Showing papers in "Quarterly of Applied Mathematics in 1983"

A decreasing property of solutions of parabolic equations with applications to thermoelasticity

Abstract: which describe the behavior of a slab — l

The form of interfacial surfaces in Korteweg's theory of phase equilibria

Abstract: On deduit les equations differentielles de la theorie de l'equilibre de Korteweg et on applique le resultat de Pucci pour determiner le comportement local des solutions. On traite le comportement global des interfaces de phase

On the existence of the rates of stress and displacement for Prandtl-Reuss plasticity

Abstract: Introduction. In this paper we shall consider an isotropic and homogeneous elasto-perfectly plastic body, subject to the Prandtl-Reuss constitutive law with the von Mises yield condition. We shall be concerned with the existence of the rate of stress and of the rate of displacement corresponding to a given state of stress, to a given rate of the force density, and to given boundary conditions. We use the following notation. The body, in its nondeformed state, occupies a bounded connected open set S2 C R3; TD and TA, are subsets of 3fi such that ro U FA, = 3fi, ro n Tv = 0; n(x) is the outward unit normal vector to at x. We are given a state of stress a = {a,■},■ •_ , 2j3: ̂ -» R9 with oIJ = aji and such that | aD |< y/lK on £2, where aD is the deviator of a and AT is a positive constant called the yield constant. The functions f(x): S2 IR3, F(x): TN -> R3, g(x): -» R3 are given. We shall consider the following problem (see [7]): Problem (P.l). Find under what conditions for the data there exists, in some space of functions or distributions, a triple {«, a, X}, where

New third-order bounds on the effective moduli of -phase composites

Abstract: We develop some new bounds on the effective moduli of TV-phase composites. These new bounds are accurate up to and including terms of third order in 0(| — Kj\\, |^ — Hj\\), where Kt and n, are the bulk and shear modulus, respectively, of phase i. These bounds use the same statistical information as McCoy's and BeranMolyneux's bounds but are tighter than, or at worst coincident with, the latter bounds. We also present in the appendix a new perturbation solution for the effective moduli which only requires that | <5|i | = 0(| — fij |) be small.

Analysis of a free boundary problem in partial lubrication

Abstract: Hydrodynamic lubrication is concerned with a particular form of creeping flow between surfaces in relative motion where cavitation takes place. The determination of the free boundary of the cavitation area is then of fundamental importance for the computation of the characteristics of the mechanisms. Different conditions at the free boundary have been introduced. We study two of them and compare corresponding solutions with respect to film extent and pressure repartition. Introduction. The resolution of the so-called Reynolds elliptic equation using a variational inequation modeling is well known [1, 2, 3, 4], both from mathematical and numerical aspects. Unfortunately, since the cavitation area is restricted to rest in the divergent portion of the bearing, the solution obtained in this way may be unrealistic and does not always respect the mass flow conservation law in the cavitation area, especially when the supply line is not located at the maximum gap. Numerous models have been introduced in order to explain the various aspects of the cavitation phenomena [5, 6, 7, 8], If tensile strength and inertial effects are neglected, [5] gives a good basis for further developments by relating, in a one-dimensional cavitated convex slider bearing, the mass flow, the supply pressure and various breakdown conditions, the supply position being located at infinity. If the supply position and the supply pressure are given, as usually in a journal bearing, the differential problem studied in [5] becomes a two-point boundary value problem where not only the breakdown position but also the beginning of the oil film are unknown. It is common practice that a regular condition on the gradient of the pressure is taken at the film rupture whereas an eventual discontinuity is allowed for this gradient at film reformation (as in the articles by Elrod [6 p. 37, Floberg [6 p. 31,7 p. 138 and 9,10]). In the present paper, we study the mathematical aspects of this modeling (problem (P)) for an infinitely long journal bearing with zero supply and cavitation pressure whatever the eccentricity e and the supply position (f>. It will be noted that we must recall the variational inequation modeling (problem (PV)), especially the study of the film extent, before giving an existence and uniqueness theorem for the problem (P). It is of interest to notice that the solution of problem (P) is always less than that of problem (PV). The coincidence of both solutions is possible only under precise operating * Received June 12, 1980; revised version received March 13, 1981. 370 G. BAYADA AND M. CHAMBAT conditions; i.e., the input mass flow at the supply line must be the greatest compatible with the external boundary conditions. 1. Physical problem and preliminary results. The lubricating region Q = ]0, 2n[ of the bearing can be divided into two distinct zones. In the first zone i) +, where the fluid film is complete, the usual Reynolds equation (1) applies and the pressure p(x) is positive. In the second zone , which is cavitated, only an unknown fraction 0(x) of the film gap h(x) is assumed to be occupied by the fluid and the pressure is assumed to be zero (see Fig1). The unknown boundary (ct) between Q + and fi0 is the free boundary. Applying the continuity equation to the dimensionless mass flow [6], 9h — h3 (dp/dx), we obtain the following problem (P) with (0, p, a) as unknowns: Problem (P): on Q +: j(h3 = y, 0=1, p > 0; (1) ax \\ ax J ax on Q0: — (Oh) = 0, 0 < 0 < 1, p = 0; (2) dx on (a): h3 ̂ = (1 0)h; (3) dx on the supply line: p(0) = p(2n) = 0. (4) For a journal bearing, h(x) is defined by (5) where e and are geometrical data: h{x) = 1 — e cos (x — 0); 0 < e < 1, 0 <

