Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Two-dimensional spreading of a granular avalanche down an inclined plane Part I. theory

Abstract: This paper is concerned with the motion of an unconfined finite mass of a granular material released from rest on an inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth averaged equations are deduced from the three-dimensional dynamical equations by scaling the equations and imposing the shallowness assumption that the moving piles are long and wide but not deep. Several distinguished limits for small depth to length and depth to width ratios can be analysed. We develop an approximate theory based upon the full dynamical equations parallel to the inclined plane and imposed hydrostatic pressure conditions perpendicular to it. The resulting model equations are then applied to construct either yet simpler model equations or else solutions for particular cases. In a first application the transverse distributions of the velocity fields and of the depth profile are prescribed, while representative values of these functions (such as the cross sectional averages or maxima) as functions of time and the downhill coordinate are left unspecified. For these quantities evolution equations are obtained from a lateral averaging of the vertically averaged equations. In a second application approximate similarity solutions of the spatially two-dimensional equations are derived. The depth and velocity profiles for the moving mass are determined in analytical form, and the evolution equation for the total length and the total width of the pile is integrated numerically. A parameter study illustrates the performance of the model.

Similarity solutions for MHD thermosolutal Marangoni convection over a flat surface in the presence of heat generation or absorption effects

Abstract: The problem of steady, laminar, thermosolutal Marangoni convection flow of an electrically-conducting fluid along a vertical permeable surface in the presence of a magnetic field, heat generation or absorption and a first-order chemical reaction effects is studied numerically. The general governing partial differential equations are converted into a set of self-similar equations using unique similarity transformations. Numerical solution of the similarity equations is performed using an implicit, iterative, tri-diagonal finite-difference method. Comparisons with previously published work is performed and the results are found to be in excellent agreement. Approximate analytical results for the temperature and concentration profiles as well as the local Nusselt and sherwood numbers are obtained for the conditions of small and large Prandtl and Schmidt numbers are obtained and favorably compared with the numerical solutions. The effects of Hartmann number, heat generation or absorption coefficient, the suction or injection parameter, the thermo-solutal surface tension ratio and the chemical reaction coefficient on the velocity, temperature and concentration profiles as well as quantitites related to the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are presented in graphical and tabular form and discussed. It is found that a first-order chemical reaction increases all of the wall velocity, Nusselt and Sherwood numbers while it decreases the mass flow rate in the boundary layer. Also, as the thermo-solutal surface tension ratio is increased, all of the wall velocity, boundary-layer mass flow rate and the Nusselt and Sherwood numbers are predicted to increase. However, the exact opposite behavior is predicted as the magnetic field strength is increased.

Axisymmetric Surface Diffusion: Dynamics and Stability of Self-Similar Pinchoff

Abstract: The dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface Laplacian of its mean curvature. This flow conserves the volume enclosed inside the surface while minimizing its surface area. We review the axisymmetric equilibria: the cylinder, sphere, and the Delaunay unduloid. The sphere is stable, while the cylinder is long-wave unstable. A subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions. We present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite-time pinchoff. We examine the structure of the pinchoff, showing it has self-similar structure, using asymptotic, numerical, and analytical methods. In addition to a previously known similarity solution, we find a countable set of similarity solutions, each with a different asymptotic cone angle. We develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution. We also consider similarity solutions describing the dynamics after the topological transition.

One-dimensional kinetic models of granular flows

Abstract: We introduce and discuss a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. Then, the behavior of the Boltzmann equation in the quasi elastic limit is investigated for a wide range of the rate function. By this limit procedure we obtain a class of nonlinear equations classified as nonlinear friction equations. The analysis of the cooling process shows that the nonlinearity on the relative velocity is of paramount importance for the finite time extinction of the solution.