Julio Gratton

Self-similar viscous gravity currents: phase-plane formalism

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V , which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV /d Z = f ( Z, V ), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

Self-similar gravity currents with variable inflow revisited : plane currents

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of the intruding fluid varies in time according to a power law of the type t α . The resistance of the ambient fluid is taken into account by a boundary condition of the von Karman type, that depends on a parameter β that is a function of the density ratio of the fluids. The flow is characterized by β, α, and the Froude number [Fscr ] 0 near the source. We find four kinds of self-similar solutions: subcritical continuous solutions (Type I), continuous solutions with a supercritical-subcritical transition (Type II), discontinuous solutions (Type III) that have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The current is always subcritical near the front, but near the source it is subcritical ([Fscr ] 0 0 > 1) for Types II, III, and IV. Type I solutions have already been found by other authors, but Type II, III, and IV currents are novel. We find the intervals of parameters for which these solutions exist, and discuss their properties. For constant-volume currents one obtains Type I solutions for any β that, when β > 2, have a ‘dry’ region near the origin. For steady inflow one finds Type I currents for O 0 is sufficiently large.

Self-similar solution of the second kind for a convergent viscous gravity current

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 obser...

Navier-Stokes solutions for parallel flow in rivulets on an inclined plane

We investigate the solutions of the Navier–Stokes equations that describe the steady flow of rivulets down an inclined surface. We find that the shape of the free surface is given by an analytic formula obtained by solving the equation that expresses the condition of static equilibrium under the action of gravity and surface tension, independently of the velocity field and of any assumption concerning the rheology of the liquid. The velocity field is then obtained by solving (in general numerically) a Poisson equation in the domain defined by the cross-section of the rivulet. The isovelocity contours are perpendicular to the free surface. Various properties of the solutions are given as functions of the parameters of the problem. Two special analytic solutions are presented. The exact solutions suggest that the lubrication approximation, frequently employed to investigate problems similar to the present one, predicts reasonably well the global properties of the rivulet provided the static contact angle is not too large.

Effect of compressibility on the stability of a vortex sheet in an ideal magnetofluid

The hydromagnetic stability of a vortex sheet with respect to all modulations is examined. A domain of absolute stability not previously noted is found for high compressibility. Stability criteria are given. Onset of instability is related to radiation of hydromagnetic waves.

Magnetoacoustic surface gravity waves

The effect of gravity on the surface magnetoacoustic waves may be important when considering applications to solar and laboratory plasmas. The linear magnetoacoustic waves, which may appear in a configuration with an interface between two plasmas or a plasma and an ordinary gas, are studied. Compressibility and gravity are taken into account. The different types of couplings between internal and surface modes are analyzed. Magnetoacoustic surface waves are studied in detail in a configuration consisting of an interface between an isothermal plasma and an ordinary gas. The possible regions where these modes may exist are discussed. A general way of grouping and classifying the complicated spectra of modes is presented. New groups of modes appear as a consequence of gravity and stratification, in addition to those already present in the absence of gravity. The results may be of help in studying more complicated cases.

Hydromagnetic oscillations of a tangential discontinuity in the Chew, Goldberger, and Low approximation

The differential equation for linear modes of oscillation of plane parallel flows of plasmas along an external magnetic field in the Chew, Goldberger, and Low approximation is obtained. Properties of modes for a tangential discontinuity are studied for the case when the surface is modulated along the magnetic field. Overstable modes found by other authors are shown to be spurious. Regions of existence of modes, proper frequencies, and spatial dependence of the perturbation are given. It is found that, broadly speaking, low β plasmas should be free of surface instabilities for all values of the flow velocity, whereas high β plasmas can be unstable if the flow velocity is nearly sonic. Changes in the anisotropy do not substantially affect the general picture of the problem.

Exact solutions for two-dimensional steady flows of a power-law liquid on an incline

Under assumptions that are not too restrictive it is possible to reduce the equations that describe steady viscous gravity flows of a power-law liquid on an inclined plane to an equivalent problem consisting of an unsteady one-dimensional nonlinear diffusion process. In a paper dealing with the steady spreading flow of a Herschel–Buckley liquid, Wilson and Burgess [“The steady, spreading flow of a rivulet of mud,” J. Non-Newtonian Fluid Mech. 79, 77 (1998)] noticed a formal analogy between the steady, two-dimensional viscous gravity flows of a power-law liquid on an incline and a one-dimensional time-dependent nonlinear diffusion phenomena; however, they did not pursue the matter further. Here we develop the analogy and show how it can be used to find a large number of exact solutions representing steady two-dimensional flows of power-law liquids, based on the available knowledge concerning nonlinear diffusion. We describe flows whose widths stay constant until a certain distance from the source, which ar...

Radiation of Hydromagnetic Waves from a Tangential Velocity Discontinuity

An outgoing group velocity criterion is used to determine the regions of existence of radiation modes for a tangential velocity discontinuity in a collisionless hydromagnetic regime. It is shown how previously published results must be corrected.

Convergent flow in a two-layer system and mountain building

With the purpose of modeling the process of mountain building, we investigate the evolution of the ridge produced by the convergent motion of a system consisting of two layers of liquids that differ in density and viscosity to simulate the crust and the upper mantle that form a lithospheric plate. We assume that the motion is driven by basal traction. Assuming isostasy, we derive a nonlinear differential equation for the evolution of the thickness of the crust. We solve this equation numerically to obtain the profile of the range. We find an approximate self-similar solution that describes reasonably well the process and predicts simple scaling laws for the height and width of the range as well as the shape of the transversal profile. We compare the theoretical results with the profiles of real mountain belts and find an excellent agreement.