Similarity solution

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

Analytical Solution for Three-Dimensional Steady Flow of Condensation Film on Inclined Rotating Disk by Optimal Homotopy Analysis Method

Abstract: In this paper, the Optimal Homotopy Analysis Method (OHAM) has been used to derive a highly accurate analytic solution for the steady three-dimensional problem of a condensation film on an inclined rotating disk. With a similarity solution method, the governing equations can be reduced to a system of nonlinear ordinary differential equations (ODEs). The control parameter (ħ) in the HAM is derived by using the averaged residual error method. Using the optimal control parameter provides a superior control on the convergence and accuracy of the analytical solution. The velocity and temperature profiles are shown and the influence of Prandtl number on the temperature profiles is discussed in detail. The validity of the obtained solutions is verified by the numerical results.

Granular matter in a silo

Abstract: A dynamic model for slow accumulation of granular matter, with standing and rolling layers, is adapted to the geometry of a silo with vertical walls and bounded cross section. For sources constant in time the typical behavior of a rising level of matter with constant surface profile is described by a similarity solution. This solution can be characterized by a nonlinear boundary value problem. For the case of a circular silo with central point source and the corresponding 1D problem of an interval with central point source the profiles of the standing and rolling layers are explicitly computed. The computed shapes agree qualitatively with experimental observations and seem better justified than results based on the assumption of constant speed of the rolling layer.

Symmetry group analysis and exact solutions of isentropic magnetogasdynamics

Abstract: In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities.

Vortex models based on similarity solutions of the two-dimensional diffusion equation

Abstract: In this paper, a class of two-dimensional (2D) vortex models is analyzed, which is based on similarity solutions of the diffusion equation. If the nonlinear advective term is neglected, the 2D Navier-Stokes equation reduces to a linear problem, for which a complete orthonormal set of eigenfunctions is known on an unbounded 2D domain. Some of the basic modes represent models for diffusing monopoles, dipoles, and tripolar vortices, which evolve self-similarly in time. Here, we mainly confine ourselves to an analysis of the dipole solution. In several respects, especially the decay and, to a lesser extent, the lateral expansion properties, the dipole model appears to be in fair agreement with the real evolution of dipolar vortices for finite Reynolds number, as obtained from numerical simulations of the full 2D Navier-Stokes equations. However, the simulations reveal that nonlinear effects result in small differences compared to the evolution according to the model. The most important nonlinear effect that was observed is the formation of “tails” of vorticity in the wake of the dipole. After a while, any initial condition leads to a vorticity distribution lying in between the viscous similarity solution and the Lamb dipole solution, which represents the limit of a stationary, inviscid flow. The exact form of the vorticity distribution is believed to be determined by an equilibrium between diffusion of vorticity through the separatrix and advection of vorticity into the wake of the dipole, which results in the formation of vorticity tails. A comparison revealed profound qualitative agreements between the model together with the simulations and dipolar vortex structures that were studied by laboratory experiments in stratified fluids. © 2004 American Institute of Physics. [DOI: 10.1063/1.1804548] An important feature of two-dimensional (2D) turbulent flows is the emergence of vortices. Numerical simulations of 2D flows and several laboratory experiments have shown that in (nearly) planar flows, coherent vortices may form spontaneously from an initially turbulent flow field, 1‐3 a process which is due to the inverse energy cascade in 2D flows and which is commonly referred to as self-organization. These coherent vortices are abundant in quasi-2D flows and play an important role in the evolution, the dynamics, and the transport properties of such flows. During the past decades, a lot of research has been devoted to 2D turbulence and the dynamics of vortices, not in the least for their relevance in the field of geophysical fluid dynamics. The flows in the Earth’s atmosphere and oceans can be considered as approximately two-dimensional due to the rotation of the Earth, the presence of a density stratification in the oceans and in the atmosphere, and also the geometrical confinement of the flow. Several types of vortices can be observed in nature, and have been studied in laboratory experiments and numerical simulations. The most common type is the monopolar vortex, which is defined as a swirling flow with one center of rotation that can be circular as well as elliptical in shape. Typical