Search

A singular perturbation analysis for the Rolie-Poly model is presented for the limit of large relaxation time. The model describes entangled linear polymer melts and time scales for a Rouse time and reptation time. First, the case of the Rouse time approaching zero is named the nonstretching model, and the addition of a solvent model gives the small parameter to be the ratio of the retardation time to the slow reptation time. This analysis is instructive in formulating the perturbation analysis for the full Rolie-Poly model without the solvent, where the small parameter is the ratio of Rouse time to reptation time. The resulting formulae elucidate the characteristic features of thixotropic yield stress fluids, including yield stress hysteresis, delayed yielding and long term persistence of a decreased viscosity after cessation of flow. Two fundamental initial value problems are solved: startup of shear flow and cessation of shear flow. The scalings lead naturally to simplified dynamical systems that describe regimes of fast, slow and yielded dynamics, and show how the combination of these regimes can be used to describe the entire flow evolution.

In this paper, we consider the Stokes equations in a two-dimen- sional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Muntz-Szasz Theorem., Accepted in Evolution Equations and Control Theory, 2018

We prove results on approximate controllability for linear viscoelastic flows, with a localized distributed control in the momentum balance equation. The constitutive law is a multimode Maxwell or Jeffreys model with an infinite number of relaxation modes.

We consider compressible flow with periodic boundary conditions. In a neighborhood of a state of constant density and nonzero velocity, we prove exact null controllability of the system with a control localized in space and acting only on the momentum equation.

We consider a generalized Newtonian fluid which exhibits a yield stress and spurt when the yield stress is reached. The constitutive behavior above the yield stress is assumed linear. We solve the associated Rayleigh problem. The solution is not self-similar and exhibits a propagating front separating yielded and unyielded regions. The behavior near t = 0 is crucial for starting the numerical simulation. We find that the position of the front behaves like − t ln t .

A viscoelastic constitutive model which combines the partially extending strand convection model and a Newtonian solvent is used in the regime of large relaxation time. Prior work on one dimensional time-dependent solutions at prescribed shear stress predicts some of the features expected of thixotropic yield stress fluids, such as delayed yielding. In this paper, we present the linear stability analysis of two-dimensional plane Couette flow, for parameter regimes that support a two-layer arrangement consisting of an unyielded layer and a yielded layer. Asymptotic analysis and computational techniques are applied. We find that the one layer yielded flow can have bulk instabilities which also emerge in the two-layer flow. Bulk instabilities in the yielded phase appear not to have been observed in prior literature. For some parameters, an interfacial mode is unstable and is driven by the normal stress difference across the interface. The yielded zone has the higher first normal stress difference, as for the well-studied Johnson–Segalman model. In order to assess the importance of the sign of the first normal stress difference at the interface, we specifically design a modification to the model to reverse the sign. It is found that instabilities still occur.

We consider the equations of a linear Maxwell fluid with spatially varying coefficients. Pure stress modes are solutions with zero velocity but nonzero stresses. We derive equations to characterize such solutions. In two dimensions, we find that under generic hypotheses only certain “trivial” solutions exist. In three dimensions, on the other hand, there exist nontrivial solutions. To get them, we derive a system of partial differential equations whose type (elliptic or hyperbolic) depends on the sign of the Gauss curvature of level surfaces of the relaxation time.

We review progress and open challenges for a number of problems related to the Oldroyd B model which are of current interest to the mathematical community. We discuss existence results for initial value problems and steady flows, flow stability, and the problem of understanding the asymptotics of the high Weissenberg number limit. We also discuss efforts and challenges in numerical simulation, exemplified by three problems: extensional flow in a four roll mill geometry, flow past a confined cylinder, and locomotion of microorganisms in a viscoelastic fluid.

We consider linear viscoelastic flow of a multimode Maxwell or Jeffreys fluid in a bounded domain with smooth boundary, with a distributed control in the momentum equation. We establish results on approximate and exact controllability.

The PEC (partially extending strand convection) model of Larson is able to describe thixotropic yield stress behavior in the limit where the relaxation time is large relative to the retardation time. In this paper, we discuss the development of shear bands in a Poiseuille flow which is started up from rest with an imposed pressure gradient. We analyze the asymptotic limit of large relaxation time; the small parameter ϵ measures the ratio of retardation time to relaxation time. We determine the position and width of shear bands as a function of time. We identify an initial phase of “fast yielding” during which the width of the transition between high and low shear rate regions behaves like t − 3 . This continues until t (measured on the scale of the retardation time) is on the order of ϵ − 1 / 3 . Then there is a phase of “delayed yielding” during which the width of the transition is of order ϵ. Eventually, the width sharpens as 1/(ϵ 2 t 3 ). We also show how these results are modified by introducing Korteweg stresses which prevent the transition from becoming infinitely sharp and also change the location where the transition takes place. This paper is dedicated to Roger Tanner on the occasion of his eighty-second birthday.

[Received on ] Thixotropic yield stress fluids have a yield stress and viscosity which change slowly over time as the fluid undergoes structural changes. This article reviews recent work which shows how such behavior can arise as a limit of viscoelastic flow when a relaxation time becomes large. The large relaxation time introduces a small parameter which can be used as a basis for singular perturbation studies. A simple prototype for a viscoelastic model will be used to illustrate the analysis. We review the behavior in startup of shear flow, oscillatory shear flow and elongational flow and identify potential directions for future research.

It has recently been shown that the PEC (partially extending strand convection) model of Larson is able to describe thixotropic yield stress behavior in the limit where the relaxation time is large. In this paper, we discuss the behavior of the model under an imposed periodic shear force τ ( t ) = A + B sin ( ω t - Φ ) . We identify regimes of fast, slow and yielded dynamics and discuss the overall behavior of the system in dependence of the parameters A , B , ω and Φ . The method of averaging turns out to be a crucial mathematical tool for the analysis. The scale of the period of the oscillation relative to the times over which yielding and unyielding occur has a crucial influence on the qualitative dynamics.

