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.

Topics
Soft matter, Porous media, Thin films, Functions and functionals, Partial differential equations, Fluid bubbles, Hydrostatics, Capillarity, Complex fluids
1.
Cohen-Addad
,
S.
,
Höhler
,
R.
 and
Pitois
,
O.
, “
Flow in foams and flowing foams
,”
Annu. Rev. Fluid Mech.
45
,
241
–
267
(
2013
).
https://doi.org/10.1146/annurev-fluid-011212-140634
Google Scholar
Crossref
Search ADS
 
2.
Colin
,
T.
 and
Fabrie
,
P.
, “
A free boundary problem modeling a foam drainage
,”
Math. Models Methods Appl. Sci.
10
,
945
–
961
(
2000
).
https://doi.org/10.1142/s0218202500000483
Google Scholar
Crossref
Search ADS
 
3.
Goldfarb
,
I. I.
,
Kann
,
K. B.
 and
Shreiber
,
I. R.
, “
Liquid flow in foams
,”
Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza
2
,
103
–
108
(
1988
) (in Russian).
Google Scholar
 
4.
Goldshtein
,
V. M.
Goldfarb
,
I. I.
 and
Shreiber
,
I. R.
, “
Drainage waves structure in gas-liquid foam
,”
Int. J. Multiphase Flow
22
,
991
–
1003
(
1996
).
https://doi.org/10.1016/0301-9322(96)00030-4
Google Scholar
Crossref
Search ADS
 
5.
Kraynik
,
A. M.
, “
Foam flows
,”
Annu. Rev. Fluid Mech.
20
,
325
–
357
(
1988
).
https://doi.org/10.1146/annurev.fluid.20.1.325
Google Scholar
Crossref
Search ADS
 
6.
Ladyženskaja
,
O. A.
,
Solonnikov
,
V. A.
 and
Ural′ceva
,
N. N.
,
Linear and Quasi-Linear Equations of Parabolic Type
(
M. Izd-vo Nauka
,
1967
) (in Russian).
Google Scholar
 
7.
Solonnikov
,
V. A.
, “
On boundary value problems for linear parabolic systems of differential equations of general form
,”
Trudy Math. Inst. Steklov
83
,
3
–
162
(
1965
) (Russian).
Google Scholar
 
8.
Tani
,
A.
, “
On the first initial-boundary value problem of the generalized Burgers’ equation
,”
Publ. RIMS Kyoto Univ.
10
,
209
–
233
(
1974
).
https://doi.org/10.2977/prims/1195192178
Google Scholar
Crossref
Search ADS
 
9.
Tani
,
A.
, “
On the free boundary value problem for compressible viscous fluid motion
,”
Kyoto J. Math.
21
,
839
–
859
(
1981
).
https://doi.org/10.1215/kjm/1250521916
Google Scholar
Crossref
Search ADS
 
10.
Tani
,
A.
, “
Two-phase free boundary problem for compressible viscous fluid motion
,”
Kyoto J. Math.
24
,
243
–
267
(
1984
).
https://doi.org/10.1215/kjm/1250521328
Google Scholar
Crossref
Search ADS
 
11.
Tani
,
A.
 and
Tani
,
M.
, “
Classical solvability to the initial boundary value problem for a forced foam drainage equation
,”
J. Math. Anal. Appl.
504
,
125573
(
2021
).
https://doi.org/10.1016/j.jmaa.2021.125573
Google Scholar
Crossref
Search ADS
 
12.
Tani
,
A.
 and
Tani
,
M.
, “
Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom
,”
J. Math. Phys.
65
,
081506
(
2024
).
Google Scholar
 
13.
Weaire
,
D.
,
Hutzler
,
S.
,
Verbist
,
G.
 and
Peters
,
E.
, “
A review of foam drainage
,”
Adv. Chem. Phys.
102
,
315
–
374
(
1997
).
https://doi.org/10.1002/9780470141618.ch5
Google Scholar
Crossref
Search ADS
 
© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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