Lewis Wheeler

Spatial Decay Estimates for the Navier–Stokes Equations with Application to the Problem of Entry Flow

The development of velocity profiles in the inlet region of channels or pipes is a classic problem of laminar flow theory which has given rise to an extensive literature. Most previous work on this entry flow problem has involved some degree of simplification either in flow geometry or in the governing equations. Here we treat the flow development within the general framework of the Navier–Stokes equations governing the steady laminar flow of an incompressible viscous fluid in a cylindrical pipe of arbitrary cross-section.The problem to be treated is that of an “end effect” involving comparison between two distinct solutions of the Navier–Stokes equations. Thus we consider the spatial evolution of the difference between the base flow and the fully developed solution. The corresponding velocity difference clearly satisfies a condition of zero net inflow. In this way, we draw an analogy between the issue of concern here and the celebrated “Saint-Venant’s Principle” of elasticity theory involving the effect ...

On the role of constant-stress surfaces in the problem of minimizing elastic stress concentration

Abstract This paper is concerned with conditions under which surfaces of constant stress magnitude serve as optimal from the standpoint of minimizing stress. Such conditions are established for elastic solids in the cases of antiplane shear deformation, axisymmetric torsional deformation, and plane deformation

Wave propagation aspects of the generalized theory of heat conduction

SummaryIt is well known that the classical theory of heat conduction, which is based upon Fouriers law, leads to infinite propagation speeds for thermal disturbances. In a recent investigation [1], Gurtin and Pipkin devised a theory appropriate to rigid heat conductors with memory, and put forth evidence that their theory gives rise in general to finite wave speeds. The present paper is concerned with the linearized version of the theory presented in [1], in the form it assumes for isotropic conductors. We arrive at conditions upon the material response functions that ensure the finiteness of the wave speeds. In addition, we establish uniqueness of solutions for a class of history-value problems suggested by the linearized theory.ZusammenfassungBekanntlich führt Fouriers klassische Theorie der Wärmeleitfähigkeit zu einer unendlich großen Ausbreitungsgeschwindigkeit lokaler Temperaturstörungen. Gurtin und Pipkin haben eine Theorie für starre Wärmeleiter mit Gedächtnis eingeführt und haben auch einen Beweis dafür gegeben, daß ihre Theorie auf eine endliche Ausbreitungsgeschwindigkeit führt. Die vorliegende Arbeit bezieht sich auf die linearisierte Form der Theorie von Gurtin und Pipkin für isotrope Leiter. Es werden Bedingungen für endliche Ausbreitungsgeschwindigkeit angegeben. Ferner wird die Eindeutigkeit der Lösungen für eine Klasse von history-value-Problemen angegeben, die durch die lineare Theorie nahegelegt werden.

On voids of minimum stress concentration

Abstract An isotropic elastic medium containing a void is loaded at infinity by given stresses. The problem of finding a minimizing void shape for the stress concentration is formulated. It is proved that a sufficient condition for a surface to be a minimizer is that the two surface principal stresses be constant and equal. A class of ellipsoids having this property is exhibited and relations between the applied stresses and the ellipsoid parameters are established.

A three-dimensional inverse problem for inhomogeneities in elastic solids

The Newtonian potential is used to solve an inverse problem in which we seek the shape of an inhomogeneity in an infinite elastic matrix under uniform applied stresses at infinity such that certain stress components are uniform on the boundary of the inhomogeneity. It is shown that ellipsoids furnish the solution of this inverse problem. Exact and general expressions for the stress and displacement are given explicitly for points in the elastic matrix outside the inhomogeneity. The solution of the corresponding plane deformation problem is found as a limiting case. Several applications are presented, and results from the literature are confirmed as special cases.

Stress bounds for bars in torsion

Upper bounds for the maximum shear stress in the St. Venant torsion problem are derived with the aid of the theory of subharmonic functions. The main result is a bound that is determined in a simple manner by the magnitude of the applied twisting moment and two parameters peculiar to the cross section: the radius of the largest circle contained in it and the minimum curvature of the curve that bounds it.ZusammenfassungMit Hilfe subharmonischer Funktionen werden obere Grenzen in dem Torsionsproblem von St. Venant erhalten. Das Hauptergebnis ist eine Grenze, die auf einfache Weise vom Drehmoment und zwei nur vom Querschnitt abhängigen Parametern bestimmt ist, und zwar von dem Radius des grössten eingeschriebenen Kreises und von der Minimalkrümmung der Begrenzungskurve des Querschnitts.

On the derivatives of the stretch and rotation with respect to the deformation gradient

In this paper, a result involving the eigenprojections of the right stretch and its derivative with respect to the deformation gradient is derived, and a related result is found for the rotation. As an application, the form of the constitutive law for an isotropic hyperelastic material in the case when the strain energy function is expressed in terms of the right stretch, is shown to follow at once.

On rigid inclusions of minimum stress concentration

Abstract T he three-dimensional problem of finding the shape of minimum stress concentration for a rigid inclusion imbedded in an elastic matrix is analyzed and solved. The matrix extends to infinity, filling the space exterior to the inclusion. Loading consists of uniform stress applied at infinity, so that in the absence of the inclusion the medium would be homogeneously stressed. The optimum inclusions are found to be ellipsoidal in shape, and conditions on the loading are found under which these ellipsoids can be rigorously proven to be optimal.

A saint-venant principle for the gradient in the Neumann problem

The maximum principle for subharmonic functions is used to obtain upper bounds for the gradient in the Neumann problem of potential theory. These bounds, which concern a curvilinear strip domain having nonzero boundary data only on an end, entail an exponential decay of the gradient magnitude with distance from that end.ZusammenfassungUnter Benutzung des Maximumprinzips für subharmonische Funktionen werden obere Schranken angegeben für den Gradienten des Neumannschen Problems der Potential-Theorie. Diese Schranken betreffen einem gekrümmten Streifenbereich mit von Null verschiedenen Grenzdaten an einem Ende und haben zur Folge, dass der Betrag des Gradienten exponentiell mit dem Abstand von diesem Ende abfällt.

Uniqueness theorems for finite elastodynamics

Described in this paper is a study of the uniqueness of solutions to the boundary-initial value problems of nonlinear dynamical elasticity. Particular consideration is given to the uniqueness implications of certain well-known a priori restrictions on the material response.ZusammenfassungIn diesem Werk wird die Eindeutigkeit der Lösungen von Anfangswert-Randwertproblemen mit nichtlinearer dynamischer Elastizität beschrieben. Besondere Betrachtung ist den Folgerungen der Eindeutigkeit von bestimmten bekannten a priori Beschränkungen auf materielle Reaction gegeben.