Third-grade Fluid Research Articles

Application of homotopy perturbation method on special third grade fluid flow with viscous dissipation effect over a stretching sheet

This paper investigates the flow and heat transfer of special third-grade fluid with a viscous dissipation effect over a stretching sheet. This model, adequate for many non-Newtonian fluids, is used to characterize the behavior of the fluids domain. The governing momentum and energy equation are reduced to ordinary nonlinear differential (self-similar) equations via the Lie group transformation method. The Homotopy Perturbation Method (HPM) is applied to solve these obtaining results. For validation, current results have been compared with the fourth-order Runga method (RK4) and shooting technique. The effects of physical parameters on fluid velocity and temperature profile were investigated with the aid of figures and tables by simply altering a single parameter while keeping the others constant. It is observed that both the non-Newtonian parameter and the Prandtl number have the effect of decreasing the temperature of the stretching surface, while the opposite behavior was found for the Eckert number.

Electroosmosis Augmented MHD Third-Grade Fluid with Slip and Variable Properties: An Application for Blood Flow in Arteries

The electroosmotic force effect on the peristaltic motion of the third-grade fluid is considered in a uniform channel. The governing equations that supplement the flow are designed for long wavelengths and low Reynolds numbers. Solutions are obtained for velocity, temperature, concentration, and trapping by considering the variable liquid properties for analyzing the various parameter effects. These effects are depicted through graphs and the relevance is discussed. The variable fluid properties have a declining impact on the velocity and temperature fields. Increasing the Helmholtz–Smoluchowski velocity values decreases the velocity field. Temperature decreases as the Deborah number increases. The velocity slip characteristics rise, and the trapping bolus’s size shrinks. The results of this paper may be beneficial in understanding the control of microvascular transport in the time of fractionation of blood into plasma and erythrocytes.

Development and theoretical analysis of slippery walls flow of third-grade fluid through the convergent symmetric channel

The present investigation is a study of third-grade fluid muddled with tiny-sized round-shaped hafnium particles drifting through a convergent geometry under the influence of slip boundary conditions. The impact of electro-osmotic is also examined in this investigation. The Debye Hückel linearization is used to investigate the current problem. Fluid and particulate phases are illustrated in convergent geometry having momentous utilizations in industry and different strolls of science. The problem is modeled in terms of nonlinear complex differential equations by using the fluid and particle phase models. The normalized quantities are used to convert the dimensional equations into dimensionless form and the perturbation method is selected to obtain the solution in analytical form. The effects of pertinent parameters on each type of flow are also presented graphically. Promotion in the velocity profile is observed for the volumetric flow parameter. The slip parameter boost motion of the fluid particle inside the channel. The current study can be utilized in the chemical industry, reservoir engineering, and microfabrication technique in which mass exchanges and electro-osmotic energy play a vital role.

Impact of thermal radiation on two-dimensional unsteady third-grade fluid flow over a permeable stretching Riga plate

In many fields, there are various applications of non-Newtonian fluids. Various complicated fluids (polymer melts, clay coatings and oil) belong to the category of non-Newtonian fluids. The third-grade fluid is one of the most important non-Newtonian fluid models. This paper has the primary object of heat transfer mechanism and boundary layer third-grade fluid flow under the effects of thermal radiation. The time-dependent two-dimensional flow is considered to flow above a permeable stretchable vertical Riga plate. For numerical solutions, the setup of ordinary differential equations (ODEs) is acquired by converting nonlinear governing equations through relevant similarity transformations. The nonlinear setup of ODEs is numerically solved with the aid of a suitable software such as MATLAB via its bvp4c technology. Graphs are sketched to discuss the various flow parameters’ significance for the expression of velocity and temperature fields. Tabulated values of surface drag force and heat transfer rate corresponding to the numerous pertinent parameters are described. The current analysis of the concerned flow mechanism concludes that the fluid parameters descend the temperature distribution but amplify the profile of the fluid velocity. The radiation parameter escalates the temperature field.

