David S. Malkus

Spurt phenomena of the Johnson-Segalman fluid and related models

Abstract We examine the behavior of viscoelastic fluid models which exhibit local extrema of the steady shear stress. For the Johnson-Segalman and Giesekus models, a variety of steady singular solutions with jumps in shear rate are constructed and their stability to one dimensional disturbances analyzed. It is found that flow-rate versus imposed stress curves in slit-die flow fit experimental observation of the “spurt” phenomenon with some precision. The flow curves involve linearly stable singular solutions, but some assumptions on the dynamics of the spurt process are required. These assumptions are tested by a semi-implicit finite element solution technique which allows solutions to be efficiently integrated over the very long time-scale involved. The Johnson-Segalman model with added Newtonian viscosity is used in the calculations. It is found that the assumptions required to model spurf are satisfied by the dynamic model. The dynamic model also displays a characteristic “latency time” before the spurt ensues and a characteristic “shape memory” hysteresis in load/unload cycles. These as well as other features of the computed solutions should be observable experimentally. We conclude that constitutive equations with shear stress extrema are not necessarily flawed, that their predicted behavior may appear to be arrested “wall slip”, and that such behavior may actually have been observed already.

Dynamics of shear flow of a non-Newtonian fluid

Abstract Viscoelastic materials with fading memory, e.g., polymers, suspensions, and emulsions, exhibit behavior that is intermediate between the nonlinear hyperbolic response of purely elastic materials and the strongly diffusive, parabolic response of viscous fluids. Many popular numerical methods used in the computation of steady viscoelastic flows fail in important flow regimes, and thus do not capture significant non-Newtonian phenomena. A key to satisfactory explanation of these phenomena is the study of the full dynamics of the flow. This paper studies the dynamics of shear flow, presenting a description of non-Newtonian phenomena caused by a non-monotone relation between the steady shear stress and shear strain rate. Analytical results for such phenomena are surveyed, and three distinct numerical methods are developed to accurately compute the dynamics. The computations reproduce experimental measurements of non-Newtonian “spurt” in shearing flow through a slit die. They also predict related phenomena (such as hysteresis and shape memory); experiments are suggested to verify these predictions.

Analysis of new phenomena in shear flow of non-Newtonian fluids

Phase-plane and small-parameter asymptotic techniques are used to analyze systems of ordinary differential equations that describe the transient behavior of non-Newtonian fluids in shear flow. These systems approximate the partial differential equations that derive from three-dimensional balance laws and from differential constitutive models for highly elastic liquids. Two models are considered: one with a single relaxation time and small Newtonian viscosity; the other with two relaxation times and no Newtonian viscosity. Both possess the key feature that the variation of steady shear stress with strain rate is not monotone.The analysis shows that both models exhibit several distinctive phenomena: spurt, shape memory, hysteresis, latency, and normal stress oscillations. The predictions for the spurt phenomenon agree quantitatively with experimental results for polymer melts; the other new phenomena, which were discovered recently in numerical simulation, should also be observable in rheological experiments.

Zero and negative masses in finite element vibration and transient analysis

Abstract Mass matrix lumping by quadrature is considered. Accuracy requirements seem to dictate the use of zero or negative masses for multidimensional higher-order elements. It is shown that the zero and/or negative masses do not destroy the essential algebraic properties of the discrete eigenproblem, in spite of the negative or infinite eigenvalues which may result. Explicit transient methods require positive-definite lumping which, for some elements, may only be achieved by sacrificing accuracy to avoid the negative or zero masses that would render the lumping indefinite. An implicit-explicit time integration method based on quadratic triangles with optimal lumping is devised, analyzed, and tested. It treats the nodes with nonzero masses explicitly and the nodes with zero masses implicitly. Analysis and numerical tests show that this formulation is optimally accurate and less costly than a similar method with nonzero masses, based on optimally lumped biquadratic rectangles. The method is also found to be substantially more accurate than the fully explicit method based on lumping the triangular elements in an ad hoc fashion to retain nonzero masses.

