Giesekus Fluid Research Articles

A material point method for simulation of viscoelastic flows

A novel variant of the Material Point Method is proposed to solve flow problems for incompressible viscoelastic fluids. As in the original spirit of the method, material points possess all necessary information about the material constitutive behavior and move with the flow. They are additionally employed as integration points, whose weights are computed as the system evolves in order to ensure exact integration of the discretized conservation equations. The method is shown to achieve quadratic convergence for transient start-up flow of the Oldroyd-B fluid. In addition, we apply the method to flow of a Giesekus fluid in concentric rotating cylinders, and in flow through an abrupt 4:1 contraction, comparing favorably to available analytical, experimental, and simulation results as appropriate.

Numerical solution of the Giesekus model for incompressible free surface flows without solvent viscosity

We present a numerical method for solving the Giesekus model without solvent viscosity. This paper is concerned with incompressible two-dimensional free surface flows and employs the finite difference method to solve the governing equations. The methodology involves solving the momentum equation using the implicit Euler scheme and an implicit technique for computing the pressure condition on the free surface. The nonlinear Giesekus constitutive equation is computed by a second order Runge–Kutta method. A novel analytic solution for channel flow is developed and is used to verify the numerical technique presented herein. Mesh refinement studies establish the convergence of the method for complex free surface flows. To demonstrate that the technique can deal with complicated free surface flows, the time-dependent flow produced by a fluid jet flowing onto a rigid surface is simulated and the influence of the parameter α on the jet buckling phenomenon is investigated. In addition, the simulation of the extrudate swell of a Giesekus fluid was carried out and the effect of the parameter α on the flow was similarly examined.

Forced convection heat transfer of Giesekus fluid with wall slip above the critical shear stress in pipes

Forced convective heat transfer in pipes is investigated for viscoelastic fluids obeying the Giesekus constitutive equation including effect of slip condition by an approximated analytical method. The slip equation at wall is considered as a nonlinear Navier model with non-zero slip critical shear stress. The problem under consideration is steady, laminar and fully developed. Thermal boundary conditions are assumed peripherally and axially constant heat flux at wall. The fluid heating and cooling cases are considered for analysis. Dimensionless temperature profiles and Nusselt number are obtained by solving governing equations and the effects of slip parameters, viscous dissipation and fluid elasticity are discussed. Results show that Nusselt number increases by increasing slip effect but decreases by increasing Brinkman number for the case of fluid heating. However, for the cooling case, the heat generated by viscous dissipation can overcome the effect of wall cooling at first critical Brinkman number and fluid starts to warm up. Also the Nusselt curve shows a singularity in a second critical Brinkman number.

A numerical study on nonlinear dynamics of three-dimensional time-depended viscoelastic Taylor-Couette flow

In this paper, three-dimensional viscoelastic Taylor-Couette instability between concentric rotating cylinders is studied numerically. The aim is to investigate and provide additional insight about the formation of time-dependent secondary flows in viscoelastic fluids between rotating cylinders. Here, the Giesekus model is used as the constitutive equation. The governing equations are solved using the finite volume method (FVM) and the PISO algorithm is employed for pressure correction. The effects of elasticity number, viscosity ratio, and mobility factor on various instability modes (especially high order ones) are investigated numerically and the origin of Taylor-Couette instability in Giesekus fluids is studied using the order of magnitude technique. The created instability is simulated for large values of fluid elasticity and high orders of nonlinearity. Also, the effect of elastic properties of fluid on the time-dependent secondary flows such as wave family and traveling wave and also on the critical conditions are studied in detail.

Analytical solution for axial flow of a Giesekus fluid in concentric annuli

An exact analytical solution for a Poiseuille flow of a Giesekus fluid in an annulus is obtained, taking into account the effects of the non-linearity of the constitutive equation. The fluid velocity is given in terms of Legendre elliptic integrals and Jacobi elliptic functions. The fluid behaviour is described using the Deborah number along the mobility factor and the radius ratio, while the tangential and the normal stresses are also evaluated.The shear-rate, the velocity and the stress tensor components as function of the involved parameters are graphically represented and analysed in detail.

