Oldroyd-B Fluid Research Articles

Analytical study of Cattaneo–Christov heat flux model for a boundary layer flow of Oldroyd-B fluid**Project supported by the Deanship of Scientific Research (DSR) King Abdulaziz University, Jeddah, Saudi Arabia (Grant No. 32-130-36-HiCi).

We investigate the Cattaneo–Christov heat flux model for a two-dimensional laminar boundary layer flow of an incompressible Oldroyd-B fluid over a linearly stretching sheet. Mathematical formulation of the boundary layer problems is given. The nonlinear partial differential equations are converted into the ordinary differential equations using similarity transformations. The dimensionless velocity and temperature profiles are obtained through optimal homotopy analysis method (OHAM). The influences of the physical parameters on the velocity and the temperature are pointed out. The results show that the temperature and the thermal boundary layer thickness are smaller in the Cattaneo–Christov heat flux model than those in the Fourier’s law of heat conduction.

Flow dynamics of a viscoelastic fluid squeezed and extruded between two parallel plates

We investigate the flow dynamics of a viscoelastic fluid squeezed and extruded between two parallel plates. We use the Oldroyd-B fluid model to describe the fluid rheology, in an effort to capture the essential flow features. We unveil the force distribution in the flow-field, as a function of the extrusion and squeezing parameters. We attribute alterations in these forces as a combined consequence of the stress modifications under the simultaneous action of squeezing and extrusion and its interplay with the elastic stress originating from the viscoelastic effects. We believe that the inferences obtained from the present study may bear significant consequences in the design of various processes such as compression moulding, for potential use in microscale and nanoscale patterning.

Model of heap formation in vibrated gravitational suspensions.

In vertically vibrated dense suspensions, several localized structures have been discovered, such as heaps, stable holes, expanding holes, and replicating holes. Because an inclined free fluid surface is difficult to maintain because of gravitational pressure, the mechanism of those structures is not understood intuitively. In this paper, as a candidate for the driving mechanism, we focus on the boundary condition on a solid wall: the slip-nonslip switching boundary condition in synchronization with vertical vibration. By applying the lubrication approximation, we derived the time evolution equation of the fluid thickness from the Oldroyd-B fluid model. In our model we show that the initially flat fluid layer becomes unstable in a subcritical manner, and heaps and convectional flow appear. The obtained results are consistent with those observed experimentally. We also find that heaps climb a slope when the bottom is slightly inclined. We show that viscoelasticity enhances heap formation and climbing of a heap on the slope.

On Cattaneo–Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous–heterogeneous reactions

This paper investigates the steady two-dimensional magnetohydrodynamic (MHD) flow of an Oldroyd-B fluid over a stretching surface with homogeneous–heterogeneous reactions. Characteristics of relaxation time for heat flux are captured by employing new heat flux model proposed by Christov. A system of ordinary differential equations is obtained by using suitable transformations. Convergent series solutions are derived. Impacts of various pertinent parameters on the velocity, temperature and concentration are discussed. Analysis of the obtained results shows that fluid relaxation and retardation time constants have reverse behavior on the velocity and concentration fields. Also temperature distribution decreases for larger values of thermal relaxation time.

Dynamic wetting of viscoelastic droplets.

We conduct numerical experiments on spreading of viscoelastic droplets on a flat surface. Our work considers a Giesekus fluid characterized by a shear-thinning viscosity and an Oldroyd-B fluid, which is close to a Boger fluid with constant viscosity. Our results qualitatively agree with experimental observations in that both shear thinning and elasticity enhances contact line motion, and that the contact line motion of the Boger fluid obeys the Tanner-Voinov-Hoffman relation. Excluding inertia, the spreading speed shows strong dependence on rheological properties, such as the viscosity ratio between the solvent and the polymer suspension, and the polymeric relaxation time. We also discuss how elasticity can affect contact line motion. The molecular migration theory proposed in the literature is not able to explain the agreement between our simulations and experimental results.

