Viscoelastic Fluid Model Research Articles

Analysis of thermally radiative magnetized viscoelastic (second-grade) fluid by impermeable vertical cylinder

We made an attempt to disclose the phenomenon of dual stratification in magnetized viscoelastic second-grade fluid model generated by the movement of vertical cylinder. Mass and heat transportation is accounted under the aspects of Joule and Ohmic dissipations. The flow is coupled due to the consideration of nonlinear convected terms. Boundary-layer concept is implemented for the simplification of mathematical system. This system is made into the non-dimensional forms by reporting the similarity variables. The system of non-dimensional equations is executed with the use of homotopic criteria. The convergent region of solutions is reported and discussed. The convergent results are shown by graphical illustrations by considering distinct emerging constraint values. An increment in Darcy and inertia coefficient parameters corresponds to a decay in the flow. Temperature distribution is an increasing function of thermal Biot number and radiation parameter while it diminishes when thermal stratification parameter increases. Concentration is reduced for larger Schmidt number along with solutal stratification parameter. Besides the heat–mass transportation rates are increasing functions of thermo-solutal Biot numbers. The considered problem finds relevance in functional smart electro conductive enrobing coating processes in manufacturing. This is a principal application of the model studied where external axisymmetric boundary layers occur with combined magnetic, thermal and rheological effects.

Fully discrete finite element error analysis of a discontinuous Galerkin method for the Kelvin-Voigt viscoelastic fluid model

Fully discrete finite element error analysis of a discontinuous Galerkin method for the Kelvin-Voigt viscoelastic fluid model

Visco-Elastic Fluid Model in an Inclined Porous Stenosed Artery with Slip Effect and Body Acceleration

The present paper analyzes an unsteady magnetohydrodynamic blood flow model of an visco-elastic fluid through an inclined porous stenosed artery with body acceleration and slip effect. Navier-Stokes equations have been used to describe the blood flow model. The governing equation of blood flow is solved by an analytic method by considering blood as an incompressible, visco-elastic fluid, and suspension of RBC’s in plasma. Axial velocity, blood acceleration, flow rate, and shear stress are derived numerically by using the finite Laplace and Hankel transformation and their inverse. The effect of parameters such as the visco-elasticity parameter, Womersley number, Hartmann number, inclination angle, parameter of slip, and body acceleration frequency is analyzed. Axial velocity reduces as the Hartmann number and visco-elasticity parameter enhance and it enhances with the enhancement of the slip parameter and inclination angle. The study is beneficial for finding the effect of slip parameter, porosity factor and Hartmann number when a human body is exposed to MRI and CT scan.

On planar flows of viscoelastic fluids of Giesekus type

Viscoelastic rate-type fluid models of higher order are used to describe the behaviour of materials with complex microstructure: geomaterials like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as styrene butadiene rubber. A standard model that belongs to the category of viscoelastic rate-type fluid models of the second order is the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the first order. This viewpoint allows one to develop the whole hierarchy of generalized models of the Burgers type. We study one such generalization that can be viewed as a combination (mixture) of two Giesekus viscoelastic models having in general two different relaxation mechanisms. We prove, in two spatial dimensions, long-time and large-data existence of weak solutions to the considered generalization of the Burgers model subject to no-slip boundary condition. We also provide, as a particular case, a complete proof of global-in-time existence of weak solutions to the Giesekus model in two spatial dimensions.

Numerical Simulation of Rheological Models for Complex Fluids Using Hierarchical Grids.

In this work, we implement models that are able to describe complex rheological behaviour (such as shear-banding and elastoviscoplasticity) in the HiGTree/HiGFlow system, which is a recently developed Computational Fluid Dynamics (CFD) software that can simulate Newtonian, Generalised-Newtonian and viscoelastic flows using finite differences in hierarchical grids. The system uses a moving least squares (MLS) meshless interpolation technique, allowing for more complex mesh configurations while still keeping the overall order of accuracy. The selected models are the Vasquez-Cook-McKinley (VCM) model for shear-banding micellar solutions and the Saramito model for viscoelastic fluids with yield stress. Development of solvers and numerical simulations of inertial flows of these models in 2D channels and planar-contraction 4:1 are carried out in the HiGTree/HiGFlow system. Our results are compared with those predicted by two other methodologies: the OpenFOAM-based software RheoTool that uses the Finite-Volume-Method and an in-house code that uses the Vorticity-Velocity-Formulation (VVF). We found an excellent agreement between the numerical results obtained by these three different methods. A mesh convergence analysis using uniform and refined meshes is also carried out, where we show that great convergence results in tree-based grids are obtained thanks to the finite difference method and the meshless interpolation scheme used by the HiGFlow software. More importantly, we show that our methodology implemented in the HiGTreee/HiGFlow system can successfully reproduce rheological behaviour of high interest by the rheology community, such as non-monotonic flow curves of micellar solutions and plug-flow velocity profiles of yield-stress viscoelastic fluids.

