Similarity solution

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

Oscillatory Solutions of the Falkner-Skan Equation

Abstract: The Falkner-Skan equation for similarity solutions of the Prandtl boundary-layer equations for incompressible flow is analysed for both positive and negative values of the parameter $\beta$. For $\beta$ 1 we have proved that there is a periodic solution. We conjecture that for $\beta$ > 2 there are infinitely many periodic solutions and that a form of 'symbolic dynamics', of the kind associated with a Smale 'horseshoe map' can be constructed. We have shown this rigorously for $\beta$ close to 2.

Transient non-Darcy free convection between parallel vertical plates in a fluid saturated porous medium

Abstract: Transient non-Darcy free convection between two parallel vertical plates in a fluid saturated porous medium is investigated using the generalized momentum equation proposed by Vafai and Tien. The effects of porous inertia and solid boundary are considered in addition to the Darcy flow resistance. Exact solutions are found for the asymptotic states at small and large times. The large time solutions reveal that the velocity profiles are rather sensitive to the Darcy number Da when Da<1. It has also been found that boundary friction alters the velocity distribution near the wall, considerably. Finite difference calculations have also been carried out to investigate the transient behaviour at the intermediate times in which no similarity solutions are possible. This analytical and numerical study reveals that the transient free convection between the parallel plates may well be described by matching the two distinct asymptotic solutions obtained at small and large times.

A Simplified Model and Similarity Solutions for Interfacial Evolution of Two-Phase Flow in Porous Media

Abstract: For two-phase flows of immiscible displacement processes in porous media, we proposed a simplified model to capture the interfacial fronts, which is given by explicit expressions and satisfies the continuity conditions of pressure and normal velocity across the interface. A new similarity solution for the interfacial evolution in the rectangular coordinate system was derived by postulating a first-order approximation of the velocity distribution in the region that the two-phase fluids co-exist. The interfacial evolution equation can be explicitly expressed as a linear function, where the slope of the interfacial equation is simply related to the mobility ratio of two-phase fluids in porous media. The application of the proposed solutions to predictions of interfacial evolutions in carbon dioxide injected into saline aquifers was illustrated under different mobility ratios and operational parameters. For the purpose of comparison, the numerical solutions obtained by level set method and the similarity solutions based on the Dupuit assumptions were presented. The results show that the proposed solution can give a better approximation of interfacial evolution than the currently available similarity solutions, especially in the situation that the mobility ratio is large. The proposed approximate solutions can provide physical insight into the interfacial phenomenon and be readily used for rapidly screening carbon dioxide storage capacity in subsurface formations and monitoring the migration of carbon dioxide plume.

Explicit solution for the one-phase stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face

Abstract: An explicit solution of a similarity type is obtained for a onephase Stefan problem in a semi-infinite material using Kummer functions. It is considered a phase-change problem with a latent heat defined as a power function of the position with a non-negative real exponent and a convective boundary condition at the fixed face x = 0. Existence and uniqueness of the solution is proved. Relationship between this problem and the problems with temperature and flux boundary condition is also analysed. Furthermore it is studied the limit behaviour of the solution when the coefficient which characterizes the heat transfer at the fixed boundary tends to infinity. Computing this limit allows to demonstrate that the problem proposed in this paper with a convective boundary condition generalizes the problem with Dirichlet boundary condition. Numerical computation of the solution is done over certain examples, with a view to comparing this results with those obtained by general algorithms that solve Stefan problems. AMS Subject Classification: 35R35, 80A22, 35C05