Unsteady rotational flows of an Oldroyd-B fluid due to tension on the boundary

Unsteady Taylor–Couette flows of an Oldroyd-B fluid, which fills a straight circular cylinder of radius R, are studied. Flows are generated by the oscillating azimuthal tension which is given on the cylinder surface. As a novelty, authors used in this paper the governing equation related to the tension field. The closed forms of the shear stress and velocity fields corresponding to the flow problems are obtained by means of the integral transforms method. Expressions for the azimuthal tension and fluid velocity were written as sums between the “permanent component” (the steady-state component) and the transient component. By customizing values of parameters from the mathematical model were obtained the corresponding solutions of other types of fluids, namely, Maxwell fluids. By using numerical simulations and diagrams of the azimuthal stress, the fluid behavior has been analyzed. The necessary time to achieve the “steady-state” was, also, determined.

The rheology and microstructure of branched micelles under shear

The rheology and shear-induced structures of a series of self-assembled surfactant wormlike micelles (WLMs) with varying levels of branching are measured using rheo- and flow-small angle neutron scattering (SANS). The degree of branching in the mixed cationic/anionic surfactant (cetyltrimethylammonium tosylate/sodium dodecyl benzene sulfonate) WLMs is controlled via the addition of the hydrotropic salt sodium tosylate and verified by cryo-transmission electron microscopy. The linear viscoelasticity of the low salt (linear) micellar solutions is well described as an extended Maxwell (Oldroyd-B) fluid, and samples exhibit shear banding under steady-shear flow. The linear viscoelasticity of more highly branched solutions deviates from Maxwellian behavior, where the plateau in G′ gradually increases in slope with increasing salt content. The higher salt solutions exhibit a shear thinning regime, followed by a shear thickening regime at high shear rates. Micelle segmental alignment in the flow-gradient plane i...

Lattice Boltzmann method for Oldroyd-B fluids

A lattice method for modeling a solution of dumbbells in a Newtonian fluid, introduced by Onishi et al. (2005) [1], is implemented within an existing lattice Boltzmann framework (Körner et al. (2005)) [2]. The numerical scheme is compared with the analytical solution of steady state and start up simple shear flow and an oscillating velocity field under uniform motion. Furthermore, a new transport model is introduced and validated for an oscillating velocity field under uniform motion. Both tests are in good agreement with the analytical solution. Additionally, a 4 to 1 planar contraction is simulated to test if the method is capable of reproducing complex viscoelastic flows. A lip vortex at the entrance corner is observed. A second benchmark is the four-roller setup. The overall results are in good agreement with previous studies.

Numerical solution of viscous and viscoelastic fluids flow through the branching channel by finite volume scheme

This work deals with the numerical modelling of steady flows of incompressible viscous and viscoelastic fluids through the three dimensional channel with T-junction. The fundamental system of equations is the system of generalized Navier-Stokes equations for incompressible fluids. This system is based on the system of balance laws of mass and momentum for incompressible fluids. Two different mathematical models for the stress tensor are used for simulation of Newtonian and Oldroyd-B fluids flow. Numerical solution of the described models is based on cetral finite volume method using explicit Runge-Kutta time integration.

Numerical Investigation of the Effect of Unsteadiness on Three-Dimensional Flow of an Oldroyb-B Fluid.

A spectral relaxation method used with bivariate Lagrange interpolation is used to find numerical solutions for the unsteady three-dimensional flow problem of an Oldroyd-B fluid with variable thermal conductivity and heat generation. The problem is governed by a set of three highly coupled nonlinear partial differential equations. The method, originally used for solutions of systems of ordinary differential equations is extended to solutions of systems of nonlinear partial differential equations. The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs. Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate. Numerical analysis of the error and convergence properties of the method are also discussed. One of the benefits of the proposed method is that it is computationally fast and gives very accurate results after only a few iterations using very few grid points in the numerical discretization process.