Finite element analysis of viscoelastic media used in abrasive flow machining process

The surface roughness of a part is the most important parameter in view of tribological applications and it also affects the working life of the part during application. The abrasive flow machining (AFM) process is an advanced non-conventional finishing process, used to deburr, polish, and to remove the recast layer from the surface as well as at the edges of the components. In AFM viscoelastic media is used to finish the workpiece with close dimensional tolerance and precision. The viscoelastic media used in the AFM process is laden with abrasive particles. In the present work, a finite element analysis of viscoelastic abrasive media is performed considering the AFM process. A mixture of polyborosiloxane and silicon carbide is used as viscoelastic abrasive media and the AFM process is modeled using ANSYS Polyflow. In the analysis, the flow of viscoelastic abrasive media is assumed to follow the Maxwell model of viscoelastic fluid. The simulations were performed for varying the extrusion pressure for the finishing of an internal cylindrical surface. The results of the simulations were validated with the experimental observation and found in good agreement.

Numerical implementation and error analysis of nonlinear coupled fractional viscoelastic fluid model with variable heat flux

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

Finite volume simulations of particle-laden viscoelastic fluid flows: application to hydraulic fracture processes

Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid–particle force $$\langle F \rangle$$ on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix, parameterized through the Weissenberg number Wi measuring the relative magnitude of elastic to viscous stresses in the fluid. Here we use direct numerical simulations (DNS) to study the creeping flow ( $$Re\ll 1$$ ) of viscoelastic fluids through static random arrays of monodisperse spherical particles using a finite volume Navier–Stokes/Cauchy momentum solver. The numerical study consists of $$N=150$$ different systems, in which the normalized average fluid–particle force $$\langle F \rangle$$ is obtained as a function of the volume fraction $$\phi$$ $$(0 < \phi \le 0.2)$$ of the dispersed solid phase and the Weissenberg number Wi $$(0 \le Wi \le 4)$$ . From these predictions a closure law $$\langle F(\phi,Wi) \rangle$$ for the drag force is derived for the quasi-linear Oldroyd-B viscoelastic fluid model (with fixed retardation ratio $$\beta = 0.5$$ ) which is, on average, within $$5.7\%$$ of the DNS results. In addition, a flow solver able to couple Eulerian and Lagrangian phases (in which the particulate phase is modeled by the discrete particle method (DPM)) is developed, which incorporates the viscoelastic nature of the continuum phase and the closed-form drag law. Two case studies were simulated using this solver, to assess the accuracy and robustness of the newly developed approach for handling particle-laden viscoelastic flow configurations with $$O(10^5-10^6)$$ rigid spheres that are representative of hydraulic fracture operations. Three-dimensional settling processes in a Newtonian fluid and in a quasi-linear Oldroyd-B viscoelastic fluid are both investigated using a rectangular channel and an annular pipe domain. Good agreement is obtained for the particle distribution measured in a Newtonian fluid, when comparing numerical results with experimental data. For the cases in which the continuous fluid phase is viscoelastic we compute the evolution in the velocity fields and predicted particle distributions are presented at different elasticity numbers $$0\le El \le 30$$ (where $$El=Wi/Re$$ ) and for different suspension particle volume fractions.

