Se usa la teoría desarrollada en el trabajo anterior para estudiar las corrientes de gravedad planas autosemejantes. Hay cuatro tipos de soluciones: (I) continuas subcríticas, (II) continuas con transición crítica, (III) discontinuas, y (IV) discontinuas con transición crítica. La corriente es siempre subcrítica cerca del frente, pero cerca de la fuente es subcrítica para corrientes Tipo I, y supercrítica en el resto. Las corrientes Tipo I eran conocidas, pero los tipos, II, III y IV son novedosos. Se determinan los intervalos de parámetros donde hay soluciones y se discuten sus propiedades.

Se estudian mediante simulaciones numéricas corrientes viscogravitatorias con tiempo de espera producidas a partir de perfiles iniciales en que el espesor del fluido es proporcional a una cierta potencia (p) de la distancia al frente. Para este tipo de condiciones iniciales, Kath y Cohen demostraron teóricamente que: (a) si p<2/3 el frente se pone en movimiento de inmediato (es decir no hay tiempo de espera), (b) si p=2/3 el frente permanece inmóvil por un tiempo finito, (c) si p>2/3 se obtienen soluciones con tiempo de espera en las que aparece un "comer shock" (CS) móvil detrás del frente, que se pone a su vez en movimiento cuando es alcanzado por el CS. Sin embargo, salvo para el caso p=2/3 no hay estimaciones teóricas del tiempo de espera. Este trabajo apunta principalmente a: (1) determinar cómo dependen de las condiciones iniciales el tiempo de espera y la posición y movimiento del es, (2) determinar la asintótica de la solución en el entorno del frente y cerca del instante en que éste se pone en movimiento para encontrar la relación d=d(p), esto es, a cuál de las soluciones auto-semejantes de segunda especie LOT (estudiadas en otro trabajo) tiende. Se encuentra que (si p>2/3) sólo se pueden obtener soluciones con 113/10.

Se aplica nuestra teoría de la inestabilidad de Kelvin-Helmholtz compresible en presencia de cizalla magnética y diferencias de densidad y temperatura de ambos lados de la discontinuidad de velocidad, para estudiar la estabilidad de configuraciones típicas de la magnetopausa terrestre. Se muestra que la inclusión de los efectos de la compresibilidad lleva a desestabilizar el plasma para bajas velocidades relativas en situaciones que son estables en el límite incompresible, lo que muestra que la teoría incompresible da predicciones incorrectas acerca de la estabilidad.

Se estudian las corrientes de gravedad autosemejantes que describen la intrusión de un fluido denso debajo de un fluido ambiente más liviano que descansa sobre un plano. Las corrientes tienen simetría plana y son producidas por una fuente de caudal variable; su volumen varía como ta. La resistencia del fluido ambiente se describe mediante una condición de contorno que depende de un parámetro β que es función de la razón de densidades de fluidos. El flujo está caracterizado por β, a y el número de Froude de la fuente. Se desarrolla el formalismo del plano de fase y se discute la construcción de las soluciones.

Se realizaron experimentos con fluí dos muy viscosos (aceites siliconados) con el objeto de verificar una solución autosimilar de segunda especie obtenida teóricamente. Se consideró el flujo plano convergente que se genera al llenarse radialmente una región inicialmente libre de aceite. Los parámetros de los experimentos se escogieron de forma tal que el flujo se desarrolla dentro de la validez de la aproximación de lubricación. Se midieron la posición del frente en función del tiempo y la forma del perfil de altura del fluí do. Se encontró que el flujo convergente sigue la solución de segunda especie sólo en una región espacial limitada.

Se estudió experimentalmente la evolución de flujos viscosos de gravedad (corrientes viscogravitatorias) sobre planos horizontales en situaciones tales que el frente permanece inmóvil durante un tiempo finito, denominado tiempo de espera. Esto se logró, en nuestro caso, a partir de una condición inicial en la cual el fluido tiene forma de prisma triangular. Se emplearon fluidos con viscosidades muy diferentes: aceite siliconado (v=4.75 stokes) y silicona ( v=1.10 105 stokes). Se compararon estos resultados con los obtenidos por medio de un código numérico que emplea la aproximación de lubricación, la cual proporciona una excelente descripción para relaciones espesor extensión del prisma de hasta 0.5, pero la relación entre los tiempos de espera experimentales y numéricos es aproximadamente 1.5. La medición del tiempo de espera puede resultar un método de interés práctico para determinar la viscosidad de fluidos muy viscosos a muy bajas tasas de deformación o de fluidos cuyo comportamiento reológico depende de la historia previa, condiciones en las que los instrumentos convencionales no resultan adecuados.

The hydromagnetic Kelvin–Helmholtz instability is relevant in many complex situations in astrophysical and laboratory plasmas. Many cases of interest are very complicated, since they involve the combined roles of velocity shear, density and magnetic field stratification, and various geometries in compressible plasmas. The present work is part of a systematic investigation of the influence of the various physical and geometrical parameters characterizing the plasmas on the Kelvin–Helmholtz modes. The general dispersion relation for ideal compressible MHD modes localized near a velocity discontinuity between two uniform plasmas is derived. The existence and characteristics of the modes and their stability are studied analytically for any relative orientation of B, u and k, for continuous B and ρ. It is shown that the stability of a given configuration cannot be determined by considering only special orientations of k (say flute or parallel modes). The results obtained here may serve as a guide to interpret results in more complicated and realistic situations, such as those occurring in experiments and natural plasmas.

