Spectral calculations of viscoelastic flows: evaluation of the Giesekus constitutive equation in model flow problems (original) (raw)
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Modelling of Shear Banding in Plane Couette Flow of Non-linear Viscoelastic Fluids
International Journal of Advances in Scientific Research and Engineering (ijasre), 2020
We investigate shear banding phenomena in the plane Couette flow of viscoelastic fluids.The viscoelastic fluids are modelled via the Giesekus constitutive equations with stress diffusion. The nonlinear and coupled systems of equations, comprising the momentum equation and the constitutive equations, are solved numerically using semi-implicit finite difference methods. The effects of the fluid parameters on the flow variables are also investigated. Under certain values of the material parameters, we observe formation of shear bands in plane Couette flow.
International Journal for Numerical Methods in Fluids, 1993
The time evolution of finite amplitude axisymmetric perturbations (Taylor cells) to the purely azimuthal, viscoelastic, cylindrical Couette flow was numerically simulated. Two time integration numerical methods were developed, both based on a pseudospectral spatial approximation of the variables, efficiently implemented using fast Poisson solvers and optimal filtering routines. The first method, applicable for finite Re numbers, is based on a time-splitting integration with the divergence-free condition enforced through an influence matrix technique. The second one, is based on a semi-implicit time integration of the constitutive equation with both the continuity and the momentum equations enforced as constraints. Stability results for an upper convected Maxwell fluid were obtained for the supercritical bifurcations, either steady or time-periodic, developed after the onset of instabilities in the primary flow. At small elasticity values, E = De/Re, the time integration of finite amplitude disturbances confirms the stability of the single branch of steady Taylor cells. At intermediate E values the rotating wave family of time-periodic solutions developed at the onset of instability is stable, whereas the standing wave is found to be unstable. At high E values, and in particular for the limit of creeping flow ( E = a ) , the present study shows that the rotating wave family is unstable and the standing (radial) wave is stable, in agreement with previous finite-element investigations. It is thus shown that spectral techniques provide a robust and computationally efficient method for the simulation of complex, non-linear, time-dependent viscoelastic flows.
A purely elastic instability in Taylor–Couette flow
Journal of Fluid Mechanics, 1990
A non-inertial (zero Taylor number) viscoelastic instability is discovered for Taylor-Couette flow of dilute polymer solutions. A linear stability analysis of the inertialess flow of an Oldroyd-B fluid (using both approximate Galerkin analysis and numerical solution of the relevant small-gap eigenvalue problem) show the growth of an overstable (oscillating) mode when the Deborah number exceeds f(S) €7, where E is the ratio of the gap to the inner cylinder radius, and f(S) is a function of the ratio of solvent to polymer contributions to the solution viscosity. Experiments with a solution of 1000 p.p.m. high-molecular-weight polyisobutylene in a viscous solvent show an onset of secondary toroidal cells when the Deborah number De reaches 20, for B of 0.14, and a Taylor number of in excellent agreement with the theoretical value of 21. The critical De was observed to increase as E decreases, in agreement with the theory. At long times after onset of the instability, the cells become small in wavelength compared to those that occur in the inertial instability, again in agreement with our linear analysis. For this fluid, a similar instability occurs in coneand-plate flow, as reported earlier. The driving force for these instabilities is the interaction between a velocity fluctuation and the first normal stress difference in the base state. Instabilities of the kind that we report here are likely to occur in many rotational shearing flows of viscoelastic fluids.
Nonlinear stability analysis of viscoelastic Taylor–Couette flow in the presence of viscous heating
Physics of Fluids, 2002
Recently, based on a linear stability analysis we demonstrated the existence of a new thermoelastic mode of instability in the viscoelastic Taylor-Couette flow ͓Al-Mubaiyedh et al., Phys. Fluids 11, 3217 ͑1999͒; J. Rheol. 44, 1121 ͑2000͔͒. In this work, we use direct time-dependent simulations to examine the nonlinear evolution of finite amplitude disturbances arising as a result of this new mode of instability in the postcritical regime of purely elastic ͑i.e., Reϭ0͒, nonisothermal Taylor-Couette flow. Based on these simulations, it is shown that over a wide range of parameter space that includes the experimental conditions of White and Muller ͓Phys. Rev. Lett. 84, 5130 ͑2000͔͒, the primary bifurcation is supercritical and leads to a stationary and axisymmetric toroidal flow pattern. Moreover, the onset time associated with the evolution of finite amplitude disturbances to the final state is comparable to the thermal diffusion time. These simulations are consistent with the experimental findings.
Stability of rheometric viscoelastic flow
Flow between two plates is considered for a fluid obeying the DeWitt rheological equation of state with the Jaumann derivative. It is found analytically that the steady-state Couette flow is stable or unstable with respect to plane shear perturbations when the Weissenberg numbers are less or greater than unity, respectively. The flow acceleration stage is studied analytically and numerically, a comparison with the case of an Oldroyd fluid is carried out, and the neutral stability curves are constructed. The fundamental role of perturbations of the type considered among the set of instability types which can act on the fluid in such a flow is noted. We note that for all three models the derivatives can be obtained from the expression for the derivative in the generalized Oldroyd model (2.4) [7] when a = 0, 1, and −1, respectively.