Magnetized supercritical third-grade nanofluid flow from a vertical cylinder using a Crank–Nicolson implicit scheme

The current study investigates the supercritical convective radiative buoyancy-driven flow and heat transfer in external coating flow of an electrically conducting viscoelastic third-grade nanofluid from hotmoving/stationary cylinder under constant radial magnetic field. A new computational thermodynamic model has been proposed to study the supercritical behavior of third-grade aqueous nanofluid in terms of supercritical water. The Redlich–Kwong equation of state (RK-EOS) has been deployed to calculate thermal expansion coefficient for free convective flow of supercritical nanofluid in terms of temperature, compressibility factor and pressure. A validation test has been conducted for RK-EOS with available experimental results. A well-tested unconditionally stable Crank–Nicolson scheme has been implemented to obtain numerical solutions. Graphical results for flow variables, heat transfer and skin-friction coefficient distributions are presented for variation of nanoscale, rheological, magnetic, radiative and thermodynamic parameters. Transient velocity is reduced whereas temperature is elevated with amplified values of third-grade fluid parameter, reduced pressure, reduced temperature, and decreased values of volume fraction of nanofluid for a stationary cylinder under supercritical conditions. Special cases of the current model validated against previously published results. Important applications of the current study include nuclear reactor vessels, deposition of smart (functional magnetic) nano-coatings and solar collector energy systems.

Special Non-Newtonian Third-Grade Fluid Flow with Magnetic Field: A Numerical Study

The Buongiorno model is utilized for the non-Newtonian third-grade fluid. With non-linear radiation and magnetic action, it incorporates both Brownian motion and thermophoresis processes. Aim this research is to look at non-Newtonian third-grade fluid boundary layer flow with non-linear solar energy thermal radiation and the magnetic effect. The similarity transformation is used to convert nonlinear PDEs into ODEs. Then numerically solved by shooting technique and the sixth-order R-K method. For various physical parameters, derived findings are graphically discussed. We believe that the current findings can be utilized to explain the non-Newtonian properties of second and third-grade fluids, as well as for applications in tribology, the automobile industry, and other fields.

Characterization of the Induced Magnetic Field on Third-Grade Micropolar Fluid Flow Across an Exponentially Stretched Sheet

The current research article discusses the two-dimensional, laminar, steady, and incompressible third-grade viscoelastic micropolar fluid flow along with thermal radiation caused by an exponentially stretched sheet. The primary goal of this extensive study is to improve thermal transportation. Thermophoresis and Brownian motion are two key causes of nanoparticle migration in nanofluids, and their impacts on the thermophysical properties of nanofluids are significant. Micropolar fluids are investigated due to their micro-motions that are significant in convective thermal and mass transport polymer formation, nanotechnology, and electronics. The consequences of third-grade fluid parameters, thermophoresis and Brownian motion, induced magnetic field, micro-polarity, and micro-inertia density on the stream of an electrically conductive fluid are analyzed. A homogeneous magnetic field is supplied perpendicularly to the surface, and the liquid is believed to be electrically conducting. As the flow has a significant magnetic Reynolds number, the contribution of the evoked magnetic field is properly accounted in the governing equations. A mathematical model in the form of partial differential equations (PDEs) is built under certain assumptions. By invoking the suitable similarity transformation, the non-linear PDEs are modified into dimensionless coupled ordinary differential equations (ODEs). The MATLAB numerical technique bvp4c is employed to settle the subsequent ODEs together with the boundary constraints. The consequences of numerous physical parameters on the non-dimensional concentration, temperature, micropolar, velocity, and induced magnetic field profiles are portrayed in graphs. It is found that the concentration boundary layer, thermal boundary layer, and micropolar boundary layer thickness decelerate with the increment in the micro-polarity of the fluid.