Recently several experiments on creeping gravity currents have been performed, using highly viscous silicone oils and putties. The interpretation of the experiments relies on the available theoretical results that were obtained by means of the lubrication approximation with the assumption of a Newtonian rheology. Since very viscous fluids are usually non-Newtonian, an extension of the theory to include non-Newtonian effects is needed. We derive the governing equations for unidirectional and axisymmetric creeping gravity currents of a non-Newtonian liquid with a power-law rheology, generalizing the usual lubrication approximation. The equations differ from those for Newtonian liquids, being nonlinear in the spatial derivative of the thickness of the current. Similarity solutions for currents whose volume varies as a power of time are obtained. For the spread of a constant volume of liquid, analytic solutions are found that are in good agreement with experiment. We also derive solutions of the waiting-time type, as well as those describing steady flows from a constant source to a sink. General traveling-wave solutions are given, and analytic formulas for a simple case are derived. A phase plane formalism that allows the systematic derivation of self-similar solutions is introduced. The application of the Boltzmann transform is briefly discussed. All the self-similar solutions obtained here have their counterparts in Newtonian flows, as should be expected because the power-law rheology involves a single-dimensional parameter as the Newtonian constitutive relation. Thus one finds similarity solutions whenever the analogous Newtonian problem is self-similar, but now the spreading relations are rheology-dependent. In most cases this dependence is weak but leads to significant differences easily detected in experiments. The present results may also be of interest for geophysics since the lithosphere deforms according to an average power-law rheology.

We study the filling of a dry region (cavity) within a viscous liquid layer on a horizontal plane. In our experiments the cavities are created by removable dams of various shapes surrounded by a silicon oil, and we measure the evolution of the cavity's boundaries after removal of the dams. Experimental runs with circular, equilateral triangular, and square dams result in circular collapse of the cavities. However, dams whose shapes lack these discrete rotational symmetries, for example, ellipses, rectangles, or isosceles triangles, do not lead to circular collapses. Instead, we find that near collapse the cavities have elongated oval shapes. The axes of these ovals shrink according to different power laws, so that while the cavity collapses to a point, the aspect ratio is increasing. The experimental setup is modeled within the lubrication approximation. As long as capillarity is negligible, the evolution of the fluid height is governed by a nonlinear diffusion equation. Numerical simulations of the experiments in this approximation show good agreement up to the time where the cavity is so small that surface tension can no longer be ignored. Nevertheless, the noncircular shape of the collapsing cavity cannot be due to surface tension which would tend to round the contours. These results are supplemented by numerical simulations of the evolution of contours which are initially circles distorted by small sinusoidal perturbations with wave numbers k greater than or equal to 2. These nonlinear stability calculations show that the circle is unstable in the presence of the mode k=2 and stable in its absence. The same conclusion is obtained from the linearized stability analysis of the front for the known self-similar solution for a circular cavity. [S1063-651X(98)04711-4].

We study numerically the Fourier pseudomodes of complex frequency, that represent leaky waves, for two configurations consisting of three layers of uniform plasma in equilibrium, using the methods developed in the previous papers. The spectra consist of an infinite set of branches. We introduce a shorthand notation that allows us to identify the branches and we discuss their behavior. We show examples of leaky waves that in the neighborhood of certain limiting cases correspond to some of the four basic mechanisms discussed in a previous paper.

In a three layer compressible plasma we study MHD leaky waves described by Fourier modes of complex frequency. We follow a matrix method that yields four transcendental dispersion relations. The roots (real or complex) make up a spectrum of pseudomodes, consisting of a complicate set of infinite branches. We derive analytical properties that allow to study the topology of the spectrum, the conditions of existence of real and complex roots, the contacts and crossings of branches of different types, etc. Our results allow to analyze any particular case of interest.

We investigate the existence of MHD leaky waves in compressible and layered plasmas. We consider perturbations of complex frequency in a slab model with three layers. We develop a general method that includes the 'cubic modes' as well as other kinds of leaky waves, and allows us to derive many results in closed form. There are several physical mechanisms that can produce leakage. We give two numerical examples to exhibit the different behaviours that can arise. Finally we comment on the connection of the present results with those obtained by other authors, and correct some errors.

The application of Fourier analysis to study leaky waves has the advantage of simplicity, but it is not clear why the complex roots of the dispersion relations represent leaky waves, nor how the leakage occurs. We investigate the different kinds of leakage that can occur in a three layer plasma, and to which Fourier pseudomodes they are associated. We find four basic mechanisms, called “surface mode ...”, “single interface ...”, “trapped wave ...” and “lost insulation leakage”. These mechanisms appear in pure form only near certain limiting cases, in which the parameters of the problem take some special values. As soon as the parameters depart significantly from the limiting values, the behavior of the leaky wave complicates and mixtures of the mechanisms occur, in varying amounts. In consequence different points of the same complex branch of the spectrum may correspond to different mechanisms. All the complex branches of the spectrum correspond to leaky waves, but in general it is not possible to classify them according to type of leakage, except close to a limiting case. Since in a three layer configuration there are many of these, the spectrum of leaky waves is very complicated.

