On asymptotic solutions of boundary value problems defined on thin domains
Authors:
Gerald W. Young and Stephen H. Davis
Journal:
Quart. Appl. Math. 42 (1985), 403-409
MSC:
Primary 35B40; Secondary 35J05, 76D10
DOI:
https://doi.org/10.1090/qam/766877
MathSciNet review:
766877
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Abstract: The solution of the Poisson equation subject to Dirichlet conditions is examined asymptotically on thin domains. The evolution of the structure of the solution is followed as the shape of the domain changes. It is found that the “end wall” boundary layers present when the domain is rectangular, shrink and weaken as the endwalls become less sloped and vanish when the domain slope is uniformly bounded. Such structural changes are important in certain viscous flows containing moving contact lines.
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- Bernard J. Matkowsky, ASYMPTOTIC SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS IN THIN DOMAINS, ProQuest LLC, Ann Arbor, MI, 1966. Thesis (Ph.D.)–New York University. MR 2616044
D. E. Cormack, L. G. Leal and J. Imberger, Natural convection in a shallow cavity with differentially heated end walls. Part 1. Asymptotic theory, J. Fluid Mech. 65, 209–229 (1974)
A. K. Sen and S. H. Davis, Steady thermocapillary flows in two-dimensional slots, J. Fluid Mech. 121, 163–186 (1982)
G. D. Towell and L. B. Rothfeld, Hydrodynamics of rivulet flow, AIChE J. 12 972–980 (1966)
R. F. Allen and C. M. Biggin, Longitudinal flow of a lenticular filament down an inclined plane, Phys. Fluids 17, 287–291 (1974)
H. P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech. 84, 125–143 (1978)
- L. M. Hocking, Sliding and spreading of thin two-dimensional drops, Quart. J. Mech. Appl. Math. 34 (1981), no. 1, 37–55. MR 610748, DOI https://doi.org/10.1093/qjmam/34.1.37
N. Fox, On asymptotic expansions in plate theory, Proc. Roy Soc. A 278, 228–233 (1964)
D. R. Westbrook, Applications of asymptotic integration to potential problems for shells, SIAM J. Appl. Math. 14, 131–146 (1966)
B. J. Matkowsky, Asymptotic solution of partial differential equations in thin domains, Ph. D. dissertation, New York University, New York, 1966
D. E. Cormack, L. G. Leal and J. Imberger, Natural convection in a shallow cavity with differentially heated end walls. Part 1. Asymptotic theory, J. Fluid Mech. 65, 209–229 (1974)
A. K. Sen and S. H. Davis, Steady thermocapillary flows in two-dimensional slots, J. Fluid Mech. 121, 163–186 (1982)
G. D. Towell and L. B. Rothfeld, Hydrodynamics of rivulet flow, AIChE J. 12 972–980 (1966)
R. F. Allen and C. M. Biggin, Longitudinal flow of a lenticular filament down an inclined plane, Phys. Fluids 17, 287–291 (1974)
H. P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech. 84, 125–143 (1978)
L. M. Hocking, Sliding and spreading of thin two-dimensional drops, Quart. J. Mech. Appl. Math. 24 37–55 (1981)
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Article copyright:
© Copyright 1985
American Mathematical Society
