Asymptotics and numerics for the upper-convected Maxwell model describing transient curved viscoelastic jets

This work deals with the modeling and simulation of non-Newtonian jet dynamics as it occurs in fiber spinning processes. Proceeding from a three-dimensional instationary boundary value problem of upper-convected Maxwell equations, we present a strict systematic derivation of a one-dimensional viscoelastic string model by using asymptotic analysis in the slenderness ratio of the jet. The model allows for the unrestricted motion and shape of the jet’s curve, and its deduction extends the hitherto existing uniaxial asymptotic approaches. However, the system of partial differential equations with algebraic constraint has a varying character (hyperbolic, hyperbolic–elliptic, parabolic deficiency). Its applicability range turns out to be limited depending on the physical parameters and the boundary conditions (i.e. singular perturbation). Numerical results are discussed for the hyperbolic regime of gravitational inflow–outflow set-ups which become relevant in drawing and extrusion processes. The simulations are performed with a normal form total upwind scheme in space and an implicit time-integration ensuring convergence of first order.

Corotating or Codeforming Models for Thermoforming: Free Forming

Our previous work [J Pol Eng, 32, 245 (2012)] explores the role of viscoelasticity for the simplest relevant problem in thermoforming, the manufacture of cones. In this previous work, we use a differential model employing the corotational derivative [the corotational Maxwell model (CM)] for which we find an analytical solution for the sheet deformation as a function of time. This previous work also identifies the ordinary nonlinear differential equation corresponding to the upper convected Maxwell model (UCM), for which she finds no analytical solution. In this paper, we explore the role of convected derivative by solving this UCM equation numerically by finite difference. We extend the previous work to include sag by incorporating a finite initial sheet curvature. We treat free forming step in thermoforming and find that the convected derivative makes the free forming time unreasonably sensitive to the initial curvature. Whereas, for the CM model, we get a free forming time that is independent of initial sheet curvature, so long as the sheet is nearly flat to begin with. We cast our results into dimensionless plots of thermoforming times versus disk radius of curvature.

A NEW MODEL FOR SHALLOW VISCOELASTIC FLUIDS

We propose a new reduced model for gravity-driven free-surface flows of shallow viscoelastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for viscoelastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. In addition to the classical layer depth and velocity in shallow models, our system describes also the evolution of two components of the stress. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver.

Velocity Distribution of Viscoelastic Fluid Axial Flowing in Eccentric Annulus Based on the Upper-Convected Maxwell Model

After polymer flooding has been put into effect in Daqing oilfield, eccentric wear of sucker rod and tubing has been so serious that it made the rods break. In order to analyze the cause of eccentric wear based on the upper-convected Maxwell model, the flow equation of viscoelastic fluid in eccentric annulus was established in cylinder coordinate system. Then the equation was transformed in bipolar coordinate system and dispersed by control-volume method. The velocity distribution was solved by ADI methods and example was calculated which provides theory basis for solving eccentric wear problems.

Geometric scaling of a purely elastic flow instability in serpentine channels

A combined experimental, numerical and theoretical investigation of the geometric scaling of the onset of a purely elastic flow instability in serpentine channels is presented. Good qualitative agreement is obtained between experiments, using dilute solutions of flexible polymers in microfluidic devices, and three-dimensional numerical simulations using the upper-convected Maxwell model. The results are confirmed by a simple theoretical analysis, based on the dimensionless criterion proposed by Pakdel & McKinley (Phys. Rev. Lett., vol. 77, 1996, pp. 2459–2462) for onset of a purely elastic flow instability. Three-dimensional simulations show that the instability is primarily driven by the curvature of the streamlines induced by the flow geometry and not due to the weak secondary flow in the azimuthal direction. In addition, the simulations also reveal that the instability is time-dependent and that the flow oscillates with a well-defined period and amplitude close to the onset of the supercritical instability.

Numerical Simulation of Confined Swirling Flows of Oldroyd Fluids

The effect of a fluid’s elasticity has been investigated on the vortex breakdown phenomenon in confined swirling flow. Assuming that the fluid obeys upper-convected Maxwell model as its constitutive equation, the finite volume method together with a collocated mesh was used to calculate the velocity profiles and streamline pattern inside a typical lid-driven swirling flow at different Reynolds and Weissenberg numbers. The flow was to be steady and axisymmetric. Based on the results obtained in this work, it can be concluded that fluid’s elasticity has a strong effect on the secondary flow completely reversing its direction of rotation depending on the Weissenberg number. Even in swirling flows with low ratio of elasticity to inertia, vortex breakdown is postponed to higher Reynolds numbers. Also, the effect of retardation ratio on the flow structure of viscoelastic fluid with the Weissenberg number being constant was surveyed. Based on our results, by decreasing the retardation ratio the flow becomes Newtonian like.