摘要:

An evolution equation in a finite depth fluid for weakly nonlinear long internal waves is derived in a stratified and sheared medium. The equation reduces to the Korteweg‐deVries equation when the depth is small compared to the wavelength, and to the Benjamin‐Ono equation when the depth is large compared to the wavelength. Both the cases with and without critical levels are investigated. Numerical solutions to the evolution equation are presented to illustrate the effect of shear on the evolution of a wavef

ISSN:0022-2526    

DOI:10.1002/sapm1981653189

年代:1981

数据来源: WILEY

摘要:

The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer.

ISSN:0022-2526    

DOI:10.1002/sapm1981653223

年代:1981

数据来源: WILEY

摘要:

We investigate the spinup from rest of two immiscible fluids of different densities and kinematic viscosities in a vertically mounted cylinder. Our attention is restricted to small internal rotational Froude numbers, in which case the interface remains essentially horizontal. By requiring small enough Ekman numbers, Wedemeyer's (1964) approximation may be used to obtain partial differential equations describing the inward convection and diffusion of the azimuthal velocity in both layers. Solutions are found illustrating the effects of varying the density, viscosity, and height of each layer. Qualitative agreement with experiment is reported.

ISSN:0022-2526    

DOI:10.1002/sapm1981653249

年代:1981

数据来源: WILEY

摘要:

LetSbe a finite set, and fixK>2. LetFbe a family of subsets ofSwith the property that wheneverA1,...,Akare sets inF, not necessarily distinct, andA1⋂ ⋯ ⋂Ak= ∅, thenA1⋃ ⋯ ⋃Ak=S. We prove here that the maximum size of such a family is 2|S|−1+ 1. If we require that the setsA1,...,Akbe distinct, then the maximum size ofFis again 2|S|−1+ 1, provided that

ISSN:0022-2526    

DOI:10.1002/sapm1981653269

年代:1981

数据来源: WILEY

ISSN:0022-2526    

DOI:10.1002/sapm1981653277

年代:1981

数据来源: WILEY

ISSN:0022-2526    

DOI:10.1002/sapm1981653279

年代:1981

数据来源: WILEY