A Theoretical Study of Reverse Roll Coating for a Non-Isothermal Third-Grade Fluid under Lubrication Approximation Theory

The flow of non-Newtonian fluids is extremely important in a variety of industrial applications and processes, including painting, printing, and coating. The goal of this paper is to develop a mathematical model for an incompressible, non-isothermal third-grade fluid as it travels through a minor gap between two heated rolls that are revolving in the opposite direction. Using appropriate dimensionless parameters, the dimensionless version of the governing equations is constructed. To simplify the nondimensional form of the governing equations, the LAT (lubrication approximation theory) is applied. It is mentioned that the perturbation approach should be used. It is a very important tool in modern mathematical physics and practical fluid mechanics, and it is used a lot. Through perturbation techniques, exact solutions for velocity distribution, flow rate, temperature distribution, and pressure gradient are offered. To observe the flow pattern, streamlines are also drawn. Numerical outcomes of some substantial engineering quantities like flow rate, roll separation force, power input, and pressure distribution are found. Some outcomes are displayed graphically although others are presented in tabular form. We have observed that the third-grade parameters give a way to control velocity, pressure gradients, flow rate, and coating thickness. Further, increasing the velocities ratio values lowers the coating thickness on the web. Furthermore, the Brickman number has a significant influence on the temperature profile and has increased with the increase in the Brickman number. Both the model and the numerical calculations are in agreement with the viscous fluid model that is already present in the literature.

Impact of nano metallic particles and magnetic force on multi-phase flow of third-grade fluid in divergent channel: analytical study

ABSTRACT The two-phase elector-osmotic (EoF) flow of a non-Newtonian fluid is the major focus of this article. Spherically homogenous metallic particles are considered to obtain a multiphase suspension. Third-grade fluid depicting non-Newtonian fluid characteristics is considered for the two-phase suspension through a divergent channel. A semi-analytic solution is obtained for the nonlinear flow dynamics which reveals that the third-grade fluid parameter enhances the inertial forces which result to increase the motion of each phase. However, the contribution of magnetic fields is quite opposite. Moreover, the obtained semi-analytical solution is in good agreement with the one reported in the literature.

Impact of gold and silver nanoparticles in highly viscous flows with different body forces

ABSTRACT The present investigation is relevant to the multiphase flow of non-Newtonian fluid passing through the steep channel. Third-grade fluid is taken as the base liquid, while two different types of metallic particles are considered to form nano-suspensions. Two different plane flows are devised based on the development of rigid walls of the inclined channels. A comprehensive apprehension regarding the nonlinear flow dynamics is carried out with the help of a regular perturbation method, which is effectively less time-consuming and more analogous to analytic. Moreover, a comparative analysis depicts complete harmony with the existing literature. In addition, the current study reduces back to the customary nanofluid problem for the limiting case.

Soret and Dufour aspects of the third-grade fluid due to the stretching cylinder with the Keller box approach

The convective transport of the third-grade fluid with thermo-diffusion features has to be worked out. The impact of the Soret and Dufour properties, as well as the radiative phenomena, is noticed. The assumed flow is due to the stretched cylinder. The partial differential equations indicating the flow model are first worked out into the dimensionless nature using similarity variables. The most effective numerical technique (the Keller Box method) has been used to solve nonlinear ODEs for obtaining the numerical data. The solution accuracy is validated using already claimed studies. The exploration subject to the flow pattern against parameters is presented. The simultaneous increase in Dufour and Soret numbers assists the temperature and concentration in the boundary layer region around the cylinder. With the increasing fluctuation in Dufour and reducing Soret numbers, the revised change in heat and mass transportation phenomenon is depicted. Furthermore, the radiation effects are elucidated by varying the effective Prandtl number.

Unsteady MHD third-grade fluid past an absorbent high-temperature shrinking sheet packed with silver nanoparticles and non-linear radiation

This investigation peruses the features of temperature and mass transport of the non-Newtonian third-order liquid over an absorbent convective temperature shrinking sheet. The sheet is packed with silver nanoparticles. The Buongiornos modelling uses a particular non-Newtonian third-order liquid with the Brownian movement and the thermophoresis consequences using the non-linear radiation. The non-linear partial differential equations are changed to the ordinary differential equations with similarity transformations. The changed system of equations is then resolved using the numerical Shooting method and the sixth-order Runge-Kutta’s method. The numerically obtained solutions of the velocity profiles, temperature distributions, and concentrations of nanoparticles are discussed graphically. Also, the non-Newtonian parameter reduced the velocity of the liquid, increased the temperature and the concentration profiles throughout the fluid. Ultimately, the qualified systematic analysis is built by the preceding study in restrictive cases and displaingy the best correlation.