Influence of elastic instability and elastic turbulence on mixed convection of viscoelastic fluids in a lid-driven cavity

The addition of a minute amount of solid polymers into a Newtonian solvent like water can originate an elastic instability as the Weissenberg number (ratio of the elastic forces to that of the viscous forces) gradually increases. These instabilities further transit to elastic turbulence (ET) regime at high Weissenberg numbers. In this study, we investigate how these elastic instabilities and elastic turbulence would tend to influence the mixed convective heat transfer phenomena in a lid-driven cavity, which is one of the widely studied problems in the domain of transport phenomena. In doing so, extensive two-dimensional time-resolved numerical simulations have been conducted by using an open-source CFD code OpenFOAM. The Oldroyd-B viscoelastic constitutive model is used in this study to realize the rheological behaviour of a constant shear-viscosity viscoelastic fluid or the so-called Boger fluid. To investigate the effect of the shear-thinning behaviour of a viscoelastic fluid on the elastic instability and elastic turbulence phenomena, we have further conducted limited simulations using the FENE-P (finitely extensible non-linear elastic spring with the Peterlin’s approximation) viscoelastic fluid model. A detailed discussion is presented on how the flow dynamics and heat transfer mechanism inside the cavity can be influenced by the Weissenberg number along with the Reynolds, Prandtl, and Richardson numbers. We show that the heat transfer rate inside the cavity can be increased by more than 100% under appropriate conditions if one uses a constant shear-viscosity Oldroyd-B fluid instead of a simple Newtonian fluid due to the presence of elastic instability and elastic turbulence in the former fluid. Furthermore, we have found that the shear-thinning behaviour of a FENE-P viscoelastic fluid tends to suppress these instabilities and turbulence and hence decreases the heat transfer rate inside the cavity. Therefore, one can use the present approach of using a constant shear-viscosity Oldroyd-B fluid as an efficient way to increase the rate of heat and mass transport processes in various systems. Our approach can be served as an alternative possibility to the existing approaches such as the application of an electric or a magnetic field or the use of a nanofluid. However, one needs to be careful in the selection of the polymer type and its degradation behaviour as well as its appropriate concentration in the solution when choosing this approach.

Numerical stability of four positive (semi-)definite reformulations for viscoelastic fluid models in benchmark flows

In this paper we study the numerical stability of four positive (semi-)definite formulations for viscoelastic fluid flows: the LCR formulation of Fattal and Kupferman (2004), the Cholesky-log formulation of Vaithianathan and Collins (2003), the b-symmetric formulation of Balci et al. (2011) and the CDT formulation of Carrozza et al. (2019). The numerical stability is tested by evaluating the critical time step in four different flow problems: the flow around a cylinder confined between two walls, planar extrudate swell, an axisymmetric swell problem and the extension of the axisymmetric swell problem to three dimensions. Although LCR is a good performer throughout, the CDT formulation turns out to be the most stable/robust one. The numerical stability of the Cholesky-log and the b-symmetric formulation turns out to be much lower in at least one benchmark. This supports our hypothesis that a vector-like formulation increases numerical stability at least as much as the LCR formulation and that positive definiteness is not the main contribution to numerical stability.

Cattaneo–Friedrich and Crank–Nicolson analysis of upper-convected Maxwell fluid along a vertical plate

The concept of fractional derivative is used to solve a variety of viscoelastic fluid problems. However, researchers mostly overlooked the consequences of nonlinear convection in the fractional viscoelastic fluid models and were concerned only with situations where the governing equations are linear. Most importantly, the nonlinear fluid models, whether classical or fractional, are solved for steady-state conditions. To overcome these limitations, this research presents unsteady flow and heat transfer of nonlinear fractional upper-convected Maxwell (UCM) viscoelastic fluid along a vertical plate. The governing equations of the fractional Maxwell fluid are developed by introducing Friedrich shear stress and Cattaneo heat flux models to the classical UCM fluid model. An additional feature to the invention of the constructed fractional model is the consequence of an external magnetic field. Moreover, the considered model comprises nonlinear, coupled, fractional partial differential equations. Therefore, a numerical scheme is developed with the aid of the L1-approximation of Caputo derivative and the Crank–Nicolson method. The effects of different regulating parameters on fluid features have been thoroughly investigated. The obtained results are exhibited graphically and discussed in detail. It is observed that the skin friction increases for the velocity relaxation time parameter, but an opposite behavior is observed against the velocity fractional derivative parameter. Moreover, a significant enhancement is noticed in the Nusselt number for increasing estimates of the Prandtl number.

