Degenerate solutions of electron-beam equations

In this paper, exact solutions are constructed for stationary election beams that are degenerate in the Cartesian (x,y,z), axisymmetric (r,θ,z), and spiral (in the planes y=const (u,y,v)) coordinate systems. The degeneracy is determined by the fact that at least two coordinates in such a solution are cyclic or are integrals of motion. Mainly, rotational beams are considered. Invariant solutions for beams in which the presence of vorticity resulted in a linear dependence of the electric-field potential ? on the above coordinates were considered in [1], In degenerate solutions, the presence of vorticity results in a quadratic or more complex dependence of the potential on the coordinates that are integrals of motion. In [2] and in a number of papers referred to in [2], the degenerate states of irrotational beams are described. The known degenerate solutions for rotational beams apply to an axisymmetric one-dimensional (r) beam with an azimuthal velocity component [3] and to relativistic conical flow [1]. The equations used below follow from the system of electron hydrodynamic equations for a stationary relativistic beam $$\begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left[ {\sqrt \gamma g^{\beta \beta } g^{\alpha \alpha } \left( {\frac{{\partial A_\alpha }}{{\partial q^\beta }} - \frac{{\partial A_\beta }}{{\partial q^\alpha }}} \right)} \right]} = 4\pi \rho \sqrt \gamma g^{\alpha \alpha } u_\alpha ,} \\ {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}\left( {\sqrt \gamma g^{\beta \beta } \frac{{\partial \varphi }}{{\partial q^\beta }}} \right)} = 4\pi \rho \sqrt {\gamma u} ,\sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta ^2 + 1 = u^2 } } \\ \begin{gathered} \frac{\eta }{c}u\frac{{\partial \mathcal{E}}}{{\partial q^\alpha }} = \sum\limits_{\beta = 1}^3 {g^{\beta \beta } u_\beta } \left( {\frac{{\partial p_\beta }}{{\partial q^\alpha }} - \frac{{\partial p_\alpha }}{{\partial q^\beta }}} \right), \hfill \\ \begin{array}{*{20}c} {\sum\limits_{\beta = 1}^3 {\frac{\partial }{{\partial q^\beta }}(\sqrt \gamma g^{\beta \beta } \rho u_\beta ) = 0,u \equiv \frac{\eta }{{c^2 }}(\varphi + \mathcal{E}) + 1,} } \\ {cu_\alpha \equiv \frac{\eta }{c}A_\alpha + p_\alpha ,\alpha ,\beta = 1,2,3,\gamma \equiv g_{11} g_{22} g_{33} } \\ \end{array} \hfill \\ \end{gathered} \\ \end{array} $$ where q^β denotes orthogonal coordinates with the metric tensor g^ββ (β=1,2,3); A_α is the magnetic potential; A_α = (u_α/u)c is the electron velocity; ρ is the scalar space-charge density (ρ > 0); is the energy in eV; p_α is the generalized momentum of an electron per unit mass; η is the electron charge-mass ratio.     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

Improvement on stability and convergence of A. D. I. schemes

IntroductionAlternatingdirectionimplicit(A.D.I.)schemeswhichwasdiscoveredin1950',hasbecomeoneofthemostimportantmethodsintheapproximationofthesolutionsofparabolicpartialdifferentialequationsinmulti-dimensionalspace.Someofresultsaboutstabilityandconvergencearetooweakandincomplete,we'lltrytoimprovetheminthispaper.Considerinitial-boundaryvalueproblemintwospacevariablesLetΩhbeauniformrectangularmeshofO.h>0isthespacestepinxandydireehon*ProjectsupPOI'tedbytheNahonalNaturalScienceFoundahonofChi…     

Regularity for the stationary Navier-Stokes equations in bounded domain

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain ^N, 5N . If u, p satisfy the additional conditions

Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients

In order to capture the complexities of two-phase flow in heterogeneous porous media, we have used the method of large-scale averaging and spatially periodic models of the local heterogeneities. The analysis leads to the large-scale form of the momentum equations for the two immiscible fluids, a theoretical representation for the large-scale permeability tensor, and a dynamic, large-scale capillary pressure. The prediction of the permeability tensor and the dynamic capillary pressure requires the solution of a large-scale closure problem. In our initial study (Quintard and Whitaker, 1988), the solution to the closure problem was restricted to the quasi-steady condition and small spatial gradients. In this work, we have relaxed the constraint of small spatial gradients and developed a dynamic solution to the closure problem that takes into account some, but not all, of the transient effects that occur at the closure level. The analysis leads to continuity and momentum equations for the-phase that are given by

Phragmén-Lindelöf theorems for nonlinear elliptic equations

We obtain theorems of Phragmén-Lindelöf type for the following classes of elliptic partial differential inequalities in an arbitrary unbounded domain \(\Omega \subseteq \mathbb{R}^n ,{\text{ }}n \geqq 2\) (A.1) $$\sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} 9(x)\frac{{\partial u}}{{\partial xj}}} \right)} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a _ij are elliptic in Ω and b _i ε L^∞(Ω) and where also a _ij are uniformly elliptic and Holder continuous at infinity and b _i = O(|x|⁺¹) as x → ∞; (A.2) $${\text{(A}}{\text{.2) }}\sum\limits_{i,j = 1}^n {a_{ij} (x,{\text{ }}u,{\text{ }}\nabla u)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a_ijare uniformly elliptic in Ω and b _iε L^∞(Ω); and finally (A.3) $${\text{div(}}\nabla u^p \nabla u {\text{)}} \geqq f{\text{(}}u{\text{), }}p > - 1,$$ where the operator on the left is the so-called P-Laplacian. The function f is always supposed positive and continuous. Moreover u is assumed throughout to be in the natural weak Sobolev space corresponding to the particular inequality under consideration, namely u ε. W _loc ^1,2 (Ω) ∩L _loc ^t8 (Ω) for (A.I), W _loc ^2,n(Ω) for (A.2), and W _loc ^1,p+2 (Ω) ∩ L _loc ^t8 (Ω) for (A.3). As a consequence of our results we obtain both non-existence and Liouville theorems, as well as existence theorems for (A.1).     

On the change in the region of localization of perturbations in nonlinear transport processes

The quasilinear parabolic equation $$\frac{{\partial u}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {\left| {\frac{{\partial u^k }}{{\partial x}}} \right|^{n - 1} \frac{{\partial u^k }}{{\partial x}}} \right) - \gamma u^m , k,\gamma > 0, m \geqslant 0, kn > 1$$ can be regarded as the generalized form of many well-known transport equations with transport coefficients that depend on the transported quantity. For example, the case n = 1 corresponds to heat transport in a medium with thermal conductivity and sinks which depend on the temperature in accordance with a power law [1, 2]; the case k = m = 1 describes the flow of a conducting non-Newtonian fluid in a transverse magnetic field [3]; the case k = 2, m = 0 corresponds to the magnetohydrodynamic flow of the same fluid in a transverse magnetic field in a laminar boundary layer [4]. In the general case, Eq. (0.1) describes turbulent flow in a porous medium with nonlinear sinks [5]. A characteristic feature of the processes described by Eq. (0.1) is the possibility of existence of a front x = x_f(t) which strictly separates regions with u(x, t) = 0 from regions with u(x, t) > 0 in which the perturbations of the transported quantity are localized [6]. In the present paper, the change in the region of localization of the perturbations of the transported quantity is investigated in the Cauchy problems for Eq. (0.1).     

