Carlos Alberto Perazzo

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.

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...

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.

Asymptotic regimes of ridge and rift formation in a thin viscous sheet model

We numerically and theoretically investigate the evolution of the ridges and rifts produced by the convergent and divergent motions of two substrates over which an initially uniform layer of a Newtonian liquid rests. We put particular emphasis on the various asymptotic self-similar and quasi-self-similar regimes that occur in these processes. During the growth of a ridge, two self-similar stages occur; the first takes place in the initial linear phase, and the second is obtained for a large time. Initially, the width and the height of the ridge increase as t1∕2. For a very large time, the width grows as t3∕4, while the height increases as t1∕4. On the other hand, in the process of formation of a rift, there are three self-similar asymptotics. The initial linear phase is similar to that for ridges. The second stage corresponds to the separation of the current in two parts, leaving a dry region in between. Last, for a very large t, each of the two parts in which the current has separated approaches the self...

Applying dimensional analysis to wave dispersion

We show that dimensional analysis supplemented by physical insight determines if a wave has dispersion, without recourse to sophisticated mathematical tools.

Self–similar asymptotics in non–symmetrical convergent viscous gravity currents

We investigate the evolution of the ridge produced by the non–symmetrical convergent motion of two substrates over which an initially uniform layer of a Newtonian liquid rests. The lack of symmetry of the flow arises because the substrates move with different velocities. We focus on the self–similar regimes that occur in this process. For short times, within the linear regime, the height and the width increase as t1/2 and the profile is symmetric, independently of degree of asymmetry of the motion of the substrates. In the self–similar regime for large time, the height and the width of the ridge follow the same power laws as in the symmetric case, but the profiles are asymmetric.

Self–similar asymptotics in convergent viscous gravity currents of non–Newtonian liquids

We investigate the evolution of the ridge produced by the convergent motion of two substrates, on which a layer of a non–Newtonian power–law liquid rests. We focus on the self–similar regimes that occur in this process. For short times, within the linear regime, the height and the width increase as t1/2, independently of the rheology of the liquid. In the self– similar regime for large time, the height and the width of the ridge follow power laws whose exponents depend on the rheological index.

Convergent flow in a two-layer system and plateau development

In order to describe the development of plateaus such as the Tibet and the Altiplano we extend the two-layer model used in a previous paper [C. A. Perazzo and J. Gratton, Phys. Fluids 22, 056603 (2010)] to reproduce the evolution of mountain ranges. As before, we consider the convergent motion of a system of two liquid layers to simulate the crust and the upper mantle that form a lithospheric plate, but now we assume that the viscosity of the crust falls off abruptly at a specified depth. We derive a nonlinear differential equation for the evolution of the thickness of the crust. The solution of this equation shows that the process consists of a first stage in which a peaked range is formed and grows until its root reaches the depth where its viscosity drops. After that the range ceases to grow in height and a flat plateau appears at its top. In this second stage the plateau width increases linearly with time as its sides move outward as traveling waves. We derive simple approximate formulas for various p...

Topography evolution in convergent flows of two liquid layers

We have recently developed a two-layer model that considers the convergent motion of two initially uniform liquid layer with different densities and viscosities and assumes that the flow is due to the basal traction that acts at the bottom of the lower layer. We have used this model to describe successfully the evolution of mountains belts (Perazzo & Gratton, Phys. Fluids 22, 056603, 2010). In this work we discuss how to modify our model to also describe the formation of plateaus. To this end we assume that below of a given level the viscosity of the upper layer drops abruptly, and in consequence the flow of this layer becomes decoupled of the motion of the lower region of the system.

Catastrophic bolide impacts on the Earth: Some estimates

We analyze the impact of large high velocity bodies on the surface of the Earth and obtain estimates of the size of the crater. We show that the atmosphere does not stop large bolides and their mass loss and heating during entry in the atmosphere can be neglected.