The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow

The viscoelastic analogue to the Newtonian Orr amplification mechanism is examined using linear theory. A weak, two-dimensional Gaussian vortex is superposed onto a uniform viscoelastic shear flow. Whilst in the Newtonian solution the spanwise vorticity perturbations are simply advected, the viscoelastic behaviour is markedly different. When the polymer relaxation rate is much slower than the rate of deformation by the shear, the vortex splits into a new pair of co-rotating but counter-propagating vortices. Furthermore, the disturbance exhibits a significant amplification in its spanwise vorticity as it is tilted forward by the shear. Asymptotic solutions for an Oldroyd-B fluid in the limits of high and low elasticity isolate and explain these two effects. The splitting of the vortex is a manifestation of vorticity wave propagation along the tensioned mean-flow streamlines, while the spanwise vorticity growth is driven by the amplification of a polymer torque perturbation. The analysis explicitly demonstrates that the polymer torque amplifies as the disturbance becomes aligned with the shear. This behaviour is opposite to the Orr mechanism for energy amplification in Newtonian flows, and is therefore labelled a ‘reverse-Orr’ mechanism. Numerical evaluations of vortex evolutions using the more realistic FENE-P model, which takes into account the finite extensibility of the polymer chains, show the same qualitative behaviour. However, a new form of stress perturbation is established in regions where the polymer is significantly stretched, and results in an earlier onset of decay.

Lees–Edwards boundary condition for simulation of polymer suspension with dissipative particle dynamics method

The Lees–Edwards boundary condition (LEbc) is widely used in particle-based simulation for producing shear flow. Application of traditional LEbc in dissipative particle dynamics (DPD) method may encounter certain problems, e.g. it will destroy the momentum conservation law at the near boundary region, and the coordinate system gives an incorrect end-to-end vector for polymer beads. Special treatments of the implementation of LEbc in DPD method are introduced in this paper. A single side ghost layer is used to keep the momentum conservation, and the global coordinate system is employed to obtain a correct calculation of the spring force between polymer beads. The simulation results give a good prediction of velocity profile and system temperature, and the elastic dumbbell model for current method can well represent the Oldroyd-B fluid.

Unsteady MHD Thin Film Flow of an Oldroyd-B Fluid over an Oscillating Inclined Belt.

This paper studies the unsteady magnetohydrodynamics (MHD) thin film flow of an incompressible Oldroyd-B fluid over an oscillating inclined belt making a certain angle with the horizontal. The problem is modeled in terms of non-linear partial differential equations with some physical initial and boundary conditions. This problem is solved for the exact analytic solutions using two efficient techniques namely the Optimal Homotopy Asymptotic Method (OHAM) and Homotopy Perturbation Method (HPM). Both of these solutions are presented graphically and compared. This comparison is also shown in tabular form. An excellent agreement is observed. The effects of various physical parameters on velocity have also been studied graphically.

Temperature and Concentration Stratification Effects in Mixed Convection Flow of an Oldroyd-B Fluid with Thermal Radiation and Chemical Reaction.

This research addresses the mixed convection flow of an Oldroyd-B fluid in a doubly stratified surface. Both temperature and concentration stratification effects are considered. Thermal radiation and chemical reaction effects are accounted. The governing nonlinear boundary layer equations are converted to coupled nonlinear ordinary differential equations using appropriate transformations. Resulting nonlinear systems are solved for the convergent series solutions. Graphs are plotted to examine the impacts of physical parameters on the non-dimensional temperature and concentration distributions. The local Nusselt number and the local Sherwood number are computed and analyzed numerically.

Comment on “The velocity field due to an oscillating plate in an Oldroyd-B fluid” by C.C. Hopkins and J.R. de Bruyn [Can. J. Phys. 92, 533 (2014)

We correct certain errors and ambiguities in the recent pedagogical article by Hopkins and de Bruyn (Can. J. Phys. 92, 533 (2014). doi:10.1139/cjp-2013-0334 ). The early-time asymptotics of the solution to the transient version of Stokes’ second problem for an Oldroyd-B fluid in a half-space is presented, as Appendix A , to complement the late-time asymptotics given by Hopkins and de Bruyn.

Pulsatile Flow of Blood in a Vessel Using an Oldroyd-B fluid

Abstract This paper is concerned with the effects of body acceleration on an unsteady pulsatile flow of blood in a vessel. The non-Newtonian behavior of blood is modeled by the constitutive equation of an Oldroyd-B fluid. The behavior of the blood is assumed as an Oldroyd-B fluid in the core and a viscous fluid in the periphery. This new model predicts the relaxation and retardation phenomena in a pulsatile flow. Our interest lies in investigating the effects of stress relaxation and retardation in unsteady pulsatile flow of an Oldroyd-B fluid. The modeled equations are normalized and solved numerically using finite difference method. The results are graphically displayed and influence of the pertinent parameter on various physical quantities of interest is analyzed.

