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Asymptotic Solutions for the Differential Equation Describing Solute Drag by Dislocations
Author(s) -
FuentesSamaniego R.,
Hirth J. P.
Publication year - 1984
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.2221210109
Subject(s) - bessel function , drag , mathematical analysis , limit (mathematics) , differential equation , mathieu function , partial differential equation , mathematics , physics , function (biology) , asymptotic analysis , line (geometry) , classical mechanics , mechanics , geometry , evolutionary biology , biology
The differential equation describing the drag of a Cottrell atmosphere by a dislocation appears in a number of other physical moving‐frame‐of‐reference problems related to line sources of vector fields, such as vortices in fluid flow and line sources of heat. It reduces to the so‐called modified Mathieu equation. This equation is solved and the asymptotic behavior of the modified Mathieu functions is determined in the limiting situations of small velocity and small or large distance from the line source. A specific asymptotic form of the function of the second kind is found to behave like a K‐Bessel function in the limit of the argument z approaching ∞ and like an I‐Bessel function in the limit z → ∞. The solution can be used to obtained analytical results for the Cottrellatmosphere problem.