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Suppression of the Saffman-Taylor Instability in Hele-Shaw Cells: A Time-dependent Strategy
Vaquero-Stainer, Christian Paul
[Thesis]. Manchester, UK: The University of Manchester; 2019.
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Abstract
Displacement of a viscous fluid by a less viscous fluid, such as air, in a narrow gap between two parallel plates (a Hele-Shaw cell) produces the well-known Saffman- Taylor interfacial instability [1]. In recent literature, a number of possibilities for suppressing the instability by altering the geometry of the cell have been explored theoretically and experimentally. Here, we focus on a time-dependent power-law plate separation, b(t) = b_1 t^1/7 [2], where b(t) is the gap width between parallel plates. We present the results of linear stability analysis and of direct numerical simulations, which employ a finite-element method to solve the depth- averaged lubrication equations. We confirm that the lifting strategy is capable of stabilising the air-liquid interface. We explore regimes beyond the validity of the linear stability analysis and show that the nonlinear interaction of unstable modes leads to patterns which are quantitatively different form those predicted by linear theory.
Keyword(s)
Finite Element Method; Hele-Shaw cell; Saffman-Taylor instability; fluid dynamics; fluid mechanics; interfacial instability; oomph-lib