Viscoelastic mechanical behavior of periodontal ligament: Creep and relaxation hyper-viscoelastic constitutive models

In this contribution, creep and relaxation hyper-viscoelastic constitutive models of periodontal ligament (PDL) was proposed to define PDL as a steady state rheology (linear)-nonlinear viscoelastic fluid biomaterial. The physical elements of the hyperelastic spring, non-Newtonian fluid damper and strain gap were proposed. The hyperelastic spring, linear elastic spring, linear viscous damper, stress lock, and strain gap constituted the creep and relaxation models. The qualitative energy analysis was made in the viscoelastic zone between the instantaneous elastic element and the infinite elastic element in the creep and relaxation models. The mechanical behavior of viscoelastic materials is qualitatively unified. The universal static, creep, and relaxation mechanical behaviors of materials were explained from the energy point of view (graphische viscoelastic). After defining a virtual “infinite elastic curve of viscoelastic solid” in the viscoelastic fluid model, the universal creep recovery mechanical behavior was analyzed. The unity of the opposites between viscoelastic solid and viscoelastic fluid was clarified. In addition, A physical thinking of “anchored” was proposed to qualitatively modify the Boltzmann superposition principle. Instantaneous/infinite elastic-plastic-fracture curve, yield curve and fracture curve were defined to distinguish viscoelastic zone and viscoplastic zone. Finally, the unsteady-rheological (nonlinear) behavior of nonlinear viscoelastic fluid materials and the hyper-viscoelastic-plastic behavior of nonlinear viscoelastic solid materials were predicted.

Numerical investigation of heat transport in Alumina–Silica hybrid nanofluid flow with modeling and simulation

The current numerical study investigates the thermal performance of four mono-component (i.e., blade-shaped Al2O3, brick-shaped Al2O3, cylinder-shaped Al2O3 or platelet-shaped Al2O3 nanoparticles in 50:50 water–ethylene glycol (EG) base fluid) and four bi-component nanofluids (i.e., blade-shaped Al2O3+SiO2/EG–water, cylinder-shaped Al2O3+SiO2/EG–water, brick-shaped Al2O3+SiO2/EG–water, and platelet-shaped Al2O3+SiO2/EG–water hybrid nanofluids) with the help of mathematical modeling. The non-Newtonian nature of these nanofluids is modeled using second grade viscoelastic fluid model accompanied by the energy equation. The highly non-linear coupled differential equations are solved by shooting method accompanied by 4th order Runge–Kutta method in MATLAB software. The numerical simulations are performed by procuring the data for thermophysical properties from the experimental studies. The effective thermal conductivity and viscosity of the four distinct alumina nanofluids are calculated by the models obtained from the experimental studies. The numerical results reveal that the nanofluid with platelet-shaped Al2O3 nanoparticles is the best heat transporter among the four alumina nanofluids The addition of SiO 2 nanoparticles further accelerates the rate of heat transfer in all four alumina nanofluids. The numerical results that illustrate the influence of nanoparticle volume fraction and thermofluidic parameters on the heat transport phenomenon are also discussed. The MATLAB code developed for the numerical solution is validated against the substantial published literature.

Rigid fiber motion in slightly non-Newtonian viscoelastic fluids

The perturbation technique based on the retardation-motion expansion is a simple method to obtain flow solutions at low Weissenberg number. In this context, this perturbation analysis is used to develop simple expressions for the motion of fibers suspended in viscoelastic fluids. In particular, the suspending fluid is characterized by a second-order fluid, Giesekus and PPT (Phan–Thien–Tanner) models, and their derivatives, such as the upper and lower convected Maxwell models. The first-order perturbation results in a similar effective velocity gradient that is exploited to express the translation and rotational motion of a single fiber and the associated extra stress tensor. In terms of a parameter related to the various viscoelastic fluid models, it is found that a fiber aligns along the vorticity direction when subjected to a shear flow. However, when a lower convected Maxwell model is considered, the elongated particle orients in the flow direction, as basically predicted by the Jeffery solution for a Newtonian suspending fluid. Furthermore, the conservation equation for particle concentration leads to particle migration in a pressure-driven flow channel and good agreement is observed with experimental data.

Numerical Modelling of Multicellular Spheroid Compression: Viscoelastic Fluid vs. Viscoelastic Solid

Parallel-plate compression of multicellular spheroids (MCSs) is a promising and popular technique to quantify the viscoelastic properties of living tissues. This work presents two different approaches to the simulation of the MCS compression based on viscoelastic solid and viscoelastic fluid models. The first one is the standard linear solid model implemented in ABAQUS/CAE. The second one is the new model for 3D viscoelastic free surface fluid flow, which combines the Oldroyd-B incompressible fluid model and the incompressible neo-Hookean solid model via incorporation of an additional elastic tensor and a dynamic equation for it. The simulation results indicate that either approach can be applied to model the MCS compression with reasonable accuracy. Future application of the viscoelastic free surface fluid model is the MCSs fusion highly-demanded in bioprinting.

