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Method of matched asymptotic expansions

About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.


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TL;DR: In this article, the authors considered the differential equation x" + a(t)f(x) = 0, t > 0, under the condition that lim,.x fl'a(s)ds exists and is finite, and necessary and/or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions cl + c2t.
Abstract: The differential equation x" + a(t)f(x) = 0, t > 0, is considered under the condition that lim,.x fl'a(s)ds exists and is finite, and necessary and/or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions cl + c2t.

41 citations

Journal ArticleDOI
TL;DR: In this paper, the authors used matched asymptotic expansions to derive the leading-order uniform solution of the classical dam-break problem, which is performed with respect to a small parameter which characterizes the short duration of the stage under consideration.
Abstract: The liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy–Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.

41 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that inertia is crucial in the development of an asymptotic solution for the temperature field, and that the singular behavior of the outer temperature field can be traced to the interaction of the slowly decaying Stokeslet, arising from the gravitational contribution to the motion of the drop.
Abstract: When a drop moves in a uniform vertical temperature gradient under the combined action of gravity and thermocapillarity at small values of the thermal Peclet number, it is shown that inclusion of inertia is crucial in the development of an asymptotic solution for the temperature field. If inertia is completely ignored, use of the method of matched asymptotic expansions, employing the Peclet number (known as the Marangoni number) as the small parameter, leads to singular behaviour of the outer temperature field. The origin of this behaviour can be traced to the interaction of the slowly decaying Stokeslet, arising from the gravitational contribution to the motion of the drop, with the temperature gradient field far from the drop. When inertia is included, and the method of matched asymptotic expansions is used, employing the Reynolds number as a small parameter, the singular behaviour of the temperature field is eliminated. A result is obtained for the migration velocity of the drop that is correct to O(Re 2 log Re)

41 citations

Journal ArticleDOI
TL;DR: In this article, a system of equations which models the formation of clusters by coagulation, with particles of unit size being injected at a time-dependent rate, was studied, and the criteria under which gelation occurs were the same as for the constant mass and constant monomer cases, which have been studied previously.
Abstract: We study a system of equations which models the formation of clusters by coagulation, with particles of unit size being injected at a time-dependent rate We observe that the criteria under which gelation occurs are the same as for the constant mass and constant monomer cases, which have been studied previously We identify a variety of types of behaviour in the large-time limit, depending on the coagulation kernel and on the rate at which monomer is introduced into the system The results are obtained by means of exact (generating function) techniques, matched asymptotic expansions and numerical simulations

41 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202321
202244
202110
202023
201913
201835