Stresses of PTT, Giesekus, and Oldroyd-B fluids in a Newtonian velocity field near the stick-slip singularity

We characterise the stress singularity of the Oldroyd-B, Phan-Thien–Tanner (PTT), and Giesekus viscoelastic models in steady planar stick-slip flows. For both PTT and Giesekus models in the presence of a solvent viscosity, the asymptotics show that the velocity field is Newtonian dominated near to the singularity at the join of the stick and slip surfaces. Polymer stress boundary layers are present at both the stick and slip surfaces. By integrating along streamlines, we verify the polymer stress behavior of r−4/11 for PTT and r−5/16 for Giesekus, where r is the radial distance from the singularity. These asymptotic results for PTT and Giesekus do not hold in the limit of vanishing quadratic stress terms for Oldroyd-B. However, we can consider the Oldroyd-B model in the fixed kinematics of a prescribed Newtonian velocity field. In contrast to PTT and Giesekus, this is not the correct balance for the momentum equation but does allow insight into the behavior of the Oldroyd-B equations near the singularity. A three-region asymptotic structure is again apparent with now a polymer stress singularity of r−4/5. The high Weissenberg boundary layer equations are found to manifest themselves at the stick surface and are of thickness r3/2. At the slip surface, dominant balance between the upper convected stress and rate-of-strain terms gives a slip boundary layer of thickness r2. The solution of the slip boundary layer shows that the polymer stress is now singular along the slip surface. These results are supported through numerical integration along streamlines of the Oldroyd-B equations in a Newtonian velocity field. The Oldroyd-B model thus extends the point singularity at the join of the stick and slip surfaces to the whole of slip surface. As such, it does not have a physically meaningful solution in a Newtonian velocity field. We would expect a similar stress behavior for this model in the true viscoelastic velocity field.

Flow and heat transfer of a Giesekus fluid in plane and 3D ducts

AbstractThis study aims to investigate the Graetz problem of Newtonian and viscoelastic fluid obeying Giesekus model using ANSYS Polyflow solver. The non‐isothermal flow in straight ducts of circular and noncircular cross‐sections under the constant heat flux boundary conditions is considered. The effect of the mobility parameter (α), fluid elasticity defined by Weissenberg number (We) and Reynolds number (Re) on the flow field, secondary flows, and the fully developed and developing Nusselt number along the ducts length are investigated for all geometries. The obtained results are of great importance for practical application in the polymer industries such as polymer melt.

Heat transfer for a Giesekus fluid in a rotating concentric annulus

Annular flow of a viscoelastic fluid described by the Giesekus model has received some attention over the years, both concerning the fluid mechanical and thermal aspects, yet no investigation has been carried out using the analytical solution for the velocity profile in the energy equation to determine temperature distribution and heat transfer characteristics. Moreover, viscous dissipation, when accounted for, is usually defined by a formulation of the Brinkman number which proves inconsistent when applied to non-Newtonian fluids.In this work the purely tangential flow of a Giesekus fluid in an annulus with rotating inner wall subject to thermal boundary conditions of the first kind (imposed temperature) at the walls is investigated employing the analytical solution for the velocity profile. Viscous dissipation is accounted for by first deriving a consistent Brinkman number which is easily related to the one usually employed in previous studies and the temperature profiles are obtained for different values of the Deborah number and the non-dimensional mobility factor. Results for the Nusselt numbers at the inner and outer wall are also discussed.

Numerical simulations of deformable particle lateral migration in tube flow of Newtonian and viscoelastic media

The deformation and cross-streamline migration of an elastic particle in pressure-driven flows of Newtonian and viscoelastic (Oldroyd-B, Giesekus) fluids in a cylindrical tube are studied through 3D finite element method numerical simulations. The dependences of particle deformation and migration on geometric confinement, flow strength, and fluid rheology are investigated. If the particle is initially not at the channel axis, it attains an asymmetric shape and migrates. In a Newtonian liquid, the migration is always directed towards the tube axis. A project equation is proposed for the design of a microfluidic cylindrical device aimed at focusing elastic particles on the cylinder centerline. In a viscoelastic liquid, the migration direction and velocity depend on the competition among particle deformability, fluid elasticity, and fluid viscosity shear thinning (if any). In a certain range of parameters, an unstable radial position appears, which separates the region where the migration is directed towards the axis from the region where it is directed towards the wall.

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.

Numerical study of vortex shedding in viscoelastic flow past an unconfined square cylinder

In this paper, the periodic viscoelastic shedding flow of Giesekus fluid past an unconfined square cylinder is investigated numerically for the first time. The global quantities such as lift coefficient, Strouhal number and the detailed kinetic and kinematic variables like normal stress differences and streamlines have been obtained in order to investigate the flow pattern of viscoelastic flow. The effects of Reynolds number and polymer concentrations have been clarified in the periodic viscoelastic flow regime. Our particular interest is the effect of mobility parameter on the stability of two dimensional viscoelastic flows past an unconfined square cylinder. To fulfill this aim, the mobility parameter has been increased from 0 to 0.5 for different polymer concentrations. Results reveal that mobility factor noticeably affects the amplitude of lift coefficient and shedding frequency more strongly at higher polymer concentrations.