Modified dissipativity for a non-linear evolution equation arising in turbulence

We are concerned with the regularity properties for all times of the equation $$\frac{{\partial U}}{{\partial t}}\left( {t,x} \right) = - \frac{{\partial ^2 }}{{\partial x^2 }}\left[ {U\left( {t,{\text{0}}} \right) - U\left( {t,x} \right)} \right]^2 - v\left( { - \frac{{\partial ^2 }}{{\partial x^2 }}} \right)^\alpha U\left( {t,x} \right)$$ which arises, with α=1, in the theory of turbulence. Here U(t,·) is of positive type and the dissipativity α is a non-negative real number. It is shown that for arbitrary ν≧0 and ?>0, there exists a global solution in \(L^\infty [0,\infty ;H^{\tfrac{3}{2} - \varepsilon } (\mathbb{R})]\) . If ν>0 and \(\alpha > \alpha _{cr} = \tfrac{1}{2}\) , smoothness of initial data persists indefinitely. If 0≦α<α _cr, there exist positive constants ν₁(α) and ν₂(α), depending on the data, such that global regularity persists for ν>ν₁(α), whereas, for 0≦ν<ν₂(α), the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of α _cr, similar results hold for the Navier-Stokes equation.     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

On circulation in relativistic hydrodynamics

In this paper it is shown that the relativistic analogue of the circulation in classical hydrodynamics is the integral

$$C = \int {\left( {{\text{1 + }}\frac{i}{{c^{\text{2}} }}} \right)} {\text{ }}U_\alpha dx^\alpha ,$$     

On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation

We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12W_e^￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},     

Analytische NÄherungslösung für die laminare Zweistoff-Grenzschichtströmung lÄngs eines ebenen verdunstenden Flüssigkeitsfilms

Zusammenfassung Die Auslegung eines Filmverdampfers für Brennkammern erfordert die Kenntnis des verdunstenden Massenstroms; dieser wird bestimmt durch den gekoppelten WÄrme- und Stoffübergang in der Strömungsgrenzschicht. Die Ergebnisse numerischer Untersuchungen der Massenstromdichte in laminaren Grenzschichten werden herangezogen, um die Genauigkeit einer einfach auszuwertenden analytischen NÄherungslösung zu überprüfen, wobei verÄnderliche Stoffwerte berücksichtigt werden. Die gute übereinstimmung der analytischen NÄherung mit der numerischen Lösung für Benzol zeigt die allgemeine Brauchbarkeit des Verfahrens.

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Analytical approximation for the laminar binary boundary-layer flow along a vaporizing liquid layer

For the design of a liquid film vaporizer the knowledge of the vaporizing mass flow is necessary. This is determined by the coupled heat and mass transfer. The results of numerical studies of the mass flow rates in laminar boundary layers are taken to test the accuracy of a simple analytical approximation taking variable transport properties into account. The analytical and numerical results for benzole agree rather well pointing out thereby the general validity of this method.

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Bezeichnungen c Massenkonzentration - c_p spezifische WÄrme - D₁₂ binÄrer Diffusionskoeffizient - f dimensionslose Stromfunktion - h₁,h₂ Enthalpie der Komponenten - K,^* von der Temperatur abhÄngige Koeffizienten (Gl.(11)) - M Molmasse - m^* Massenstromdichte - p Druck - r VerdampfungswÄrme - T Temperatur - u,v Geschwindigkeitskomponenten - x, y, Y Ortskoordinaten - normierte Konzentration - normierte Temperatur - dynamische ZÄhigkeit - Dichte-ZÄhigkeitsverhÄltnis - WÄrmeleitfÄhigkeit Kennzahlen Prandtl-Zahl - Schmidt-Zahl - Reynoldszahl Indizes o Filmoberseite - u Plattenunterseite - 1 Gas 1 (Benzoldampf) - 2 Gas 2 (Luft) - Au\enrand der Grenzschicht     

Approximating equations of a plane gas flow and their integration

It is known that the nonlinear system of equations of plane steady isentropic potential gas flow can be linearized and transformed to a single equivalent linear differential equation of second order. For the case of a perfect gas this equation has the form [1]

$$\begin{gathered} \frac{{1 - \tau ^2 }}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial ^2 \Phi }}{{\partial \theta ^2 }} + \frac{{\partial ^2 \Phi }}{{\partial \tau ^2 }} + \frac{{\tau (1 - \tau ^2 )}}{{\tau ^2 (1 - \alpha \tau ^2 )}} \frac{{\partial \Phi }}{{\partial \tau }} = 0, \hfill \\ (\tau = w/c_k , w = \sqrt {u^2 + \upsilon ^2 } , \alpha = (\gamma - 1)/(\gamma + 1); \gamma = c_p /c_\upsilon ). (0.1) \hfill \\ \end{gathered} $$     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator     

