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.
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.
Using the lubrication approximation we investigate the self-similar axisymmetric flow of a power-law liquid towards a central circular cavity. It is shown that this problem has a self-similar ...solution of the second kind. The self-similarity exponent is found by solving a non-linear eigenvalue problem arising from the requirement that the integral curve that represents the solution must join the appropriate singular points in the phase plane of the governing equation. The eigenvalues for different values of the rheological index are computed. Numerical integration of the equations allows us to determine the shape of the solution in terms of the physical variables. We make a detailed analysis of the influence of the rheology on the properties of the solutions.
We investigate exact solutions of the Navier–Stokes equations for steady rectilinear pendent rivulets running under inclined surfaces. First we show how to find exact solutions for sessile or hanging ...rivulets for any profile of the substrate (transversally to the direction of flow) and with no restrictions on the contact angles. The free surface is a cylindrical meniscus whose shape is determined by the static equilibrium between gravity and surface tension, by the shape of the solid surface, and by the contact angles on both contact lines. Given this, the velocity field can be obtained by integrating numerically a Poisson equation. We then perform a systematic study of rivulets hanging below an inclined plane, computing some of their global properties, and discussing their stability.
The slow flow of thin liquid films on solid surfaces is an important phenomenon in nature and in industrial processes, and an intensive effort has been made to investigate it. So far research has ...been focused mainly on Newtonian fluids, notwithstanding that often in the real situations as well as in the experiments, the rheology of the involved liquid is non-Newtonian. In this paper we investigate within the lubrication approximation the family of traveling wave solutions describing the flow of a power-law liquid on an incline. We derive general formulae for the traveling waves, that can be of several kinds according to the value of the propagation velocity
c and of an integration constant
j
0 related to the difference between
c and the averaged velocity of the fluid
u. There are exactly 17 different kinds of solutions. Five of them are the steady solutions (
c=0). In addition there are eight solutions that correspond to different downslope traveling waves, and four that describe waves traveling upslope.
The compressible magnetohydrodynamic Kelvin-Helmholtz instability occurs in two varieties, one that can be called incompressible as it exists in the limit of vanishing compressibility (primary ...instability), while the other exists only when compressibility is included in the model (secondary instability). In previous work we developed techniques to investigate the stability of a surface of discontinuity between two different uniform ows. Our treatment includes arbitrary jumps of the velocity and magnetic fields as well as of density and temperature, with no restriction on the wave vector of the modes. Then it allows stability analyses of complex configurations not previously studied in detail. Here we apply our methods to investigate the stability of various typical situations occurring at different regions of the front side, and the near anks of the magnetopause. The physical conditions of the vector and scalar fields that characterize the equilibrium interface at the positions considered are obtained both from experimental data and from results of simulation codes of the magnetosheath available in the literature. We give particular attention to the compressible modes in configurations in which the incompressible modes are stabilized by the magnetic shear. For configurations of the front of the magnetopause, which have small relative velocities, we find that the incompressible MHD model gives reliable estimates of their stability, and compressibility effects do not introduce significant changes. However, at the anks of the magnetopause the occurrence of the secondary instability and the shift of the boundary of the primary instability play an important role. Consequently, configurations that are stable if compressibility is neglected turn out to be unstable when it is considered and the stability properties are quite sensitive on the values of the parameters. Then compressibility should be taken into account when assessing the stability properties of these configurations, since the estimates based on incompressible MHD may be misleading. A careful analysis is required in each case, since no simple rule of thumb can be given.
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