Analytical solution for a Couette flow of a Giesekus fluid in a concentric annulus

The present work proposes an exact analytical solution for a Couette flow of a Giesekus fluid in an annulus, taking into account the effects of the non-linearity of the constitutive equation. The fluid velocity is given as a function of the Deborah number, the mobility factor and the radius ratio. Analysis shows that if the mobility factor is greater than 0.5, there exists a limit (maximum) Deborah number above which the studied phenomenon has no valid solution. The normal, the tangential and the azimuthal stresses are evaluated; the ratio between the torque required to rotate the inner cylinder for a Giesekus and for a Newtonian fluid is investigated. Results show that an increase of the fluid elasticity and/or of the mobility factor, leads to a decrease of the friction factor ratio for all values of the radius ratio. Furthermore, the behaviour of the velocity and of the stress tensor components as function of the involved parameters is graphically represented and discussed in detail.

Axial annular flow of a Giesekus fluid with wall slip above the critical shear stress

Analytical solutions for axial annular flow of viscoelastic fluid obeying the Giesekus model are obtained including effect of slip condition at both walls. A power-law relation between wall shear stress and wall slip velocity called nonlinear Navier model was employed for the slip equation. The wall slip condition is assumed to be the same for both walls because both walls have similar properties. For the slip model it is considered that wall slip occurs only when a critical value is reached which is known as slip critical shear stress. Results indicate that slip initially occurs at the inner wall and then at the outer wall. Thus there are three flow regimes: no slip condition, slip only at the inner wall and slip at both walls. The flow regime is characterized by the value of slip critical shear stress. Results show that increasing slip effect causes pressure gradient, shear and normal stresses to decrease but, they increase by increasing slip critical shear stress. Also increasing elasticity decreases slip velocity at walls.

Determination of the shear and extensional rheology of bubbly liquids with a shear-thinning continuous phase

The effect of bubble volume fraction, ϕ, and bubble size on the rheology of rheology of κ/ι-hybrid carrageenan gum solutions and bubbly liquids with ϕ ≤ 0.25 was studied in steady shear using a parallel plate configuration and in extension using a filament stretching device. Three different bubble size distributions were prepared using (1) planetary mixing (mean bubble size 90–130 μm), (2) mixing followed by pumping through a symmetric syringe (25–38 μm), and (3) as (2) but using an asymmetric syringe (12–20 μm). Gum concentrations of 10 and 20 g/L, lying in the semi-dilute regime, were studied. The unaerated gum solutions are viscoelastic and could be modelled very well as a single-mode Giesekus fluid. The bubbly liquid behaviour in both shear and extension could also be modelled as a single-mode Giesekus fluid. The Giesekus parameters for the bubbly liquids were related to those of the unaerated solutions by monatonic increasing dependencies on ϕ. The low shear rate apparent viscosity followed the trend predicted by the Choi and Schowalter (Phys Fluids 18: 420, 1975) model for emulsions, while the mobility parameter and relaxation time were linearly dependent on ϕ. In these tests bubble size only affected filament extension noticeably, influencing the mobility parameter. Further work is required to explain the observations in terms of micromechanical models.

Effect of bubble volume fraction on the shear and extensional rheology of bubbly liquids based on guar gum (a Giesekus fluid) as continuous phase

The effect of air bubble volume fraction, ϕ, on the steady shear and extensional rheology of aqueous guar gum solutions was studied at 0⩽ϕ⩽0.25 and gum concentrations of (i) 5g/L and (ii) 10g/L, corresponding to solutions in the (i) semi-dilute and (ii) entanglement regime. The rheological response of the fluids was largely independent of bubble size but strongly dependent on ϕ. The viscous and elastic moduli increased with increasing bubble volume fraction, with elastic dominance prevalent at the higher gum concentration. Extensional rheometry, investigated using filament stretching, revealed that the thinning dynamics of the liquid thread were affected by bubble size, but the filament rupture time was primarily dependent on ϕ. The rheological behaviour in both shear and extension could be modelled as a single mode Giesekus fluid, with a single set of parameters able to describe both the shear and extensional behaviour in the semi-dilute regime. In the entanglement regime the single mode Giesekus fluid could fit the shear data or the extensional data individually, but not both. The fitted Giesekus fluid model parameters exhibited a strong dependency on ϕ, offering a way to predict the flow behaviour of these complex food fluids.