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

A new streamfunction/log-conformation formulation of incompressible viscoelastic flows is presented. The log-conformation representation guaranties the positive-definiteness of the conformation tensor and obviates the high Weissenberg number problem. The streamfunction is defined as a vector potential of the velocity field, and provides a pressureless formulation of the conservation laws, which automatically enforces the incompressibility. The resulting numerical method is free from velocity–pressure decoupling errors, and can achieve stable calculations for large Courant numbers, which improve the robustness and the efficiency of the solver. The two-dimensional flow of an Oldroyd-B fluid inside the lid-driven cavity is simulated for a large range of Weissenberg numbers. The numerical results demonstrate the second-order accuracy of our scheme, and our solutions are in good agreement with the available data from the literature for Weissenberg number 3 and below. Finally, the simulations at higher Weissenberg numbers 5 and 10 reveal a structural mechanism that sustains quasi-periodic elastic instabilities arising at the upstream corner of the moving lid.

On Hydromagnetic Flow of an Oldroyd-B Fluid Between Two Oscillating Plates

An initial value investigation is made of the motion of an incompressible, viscoelastic, electrically conducting Oldroyd-B fluid bounded by two infinite rigid non-conducting plates. The flow is generated impulsively from rest in the fluid due to rectilinear oscillations of given frequencies superimposed on the plates in their own planes in presence of an external magnetic field acting transversely to the plates. The operational method is used to derive exact solutions for the fluid velocity and the shear stress on the walls. The quantitative evaluation of the results is considered when two plates oscillate in phase but with different frequencies. The results are shown graphically for different time periods of oscillations of the plates which represent the cases: (i) the lower plate oscillates with a time period less than the upper, (ii) both the plates oscillate with the same time period and (iii) the lower plate oscillates with a time period grater than the upper. It is seen that the effect of fluid elasticity on the flow depends on the advancing and retarding motion of the plates. On the other hand, the magnetic field damps the fluid motion for all values of the time period of oscillations of the plates. The drag on the plates, for small and large time, are shown graphically when the time period of oscillation of the lower plate is small. For small values of time, the drag on the lower plate is negative with the amplitude decreasing continuously till the oscillations occur from moderately large values of time. Similar phenomenon also occurs for the drag which is positive at the upper plate for small values of time. In all cases, the amount of drag on the plates increases with the increase of the elasticity of the fluid and the magnetic field. Some particular results for the fluid velocity are derived as special cases of the present solution.

Simultaneous effects of viscoelasticity and curvature on peristaltic flow through a curved channel

We have analyzed peristaltic flow of an Oldroyd-B fluid in a curved channel. Assuming the flow to be incompressible, laminar and two-dimensional, the governing partial differential equations are reduced under long wavelength and low Reynolds number approximations into a single nonlinear ordinary differential equation in the stream function. Matlab built-in routine bvp4c is utilized to solve this nonlinear ordinary differential equation. The solution thus obtained is used to investigate the effects of curvature of the channel and Weissenberg number on important phenomena of pumping and trapping associated with peristaltic motion. It is found that for small values of Weissenberg number, the effects of curvature are dominant. However, for large values of Weissenberg number, viscoelastic effects counteract the effects of curvature and help the flow velocity and circulating bolus of fluid to regain their symmetry.

The simpler GMRES method combined with finite volume method for simulating viscoelastic flows on triangular grid

An efficient solver integrating the restarted simpler generalized minimal residual method (SGMRES(m)) with finite volume method (FVM) on triangular grid is developed to simulate the viscoelastic fluid flows. In particular, the SGMRES(m) solver is used to solve the large-scale sparse linear systems, which arise from the course of FVM on triangular grid for modeling the Newtonian and the viscoelastic fluid flows. To examine the performance of the solver for the nonlinear flow equations of viscoelastic fluids, we consider two types of numerical tests: the Newtonian flow past a circular cylinder, and the Oldroyd-B fluid flow in a planar channel and past a circular cylinder. It is shown that the numerical results obtained by the SGMRES(m) are consistent with the analytical solutions or empirical values. By comparing CPU time of different solvers, we find our solver is a highly efficient one for solving the flow equations of viscoelastic fluids.