Directed fluid sheets

1976 ◽  
Vol 347 (1651) ◽  
pp. 447-473 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Steady Motion ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Inviscid Fluids ◽  
Two Dimensional Flows

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

On the theory of water waves

1974 ◽  
Vol 338 (1612) ◽  
pp. 43-55 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
De Vries ◽  
Set Up ◽  
Propagation Of Waves

The aim of this paper is to formulate a two-dimensional theory for the propagation of fairly long water waves. The approach differs from the usual in that the theory is set up via two-dimensional postulates. Subsequently, it is shown how a simple three-dimensional approxi­mation enables us to relate the two-dimensional theory to the three-dimensional theory. The resulting equations are used to discuss the unidirectional propagation of waves. lt is shown how the results obtained from the theory proposed here are related to the results of Korteweg & de Vries (1895) and to those of Benjamin, Bona & Mahony (1972).

Membrane theory

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Three Dimensional ◽  
Thin Sheets ◽  
Two Dimensional ◽  
Leading Order ◽  
Membrane Theory ◽  
Dimensional Theory ◽  
Small Thickness

This chapter develops two-dimensional membrane theory as a leading order small-thickness approximation to the three-dimensional theory for thin sheets. Applications to axisymmetric equilibria are developed in detail, and applied to describe the phenomenon of bulge propagation in cylinders.

Comparison principles for free-surface flows with gravity

1991 ◽  
Vol 230 ◽  
pp. 231-243 ◽  
Author(s):  
Walter Craig ◽  
Peter Sternberg
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Immiscible Fluids ◽  
Steady Flows ◽  
Two Dimensional ◽  
Surface Flows ◽  
Ship Hulls ◽  
Geometrical Information ◽  
Supercritical Flows

This article considers certain two-dimensional, irrotational, steady flows in fluid regions of finite depth and infinite horizontal extent. Geometrical information about these flows and their singularities is obtained, using a variant of a classical comparison principle. The results are applied to three types of problems: (i) supercritical solitary waves carrying planing surfaces or surfboards, (ii) supercritical flows past ship hulls and (iii) supercritical interfacial solitary waves in systems consisting of two immiscible fluids.

scholarly journals Steady-State Three-Dimensional Ice Flow over an Undulating Base: First-Order Theory with Linear Ice Rheology

1987 ◽  
Vol 33 (114) ◽  
pp. 177-185 ◽  
Author(s):  
Niels Reeh
Keyword(s):  
Three Dimensional ◽  
Steady Motion ◽  
Simple Shear Flow ◽  
Finite Width ◽  
Two Dimensional ◽  
Internal Layers ◽  
Ice Flow ◽  
Depth Variation ◽  
First Order ◽  
First Order Theory

AbstractThe problem of ice flow over threedimensional basal irregularities is studied by considering the steady motion of a fluid with a linear constitutive equation over sine-shaped basal undulations. The undisturbed flow is simple shear flow with constant depth. Using the ratio of the amplitude of the basal undulations to the ice thickness as perturbation parameter, equations to the first order for the velocity and pressure perturbations are set up and solved.The study shows that when the widths of the basal undulations are larger than 2–3 times their lengths, the finite width of the undulations has only a minor influence on the flow, which to a good approximation may be considered two-dimensional. However, as the ratio between the longitudinal and the transverse wavelengthL/Wincreases, the three-dimensional flow effects becomes substantial. If, for example, the ratio ofLtoWexceeds 3, surface amplitudes are reduced by more than one order of magnitude as compared to the two-dimensional case. TheL/Wratio also influences the depth variation of the amplitudes of internal layers and the depth variation of perturbation velocities and strain-rates. With increasingL/Wratio, the changes of these quantities are concentrated in a near-bottom layer of decreasing thickness. Furthermore, it is shown, that the azimuth of the velocity vector may change by up to 10° between the surface and the base of the ice sheet, and that significant transverse flow may occur at depth without manifesting itself at the surface to any significant degree.

On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

scholarly journals Steady-State Three-Dimensional Ice Flow over an Undulating Base: First-Order Theory with Linear Ice Rheology

1987 ◽  
Vol 33 (114) ◽  
pp. 177-185 ◽  
Author(s):  
Niels Reeh
Keyword(s):  
Three Dimensional ◽  
Steady Motion ◽  
Simple Shear Flow ◽  
Finite Width ◽  
Two Dimensional ◽  
Internal Layers ◽  
Ice Flow ◽  
Depth Variation ◽  
First Order ◽  
First Order Theory

AbstractThe problem of ice flow over threedimensional basal irregularities is studied by considering the steady motion of a fluid with a linear constitutive equation over sine-shaped basal undulations. The undisturbed flow is simple shear flow with constant depth. Using the ratio of the amplitude of the basal undulations to the ice thickness as perturbation parameter, equations to the first order for the velocity and pressure perturbations are set up and solved.The study shows that when the widths of the basal undulations are larger than 2–3 times their lengths, the finite width of the undulations has only a minor influence on the flow, which to a good approximation may be considered two-dimensional. However, as the ratio between the longitudinal and the transverse wavelength L/W increases, the three-dimensional flow effects becomes substantial. If, for example, the ratio of L to W exceeds 3, surface amplitudes are reduced by more than one order of magnitude as compared to the two-dimensional case. The L/W ratio also influences the depth variation of the amplitudes of internal layers and the depth variation of perturbation velocities and strain-rates. With increasing L/W ratio, the changes of these quantities are concentrated in a near-bottom layer of decreasing thickness. Furthermore, it is shown, that the azimuth of the velocity vector may change by up to 10° between the surface and the base of the ice sheet, and that significant transverse flow may occur at depth without manifesting itself at the surface to any significant degree.

Contact Temperature in Dry Bearings. Three Dimensional Theory and Verification

1981 ◽  
Vol 103 (2) ◽  
pp. 243-251 ◽  
Author(s):  
A. Floquet ◽  
D. Play
Keyword(s):  
Fourier Transform ◽  
Boundary Conditions ◽  
Experimental Verification ◽  
Three Dimensional ◽  
Contact Temperature ◽  
3D Analysis ◽  
New Work ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Infra Red

Boundary conditions were arbitrarily specified in an earlier two dimensional (2D) analysis of contact temperature. In this new work a general three dimensional (3D) Fourier transform solution is obtained from which for specific cases, the boundary conditions can be estimated. Further, experimental verification of 3D analysis was performed using infra-red technique.

On the observability of finite-depth short-crested water waves

1996 ◽  
Vol 322 ◽  
pp. 1-19 ◽  
Author(s):  
M. Ioualalen ◽  
A. J. Roberts ◽  
C. Kharif
Keyword(s):  
Standing Wave ◽  
Water Waves ◽  
Numerical Study ◽  
Standing Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Wave Fields ◽  
Progressive Waves ◽  
Wave Limit ◽  
Narrow Bands

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

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