/ n. (5) Remarks. Due to the lack of side leakage, the input and output mass flow must be equal, but the supply line may be a discontinuity line for 6 if the film starts at x = 0. Let us note that S, = 0(0 —) = 0(271 — ); then we have ^(0) = 0(0 + )h(0) h3(0) d/(0 + ). (6) dx (3) implies that at the end (ct + ) of a non-cavitated area (i.e. dp/dx < 0) we have <0=1, dp/dx = 0 (7) whereas at the beginning (a —) of H + , 0 and dp/dx have a jump. 0+ TT °* no 2tt Fig. 1. Typical h(x), p(x), (Hx) aspects. FREE BOUNDARY PROBLEM IN PARTIAL LUBRICATION 371 Then the following results hold for the shape of the cavitated area and the existence of a lower bound for the input mass flow. Theorem 1. If ft + is not empty, ft + is a connected set and dh/dx > 0 on a +. Proof. Let [fcl5 a2] be a cavitated area between two non-cavitated zones Q, + and Q2 + ; we have, by integrating (1) in ft1+ and fi2 +: h\\x) ~ = h(x) hf, i = 1, 2 (8) where hf is the film thickness at the point where p(x) is maximum in each Q,+. From (2) and (7) we have: /if = h% = hibj = d(a2^)h(a2), and this is impossible for the given film gap (5). Corollary 1. For all £, with £h(Q) < hmin (hmin = 1 — e), problem (P) has the unique solution p = 0, 6(x) = £,h(0)/h(x). Proof. This is obvious from (2), (8) because existence of a point hmin. 2. The variational inequality modeling (Problem (PV)). We recall here the mathematical formulation [1] of problem (PV) and study the shape of the non-cavitated area S1R+. Let Hl0(Q) be the Sobolev space of square-integrable functions with square-integrable derivatives, which are zero for x = 0 and x = 2k. Let K be the closed convex set defined by: K = { e //J(fi), 0 a.e. in Q}. It is well known [1, 2, 3] that there is a unique function pRe K which is a solution of the variational inequality: h3 ̂ 2A dx > n dx dx dh (

, e K. n dx pR is continuously differentiable and satisfies: dh on CiR0 = {x e Cl/pR(x) 0}, pR = 0, — > 0, (9) d f dpR on QR+ = {x e Cl/pR(x) > 0}, — yh —J = dh/dx, dpR on aR (the Reynolds free boundary surface): pR = —— = 0. (10) dx Moreover, we have the following result: Theorem 2. Each connected set of QR+ has at most one free boundary, so 0^+ begins at x = 0 or ends at x = 2n. Proof. Integrating (1) on fis+ = ]a, ft[, where Eq. (10) holds, if pR is maximum for x = x*, we have dx dx dx and then h(a) = h(b) = h(x*), which is impossible from (5). So pR has one of the three patterns shown in Fig. 2. The usual one-hump pressure 372 G. BAYADA AND M. CHAMBAT

On existence of solution of the Dirichlet problem of fourth order partial differential equations with variable coefficients

Abstract: On donne des conditions suffisantes pour l'existence et l'unicite de la solution du probleme de Dirichlet pour des equations aux derivees partielles elliptiques d'ordre 4 a coefficients variables

The dynamics of simple fluids in steady circular shear

Abstract: An isotropic simple fluid of constant density p is confined between two infinite horizontal planes which rotate steadily about separate vertical (z) axes with common angular velocity a>. We show that at least one solution to the exact equations of motion is determined by the differential equation d ( du\\ pim = 0 where u(z) is a complex variable representing the horizontal velocity and t]( \\ du/dz |, cu) is a complex shear modulus. This equation represents the extension to nonlinear viscoelasticity of the previous works of Berker and of Abbott and Walters on linear viscoelastic fluids, for which r] reduces to the usual dynamic viscosity r]*(u>). We cite two examples for the form of r] which emerge from particular rheological models; and, without attempting to solve the above equation, we briefly discuss certain of its global features.