The determination of the reactions on a ladder is discussed including friction against the wall. In a recent article, this problem was studied considering elastic compression and it was argued that an additional condition holds that allows one to find all the reactions. We show that this condition is incorrect for typical ladders, when flexion is important. We clarify the issues of static determinacy by means of an analysis that takes into account both compression and flexion. We find that when the climber arrives at the top, the result depends on which deformation prevails. We conclude that the reactions can never be determined using only statics, because there is no way to ascertain which kind of deformation dominates, and so which limiting condition will be attained, if any. (C) 1996 American Association of Physics Teachers.

The hydromagnetic Kelvin‐Helmholtz instability has been extensively studied in simple plasma configurations, but more complex situations relevant for fusion research involving the combined role of effective gravity due to acceleration, velocity shear, density and magnetic field stratification has not yet been investigated for compressible plasmas. We consider an isothermal equilibrium configuration with stratified density and magnetic field in a compressible plasma. The dispersion relation can be reduced to a highly complicated algebraic equation that involves many parameters. Various cases are examined. Analytical conditions for stability are given for a stratified plasma with no density jump. When there is a density discontinuity, the conditions for the existence of unstable modes are presented. The limiting case without gravity is examined.

The hydromagnetic Kelvin–Helmholtz instability is relevant in many complex situations in astrophysical and laboratory plasmas. Many cases of interest are very complicated, since they involve the combined role of velocity shear, of density and magnetic field stratification, and of various geometries in compressible plasmas. In the present work we continue investigating the influence of various physical and geometrical parameters of the plasma on the Kelvin–Helmholtz modes. We use the general dispersion relation for the ideal compressible MHD modes localized near a velocity discontinuity between two uniform plasmas. We study analytically the existence and properties of the modes and their stability, for a velocity jump combined with a density jump, and for any relative orientation of B, u and k (B is continuous). Stability is analysed by means of a general procedure that allows discussion of any configuration and all kinds of perturbations. The boundaries between modes of different kinds are discussed. In contrast to the case of uniform density, for a density jump there are no monotonically unstable modes, only overstabilities. The unstable modes belong to two types. Those with the largest growth rates tend to monotonically unstable modes in the limit of uniform density, and are related to the torsional Alfvén mode. The other overstable modes have no analogue among the purely incompressible modes, and occur in a range of U that is stable in the incompressible limit. We derive bounds for the growth rate of the instability. The present results may serve as a guide to interpret results in more complicated and realistic situations as those occurring in laboratory and natural plasmas.

The axisymmetric flow of a very viscous fluid toward a central orifice is studied. In a recent paper, a self-similar solution for this problem has been found. The self-similarity is of the second kind and hence the flow remembers its initial condition only through a nondimensional constant which characterizes it. In this work this convergent flow is studied experimentally (using silicone oils) by measuring the front position and the height profile as a function of time. It is verified that the self-similar solution properly describes the flow within a certain interval of the cavity radius, where values are obtained for the similarity exponent δ in agreement (accounting for experimental errors) with the theoretical value 0.762... . The transition to the self-similar flow is also simulated numerically and numerical values are obtained for the time closure for different initial conditions. These simulations also show the theoretical self-similar flow after the cavity closure, which is very difficult to observe experimentally.

The stability of gravitational compressible surface modes of a plasma-vacuum interface is investigated. Stratified equilibrium profiles of density and magnetic field in the plasma are considered. The corresponding boundary conditions for magnetohydrodynamic linear perturbations are deduced. Three types of surface normal modes (slow, intermediate and fast) may appear. The slow mode is unstable below a critical wavenumber. The stability criterion is affected in general by the compressibility. The growth rate of the unstable modes is increased by compressibility. It is shown that for 'flute' or pure interchange modes with respect to the vacuum magnetic field (but not with respect to the equilibrium magnetic field in the plasma) the compressibility enlarges the instability domain of surface modes. The interval of existence of unstable surface perturbations, as well as their growth rates, are larger than those of the internal unstable modes. Intervals of non-existence of surface modes appear for the flute perturbations with respect to the plasma magnetic field. The critical values of the wavenumber below which the modes are unstable, are discussed in general.

The linear stability of accelerated plasmas is studied. It is considered an unperturbed state that allows stratification of density and magnetic field in the plasma, as well as a plasma‐vacuum interface. We consider the effect of compressibility and show that it enlarges the spectrum of unstable modes, as well as increases the growth rate. Stability criteria and growth rates are given both for internal and surface modes. On the other hand, viscous effects on solenoidal modes are considered. The limiting cases of highly collisional and strongly magnetized plasmas are analyzed, showing different behavior. General properties of the spectrum are derived by means of normal mode and variational analysis.