John A. Nohel

Energy methods for nonlinear hyperbolic volterra integrodifferential equations

Abstract : We use energy methods to study global existence, boundedness, and asymptotic behavior as t approaches infinity, of solutions of the two Cauchy problems (and related initial-boundary value problems).

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.

A Diffusion Equation with a Nonmonotone Constitutive Function

We discuss the well-posedness of the model problem ut = o(ux)x on [0,1] x [0,T], T > 0, subject to given Neumann or Dirichlet boundary conditions at x = 0 and x = 1, and to the initial condition u(x,0) = f(x); the given functions f: [0,1] → R, o: R → R are assumed to be smooth, o(0) = 0, o satisfies the coercivity assumption Ęo(Ę) > cĘ2, for some constant c > 0 and for Ę Є R, and o is assumed to be decreasing on an interval (a,b) with a > 0. We present a recent nonuniqueness result in the special case when o is piecewise linear and study a related convexified problem.

Weak solutions for a nonlinear system in viscoelasticity

We consider a one-dimensional model problem for the motion of a viscoelastic material with fading memory governed by a quasilinear hyperbolic system of integrodifferential equations of Volterra type. For given Cauchy data in , we use the method of vanishing viscosity and techniques of compensated compactness to obtain the existence of a weak solution (in the class of bounded measurable functions) in a special case

Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

Abstract : This document studies the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain- rate that results in steady states having, in general, discontinuities in the strain rate. It is shown that every solution tends to a steady state as t approaches limit of infinity, and we identify steady states that are stable.

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.

Analysis of spurt phenomena for a non-newtonian fluid

Abstract : This document discusses novel phenomena in dynamic shearing flows of non-Newtonian fluids of importance for polymer processing. A striking example is spurt which was observed experimentally in the flow of monodispersive polyisoprenes through capillaries; the volumetric flow rate increased dramatically at a critical stress independent of molecular weight. The authors show that satisfactory explanation of spurt requires studying the full dynamics of the equations of motion and constitutive relations characterized by a non- monotonic relation between the steady shear stress and strain rate. The increase in volumetric flow rate is shown to correspond to jumps in the strain rate when the driving pressure gradient exceeds a critical value. Motivated by scaling suggested by accurate numerical computations of the governing dynamic problem that yielded qualitative and quantitative agreement with experiment, we introduce a system of ordinary differential equation that approximates dynamic behavior of highly elastic and very viscous fluids. The complete dynamics of the system of odes determined by phase plane analysis. These results are then used to explain not only spurt but also shape memory, hysteresis, latency, and other effects that have also been observed in numerical simulations.

A nonlinear integral equation occurring in a singular free boundary problem

Abstract : One motivation for the study of the Cauchy problem is its similarity with the well-known one phase Stefan problem (in one space dimension). The principal motivation for the study of this problem is that it serves as a prototype of nonlinear parabolic problems which arise as monotone convexifications of nonlinear diffusion equations with nonmonotone constitutive functions phi.

Non-Newtonian Phenomena in Shear Flow

We present mathematical results that are needed to analyse novel phenomena occurring in dynamic shearing flows of highly elastic and viscous non-Newtonian fluids. The key property of solutions to the time-dependent, quasilinear partial differential equations that are used to model such flows is a non-monotonic relation between the steady shear stress and strain rate. The phenomena discussed may lead to material instabilities that could disrupt polymer processing.

Quadratic dynamical systems describing shear flow of non-newtonian fluids

Phase-plane techniques are used to analyze a quadratic system of ordinary differential equations that approximates a single relaxation-time system of partial differential equations used to model transient behavior of highly elastic non-Newtonian liquids in shear flow through slit dies. The latter one-dimensional model is derived from three-dimensional balance laws coupled with differential constitutive relations well-known by rheologists. The resulting initial-boundary-value problem is globally well-posed and possesses the key feature: the steady shear stress is a non-monotone function of the strain rate. Results of the global analysis of the quadratic system of ode’s lead to the same qualitative features as those obtained recently by numerical simulation of the governing pde’s for realistic data for polymer melts used in rheological experiments. The analytical results provide an explanation of the experimentally observed phenomenon called spurt; they also predict new phenoinena discovered in the numerical simulation; these phenomena should also be observable in experiments.