Magnetohydrodynamic Effects on Third-Grade Fluid Flow and Heat Transfer with Darcy–Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium

The major aim of the current investigations is to study the magnetohydrodynamic effects on heat and mass transfer phenomena in third-grade fluid past an inclined exponentially stretching sheet fixed in a porous medium with Darcy–Forchheimer law influence. The constitutive equations compatible for heat and mass transportation in third-grade fluid in terms of partial differential equations are modeled. These partial differential equations are then converted to ordinary differential equations by using suitable similarity variables formulation. The transformed flow model is solved by using MATLAB built-in numerical solver bvp4c. Effects of pertinent parameters on physical properties that are velocity field, temperature field and mass concentration along with skin friction coefficient, Nusselt number and Sherwood number are demonstrated in graphs and tables. The impact of dimensionless numbers on the physical properties is analyzed and discussed with a physical view point at angle α=π/6 (inclined sheet). It is seen that as the third-grade fluid parameter (0.1≤β≤11) is increased, the velocity profile increases, but the temperature field and mass concentration are decreased. It is observed that as the permeability parameter (1≤K*≤11) is raised, the velocity distribution decreases and mass concentration increases. It is concluded from the results that owing to an increase in the local inertial coefficient (0.1≤Fr≤5), the velocity profile reduces but an increment in mass concentration is noted. It is concluded that by increasing values of magnetic field parameter (0.1≤M≤10) the velocity field is delineated and temperature field is elevated exactly according to the physics of magnetic field parameters. The present results are compared with already published results and it is observed that there is good agreement between them. This good agreement ensures the validation of accuracy of the results.

Analysis of fourth-grade fluid model over a stretchable surface with Riga plate subject to permeable medium

AbstractThe current investigation is concerned with the rheological impact of fourth-grade confined by Riga surface. The flow behaviour is analysed over a Riga plate in the presence of stagnation point and porous medium. The relevant similarity variables and corresponding boundary conditions are adopted to model the current problem. The highly non-linear coupled differential system is via optimal homotopy scheme. The outcomes of relevant dimensionless parameters on the velocity profile have been visualized with physical exploration. It is observed from the obtained outcomes that the fluid velocity declines against rising estimations of modified magnetic variable and inverse Darcy number. The increasing velocity change is noted for boosting values of third-grade fluid parameter. Moreover, the velocity pattern for fourth-grade material is comparatively higher than viscous, second-grade, and third-grade materials. The comparative analysis against obtained simulations is also listed.

Levenberg Marquardt knacks-based intelligent networks for the magnetohydrodynamic third-grade fluid flow under activation energy and bio-convection

In the present study, numerical work is carried out to investigate the third-grade fluid flow model using the algorithm of Levenberg Marquardt with backpropagated neural networks. We inspect a two-dimensional, incompressible, and steady laminar flow over a stretched sheet with gyrotactic microorganisms, where the flow is electro-conductive and magnetized, and the flow is generated by the stretching. In the mathematical modeling, we used the Buongiorno nanofluid model. The governing partial differential equations are transformed into a system of ordinary differential equations. These ordinary differential equations are solved by the Homotopy Analysis Method in the Mathematica software to generate the reference dataset with the various physical parameters. The approximated solution of the considered model is computed in MATLAB software using the reference dataset. By testing, training, and validation processes, we examined the correctness and the validation of the proposed model. These stochastic techniques have many applications in medicines, nanophysics, telecommunication industries, etc. The impacts of the variation of physical parameters on the velocity, temperature, concentration, and motile density profiles are examined, where the results for Nusselt number, Entropy generation, Sherwood number and Motile density number are also discussed in the present article. The best performance values 1.34E−09, 1.03E−09, 9.22E−10, 1.34E−09, 1.18E−09, and 1.32E−08 are attained at 311, 595, 347, 297, 581 and 521 epochs, respectively, for different scenarios. The values of Gradient are 9.96E−08, 9.97E−08, 9.94E−08, 9.98E−08, 9.90E−08, and 9.93E−08, whereas the values of Mu are 1E−08, 1E−08, 1E−08, 1E−08, 1E−08 and 1E−08 at 281, 505, 459, 388, 181, and 410, respectively for various instances. The regression value is R = 1 for training, testing, and validation data. The absolute error values are 10-6 to 10-4, 10-7 to 10-4, 10-9 to 10-4, 10-7 to 10-4, 10-7 to 10-4, 10-7 to 10-3, 10-7 to 10-4, 10-7 to 10-4, 10-7 to 10-4, 10-7 to 10-2, 10-7 to 10-4, 10-7 to 10-3 and 10-7 to 10-2 for (I–XIII) scenarios.