About a condition for blow up of solutions of cauchy problem for a wave equation

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Inertia effect in conical converging flow of viscoelastic fluids

The convergent flow of viscoelastic fluids in conical nozzles has been examined. The experimentally determined streamlines agreed with those obtained from calculations with an approximation up to \(\dot V^2 \) Because of the elasticity of the fluid a rotationally symmetric eddy arises. It clings to the wall whereas the output flow proceeds in the middle. When $$\frac{\varrho }{{\eta _0 }}\left( {\frac{{\dot V^2 }}{{t_0 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} > 40$$ > 40 another eddy can arise in front of the first one. This second eddy is situated in the middle and causes the output to proceed as a flow near the wall. The higher the afore-mentioned dimensionless value is, the higher is the inclination of the flow to instability.     

Forced convection heat transfer and friction characteristics in multi-exit -rectangular package with inclined tube bundles

An experiment was carried out to investigate the characteristics of the heat transfer and pressure drop for forced convection airflow over tube bundles that are inclined relative to the on-coming flow in a rectangular package with one outlet and two inlets. The experiments included a wide range of angles of attack and were extended over a Reynolds number range from about 250 to 12,500. Correlations for the Nusselt number and pressure drop factor are reported and discussed. As a result, it was found that at a fixed Re, for the tube bundles with attack angle of 45 ° has the best heat transfer coefficient, followed by 60, 75 and 90 °, respectively. This investigation also introduces the factors which can be used for finding the heat transfer and the pressure drop factor on the tube bundles positioned at different angles to the flow direction. Moreover, no perceptible dependence of Cand C on Re was detected. In addition, flow visualizations were explored to broaden our fundamental understanding of the heat transfer for the present study.     

Laminar forced convection of power-law non-Newtonian fluids inside ducts

Thermal entrance region heat transfer for laminar forced convection of power-law fluids inside a circular tube and parallel plate channel for uniform wall temperature is solved exactly, and as many eigenvalues and eigenfunctions as needed for the solution are determined automatically and with high accuracy by using the recently advanced Sign-Count method. Results are presented for the local and average Nusselt number over a wide range of the Graetz number in both graphical and tabular forms. The present benchmark results are utilized to critically examine the accuracy of the approximate Leveque solution.

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Laminare Zwangskonvektion von nicht-Newtonschen Fluiden, die in Kanälen strömen und dem Potenzansatz folgen

Zusammenfassung Es werden exakte Lösungen für den Wärmetransport in der thermischen Einlaufzone in runden Rohren und zwischen parallelen Platten für Fluide nach dem Exponentialansatz bei laminarer Zwangskonvektion und mit gleichmäßiger Wandtemperatur angegeben. Unter Benutzung der jüngst verbesserten Sign-Count-Methode werden so viele Eigenwerte und Eigenfunktionen, wie für die Lösung benötigt, automatisch und mit großer Genauigkeit bestimmt. Ergebnisse werden in graphischer wie tabellarischer Form über einen weiten Bereich der Graetz-Zahl für die örtliche und mittlere Nusselt-Zahl vorgestellt. Die vorliegenden richtungsweisenden Ergebnisse werden dazu benützt, um die Genauigkeit der Levequeschen Näherungslösung kritisch zu prüfen.