We prove that a $C_0$-semigroup of operators $\exp(At)$ satisfies backward uniqueness if the resolvent of $A$ exists on a ray $z=re^{i\theta}$ in the left half plane ($\pi/2

We consider the wave equation $u_{tt}=\Delta u$ on a bounded domain $\Omega\subset{\mathbb R}^n$, $n>1$, with smooth boundary of positive mean curvature. On the boundary, we impose the absorbing boundary condition ${\partial u\over\partial\nu}+u_t=0$. We prove uniqueness of solutions backward in time.

Nonmonotone constitutive behavior leading to shear banding occurs in a number of fluids, such as wormlike micelles and clay suspensions. In general, shear banded flows are not unique. Higher order terms in the governing equations are often introduced to distinguish a preferred solution that arises as a steady state in the long term. In the literature on wormlike micelles, stress diffusion has been widely considered for this purpose. In this paper, we discuss a different physical mechanism, based on interfacial energies and Korteweg stresses associated with them. It is shown how this criterion leads to a form of “equal area” rule.

In this paper we consider the one-dimensional compressible Navier–Stokes equations linearized around a constant steady state (Q0, V0), Q0 > 0, V0 > 0, with periodic boundary conditions in the interval I2π := (0, 2 π) . We explore the controllability of this linearized system using a control only for the velocity equation. We prove that the linearized system with homogeneous periodic boundary conditions is null controllable in ˙ H 1 per(I2π ) × L 2 (I2π ) by a localized interior control when time is sufficiently large, where ˙ H 1 per (I2π ) denotes the Sobolev space of periodic functions with mean value zero. We show null controllability of the system by proving an observability inequality with the help of two types of Ingham inequality.

The Prandtl equations, arising naturally in the description of high Reynolds number boundary layers, have turned out to be quite difficult from the point of view of mathematical analysis. Recent work by the author has shown that the analogous problem for the upper-convected Maxwell fluid is actually better behaved, and the well-posedness of the boundary layer equations has been established. In this paper, boundary layers for the Phan-Thien–Tanner and Giesekus fluid are considered. It turns out that there are two fundamentally different types of boundary layers, which we shall call elastic and viscometric boundary layers. The elastic boundary layers will be analyzed in an analogous fashion as those for the upper-convected Maxwell fluid. On the other hand, for viscometric boundary layers, which occur only for the PTT fluid, the equations are equivalent to those for a power law fluid.

We prove the well-posedness of the Prandtl boundary layer equations for the upper convected Maxwell fluid under the assumption that the stress tensor is strictly positive definite.

We discuss boundary layers arising in the high Weissenberg number limit of viscoelastic flows and the scalings that arise in analyzing them. Two quite distinct mechanisms for the formation of viscoelastic boundary layers exist: One mechanism is analogous to that of the Prandtl boundary layer and is linked to the enforcement of the no-slip boundary condition. The other mechanism has nothing to do with the no-slip condition, but is linked to long memory which manifests itself in different stresses near the wall versus some distance from the wall. In certain situations, these stress boundary layers may be embedded within Prandtl type boundary layers.

The well-posedness of the hydrostatic equations is linked to long wave stability criteria for parallel shear flows. We revisit the Kelvin--Helmholtz instability with a free surface. In the wall-bounded case, the flow is unstable to all wave lengths. Short wave instabilities are localized and independent of boundary conditions. On the other hand, long waves are shown to be stable if the upper boundary is a free surface and gravity is sufficiently small. We also consider smooth velocity profiles of the base flow rather than a velocity jump. We show that stability of long waves for small gravity generally holds for monotone profiles U(y). On the other hand, this need not be the case if U is not monotone.

The well-posedness of the equations of fluid mechanics in the hydrostatic limit is well known to be a difficult problem. Partial results, both positive and negative, will be reviewed below. In this paper, it is shown that, for ideal magnetohydrodynamics, a magnetic field parallel to the flow direction can ensure well-posedness. The only condition required is that the flow is subalfvenic. The result has some relevance to viscoelastic flows of the upper convected Maxwell fluid, which, in the infinite Weissenberg number limit, is related to ideal MHD.

We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem locally in time. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data; after that we extend our results to some 3D flow problems with large initial data. We solve our problem using an iteration between linear subproblems. The limit of the iteration provides the solution of our original problem.

The positive-definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models’ equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.

We give a rigorous condition for stability of steady flows of multi-mode nonlinear Maxwell fluids. Specifically, we prove that a flow is linearly stable if and only if the spectrum of the linearization is in the left half plane and, in addition, a certain system of ordinary differential equations associated with short wave asymptotics is stable.

We investigate the linear stability of inviscid plane Poiseuille flow between two parallel free surfaces. We show that there are short wave instabilities with eigenfunctions localized near the free surface and derive the asymptotics of these modes.

This paper deals with well-posedness of the system modelling two-layer shallow-water flow between two horizontal rigid plates. More precisely, we generalize results obtained by Guyenne (2010 Well-posedness of the Cauchy problem for models of large amplitude internal waves Nonlinearity 23 237–75) to the non-irrotational case. The main idea is an iterative construction with a hyperbolic equation for the bottom depth h assuming the following condition on the data: , where h0 is the initial depth of the bottom layer, and denote, respectively, the initial velocity of the bottom and top layer, and ρ = ρ1/ρ2.

We consider controllability of linear viscoelastic flow with a localized control in the momentum equation. We show that, for Jeffreys fluids or for Maxwell fluids with more than one relaxation mode, exact null controllability does not hold. This contrasts with known results on approximate controllability.