Darcy–Forchheimer Relation Influence on MHD Dissipative Third-Grade Fluid Flow and Heat Transfer in Porous Medium with Joule Heating Effects: A Numerical Approach

The current investigations are carried out to study the influence of the Darcy–Forchheimer relation on third-grade fluid flow and heat transfer over an angled exponentially stretching sheet embedded in a porous medium. In the current study, the applied magnetic field, Joule heating, thermaldiffusion, viscous dissipation, and diffusion-thermo effects are incorporated. The proposed model in terms of partial differential equations is transformed into ordinary differential equations using suitable similarity transformation. The reduced model is then solved numerically with the help of MATLAB built-in function bvp4c.The numerical solutions for velocity profile, temperature profile, and mass concentration under the effects of pertinent parameters involved in the model are determined and portrayed in graphical form. The graphical effects of the skin friction coefficient, the Nusselt number, and the Sherwood number are also shown. From the displayed results, we conclude that when the Joule heating parameter is enlarged, the velocity and the temperature of the fluid are increased. We observed that while enhancing the viscous dissipation parameter (Eckert number) the fluid’s velocity and temperature increase but decreases the mass concentration. By increasing the values of the thermal-diffusion parameter, the velocity distribution, the temperature field, and the mass concentration increase. When the diffusion–thermo parameter rises, the velocity field and the temperature distribution increase, and the reverse scenario is seen in the mass concentration. The results of the current study are compared with already published results, and a good agreement is noted to validate the current study.

Convective Heat and Mass Transfer in Third-Grade Fluid with Darcy–Forchheimer Relation in the Presence of Thermal-Diffusion and Diffusion-Thermo Effects over an Exponentially Inclined Stretching Sheet Surrounded by a Porous Medium: A CFD Study

The current study aims to investigate the thermal-diffusion and diffusion-thermo effects on heat and mass transfer in third-grade fluid with Darcy–Forchheimer relation impact over an exponentially inclined stretching sheet embedded in a porous medium. The proposed mechanism in terms non-linear and coupled partial differential equations is reduced to set of ordinary differential equations by employing an appropriate similarity variable formulation. The reduced form of equations is solved by using the MATLAB built-in numerical solver bvp4c. The numerical results for unknown physical properties such as velocity profile, temperature field, and mass concentration along with their gradients such as the skin friction, the rate of heat transfer, and the rate of mass transfer at angle of inclination α=π/6 are obtained under the impact of material parameters that appear in the flow model. The solutions are displayed in forms of graphs as well as tables and are discussed with physical reasoning. From the demonstration of the graphical results, it is inferred that thermal-diffusion parameter Sr velocity, temperature, and concentration profiles are augmented. For the increasing magnitude of the diffusion-thermo parameter Df the fluid velocity and fluid temperature rise but the opposite trend in mass concentration is noted. The current results are compared with the available results in the existing literature, and there is good agreement between them that shows the validation of the present study.

Fuzzy Analysis for Thin-Film Flow of a Third-Grade Fluid Down an Inclined Plane

We examined the thin-film flow problem of a third-grade fluid on an inclined plane under a fuzzy environment. The highly nonlinear flow governing differential equations (DEs) with the boundary conditions are fuzzified using the triangular fuzzy numbers (TFNs) developed by α -cut α ∈ 0,1 . The fuzzy perturbation (FPM) method is adopted to calculate the fuzzified form of the governing equations as well as the fuzzified boundary conditions. For the validation, the present work is in good agreement as compared to existing work in the literature under the crisp form. For various values of the fluid parameter λ , inclined parameter γ and fuzzy parameter α -cut is presented in graphical form. The α -cut controls TFNs, and the variability of uncertainty is investigated using a triangular membership function (MF). Using TFNs, the middle (crisp), left, and right values of the fuzzy velocity profile are used for fuzzy linear regression analysis. The outcome of this study and the fuzzy velocity profile have the maximum rate of flow as compared to the crisp velocity profile (mid values).

