Current issue
Archive
About the Journal
About
Editorial Board
Abstract & Indexing
Contact
Editorial Policies
Authorship & COI
Ethical principles
Principles of Transparent Checklist
Open access
For Authors
Instructions to Authors
Detailing Guidelines for Authors
Review process
Review sheet
How to submit
Open Access license
ORIGINAL PAPER
The Two-Phase Hell-Shaw Flow: Construction of an Exact Solution
1
Department of Mathematics, Ohio University Athens, OH 45701, USA
Online publication date: 2013-04-19
Publication date: 2013-03-01
International Journal of Applied Mechanics and Engineering 2013;18(1):249-257
KEYWORDS
ABSTRACT
We consider a two-phase Hele-Shaw cell whether or not the gap thickness is time-dependent. We construct an exact solution in terms of the Schwarz function of the interface for the two-phase Hele-Shaw flow. The derivation is based upon the single-valued complex velocity potential instead of the multiple-valued complex potential. As a result, the construction is applicable to the case of the time-dependent gap. In addition, there is no need to introduce branch cuts in the computational domain. Furthermore, the interface evolution in a two-phase problem is closely linked to its counterpart in a one-phase problem
REFERENCES (23)
1.
Alimov M.M. (2011): Construction of exact solutions to the Muskat problem.- Lobachevskii. J. Math, vol.32, pp.404-413.
2.
Crowdy D.G. (2002): On a class of geometry-driven free boundary problems.- SIAM J. Appl Math, vol.62, pp.945-964.
3.
Crowdy D.G. (2006): Exact solutions to the unsteady two-phase Hele-Shaw problem.- Quart J. Mech. Appl. Math, vol.59, pp.475-485.
4.
Cummings L.J., Howison S.D. and King J.R. (1999): Two-dimensional Stokes and Hele-Shaw flows with free surface.- Euro. J. Appl Math., vol.10, pp.635-680.
5.
Currie I.G. (1974): Fundamental mechanics of fluids.- USA: McGraw-Hill Book Company.
6.
David P.J. (1974): The Schwarz Function and its Application.- The Mathematical Association of America p.228.
7.
Galin L.A. (1945): Unsteady filtration with a free surface.- Dokl. Akad. S.S.S.R, vol.47, pp.246-249.
8.
Gustafsson B. and Vasil’ev A. (2006): Conformal and potential analysis in Hele-Shaw cells.- Basel, Switzerland: Part of Springer Science and Business Media.
9.
Howison S.D. (1992): Complex variable methods in Hele-Shaw moving boundary problems.- Euro. J. Appl. Math, vol.3, pp.209-224.
10.
Howison S. D. (2000): A note on the two-phase Hele-Shaw problem.- J. Fluid Mech vol.409, pp.243-249.
11.
Kang H. and Crowdy D. (2001): Squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell. - J. Nonlinear Sci, vol.11, pp.279-304.
12.
Lacey A.A. (1982): Moving boundary problems in the flow of liquid though porous media.- J. Austra. Math. Soc. (series B), vol.24, pp.171-193.
13.
Lundberg E. (2011): Problems in classical potential theory with applications to mathematical physics.- (Doctoral dissertation), Retrieved from Barnes and Nobel, ISBN-13: 9781249069386.
14.
McDonald N.R. (2011): Generalized Hele-Shaw flow: A Schwarz function approach.- Euro. J. Appl. Math, vol.22, pp.517-532.
15.
Oust B. (2009): Laplace growth patterns: A study of boundary evolution using iterated conformal maps and Loewner evolution.- Unpublished master’s thesis: University of Oslo, Norway.
16.
Polubarinova-Kochina P.Ya (1945): On the motion of an oil contour. - Dokl. Akad. S.S.S.R, vol.47, pp.254-257.
17.
Richardson S. (1972): Hele-Shaw flows with a free boundary by the injection of fluid into a narrow channel. - J. Fluid Mech, vol.56, pp.609-618.
18.
Richardson S. (1981): Some Hele-Shaw flows with time-dependent free boundaries.- J. Fluid Mech, vol.102, pp.263-278.
19.
Richardson S. (1992): Hele-Shaw flows with time-dependent free boundaries involving injuction through slits.- Studies in Applied Mathematics, vol.87, pp.175-194.
20.
Safman P.G. and Taylor G.I. (1958): The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid.- Proc. R. Soc. London, vol.245, pp.312-329.
21.
Savina T. and Nepomnyashchy A. (2011): Dynamical mother body in a Hele-Shaw problem.- Physica D, vol.240, pp.1156-1163.
22.
Shelley M.J., Tian F.R. and Wlodarski K. (1997): Hele-Shaw flow and pattern formation in a time-dependent gap.- Nonlinearity, vol.10, pp.1471-1495.
23.
Vasconcelos G.L. (1993): Exact solutions for steady bubbles in a Hele-Shaw cell with rectangular geometry. - Journal of Fluid Mechanics, vol.444, pp.175-198.
| eISSN: | 2353-9003 |
| ISSN: | 1734-4492 |
Task: edition of a scientific journal, its internationalization and ensuring open access to the journal via the Internet
© 2006-2024 Journal hosting platform by Bentus
We process personal data collected when visiting the website. The function of obtaining information about users and their behavior is carried out by voluntarily entered information in forms and saving cookies in end devices. Data, including cookies, are used to provide services, improve the user experience and to analyze the traffic in accordance with the Privacy policy. Data are also collected and processed by Google Analytics tool (more).
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
You can change cookies settings in your browser. Restricted use of cookies in the browser configuration may affect some functionalities of the website.
