Analytic theory for the linear stability of the Saffman--Taylor finger
An analytic theory is presented for the linear stability of the Saffman--Taylor finger in a Hele--Shaw cell. Eigenvalues of the stability operator are determined in the limit of zero surface tension and it is found that all modes for the McLean--Saffman branch of solutions (J. Fluid Mech. 102, 455 (1980)) are neutrally stable, whereas other branches first calculated by Romero (Ph.D. thesis, California Institute of Technology, 1982) and Vanden-Broeck (Phys. Fluids 26, 2033 (1983)) are unstable to arbitrary infinitesimal disturbances. It is also shown that the Saffman--Taylor discrete set of eigenvalues is a special case of a continuous unstable spectrum for zero surface tension. The introduction of any amount of surface tension perturbs the corresponding eigenmodes such that the finger boundary is a nonanalytic curve in general. Only transcendentally small terms in surface tension are responsible for the nonanalyticity of the finger boundary as in the case of Saffman--Taylor steady finger solutions of arbitrary finger width.
- Research Organization:
- Department of Applied Mathematics, California Institute of Technology, Pasadena, California 91125
- OSTI ID:
- 6348335
- Journal Information:
- Phys. Fluids; (United States), Journal Name: Phys. Fluids; (United States) Vol. 30:8; ISSN PFLDA
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
ANALYTICAL SOLUTION
BOUNDARY-VALUE PROBLEMS
CONVECTION
EIGENVALUES
ENERGY TRANSFER
FLUID FLOW
FLUIDS
HEAT TRANSFER
INSTABILITY
MASS TRANSFER
SURFACE PROPERTIES
SURFACE TENSION
VISCOUS FLOW