Significance of Chemical Reaction and Lorentz Force on Third-Grade Fluid Flow and Heat Transfer with Darcy–Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium

The combined impact of a linear chemical reaction and Lorentz force on heat and mass transfer in a third-grade fluid with the Darcy–Forchheimer relation over an inclined, exponentially stretching surface embedded in a porous medium is investigated. The proposed process is mathematically expressed in terms of nonlinear and coupled partial differential equations, with the symmetry of the conditions normal to the surface. To solve the mathematical model of the proposed phenomenon, the partial differential equations are first reduced to ordinary differential equations; then, MATLAB built-in Numerical Solver bvp4c is used to obtain the numerical results of these equations. The influence of all the pertinent parameters that appeared in the flow model on the unknown material properties of interest is depicted in the forms of tables and graphs. The physical attitude of the unknown variables is discussed with physical reasoning. From the numerical solutions, it is inferred that, as Lorentz force parameter M is increased, the velocity of the fluid decreases, but fluid temperature and mass concentration increase. This is due to the fact that Lorentz force retards the motion of fluid, and the increasing resistive force causes the rise in the temperature of the fluid. It is also noted that, owing to an increase in the magnitude of chemical reaction parameter R, the velocity profile and the mass concentration decline as well, but the fluid temperature increases in a reasonable manner. It is noted that, by augmenting the values of the local inertial coefficient Fr and the permeability parameter K*, the velocity field decreases, the temperature field increases, and mass concentration also increases with reasonable difference. Increasing values of Prandtl number Pr results in a decrease in the profiles of velocity and temperature. All the numerical results are computed at the angle of inclination α=π/6. The current results are compared with the available results in the existing literature for this special case, and there is good agreement between them that shows the validation of the present study. All the numerical results show asymptotic behavior by satisfying the given boundary conditions.

Analysis of temperature-dependent viscosity effect on wire coating using MHD flow of incompressible third-grade nanofluid filled in cylindrical coating die

The convective heat and mass propagation inside dies are used to determine the characteristics of coated wire products. As a result, comprehending the properties of polymerization mobility, heat mass transport, and wall stress concentration is crucial. The wire coating procedure necessitates an increase in thermal performance. As a result, this research aims to find out how floating nanoparticles affect the mass and heat transport mechanisms of third-grade fluid in the posttreatment for cable coating processes in presence of a magnetic field with time-dependent viscosity. For nanofluids, the Buongiorno model is used. The model equations are developed using continuity, momentum, and energy in the presence of nanoparticle and time-dependent variable viscosity. We propose a few nondimensional transformations that are relevant. The numerical technique Runge-Kutta fourth method is used to generate numerical solutions for nonlinear systems. Pictorial depictions are used to examine the effects of various factors in the nondimensional flow. Furthermore, the numerical results are also verified analytically using Homotopy Analysis Method (HAM). The analytical findings of this investigation reveal that within the Reynolds modeling, the stress on the whole wire surface combined shear forces at the surface predominate the Vogel model. The contribution of nanomaterials on force on the entire surface of wire and shear forces at the surface appears positive. A non-Newtonian feature can increase the capping substance’s velocity. This research could aid in the advancement of wire coating technologies. For the very first instance, the significance of nanotechnology during wire coating evaluation is explored utilizing time-dependent variable viscosity regarding the magnetic field. For time-dependent viscosity, two alternative models are useful. The Lorentzian strength (a resistive form of force) increases in magnetic strength increases. As a result of the increased magnetic field, the motion of the polymerization in a die decreases. It is clear that increasing the intensity of Nb increases the heat transfer. The innovative fragment of the present study is to scrutinize the magnetized third-grade nanofluid for wire coating with variable viscosity inside the pressurized coating die, which still not has been elaborated in the available works to date. Consequently, in the restrictive sense, the existing work is associated with available work and originated in exceptional agreement.