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Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell

Malaikah, Khalid R.
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367417377

Abstract Details

2013, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
Laplacian growth is the interface dynamics where the normal component of velocity of a free boundary is proportional to the normal derivative of a harmonic function defined in a moving domain. The interface evolution in a Hele-Shaw cell is described by the Laplacian growth model. In this study we derive governing equations in terms of the Schwarz function of the interface for some specific Hele-Shaw flows in which the interface is not equipotential. This is a generalization of the well-known equation w_z=-(1/2) S_t derived for the free boundary one-phase Hele-Shaw problem. Here, w is the complex potential and S_t is the time derivative of the Schwarz function. The structure of the thesis is as follows: In Chapter 1 we give an introduction to the history of the problem, and discuss the methods and the state of the art. Chapter 2 is devoted to the Schwarz function equation for the two-phase Hele-Shaw flows. Here we re-derive the equations earlier obtained by D. Crowdy using a slightly different method. Our derivation is based on an introduction of a single-valued complex velocity potential. In Chapter 3 we derive the Schwarz function equation for a class of generalized Hele-Shaw flows and apply it to the case of an interior problem in a cell with the time-dependent gap. This generalizes the governing equation of the interfacial motion in a Hele-Shaw cell in the presence of an arbitrary external potential.
Tatiana Savin (Advisor)
101 p.
Mathematics
Laplace Growth; Schwarz function; two-phase Hele-Shaw; Generalized Hele-Shaw flows

Recommended Citations

Citations

  • Malaikah, K. R. (2013). Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367417377

    APA Style (7th edition)

  • Malaikah, Khalid. Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell. 2013. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367417377.

    MLA Style (8th edition)

  • Malaikah, Khalid. "Laplacian Growth: Interfacial Evolution in a Hele-Shaw Cell." Doctoral dissertation, Ohio University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1367417377

    Chicago Manual of Style (17th edition)

ohiou1367417377
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© 2013, all rights reserved.
This open access ETD is published by Ohio University and OhioLINK.