Richard J. Weinacht

Steady Poiseuille flows for a Giesekus fluid

Abstract We determine weak and classical solutions for steady Poiseuille flows of a Giesekus fluid with zero and non-zero solvent viscosity. A one-dimensional stability analysis and considerations based upon the molecular description of the model help us to determine the physically meaningful solution among multiple solutions. These solutions should prove useful in investigating some interesting phenomena such as “spurt”.

A remark on the Giesekus viscoelastic fluid

Steady planar shear flow is considered as a solution of a boundary value problem for the equation of a Giesekus fluid for various values of the mobility parameter. It is shown that if the mobility parameter is greater than 1/2 then such a solution either does not exist or is not realizable for large values of the shear rate (if there is no solvent viscosity contribution) or for an entire finite range of shear rates (if there is a small contribution from the solvent viscosity). If the mobility parameter does not exceed 1/2 then the solution exists and is realizable for all values of the shear rate. The conclusions follow from a one‐dimensional linear stability analysis of the appropriate boundary value problem and from an admissibility criterion that stems from restrictions imposed on the configuration tensor which arises from the molecular model description of the polymer liquid. We consider a solution realizable if it is stable and admissible. Thus a clearly distinct behavior of the Giesekus model is obs...

A singularly perturbed Cauchy problem

Singular perturbations are considered for the initial value problem for second order hyperbolic equations with a small positive parameter multiplying the second order time derivative term. Thus, in contrast to recent work of de Jager and Geel, the reduced equation is of the same order as the original equation but of a different type. Asymptotic expansions are constructed and shown to be uniformly asymptotically valid on sets bounded in the time direction. The proof uses energy estimates which require some delicacy due to the dependence of the characteristics on the small parameter. The problem includes that of a vibrating string in a highly viscous medium when appropriate scaling is made. The proper initial conditions differ from those treated by Zlamal and the methods employed here are different as well.

Separation and comparison theorems for classes of singular elliptic inequalities and degenerate elliptic inequalities

Separation and comparison theorems of Sturms type are given for differential inequalities involving singular elliptic operators of the form and a related class of degenerate elliptic operators. The region considered is an open connected subset of the half-space y > 0 in E m+l with an open subset of the hyperplane y< = 0 as part of the boundary. A notable feature is that data need not be prescribed on y = 0 in contrast to the regular case where data is prescribed on the entire boundary. The results are obtained by the use of Greens identity and a new Picone-type identity.

A representation formula for the wave equation revisited

We give an elementary, rigorous and self-contained derivation of a Kirchhoff representation formula for solutions of the wave equation for a bounded region in three-dimensional space for the case of non-homogeneous initial conditions. A special feature of the result is the role of spherical caps which seems not to be noticed previously. The formula is readymade for boundary integral formulations of initial/boundary-value problems for the wave operator.

A Time-Dependent Wave-Thermoelastic Solid Interaction

This paper presents a combined field and boundary integral equation method for solving time-dependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the time-dependent fluid-structure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analized with Lubichs approach for time-dependent boundary integral equations. Using the Laplace transform in terms of time-domain data existence and uniqueness results are established. Galerkin semi-discretization approximations are derived and error estimates are obtained. A full discretization based on the Convolution Quadrature method is also outlined. Some numerical experiments are also included in order to demonstrate the accuracy and efficiency of the procedure.