Effect of real and whole blood rheology on flow through an axisymmetric stenosed artery

This study presents an extensive numerical investigation on the flow phenomena of a blood analogous fluid through an axisymmetric stenosed artery under both steady and pulsatile flow conditions. Most of the previous investigations, carried out for this benchmark problem, used either a simple Newtonian fluid model or a generalized Newtonian (GNF) fluid model like the power-law or Bingham plastic fluid model to represent the rheological behaviour of blood. However, many prior rheological studies showed that the real and whole blood exhibits both the shear-thinning and viscoelastic properties. In this study, for the first time, a multi-mode sPTT (simplified Phan-Thein-Tanner) model is used to carry out the numerical computation, which accounts for both the shear-thinning and viscoelastic rheological properties of blood. Additionally, a realistic set of viscoelastic model parameters, obtained by fitting the response of real and whole blood in standard viscometric flows, is used in this study. An excellent agreement is seen between the experimentally determined apparent viscosity (in simple shear flows) of blood with that predicted by the present viscoelastic model and other prior models developed for blood. Therefore, we believe that this study presents more accurate and realistic numerical results for the flow of blood through a stenosed artery than that presented by the previous studies in the literature. A simple Newtonian fluid model is also used in the present analysis to compare and show how the flow behavior of blood can be influenced by its complex rheological properties under otherwise identical conditions. We find a significant difference in the flow characteristics (in terms of the streamline profiles, velocity magnitude, pressure drop, etc.) obtained with the Newtonian and viscoelastic fluid models. For instance, the velocity gradient is seen to be large near the artery wall for the viscoelastic fluid model than that seen for the Newtonian model; whereas, a reverse trend is seen for the pressure drop across the stenosis. Furthermore, we find that the flow dynamics is strongly modulated by the stenosis geometry, Reynolds number and flow types, i.e., whether it is steady or pulsatile.

Scott-Blair model with unequal diffusivities of chemical species through a Forchheimer medium

Owing to the fact that fractional viscoelastic models contribute significantly with application point of view, a good number of researchers have diverted their attention to this area. In the current study, we have carried out a numerical investigation of the fractional viscoelastic fluid model with unsteady convection, on an inclined plane, via Forchheimer medium. Furthermore, we have presented the generalized Scott-Bliar model to Caput-type fractional derivatives with homogeneous-heterogeneous reactions in the constitutive relations. The viscoelastic features of the proposed model are explored by introducing three memory parameters. The homogeneous-heterogeneous reactions in the flow domain cause variations in the concentration, leading to one of the long-lasting chemical species. One can witness the homogeneous reactions in the flowing region, whereas the heterogeneous reactions occur at the boundary. A broader cluster of physical and chemical processes is tackled by considering diffusion coefficients of a different order. The well-known finite difference technique combined with the “L1 algorithm” is utilized to discretize the principal nonlinear boundary layer equations. The proposed numerical scheme is being validated by performing error and convergence analysis so that a thorough investigation of the chemical reaction process through the Forchheimer medium in the viscoelastic fluid model may be carried out. The proposed mathematical approach may be regarded as a valuable technique for studying the viscoelastic model and chemical reaction processes in a fluid to formulate schemes dealing with unsteady convection. Resultantly, this model can be utilized in the chemical industry.

Mixed convection flow of viscoelastic Ag-Al2O3 /water hybrid nanofluid past a rotating disk

An exploration is carried out to inspect mixed convection flow of viscoelastic hybrid nanofluid past a rotating disk under slip and convective heating condition influences. As the hybrid nanoparticles, and silver (Ag) are considered with carboxymethyl cellulose (CMC) - water with low concentration () preferred as a base fluid. The viscoelastic (non-Newtonian) fluid model is assumed in favor of hybrid nanofluids applying magnetic field influences normal to the flow of fluid. The nonlinear ordinary differential equations get a hold from the governing equations are simplified using suitable similitude into dimensionless from and are solved via the influential method called Galerkin finite element method. The roles of physical parameters on radial and tangential velocities, temperature and concentration are exhibited graphically with their physical features. The results show that enrichment in the values of Grashof number be inclined to develop buoyancy forces which speed up the motion of fluid and tends to increases radial and tangential velocity fields but it imposes to decline temperature and concentration profile. Also, the outcome confirms that the distribution of temperature and concentration can be controlled with higher alumina and silver nanoparticles volume fraction. Moreover, the effects of thermal Grashof number, volume fraction of alumina and silver nanoparticles on skin friction coefficients, Nusselt number and Sherwood number are numerically discussed through tables. It also corroborates that 3% vol. fraction of and Ag nanoparticles has the greatest − than 1% vol. fraction of and Ag nanoparticles.

