Steady Flows Of Viscoelastic Fluids Research Articles

Finite Amplitude Stability of Internal Steady Flows of the Giesekus Viscoelastic Rate-Type Fluid

Using a Lyapunov type functional constructed on the basis of thermodynamical arguments, we investigate the finite amplitude stability of internal steady flows of viscoelastic fluids described by the Giesekus model. Using the functional, we derive bounds on the Reynolds and the Weissenberg number that guarantee the unconditional asymptotic stability of the corresponding steady internal flow, wherein the distance between the steady flow field and the perturbed flow field is measured with the help of the Bures–Wasserstein distance between positive definite matrices. The application of the theoretical results is documented in the finite amplitude stability analysis of Taylor–Couette flow.

Radiation Effect on the Mixed Convection Flow of a Viscoelastic Fluid Along an Inclined Stretching Sheet

This study concentrates on the heat transfer analysis of the steady flow of viscoelastic fluid along an inclined stretching surface. Analysis has been carried out in the presence of thermal radiation and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The equations of continuity, momentum and energy are reduced into the system of governing differential equations and solved by homotopy analysis method (HAM). The velocity and temperature are illustrated through graphs. Exact and homotopy solutions are compared in a limiting sense. It is noticed that viscoelastic parameter decreases the velocity and boundary layer thickness. It is also observed that increasing values of viscoelastic parameter reduces the thickness of momentum boundary layer and increase the heat transfer rate. However, it is found that increasing the radiation parameter has the effect of decreasing the local Nusselt number

Steady viscoelastic fluid flow between parallel plates under electro-osmotic forces: Phan-Thien–Tanner model

The electro-osmotic flow of a viscoelastic fluid between parallel plates is investigated analytically. The rheology of the fluid is described by the Phan-Thien–Tanner model. This model uses the Gordon–Schowalter convected derivative, which leads to a non-zero second normal stress difference in pure shear flow. A nonlinear Poisson–Boltzmann equation governing the electrical double-layer field and a body force generated by the applied electrical potential field are included in the analysis. Results are presented for the velocity and stress component profiles in the microchannel for different parametric values that characterize this flow. Equations for the critical shear rates and maximum electrical potential that can be applied to maintain a steady fully developed flow are derived and discussed.

Uniform flow of viscoelastic fluids past a confined falling cylinder

Uniform steady flow of viscoelastic fluids past a cylinder placed between two moving parallel plates is investigated numerically with a finite-volume method. This configuration is equivalent to the steady settling of a cylinder in a viscoelastic fluid, and here, a 50% blockage ratio is considered. Five constitutive models are employed (UCM, Oldroyd-B, FENE-CR, PTT and Giesekus) to assess the effect of rheological properties on the flow kinematics and wake patterns. Simulations were carried out under creeping flow conditions, using very fine meshes, especially in the wake of the cylinder where large normal stresses are observed at high Deborah numbers. Some of the results are compared with numerical data from the literature, mainly in terms of a drag coefficient, and significant discrepancies are found, especially for the constant-viscosity constitutive models. Accurate solutions could be obtained up to maximum Deborah numbers clearly in excess of those reported in the literature, especially with the PTT and FENE-CR models. The existence or not of a negative wake is identified for each set of model parameters.

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first-order upwind approximation for the viscoelastic stress. A non-uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non-linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd-B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

AbstractThis paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.

Regions of vortices in an axisymmetric contraction, examined at the example of a second-order fluid (SOF)

The steady flow of viscoelastic fluids across an axisymmetric contraction is of principle and technological interest. Particularly (but not only) within the entrance of such a convergent flow it is possible and useful to approximate the rheological constitutive equation as second-order-fluid constitutive equation (SOF). For that we have considered the flow across an open hyperboloid of rotation with the aid of a complete numerical simulation as well as a perturbation calculation.

Steady Flows of Viscoelastic Fluids in Domains with Outlets to Infinity

The equations governing the motion of incompressible viscoelastic fluids of Rivlin—Ericksen and Oldroyd type are investigated in domains with cylindrical and paraboloidal outlets to infinity. For sufficiently small fluxes, prescribed in each outlet, existence and uniqueness of solutions are proven in weighted Holder spaces. In domains with paraboloidal outlets the solution is obtained as a perturbation of the corresponding Navier—Stokes solution and in domains with cylindrical outlets as a perturbation of a flux carrier, constructed by joining together the exact solutions found in each outlet. These exact solutions are shown to be either rectilinear flows of Poiseuille type or flows composed of a rectilinear and of a transverse secondary component.

Three-dimensional Steady Flow of Viscoelastic Fluid past an Obstacle

We consider the three-dimensional steady flow of certain classes of viscoelastic fluids in exterior domains with non-zero velocity prescribed at infinity. We show that the solution behaves near infinity similarly as the fundamental solution to the Oseen problem.

Existence of slow steady flows of viscoelastic fluids of White-Metzner type

This work is concerned with the study of steady flows of viscoelastic fluids for which the extra stress is given by a differential constitutive equation of White-Metzner type.

Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers

We consider the stability of steady flows of viscoelastic fluids of Jeffreys type. For sufficiently small Weissenberg numbers, but arbitrary Reynolds numbers, it is proved that the flow is stable to small disturbances if the spectrum of the linearized operator is in the left half plane.