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Nomenclature b radius of circular duct or half the spacing between parallel plates - C - D _h hydraulic diameter=4b for parallel plate, 2b for circular tube - f(r), F(r) temperature distribution at the inlet, dimensional and dimensionless, respectively - g (r), G (R) energy generation, dimensional and dimensionless, respectively - h (z) heat transfer coefficient - H _i(Z) defined by Eq. (11b) - k thermal conductivity - l₀,l _N reference lengths to nondimensionalizer andz coordinates respectively (l ₀=b andl _N= D_h) - n power-law index - Nu _av average Nusselt number=h _av D_h/k - Nu (Z) local Nusselt number=h (z)D _h/k - p 0 for parallel-plate duct, 1 for circular duct - r radial or normal coordinate - R = dimensionless radial coordinate - T(r, z) fluid temperature - T _av (z) average fluid temperature - T ^* reference temperature - T reference temperature difference - U(R) = normalized velocity profile - U ^*(R) - w (r) fully developed velocity profile - w _av average velocity - z axial coordinate - Z = dimensionless axial coordinate Greek letters thermal diffusivity - (R,Z) = dimensionless temperature - _w = dimensionless wall temperature - _i eigenvalues of the eigenvalue problem (10) - (_i,R) eigenfunctions of the eigenvalue problem (10)     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Boundary conditions at the edge of a thin or thick plate bonded to an elastic support

At the clamped edge of a thin plate, the interior transverse deflection ω(x ₁, x₂) of the mid-plane x ₃=0 is required to satisfy the boundary conditions ω=?ω/?n=0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for ω must always have the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakqaabeqaaiaabEhacaqGTaWa% aSaaaeaacaGG0aGaae4vamaaCaaaleqabaGaamOqaaaaaOqaaiaaco% dadaqadaqaaiaacgdacqGHsislcaqG2baacaGLOaGaayzkaaaaaiaa% bIgadaahaaWcbeqaaiaackdaaaGcdaWcaaqaaiaabsgadaahaaWcbe% qaaiaackdaaaGccaqG3baabaGaaeizaiaabIhafaqabeGabaaajaaq% baqcLbkacaGGYaaajaaybaqcLbkacaGGXaaaaaaakiabgUcaRmaala% aabaGaaiinaiaabEfadaahaaWcbeqaaiaadAeaaaaakeaacaGGZaWa% aeWaaeaacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGOb% WaaWbaaSqabeaacaGGZaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaa% caGGZaaaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaa% saaiaacodaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0Jaaiimaiaa% cYcaaeaadaWcaaqaaiaabsgacaqG3baabaGaaeizaiaabIhaliaacg% daaaGccqGHsisldaWcaaqaaiaacsdacqqHyoqudaahaaWcbeqaaiaa% bkeaaaaakeaacaGGZaWaaeWaaeaacaGGXaGaeyOeI0IaaeODaaGaay% jkaiaawMcaaaaacaqGObWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGG% YaaaaOGaae4DaaqaaiaabsgacaqG4bqbaeqabiqaaaqcaauaaKqzGc% GaaiOmaaqcaawaaKqzGcGaaiymaaaaaaGccqGHRaWkdaWcaaqaaiaa% csdacqqHyoqudaahaaWcbeqaaiaabAeaaaaakeaacaGGZaWaaeWaae% aacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaWba% aSqabeaacaGGYaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGGZa% aaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaasaaiaa% codaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0JaaiimaiaacYcaaa% aa!993A!\[\begin{gathered}{\text{w - }}\frac{{4{\text{W}}^B }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4{\text{W}}^F }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^3 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4\Theta ^{\text{F}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\end{gathered}\]with exponentially small error as L/h→∞, where 2h is the plate thickness and L is the length scale of ω in the x ₁-direction. The four coefficients W ^B, W^F, Θ ^B , Θ ^F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h→∞ (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for Θ ^B , W ^F, we prove that these constants take large values when the support is ‘soft’ and so may still have a strong influence even when h/L is small. The coefficient W ^F is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are w=0,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaabsgacaqG% 3baabaGaaeizaiaabIhaliaacgdaaaGccqGHsisldaWcaaqaaiaacs% dacqqHyoqudaahaaWcbeqaaiaabkeaaaaakeaacaGGZaWaaeWaaeaa% caGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaSaaae% aacaqGKbWaaWbaaSqabeaacaGGYaaaaOGaae4DaaqaaiaabsgacaqG% 4bqbaeqabiqaaaqcaauaaKqzGcGaaiOmaaqcaawaaKqzGcGaaiymaa% aaaaGccqGH9aqpcaGGWaGaaiilaaaa!5DD4!\[\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} = 0,\]which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.