Conservation laws with sharp inhomogeneities

Abstract: where f(u, x) is a function which is discontinuous in x. This problem has properties of nonuniqueness which have not been previously encountered. The classical theory of conservation laws has been developed, in part, to handle the question of non-uniqueness for weak solutions. Unfortunately this theory is insufficient for our study. More specifically, when waves collide with a sharp inhomogeneity, which we will call an interface, they may be transmitted, reflected, or partially transmitted and partially reflected. Unfortunately, the classical theory cannot be used to decide how an interface will affect a wave when such a collision occurs, leaving our solution undetermined. We will consider the initial value problem for Eq. (1.1a) under the initial condition

A note on the mathematical formulation of the problem of numerical coordinate generation

Abstract: A set of second order partial differential equations for the generation of three-dimensional grids around and between arbitrary shaped bodies has been proposed. These equations basically depend on the Gauss equations for a surface and have been structured in such a way that an automatic connection is established between the succeeding generated surfaces. The vanishing of the Riemann curvature tensor has been used to isolate those fundamental equations which every coordinate system in either twoor three-dimensional Euclidean space must satisfy.

On a linear age-dependent population diffusion model

Abstract: (/? is the birth modulus), that there is no diffusion through the boundary dQ of Q, that is: d A k(a, cc)u(t, a, x) doc = 0 (3) o 8*1 Jc where d/dr] is the normal derivative, and that the initial population is known: w(0, a, x) = u0(a, x). (4) A is the maximum life expectancy of the species. The initial boundary value problem (1)—(4) will be referred to as problem (I), namely (subscripts indicate partial differentiation):

The approach to normality of the solutions of random boundary and eigenvalue problems with weakly correlated coefficients

Abstract: A general class of linear self-adjoint random boundary value problems with weakly correlated coefficients is considered. The earlier result that the distribution function of the solution approaches the normal as the correlation length e tends to zero is generalized somewhat. Correction terms are derived that yield estimates for the distribution function when e is small but nonzero. The results are also applied to the eigenvalues and eigenfunctions of a corresponding class of random eigenvalue problems. The discussion is given in terms of second-order equations, but extensions to higher-order problems are readily apparent.

Nonlinear breakup of an electrohydrodynamic jet

Abstract: On presente une theorie faible non lineaire de la rupture d'un jet maintenu par les forces capillaires en presence d'un champ electrique applique. On montre la rupture du jet en gouttes principales avec leurs satellites dont les dimensions sont sensibles au nombres d'onde et au champ electrique. On n'observe pas de gouttelettes satellites lorsque le champ electrique est absent

The nonlinear circular membrane under a vertical force

Abstract: On donne la theorie exacte de la deformation d'une membrane circulaire plane, soumise a une charge verticale. On montre que le systeme d'equations peut se ramener a une seule equation differentielle ordinaire non lineaire. On montre que l'approximation de Foppl est le premier terme d'un developpement asymptotique de la theorie exacte

The solitary wave with surface tension

Abstract: Recently Brooke Benjamin [1] has drawn attention to the paper of Korteweg and de Vries [2] in which they analysed the effect of surface tension on solitary waves. If t = T/pgh2 where T is the coefficient of surface tension for the liquid, p its density, g the gravity force per unit mass and h the depth of the liquid at rest, they show that solitary waves of elevation exist when t <* which are supercritical with speeds C > (gh)i/2. Also, when t > \\ solitary waves still exist but are waves of depression rather than waves of elevation, with subcritical speeds C < (gh),/2. Brooke Benjamin [1] remarks \"that in the exceptional case t = y, there is no solitary wave according to the approximation used—the question whether the complete hydrodynamic problem has any such solution in this case remains open.\" The three dimensional equations for water waves are not considered here. Instead some information about the case t = y is obtained by a different procedure. Equations for propagation of waves in incompressible liquids have been derived by Green, Laws and Naghdi [3] using a direct two dimensional method. Similar equations have also been obtained by Green and Naghdi [4] from the three dimensional equations by an approximation. These wave propagation equations take account of surface tension and an equation for the study of solitary type waves with surface tension was given in [3], equation (5.7). In non-dimensional form this equation is

Numerical conformal mapping and analytic continuation

Abstract: A numerical method for determination of least-square approximations of an arbitrary complex mapping function is derived here and implemented with fast Fourier transforms (FFTs). An essential feature of the method is the factoring of a discrete Hilbert transform in a pair of Fourier transforms in order to reduce the operation count of the longest computation to 0(N log N). A similar factoring of the discrete Poisson integral formula allows an explicit inversion of it in 0(N log N) operations instead of 0(N3). The resulting scheme for analytic continuation appears to be considerably more reliable than the evaluation of polynomials. Examples are treated, and APL implementations of algorithms are provided.

Near critical free surface flow past an obstacle

Abstract: Description des effets non lineaires produits par de petites bosses dans des ecoulements bidimensionnels a surface libre avec des nombres de Foude proches de 1

Free boundaries in one-dimensional flow

Abstract: On discute plusieurs problemes concernant l'ecoulement dans un canal horizontal dont le fond peut etre impermeable ou non aux infiltrations. Le probleme de frontiere libre est l'histoire du mouvement d'un piston pour lequel la hauteur d'eau a ete specifiee μ(x). Le cas μ(x)≡0 est le probleme de rupture de digue