Solution of the equations of a regular electrostatic beam in the presence of emission from an arbitrary surface

An analytic solution of the equations of a regular electrostatic beam in the presence of emission from an arbitrary surface under total space charge conditions is given. It is assumed that the emitter is a coordinate surface x¹= const in an orthogonal system xⁱ (i=1, 2, 3), and the emission-current density J is a given function J(x², x³). The solution is represented in the form of series in x with coefficients that are functions of x², x³ and determined from recurrence relations. In expansion along the length of an arc of the curvilinear axis x¹, which is orthogonal to the emitter, the first correction of the Child-Langmuir 3/2 law is determined only by the total curvature (the sum of the principal curvatures) of the emitting surface. Solution of the problem in the formulation in question permits determination of the collector shape that ensures the given distribution of the emissioncurrent density over the given surface.     

任意轴对称弹性体吸附接触的广义Maugis模型

通过线性叠加Sneddon方法和Lowengrub-Sneddon方法分别给出的解, 得到了一个弹性半空间 轴对称混合边值问题的一般解，进而研究了两个一般轴对称弹性体的正向无摩擦吸附接触问 题. 考虑任意有效的表面形状(要求中心部分首先进入接触)和任意的表面吸附作用，推广 得到了广义Maugis模型. 该模型是一个半解析的模型，它可以分解成表面形状和表面吸附 作用的分别独立影响的两部分，以及一个关联变形和吸附作用的式子. 利用Dugdale模型近 似表面吸附作用，得到了具有任意有效的表面形状的广义M-D模型. 它在强吸附或软材料条 件下的极限形式是广义JKR模型，而在弱吸附或硬材料下的另一个极限形式是广义DMT模型.     

Slow rotation of a sphere in a confined volume of a dilute gas

The axially symmetric motion of a gas in a volume confined between an external immobile surface of rotation and a coaxial surface of a rotating sphere is considered. A solution is obtained by the moment method based on the Boltzmann equation with a collision integral of Maxwellian molecules. The gas-velocity distribution and an expression for the friction torque exerted on the sphere are obtained for arbitrary Knudsen numbers and for an arbitrary shape of the outer surface. The proportionality of the gas slip velocity over the surface of the sphere to the friction strain is shown. The friction torque is investigated for specific shapes of the outer surface. The motion of a gas filling the space between concentric spheres, each of which rotates about an arbitrary axis, is treated. In the limiting case of small Knudsen numbers the expressions obtained are compared with the corresponding results for a continuous medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–124, July–August, 1978.The authors are grateful to the participants of the seminars guided by G. I. Petrov and A. M. Golovnyi for discussions concerning this work.     

Effect of surface emittance and approximate kernels in radiation-conduction interaction

Representative results are presented for nongray radiation-conduction interaction in a radiating gas bounded by two gray infinite parallel plates. The effect of the surface emittanee on the interaction of radiation with conduction for diffuse surfaces and a special case of one black surface combined with a diffusely emitting, specularly reflecting surface is investigated. Additionally, the accuracy of the one-term exponential kernel approximation is examined.     

Arbitrary motion of an elongated body in an ideal fluid

A method is considered which permits the use of a computer to calculate the fluid velocity on the surface of a solid body moving in an ideal fluid and to calculate the added mass. The method of [1, 2], developed for bodies of revolution, in which the flow is simulated by a system of sources and sinks distributed continuously over the body surface, is extended to the case of an arbitrary body. In contrast with the analogous work of Hess and Smith [3], where the fluid velocities on the surface of an arbitrary body were determined for translational motions, in the present case the basic integral equation of the problem is solved by the method of successive approximations without preliminary approximation of the equation by a system of linear algebraic equations of high order, which leads to a shortening of the computations.The results of the calculations are compared with the known exact values of the velocities and the added mass for a triaxial ellipsoid, and also with the results of the experimental determination of the pressures on the surface of an elongated body.     