A note on selection of spaces in computation of viscoelastic flows using the hp-finite element method

The stability, accuracy, and computational efficiency of higher order Galerkin ( hp-type) finite elements for steady flow of viscoelastic fluids past square arrays of cylinders and through a corrugated tube with two different polynomial approximating spaces, namely truncated and product, have been investigated. It has been shown that both spaces produce a stable discretization with an exponential convergence rate toward the exact solution without an upper Weissenberg number limitation. Based on global deviation from mass and momentum conservation, it is shown that the truncated space provides a better quality solution at a reduced computational cost for a given number of degrees of freedom. In addition, the restrictions on the relative order of approximating polynomials for stresses and velocities have been examined. It has been shown that without splitting of the stress into purely viscous and elastic components the most cost efficient solution is obtained when the stresses and velocities are approximated by the same order polynomial, while with this splitting the best computational performance is obtained when the stresses are discretized with a polynomial order of one less than the velocities.

Steady flows of viscoelastic fluids in axisymmetric abrupt contraction geometry: A comparison of numerical results

Recently two groups of researchers have reported numerical results simulating the steady flow of a KBKZ fluid in torsion free axisymmetric abrupt contraction geometry. The fluid model used in both cases was a constitutive equation chosen to match laboratory behavior of LDPE. In both cases, quadratic finite elements were used. Large corner vortices were observed in the simulations, similar to those observed in laboratory experiments. The agreement between the results in the two papers is good. We repeat the experiment, using linear finite elements. Tracking is performed via an artificial time method, and a novel ‘‘reduced velocity’’ variable is used in our finite element simulation. There is good qualitative and quantitative agreement between the results reported here and the results previously reported by others. Quantitative measures used in the comparison—vortex opening angle, Couette correction, and vortex intensity—are analyzed.

Anisotropy and Thermal Stresses in the Steady Flow of Viscoelastic Fluids

Anisotropy and Thermal Stresses in the Steady Flow of Viscoelastic Fluids

Application of higher order finite element methods to viscoelastic flow in porous media

A higher order Galerkin finite element scheme for simulation of two‐dimensional viscoelastic fluid flows has been developed. The numerical scheme used in this study gives rise to a stable discretization of the continuum problem as well as providing an exponential convergence rate toward the exact solution. Hence, with this method, spurious oscillatory modes are effectively eliminated by increasing the order of the interpolant within each subdomain. In our calculations, an upper limit for the Weissenberg number due to numerical instability was not encountered. However, the memory requirements of the discretization grow quadratically with the polynomial order. Consequently, the maximum attainable We is determined by the availability of computational resources. The algorithm was tested for flow of upper convected Maxwell and Oldroyd‐B fluids in the undulating tube problem and it was subsequently applied to viscoelastic flow past square cylindrical arrangements. The results obtained show no increase in the flow resistance with increasing elasticity. These findings are in contrast to the experimental data reported in the literature. However, recent experimental investigations have indicated that this dramatic increase in flow resistance is due to a purely elastic instability. Hence, numerical simulation based on two‐dimensional steady flow of viscoelastic fluids cannot be expected to capture this phenomenon.

Existence and stability of steady flows of weakly viscoelastic fluids

SynopsisWe consider steady flows of viscoelastic fluids for which the extrastress tensor is given by a differential constitutive equation and is such that the retardation time is large (weaklyviscoelastic fluids).We show the existence of a unique viscoelastic steady flow close to a given Newtonian flow and investigate its linear stability.As an example, we consider the Bénard problem for viscoelastic fluids and we prove that there exists a nontrivial linearly stable flow of a weakly viscoelastic fluid in a container heated from below.

Existence of steady flows of viscoelastic fluids of Jeffreys type with traction boundary conditions

Existence of steady flows of viscoelastic fluids of Jeffreys type with traction boundary conditions

Inflow boundary conditions for steady flow of viscoelastic fluids with differential constituitive laws

Abstract : Steady flows of viscoelastic fluids can not be uniquely determined by imposing boundary conditions only for the velocities as in the Newtonian case. The reason for this is that the fluids have memory, and therefore the flow inside the domain is affected by what happened before the fluid entered the domain. This leads to the need for extra boundary conditions at an inflow boundary. The nature of these inflow boundary conditions has not been analyzed previously, and it is certainly dependent on the constitutive law. In this paper, we look at the special case of differential constitutive relations with a single relaxation mode. We consider steady transverse flows across a strip which are small perturbations of a flow with constant velocity. It turns out that in this case two extra inflow boundary conditions are required in two dimensions, and four in three dimensions. This is what would be expected from an analysis of characteristics, but it contradicts the belief of many rheologists that it is possible to prescribe the extra stress at an inflow boundary. The problem studied here is of potential relevance for numerical simulations of steady flows. Many of the flows currently simulated are on infinite domains. Numerically, these domains are truncated, and on the inflow boundary of the truncated domain people usually prescribe the extra stress. According to the analysis in this paper, this is an overdetermined problem, and therefore errors must be expected from this procedure unless the artificial boundaries are chosen far enough out. Keywords: Upper Convected Maxwell Model.

Recent advances in the mathematical theory of steady flow of viscoelastic fluids

The paper reviews recent existence theorems for steady flows of viscoelastic fluids. The results concern small perturbations of either the rest state or a flow with uniform velocity. Differential as well as integral models are discussed. For differential models, we also discuss the boundary conditions which need to be prescribed at an inflow boundary; it turns out that there is a difference between Maxwell- and Jeffreys-type models in this respect. For Newtonian fluids, existence results which go much farther are known; the data need not be small and no regularity of the boundary is required. A number of problems are pointed out which make similar results for viscoelastic fluids difficult and in some cases unlikely.

Existence of Slow Steady Flows of Viscoelastic Fluids with Differential Constitutive Equations

Abstract : Questions of existence and uniqueness of steady flows of viscoelastic fluids have thus far not been understood, even for slow flows perturbing rest. This paper provides an existence result for slow flows with no in- and outflow boundaries. The fluid is assumed to be described by constitutive equations of a differential nature. The method used to prove existence is constructive and in fact very close to procedures used in numerical calculations.