Oscillations in Piston-Driven Shear Flow of a Non-Newtonian Fluid

In recent experiments on piston-driven shear flow of a highly elastic and very viscous non-Newtonian fluid, Lim and Schowalter observed nearly periodic oscillations in the particle velocity at the channel wall for particular values of the constant volumetric flow rate. Such periodicity has been characterized as a “stick/slip” phenomenon caused by the failure of the fluid to adhere to the wall. We suggest an alternative explanation for these oscillations using a dynamic mathematical model based on the Johnson-Segalman-Oldroyd constitutive relation, the key feature of which is a non-monotonic relationship between steady shear stress and strain rate. The resulting three-dimensional equations of motion and stress are reduced to one space dimension, consistent with experimental results. In the inertialess approximation, the equations governing the flow can be viewed as a continuous family of quadratic ordinary differential equations coupled by the non-local constraint that fixes the volumetric flow rate. Varying the flow rate, the numerical simulation of solutions to this system exhibits transitions to and from a regime with persistent oscillations that compare favorably with the Lim-Schowalter observations. When the time-asymptotic behavior is cyclic, large shear strain rates are observed in a thin but macroscopic layer near the wall. The transitions are explained using spectral analysis of the linear (infinite dimensional) operator resulting from linearization of the quadratic system about a discontinuous steady state with a jump discontinuity in the stress components. The persistent oscillations arise as a Hopf bifurcation to periodic orbits as the volumetric flow rate is increased beyond a critical value.

Cancellation of errors in the Higashitani-Pritchard treatment of hole pressures generated by viscoelastic liquids in creeping flow past a transverse slot

Analytical studies of the hole pressure for non-Newtonian creeping flow past a transverse slot are pursued with particular interest in the formulation of Higashitani and Pritchard (HP). To correct the flaws in the treatment of HPs original work, a modified hole-pressure relation (MHPR) is employed. Some important mathematical properties of the MHPR are presented. By studying the MHPR in streamline coordinate formulation, we find a fortuitouserror cancellation phenomenon in the derivation of the HP formula: namely, the error caused by one key flaw is fortuitously cancelled out by the error introduced through another key flaw. For second-order fluids and Tanners “viscometric model” (under certain assumptions) the cancellation of errors is proved to be exact. It is this cancellation of errors that provides a theoretical explanation for the paradox between an apparently flawed derivation and the fortunate success of the HP prediction.

Improved numerical dissipation for time integration algorithms in conduction heat transfer

Abstract The Hilber-Hughes-Taylor (HHT) α-method is applied to the direct integration of the equations of conduction heat transfer. The method is shown to possess improved algorithmic dissipation, relative to the general trapezoidal methods, while retaining second order accuracy. The α-method is compared with members of the generalized trapezoidal family of methods.

Finite rise time step strain modeling of nearly monodisperse polymer melts and solutions

The step shear strain experiment is one of the fundamental transient tests used to characterize the rheology of viscoelastic polymer melts and solutions. Many melts and solutions exhibit homogeneous deformation and stress relaxation; in these cases the transient dynamics can be modeled by completely ignoring momentum effects and imposing singular kinematics. Recently, however, it has been observed that there are certain classes of nearly monodisperse melts and solutions that exhibit anomalous nonhomogeneous deformation and stress relaxation (Morrison and Larson (1990), Larson, Khan, and Raju (1988), Vrentas and Graessley (1982), and Osaki and Kurata (1980)). We demonstrate that, for these classes, a finite rise time must be incorporated, some source of inhomogeneity must be present, and a small amount of added Newtonian viscosity is necessary. We examine five nonlinear and quasilinear models; the Johnson-Segalman, Phan Thien Tanner, Giesekus, White-Metzner, and Larson models. We determine which mathematical features of the models are necessary and/or sufficient to describe the observed experimental behavior.

Combining metamodels with rational function representations of discretization error for uncertainty quantification

Abstract Techniques for producing metamodels for the efficient Monte Carlo simulation of high consequence systems are presented. The bias of f.e.m mesh discretization errors is eliminated or minimized by extrapolation, using rational functions, rather than the power series representation of Richardson extrapolation. Examples, including estimation of the vibrational frequency of a one-dimensional bar, show that the rational function model gives more accurate estimates using fewer terms than Richardson extrapolation, an important consideration for computational reliability assessment of high-consequence systems, where small biases in solutions can significantly affect the accuracy of small-magnitude probability estimates. Rational function representation of discretization error enable the user to accurately extrapolate to the continuum from numerical experiments performed outside the asymptotic region of the usual power series, allowing use of coarser meshes in the numerical experiments, resulting in significant savings.

Reversed stability conditions in transient finite element analysis

Abstract Numerical methods which introduce artificially unstable modes are discussed. In structural and elastodynamics these results from optimal mass lumping with higher-order elements. In fluid mechanics an additional source of these modes can be a penalty function with alternating signs. These modes yield unstable modal equations; however, they do not necessarily imply unstable ttransient integration in the presence of algorithmic damping. Stable integration can be achieved by satisfying a stability condition in which the roles of space-step and time-step are reversed. Elastodynamics, the Navier-Stokes equations, and non-Newtonian fluids provide numerical examples.