation for the position of the impregnation front of the porous carbon filament influenced by an external pressure that is linearly distributed along the channel is obtained on the basis of the analytical solution of the flat-radial filtration problem under the assumption that each filament in the strand is under the same conditions in the process of crimping. The developed mathematical model allows one to calculate the pressure and temperature fields and flow kinematics, to evaluate the influence of technological parameters on the thickness of the thermoplastic layer applied to the filament, and to predict the degree of uniformity of the finished product.

In order to imitate the atherosclerosis artery disease and determine the key issues, Computational Fluid Dynamics (CFD) is able to play a leading rule in the analysis of flow physics within the clogged arteries, in particular the stenosis artery. The problem of blood flow blockage through the blood vessel has been investigated numerically within a stenosis artery. In this work, a CFD technique was employed to solve the three-dimensional, steady, laminar and non-Newtonian Carreau model blood flow through a stenosis artery using Star-CCM+ software. The shape of stenosis that has been selected is a trapezoidal with two cases (70% and 90% blockage). Shear rate, streamlines, vorticity and importance factor are examined to assess the influence of non-Newtonian model through the test section, the Carreau model was compared with Newtonian model. The clinical significance of the shear rate is reported for the examined cases, observing that the levels of non-Newtonian model are predicted to be higher in the 90% blockage than that observed within the 70%; the same finding as related with the axial velocity and vorticity. The levels of re-circulation areas and vorticity are showed to be enlarged in the Carreau model compared with the case of Newtonian.

The paper presents a study of hemodynamic processes in an artery with stenosis. The Navier–Stokes model and the Carreau model for a non-Newtonian fluid are proposed for computer simulations. Computational experiments were performed using the FlowVision© software. The results of a test simulation for blood flow in the 2D tube, as well as a simulation for the native artery model, based on medical research data, are presented. The simulation results show that in a native artery with a stenosis, the distribution of blood velocity and viscosity may be asymmetric, and their local values differ significantly from the average values observed in the artery after the presumed revascularization. The results have great potential importance for diagnosing and treating arterial stenosis.

The primary motive of this study is to examine boundary layer flow of Carreau fluid over a convectively heated disk stretching with nonlinear velocity. The flow is assumed to be two dimensional. Moreover, viscous dissipation possessions are taken into description. The dominating nonlinear differential equations involving partial derivatives are changed into nonlinear differential equations involving ordinary derivatives by applying suitable transformations. Numerical outcomes for velocity and temperature are obtained from MATLAB’s built-in solver bvp4c and presented graphically and in tabular form.

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.

Blood is a complex biological fluid and has elements in its composition, such as erythrocytes, which give it a non-Newtonian behavior. Typically, when carotid blood flow is studied, this aspect is frequently ignored and blood is modeled as a Newtonian fluid, with constant viscosity. This work demonstrates a Fluid–Structure Interaction study of carotid blood flow, with the comparison of two distinct viscosity models (Newtonian and Carreau). The results of the Independent Samples Kolmogorov–Smirnov statistical test show that the velocity in the center of geometry of the is similar for both viscosity models (p = 1.00; p = 0.71; p = 1.00) in the three vessels of the bifurcation Wall Shear Stress is dependent on the blood viscosity (p = 0.027), and is considerably higher for the Carreau model. The displacement of the arterial wall is reliant on the fluid pressure, and it is not influenced by the blood viscosity.

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 thermal features of nanoparticles have a dynamic role in medical, electrical, optical, built-up, and energy area, and compromise the potential to report the lower thermal conductivities established in conventional heat transport fluids. These specimens of nanofluids emphasize the engineers and investigators to progress newfangled approaches and methodologies in the heat transport field. Here, this notion elaborates the concept of nanoparticles for dual solutions by considering the influence of nanoparticles mass flux theory with an activation energy in Carreau fluid flow. The magnetite nanoparticles are reported subject to mixed convection in stagnation region. The heat transport aspects are conferred via thermal radiation, non-uniform heat sink-source, convective phenomenon, and Joule heating. Additionally, the chemical reaction is discussed. The apposite alterations altered the non-linear system of partial differential equations into non-linear ODEs. The achieved system of non-linear ODEs is elucidated numerically exploiting bvp4c methodology. Furthermore, for the confirmation of bvp4c technique, comparison tables are organized in this work which ensure the correctness of our numerical approach. The graphs of temperature, concentration, and velocity fields are formed to scrutinize the influence of diverse influential parameters and conferred. Enhancing behavior of the velocity field is reported for increasing estimations of the local Wessinberg number for both solutions. Additionally, the temperature field has conflicting influence for magnetic parameter and Biot number on both solutions. The skin friction coefficient is an increasing function of the local Wessinberg number.

In the present article, importance of double-diffusivity convection in nanofluids on peristaltic pumping of Carreau fluid in tapered channel with induced magnetic field and different waveforms has been theoretically discussed. Mathematical modeling of the two-dimensional and two-directional flow of a Carreau fluid are described in detail. A well-known assumption (long wave length and low Reynolds numbers) are employed in order to find the exact and analytical solution of the proposed problem. Exact solutions of temperature, concentration, and nanoparticle volume fraction are computed. Perturbation technique is used to calculate the solutions of highly nonlinear partial differential equations. With the help of Mathematica software and MATLAB, graphical results are displaced to see the behavior of temperature, concentration, nanoparticle volume fraction, pressure rise, pressure gradient, magnetic force function, and stream functions. It is observed that temperature and concentration profile increases with an increase in Nb, Nt, NCT, and NTC.

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.

PurposeThe magnetohydrodynamic (MHD) flow problems are important in the field of biomedical applications such as magnetic resonance imaging, inductive heat treatment of tumours, MHD-derived biomedical sensors, micropumps for drug delivery, MHD micromixers, magnetorelaxometry and actuators. Therefore, there is the impact of the magnetic field on the transport of non-Newtonian Carreau fluid in the presence of binary chemical reaction and activation energy over an extendable surface having a variable thickness. The significance of irregular heat source/sink and cross-diffusion effects is also explored.Design/methodology/approachThe leading governing equations are constructed by retaining the effects of binary chemical reaction and activation energy. Suitable similarity transformations are used to transform the governing partial differential equations into ordinary differential equations. Subsequent nonlinear two-point boundary value problem is treated numerically by using the shooting method based on Runge–Kutta–Fehlberg. Graphical results are presented to analyze the behaviour of effective parameters involved in the problem. The numerical values of the mass transfer rate (Sherwood number) and heat transfer rate (Nusselt number) are also calculated. Furthermore, the slope of the linear regression line through the data points is determined in order to quantify the outcome.FindingsIt is established that the external magnetic field restricts the flow strongly and serves as a potential control mechanism. It can be concluded that an applied magnetic field will play a major role in applications like micropumps, actuators and biomedical sensors. The heat transfer rate is enhanced due to Arrhenius activation energy mechanism. The boundary layer thickness is suppressed by strengthening the thickness of the sheet, resulting in higher values of Nusselt and Sherwood numbers.Originality/valueThe effects of magnetic field, binary chemical reaction and activation energy on heat and mass transfer of non-Newtonian Carreau liquid over an extendable surface with variable thickness are investigated for the first time.

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.

This work presents a transient analysis of journal bearings under thermal elastohydrodynamic lubrication (TEHL) based on the Carreau model. The effects of the elastic modulus of the bearing liner h...

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 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.