On some problems of similarity flow of fluid with a free surface

1969 ◽  
Vol 36 (4) ◽  
pp. 805-829 ◽  
Author(s):  
Z. N. Dobrovol'skaya
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Singular Integral Equation ◽  
Incompressible Fluid ◽  
Singular Integral ◽  
Classical Problem ◽  
Flow Region ◽  
Hydrodynamic Characteristics ◽  
Two Dimensional ◽  
Free Surfaces

The paper presents the method of solving a class of two-dimensional problems of the similarity flow of an incompressible fluid with a free surface. The fluid is assumed to be non-viscous and weightless. We consider two-dimensional irrotational similarity flows with dimensionless hydrodynamic characteristics depending only on the ratios x/v0t, y/v0t, where x, y are Cartesian co-ordinates, t is time and v0 is a constant of the velocity dimension.The proposed method is based upon using the function introduced by Wagner (1932) and can be applied to the problems where the flow region is bounded by free surfaces and uniformly moving (or fixed) rectilinear impermeable boundaries. Introduction of Wagner's function makes it possible to reduce each of the problems under consideration to a non-linear singular integral equation for the real function.The method is illustrated by solving the classical problem of the uniform symmetrical entry of a wedge into a half-plane of a fluid.

Elasticity Solution of Deep Beam Under Settlement of Supports

1976 ◽  
Vol 43 (4) ◽  
pp. 613-618 ◽  
Author(s):  
S. N. Prasad ◽  
P. K. Chiu
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
The Other ◽  
Tangential Displacement ◽  
Deep Beam ◽  
Two Dimensional ◽  
Singular Behavior ◽  
Elasticity Solution ◽  
Collocation Technique

The present study investigates the two-dimensional stresses and displacements in a finite rectangle whose one set of parallel edges is given a relative tangential displacement by means of rigidly attached planes. The other set of parallel edges is free from tractions. The problem is formulated in terms of a singular integral equation of the first kind, which yields the correct singular behavior of stresses at the corners. The integral equation is solved numerically by employing Gauss-Jacobi quadrature in conjunction with certain collocation technique. Numerical results of quantities of practical interests are shown graphically and also compared with the classical bending analysis.

Analysis of Branched Cracks

1978 ◽  
Vol 45 (4) ◽  
pp. 797-802 ◽  
Author(s):  
K. K. Lo
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Complex Form ◽  
Function Technique ◽  
Two Dimensional ◽  
Intensity Factors ◽  
Potential Formulation ◽  
Crack Problems

This paper presents a method for solving a class of two-dimensional elastic branched crack problems. In contrast to other approaches in the literature, the integral equation presented here enables different branched crack problems to be solved in a unified manner. Muskhelishvili’s potential formulation is used to derive, by means of a Green’s function technique, a singular integral equation in complex form. The problems of the asymmetrically, symmetrically, and doubly symmetrically branched cracks are considered. The ratio of the length of the branched crack to that of the main one may be varied arbitrarily and the limit in which this ratio goes to zero is obtained analytically. Stress-intensity factors at the branched crack tip are computed numerically and the results, where possible, are compared to those in the literature. Disagreements in the literature are discussed and clarified with the aid of the present results.

scholarly journals Two-dimensional jet aimed vertically upwards

1993 ◽  
Vol 34 (4) ◽  
pp. 393-400 ◽  
Author(s):  
Jean-Marc Vanden-Broeck
Keyword(s):  
Free Surface ◽  
Incompressible Fluid ◽  
Finite Differences ◽  
Two Dimensional ◽  
Free Surfaces ◽  
Inviscid Incompressible Fluid ◽  
Surface Profiles

AbstractA steady two-dimensional jet of an inviscid incompressible fluid rising and falling under gravity is considered. The jet is aimed vertically upwards and the flow is assumed to be bounded entirely by two free surfaces. The problem is solved numerically by finite differences. Accurate results for the free surface profiles are presented.

Exact Two-Dimensional Contact Analysis of Piezomagnetic Materials Indented by a Rigid Sliding Punch

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
Yue Ting Zhou ◽  
Kang Yong Lee
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Mixed Boundary ◽  
Piezoelectric Materials ◽  
Fundamental Solutions ◽  
Two Dimensional ◽  
Moving Contact ◽  
Cylindrical Punch

The aim of the present paper is to investigate the two-dimensional moving contact behavior of piezomagnetic materials under the action of a sliding rigid punch. Introduction of the Galilean transformation makes the constitutive equations containing the inertial terms tractable. Eigenvalues analyses of the piezomagnetic governing equations are detailed, which are more complex than those of the commercially available piezoelectric materials. Four eigenvalue distribution cases occur in the practical computation. For each case, real fundamental solutions are derived. The original mixed boundary value problem with either a flat or a cylindrical punch foundation is reduced to a singular integral equation. Exact solution to the singular integral equation is obtained. Especially, explicit form of the stresses and magnetic inductions are given. Figures are plotted both to show the correctness of the derivation of the exact solution and to reveal the effects of various parameters on the stress and magnetic induction.

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