Exponential asymptotics of free surface flow due to a line source

Lustri, Christopher J., McCue, Scott W., & Chapman, S. Jonathan (2013) Exponential asymptotics of free surface flow due to a line source. IMA Journal of Applied Mathematics, 78(4), pp. 697-713.

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Abstract

The steady problem of free surface flow due to a submerged line source is revisited for the case in which the fluid depth is finite and there is a stagnation point on the free surface directly above the source. Both the strength of the source and the fluid speed in the far field are measured by a dimensionless parameter, the Froude number. By applying techniques in exponential asymptotics, it is shown that there is a train of periodic waves on the surface of the fluid with an amplitude which is exponentially small in the limit that the Froude number vanishes. This study clarifies that periodic waves do form for flows due to a source, contrary to a suggestion by Chapman & Vanden-Broeck (2006, J. Fluid Mech., 567, 299--326). The exponentially small nature of the waves means they appear beyond all orders of the original power series expansion; this result explains why attempts at describing these flows using a finite number of terms in an algebraic power series incorrectly predict a flat free surface in the far field.

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ID Code: 57955
Item Type: Journal Article
Refereed: Yes
Additional URLs:
Keywords: Exponential asymptotics, Asymptotics beyond all orders, Free surface flow, Water waves, Line source, Periodic waves, Stokes lines, Divergent series, Optimal truncation
DOI: 10.1093/imamat/hxt016
ISSN: 1464-3634
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Approximation Theory and Asymptotic Methods (010201)
Divisions: Current > Institutes > Institute for Future Environments
Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright The Author 2013.
Copyright Statement: Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Deposited On: 10 Mar 2013 22:44
Last Modified: 30 Apr 2014 18:38

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