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

Analysis of non-Fourier heat conduction in a solid sphere under arbitrary surface temperature change

In this paper, the non-Fourier heat conduction in a solid sphere under arbitrary surface thermal disturbances is solved analytically. Four cases including sudden, simple harmonic periodic, triangular and pulse surface temperature changes are investigated step-by-step. The analytical solutions are obtained using the separation of variables method and Duhamel’s principle along with the Fourier series representation of an arbitrary periodic function and the Fourier integral representation of an arbitrary non-periodic function. Using these analytical solutions, the temperature profiles of the solid sphere are analyzed, and the differences in the temperature response between the “hyperbolic” and “parabolic” are discussed. These solutions can be applicable to all kinds of non-Fourier heat conduction analyses for arbitrary boundary conditions occurred in technology. And as application examples, particular attention is devoted to the cases of triangular surface temperature change and pulse surface temperature change. The examples presented in this paper can be used as benchmark problems for future numerical method validations.     

多层土深基坑混合支护面向对象的程序设计

由多层土构成的深基坑 ,采用多层支点的混合支护 ,在地面荷载及地下水的影响下 ,支护设计计算十分复杂。作者采用寻找任意层支点和任意层地层的通用算法并编制通用程序的方法 ,解决了混合支护设计的计算问题。作者采用面向对象的程序设计方法设计具有 Window风格的界面 ,提高了程序的实用性。     

Instability of the magnetic fluid shape in the field of a line conductor with current

The static shape of the surface of a finite magnetic fluid volume between horizontal plates is investigated theoretically. The nonuniform magnetic field is generated by a horizontal line conductor with current, which is placed above the upper plate. The variational problem of minimum energy relative to plane surface perturbations is considered for a simply connected magnetic fluid volume. The problem is solved for arbitrary magnetic fields in the noninductive approximation with account for the gravity force and surface tension. Unstable solutions are found. The possibility of stepwise behavior in response to quasi-static changes of the current in the conductor is investigated for the surface shape of a finite magnetic fluid volume.     

Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall

The slow viscous flow problem of an arbitrary solid particle in motion near a planar wall is recast into a boundary integral formulation. The present formulation employs the Green function appropriate to the planar wall problem and is developed in sufficient generality to allow calculations for arbitrary particles in any base flow which satisfies Stokes equations and no-slip on the wall. The resulting integral equations are easily discretized and solved for the particle surface tractions. Calculations are performed for axisymmetric motions of a variety of ellips?ids near the planar wall. Agreement with existing theory is excellent.     

An analysis of turbulent free convection about bodies of arbitrary geometrical configurations

A general solution method is suggested for the prediction of the turbulent free convection heat transfer from curved surfaces. The method which may be viewed as a generalized version of the Eckert-Jackson method for the isothermal plate, is designed to deal with bodies of arbitrary geometrical configurations. The surface wall temperature is also allowed to vary in the streamwise direction in an arbitrary fashion. For the illustrative purpose, the calculations are carried out for the turbulent free convection about the horizontal circular cylinder, and the results are compared with the existing empirical formula. The flow transition from laminar to turbulent is also predicted by matching the laminar and turbulent solutions.     

Viscous liquid film flow down an inclined corrugated surface. Calculation of the flow stability to arbitrary perturbations using an integral method

Viscous liquid film flow along an inclined corrugated (sinusoidal) surface has been studied. Calculations were performed using an integral model. The stability of nonlinear steady-state flows to arbitrary perturbations was examined using the Floquet theory. It has been shown that for each type of corrugation there is a critical Reynolds number for which unstable perturbations occur. It has been found that this value greatly depends on the physical properties of the liquid and geometric parameters of the flow. In particular, in the case of film flow down a smooth wall, the critical waveformation parameter depends only on the angle of inclination of the flow surface. The values of the corrugation parameters (amplitude and period) were obtained for which the film flow down a wavy wall is stable to arbitrary perturbations up to moderate Reynolds numbers. Such parameter values exist for all investigated angles of inclination of the flow surface.     

Dynamics of a rigid-plastic curvilinear plate of varying thickness with an arbitrary hole

A perfect rigid–plastic body is used as a model to develop a general procedure for analyzing the dynamic behavior of an arbitrary curvilinear plate of variable thickness with an arbitrary internal hole. The plate is subjected to an arbitrary, uniform, short-term dynamic surface load. Two plate deformation patterns are considered. Analytic formulas for plastic zones, ultimate loads, and residual deflections are presented. Numerical examples are given     

Scattering of elastic waves by an arbitrary small imperfection in the surface of a half-space