Natural Giesekus fluids: Shear and extensional behavior of food gum solutions in the semidilute regime

The shear and extensional behavior of two aqueous gum solutions, namely (1) 1–20 g/L guar gum (Torres et al., Food Hydrocolloids. 2014;40:85–95) and (2) κ/ι‐hybrid carrageenan solutions (5–20 g/L), are shown to exhibit Giesekus‐fluid behavior when in the semidilute regime. In this regime, a common set of Giesekus fluid parameters described both shear and extensional behavior. A new analytical result describing the extension of a Giesekus fluid in the filament stretching geometry is presented. This also gave reasonable predictions of the Trouton ratio. Higher concentration guar solutions, in the entangled regime, yielded different Giesekus fluid parameters for extension to those for simple shear. The extensional data for all concentrations of both gums collapsed to a common functional form, similar to that reported for cake batters (Chesterton et al., J Food Eng. 2011;105(2):332–342); the limits of the new filament thinning expression provide insight into this behavior. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3902–3915, 2014

Stick-slip singularity of the Giesekus fluid

The local asymptotic behaviour at the stick-slip singularity is determined for the Giesekus fluid in the presence of a solvent viscosity. In planar steady flow, the method of matched asymptotic expansions is used to show that it comprises a three region structure. Specifically, an outer or core region that links boundary layers at the rigid stick and free slip surfaces. In the outer region, the velocity field is shown to be Newtonian at leading order, with solvent stresses dominating the polymer stresses. In terms of the radial distance r from the singularity at the join of the stick and slip surfaces, the velocity field vanishes as O(r12). Consequently, the singular velocity gradients and solvent stresses are of O(r-12) with the less singular polymer stresses being shown to be O(r-516). The solvent and polymer stresses become comparable near the rigid stick and free slip surfaces, where boundary layers are required. These are of thickness O(r54) at the rigid stick surface and thickness O(r1714) at the free slip surface. Solutions are constructed for both stick-slip and slip-stick flow regimes. These asymptotic results do not hold for the Oldroyd-B model nor for the case when the solvent viscosity is absent.

Prandtl boundary layers for the Phan-Thien Tanner and Giesekus fluid

The Prandtl equations, arising naturally in the description of high Reynolds number boundary layers, have turned out to be quite difficult from the point of view of mathematical analysis. Recent work by the author has shown that the analogous problem for the upper-convected Maxwell fluid is actually better behaved, and the well-posedness of the boundary layer equations has been established. In this paper, boundary layers for the Phan-Thien–Tanner and Giesekus fluid are considered. It turns out that there are two fundamentally different types of boundary layers, which we shall call elastic and viscometric boundary layers. The elastic boundary layers will be analyzed in an analogous fashion as those for the upper-convected Maxwell fluid. On the other hand, for viscometric boundary layers, which occur only for the PTT fluid, the equations are equivalent to those for a power law fluid.

Peristaltic Flow of Giesekus Fluids through Curved Channels: an Approximate Solution

Peristaltic Flow of Giesekus Fluids through Curved Channels: an Approximate Solution

Simulations of deformable systems in fluids under shear flow using an arbitrary Lagrangian Eulerian technique

An arbitrary Lagrangian Eulerian finite element method based numerical code for viscoelastic fluids using well-known stabilization techniques (SUPG, DEVSS, log-conformation) is adapted to perform a 3D study of soft systems (drops, elastic particles) suspended in Newtonian and viscoelastic fluids under unbounded shear flow. Since the interface between the suspended objects and the matrix needs to be tracked, a finite element method with SUPG stabilization and second-order time discretization is defined on the interface, with the normal velocity of the interface equal to the normal component of the fluid velocity and a tangential velocity such that the elements on the interface are evenly distributed. This allows the mesh to get rid of the tank-treading motion of the particle. Both drops and elastic particles deform because of the flow and attain stationary deformed shape and orientation with respect to the flow direction. The effects of the physical parameters of the system on the phenomenon are investigated. The code is validated for drops and elastic particles in a Newtonian fluid through comparison with data from literature. New results on the deformation of elastic particles in an Upper Convected Maxwell fluid and a Giesekus fluid are presented.