<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

Sachdev, P. L.; Janna, WS

doi:10.1115/1.1421123

Cited by 31 publications

(37 citation statements)

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“…1 the ratio I converges to 1. Thus, for large t [Whitham, 1974;Sachdev, 2000]. Note that, for É(t) identically zero, I have the similarity solution of Buckley and Leverett [1942].…”

Section: Similarity Solutionmentioning

confidence: 99%

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An asymptotic solution for two‐phase flow in the presence of capillary forces

Vasco

2004

Water Resources Research

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[1] Under the assumption of smoothly varying background properties I derive an asymptotic solution for two-phase flow. This formulation partitions the modeling of two-phase flow into two subproblems: an arrival time calculation and a saturation amplitude computation. The asymptotic solution itself is defined along a trajectory though the model. If gravitational forces are not important and the flow field is independent of changes in the background saturation, the trajectory may be identified with a streamline, and the asymptotic approach provides a mathematical basis for streamline simulation. The evolution of the saturation amplitude is governed by a generalization of Burgers' equation, defined along the trajectory. If the capillary properties are uniform and gravitational forces are negligible, one can derive self-similar solutions for the three limiting behaviors of a two-phase front. Asymptotic solutions were found to deviate by less than 5% from saturations calculated using a numerical simulator. The numerical results indicate that changes in capillary properties can introduce significant variations in the arrival time of a specific aqueous fraction. A more robust definition of arrival time appears to be the time associated with the peak rate of change in aqueous phase fraction, the greatest slope of the breakthrough curve.

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“…1 the ratio I converges to 1. Thus, for large t [Whitham, 1974;Sachdev, 2000]. Note that, for É(t) identically zero, I have the similarity solution of Buckley and Leverett [1942].…”

Section: Similarity Solutionmentioning

confidence: 99%

“…[18] Self-similar and traveling wave solutions form a vast subject of investigation [Sachdev, 2000]. Such solutions have been grouped into two categories: self-similar solutions of the first and second kind [Barenblatt, 1979].…”

Section: Similarity Solutionmentioning

confidence: 99%

An asymptotic solution for two‐phase flow in the presence of capillary forces

Vasco

2004

Water Resources Research

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“…We omit the general theory of Lie algebras and groups, but texts such as [7] and [8] give detailed background on these algebraic structures. In essence, a Lie Group is a group with a well-defined action on a smooth manifold.…”

Section: Structural Identifiability In the Context Of Lie Algebrasmentioning

confidence: 99%

Structural identifiability analysis via symmetries of differential equations

2009

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Results and derivations are presented for the generation of a local Lie algebra that represents the 'symmetries' of a set of coupled differential equations. The subalgebra preserving the observation defined on the model structure is found, thus giving all transformations of the system that preserve its structure. It is shown that this is equivalent to the similarity transformation approach [1] for structural identifiability analysis and as such is a method of generating such transformations for this approach. This provides another method for performing structural identifiability analysis on nonlinear state-space models that has the possibility of extension to PDE type models. The analysis is easily automated and performed in Mathematica, and this is demonstrated by application of the technique to a number of practical examples from the literature.

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“…It is seen that there are two rectilinear parts: one linear region corresponds to the HRR-filed while the order linear part corresponds to the new intermediate asymptotic solution. The intermediate asymptotic solution is the stress and integrity distributions valid for times and distances at which the influence of fine details of initial and boundary conditions is lost [11][12][13][14]. …”

Section: Finite Difference Solution Of Mode III Crack Problemmentioning

confidence: 99%

An intermediate asymptotic solution of the coupled creep-damage crack problem in similarity variables

Степанова¹

2009

Vietnam J. Mech.

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Abstract. The class of self-similar solutions to coupled (creep-damage) crack problems is considered. The constitutive model is based on continuum damage mechanics. The conventional Kachanov-Rabotnov creep-damage theory is utilized to study the asymptotic behavior of damage in the region very near the crack tip. The totally damaged zone where the damage (integrity) parameter reaches its critical value is assumed to exist in the vicinity of the crack tip. Using the similarity variable the asymptotic solutions to mode I and mode III crack problems are obtained. The asymptotic stress, creep strain rate and damage fields near the crack tip are analyzed by solving nonlinear eigenvalue problems resulting in a new far stress distribution. The configurations of the totally damaged zone governed by the new far stress field are found and analyzed.

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Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems

Cited by 31 publications

References 0 publications

An asymptotic solution for two‐phase flow in the presence of capillary forces

An asymptotic solution for two‐phase flow in the presence of capillary forces

Structural identifiability analysis via symmetries of differential equations

An intermediate asymptotic solution of the coupled creep-damage crack problem in similarity variables

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