T , the first of two articles, is concerned with the scattering of elastic waves by arbitrary surface-breaking or near surface defects in an isotropic half-plane. We present an analytical solution, by the method of matched asymptotic expansions, when the parameter , which is the ratio of a typical length scale of the imperfection to the incident radiation's wavelength, is small. The problem is formulated for a general class of small defects, including cracks, surface bumps and inclusions, and for arbitrary incident waves. As a straightforward example of the asymptotic scheme we specialize the defect to a two-dimensional circular void or protrusion, which breaks the free surface, and assume Rayleigh wave excitation ; this inner problem is exactly solvable by conformal mapping methods. The displacement field is found uniformly to leading order in , and the Rayleigh waves which are scattered by the crack are explicitly determined. In the second article we use the method given here to tackle the important problem of an inclined edge-crack. In that work we show that the scattered field can be found to any asymptotic order in a straightforward manner, and in particular the Rayleigh wave coefficients are given to O(²).     

Effect of surroundings on the transient energy transfer in a layer of radiating gas

Summary An analysis is presented for the transient cooling or heating of a stagnant layer of hot radiating gas surrounded by a cold gas capable of absorbing and emitting radiation. Scattering of radiation is neglected, and energy transfer by conduction and convection is considered to be negligible compared with radiation. The gas is assumed to be perfect and in local thermodynamic equilibrium. The heating of a cold gas by a diffuse and a collimated radiation flux incident on the boundary of the gas from some external source is considered, and the dependence of physical and radiative properties on temperature is taken into account. The problem is formulated exactly using radiative transfer theory. A scheme is developed for the numerical solution of the nonlinear integrodifferential equations of energy conservation. Starting with arbitrary, but given, initial temperature profiles, temperature distributions and local radiative fluxes are predicted as a function of time for a wide range of physically interesting conditions.     

Velocity field generated by wing vibrations propagating along the surface

The velocity field generated by wing vibrations propagating along an elastic wing surface with finite velocity is studied.The gasdynamic problem is reduced to a mixed boundary-value problem with a moving boundary for the three-dimensional wave equation. The solution is obtained in closed form when the wing travels at supersonic velocity following an arbitrary law, the vibration propagation front is an arbitrary curve displacing along the wing surface, and the wing edges are supersonic.     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

A wave propagation problem for non-uniform screw dislocation motion in a viscoelastic half-plane

The wave propagation problem for a largely arbitrary anti-plane displacement discontinuity imposed along a line perpendicular to the surface of a stress-free linearly viscoelastic half-plane is considered. The general Laplace transform solution is obtained and then inverted for the case of a screw dislocation moving at an arbitrary speed in a Maxwell material. It is shown that the material viscoelasticity alters the coefficient of the dislocation edge stress singularity and damps the surface displacements from the elastic values. The surface damping increases with time, distance from the dislocation path and dislocation speed, whether sub- or supersonic.     

Finite-difference solution of unsteady natural convection flow past a nonisothermal vertical cone under the influence of a magnetic field and thermal radiation

An analysis is performed to study the heat transfer characteristics of natural convection over a vertical cone under the combined effects of a magnetic field and thermal radiation. The cone surface is subjected to a variable surface temperature. The fluid considered is a gray absorbing/emitting, but non-scattering medium. The boundary layer equations governing the flow are reduced to non-dimensional equations using non-dimensional quantities valid in the free-convection regime. The resulting non-dimensional governing equations are solved by an implicit finite-difference method of the Crank-Nicolson type, which is rapidly convergent and unconditionally stable. Numerical results are obtained for velocity, temperature, local and average skin friction, and local and average Nusselt numbers for various values of parameters occurring in the problem and are presented in the graphical form. Excellent agreement of the results obtained with available data is demonstrated.     

Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder

In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder are investigated. It is assumed that a transient magnetic field which is defined by an arbitrary function of time acts on the surface of the solid cylinder in the direction parallel to its surface. Fundamental equations of plane axisymmetrical electromagnetic, temperature, and elastic fields are formulated. Then, solutions of magnetic field, eddy current, temperature change and both dynamic solutions and quasi-static ones of stresses and deformations are analytically derived in the forms including the arbitrary function. The solutions of stresses are determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused by Lorentz force. For this case that the arbitrary function is given by the smoothed ramp function with sine function, the dynamic and quasi-static behaviors of the stresses are examined by numerical calculations.