A stable and convergent scheme for viscoelastic flow in contraction channels
Journal of Computational Physics, 2005
We present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels. In our approach we split the viscoelastic terms of the Oldroyd-B constitutive equation using DuhamelÕs formula and discretize the resulting PDEs using a semi-implicit finite difference method based on a Lax-Wendroff method for hyperbolic terms. In particular, we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well-posed. A projection method is used to impose the incompressibility constraint. We are able to compute the full range of unsteady elastic flows in an abrupt contraction channel -from the viscous limit to the elastic limit -in a stable and convergent manner. We demonstrate the range of our method for unsteady flow of a Maxwell fluid with and without viscosity in planar contraction channels. We also demonstrate stable and convergent results for benchmark high Weissenberg number problems at We = 1 and We = 10. Published by Elsevier Inc. PACS: 65N06; 76D05
2022
We shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertialess flows with curved streamlines. Then we describe a theory of elastic turbulence and prediction of elastic waves Re << 1 and Wi >> 1, which speed depends on the elastic stress similar to the Alfven waves in magneto-hydrodynamics and in a contrast to all other, which speed depends on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvature, so parallel shear flows are proved linearly stable, similar to Newtonian parallel shear flows. However, the linear stability of parallel shear flows does not imply their global stability. Here we switch to the main subject, namely a recent development in inertia-less parallel shear channel flow of polymer solutions. In such flow, we discover an elastically driven instability, elastic turbulence, elastic waves, and drag reduction down to relaminarization that contradict to the linear stability prediction. In this regard, we discuss shortly normal versus non-normal bifurcations in such flows, flow resistance, velocity and pressure fluctuations, and spatial and spectral velocity as function of Wi at high elasticity number.
Channel, tube, and Taylor–Couette flow of complex viscoelastic fluid models
Rheologica Acta, 2006
We show how to formulate two-point boundary value problems to compute laminar channel, tube, and Taylor-Couette flow profiles for some complex viscoelastic fluid models of differential type. The models examined herein are the Pom-Pom Model [McLeish and Larson 42:81-110, (1998)] the Pompon Model [Öttinger 40:317-321, (2001)] and the Two Coupled Maxwell Modes Model (Beris and Edwards 1994). For the two-mode Upper-Convected Maxwell Model, we calculate analytical solutions for the three flow geometries and use the solutions to validate the numerical methodology. We illustrate how to calculate the velocity, pressure, conformation tensor, backbone orientation tensor, backbone stretch, and extra stress profiles for various models. For the Pom-Pom Model, we find that the two-point boundary value problem is numerically unstable, which is due to the aphysical nonmonotonic shear stress vs shear rate prediction of the model. For the other two models, we compute laminar flow profiles over a wide range of pressure drops and inner cylinder velocities. The volumetric flow rate and the nonlinear viscoelastic material properties on the boundaries of the flow geometries are determined as functions of the applied pressure drop, allowing easy analysis of experimentally measurable quantities.
Taylor-Couette Instability of Giesekus Fluids: Inertia Effects
Nihon Reoroji Gakkaishi, 2012
Taylor-Couette instability of Giesekus fluids is investigated at large gaps using a temporal, linear instability analysis. Having superimposed axisymmetric, normal-mode perturbations to the base flow velocity and stress fields, an eigenvalue problem is obtained which is solved numerically using pseudo-spectral, Chebyshev-based, collocation method. The neutral instability curve is then plotted as a function of the Weissenberg number and also the mobility factor of the Giesekus model. Based on the results obtained in this work, it is concluded that at large gaps, a fluid's elasticity can have a stabilizing or destabilizing effect on the Couette flow depending on the Weissenberg number being smaller or larger than a critical value. The critical Weissenberg number increases by an increase in the gap size, and also by an increase in the mobility factor. Fluid's inertia is identified as the main source of instability.
Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids
Journal of Fluid Mechanics, 2013
We study the modal and non-modal linear instability of inertia-dominated channel flow of viscoelastic fluids modelled by the Oldroyd-B and FENE-P closures. The effects of polymer viscosity and relaxation time are considered for both fluids, with the additional parameter of the maximum possible extension for the FENE-P. We find that the parameter explaining the effect of the polymer on the instability is the ratio between the polymer relaxation time and the characteristic instability time scale (the frequency of a modal wave and the time over which the disturbance grows in the non-modal case). Destabilization of both modal and non-modal instability is observed when the polymer relaxation time is shorter than the instability time scale, whereas the flow is more stable in the opposite case. Analysis of the kinetic energy budget reveals that in both regimes the production of perturbation kinetic energy due to the work of the Reynolds stress against the mean shear is responsible for the ...