We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.
Skip Nav Destination
Article navigation
August 2024
Research Article|
August 06 2024
Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface
Atusi Tani
;
Atusi Tani
a)
(Formal analysis, Writing – original draft)
1
Department of Mathematics, Keio University
, Yokohama 223-8522, Japan
a)Author to whom correspondence should be addressed: tani@math.keio.ac.jp
Search for other works by this author on:
Marie Tani
Marie Tani
(Formal analysis, Writing – original draft)
2
Department of Physics, Kyoto University
, Kyoto 606-8502, Japan
Search for other works by this author on:
a)Author to whom correspondence should be addressed: tani@math.keio.ac.jp
J. Math. Phys. 65, 081505 (2024)
Article history
Received:
April 20 2023
Accepted:
May 26 2024
Citation
Atusi Tani, Marie Tani; Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface. J. Math. Phys. 1 August 2024; 65 (8): 081505. https://doi.org/10.1063/5.0155449
Download citation file:
Sign in
Don't already have an account? Register
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Pay-Per-View Access
$40.00
Citing articles via
Derivation of the Maxwell–Schrödinger equations: A note on the infrared sector of the radiation field
Marco Falconi, Nikolai Leopold
Almost synchronous quantum correlations
Thomas Vidick
Learning from insulators: New trends in the study of conductivity of metals
Giuseppe De Nittis, Max Lein, et al.
Related Content
Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom
J. Math. Phys. (August 2024)
Analysis of capillary drainage from a flat solid strip
Physics of Fluids (June 2014)
Capillary rise of a liquid into a deformable porous material
Physics of Fluids (January 2009)
PERSPECTIVE: Interfacial stresses in thin film drainage: Subtle yet significant
J. Rheol. (June 2024)
Ganglia dynamics during imbibition and drainage processes in nanoporous systems
Physics of Fluids (April 2022)