Effects of blockage and fluid inertia on drag and heat transfer of a solid sphere translating in FENE-P viscoelastic fluids in a tube

An extensive numerical investigation of the flow and heat transfer phenomena of a solid sphere translating in a cylindrical tube filled with FENE-P viscoelastic fluids is reported herein. The governing equations, namely, mass, momentum, energy, and viscoelastic constitutive equations, have been solved over the following ranges of conditions: Reynolds number, 1≤Re≤100, Weissenberg number, 0≤Wi≤10, polymer extensibility parameter, 10≤L2≤500 and blockage ratio, 0≤BR≤0.7 for a fixed value of the polymer viscosity ratio β=0.5 and Prandtl number Pr = 10. Limited simulations with the FENE-CR viscoelastic fluid model have also been carried out to make a comparison between the two viscoelastic models. At low Reynolds numbers, the velocity overshoot and/or negative wake downstream the sphere has been observed under appropriate conditions. This tendency of their appearing decreases with the increasing Reynolds number, and decreasing blockage ratio, polymer extensibility parameter and Weissenberg number. The size of the recirculation region (wake length) increases with the Weissenberg number at low values of the polymer extensibility parameter, whereas a reverse trend is seen at high values of L2. The drag coefficient decreases with the Reynolds and Weissenberg numbers, whereas it increases with the blockage ratio. On the other hand, the average Nusselt number always increases with the Reynolds number irrespective of the values of Wi, L2 and BR. However, the corresponding effect of the blockage ratio and polymer extensibility parameter is seen to be more complex, modulated by the values of Re and Wi. For instance, at high Reynolds numbers, the average Nusselt number always increases with the blockage ratio; however, at low values of it, there is a critical value of the blockage ratio present up to which the average Nusselt number increases, and beyond that, it decreases. Furthermore, the average Nusselt number always initially increases up to a certain value of the Weissenberg number, and after that, it remains almost constant or decreases depending upon the values of L2, BR and Re. Finally, simple correlations for the average Nusselt number and drag ratio are presented, which not only capture the functional dependence of the governing parameters, but also can be used for the interpolation of the present results for the intermediate values of the governing parameters in a new application.

Thermal performance of GO-MoS2/ engine oil as Maxwell hybrid nanofluid flow with heat transfer in oscillating vertical cylinder

Engine oil (EO) is used as a lubricant in the engines of different machineries. The basic need of all phenomena is the rate of heat transfer. To enhance the rate of heat transfer and to save the energy wasted due to high temperature. For this reason in the present study we have taken engine oil as base fluid and molybdenum disulphide and graphene oxide (MoS2 + GO) hybrid nano-composites are suspended in the (EO). Furthermore, the nonlinear nature of viscoelastic non-Newtonian fluids, introduce a unique challenge to physicists and mathematicians. In the past three decades, viscoelastic fluid models are focused to improve its accuracy and reliability. In this article, viscoelastic Maxwell (MoS2 + GO) hybrid nanofluid (MHNF) is considered in oscillating cylindrical tube together with heat transfer. Exact solutions are obtained by using the joint applications of the Laplace and Hankel transforms and the obtained results are portrayed through different figures. All the figures of the given flow model are constructed for unitary nanofluid (MoS2 + EO) as well as hybrid nanofluid (GO + MoS2 + EO). Effects of flow parameters on Maxwell fluid velocity have shown through graph using computational software MATHCAD. From the present study, we have concluded that hybrid nanofluid gives us more satisfactory results than unitary nanofluid. During this analysis we found that the Maxwell hybrid nanofluid (GO + MoS2 + EO) enhance the rate of heat transfer up to 23.17 %. Furthermore, it is worth noting that engine oil has many engineering and industrial applications. Keeping this fact in mind the present study will help to enhance the rate of heat transfer due to which working machines will do better performance and the loss of useful energy will be reduced. Finally, we have present a limiting case by putting Maxwell fluid parameter (λ = 0) our solutions reduced to well-known published results which validate our work.