Interfacial instability of two superimposed immiscible viscous fluids in a vertical Hele-Shaw cell under horizontal periodic oscillations

We derive an integro-differential equation for the evolution of the interface separating two immiscible viscous fluids in a Hele-Shaw cell with a channel geometry, for arbitrary viscosity contrast. Our equation differs from a previous one obtained by a vortex-sheet formulation of the problem, in that the normal component of the interface velocity is formally decoupled from the gauge-dependent tangential part. The result is thus a closed integral equation for the normal velocity. We briefly comment on the advantages of such a formulation and implement an alternative computational algorithm based on it. Preliminary numerical results confirm a highly inefficient finger competition in the zero viscosity contrast limit.

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Interfacial instability in a time-periodic rotating Hele-Shaw Cell

MATEC Web of Conferences ◽

10.1051/matecconf/20141609004 ◽

2014 ◽

Vol 16 ◽

pp. 09004 ◽

Cited By ~ 2

Author(s):

J. Bouchgl ◽

S. Aniss ◽

M. Souhar ◽

A. Hifdi

Keyword(s):

Interfacial Instability ◽

Time Periodic ◽

Hele Shaw Cell

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Interfacial instability of two inviscid fluid layers under quasi-periodic oscillations

MATEC Web of Conferences ◽

10.1051/matecconf/201928607011 ◽

2019 ◽

Vol 286 ◽

pp. 07011

Author(s):

A. Eljaouahiry ◽

A. Arfaoui ◽

M. Assoul ◽

S. Aniss

Keyword(s):

Numerical Study ◽

Mathieu Equation ◽

Immiscible Fluids ◽

Interfacial Instability ◽

Periodic Case ◽

Inviscid Fluid ◽

Helmholtz Instability ◽

Periodic Oscillations ◽

Fluid Layers ◽

The Stability

We investigate the eﬀect of horizontal quasi-periodic oscillations on the stability of two immiscible fluids of different densities. The two fluid layers are confined in a cavity of infinite extension in the horizontal directions. We show in the inviscid theory that the linear stability analysis leads to the quasi-periodic Mathieu equation, with damping, which describes the evolution of the interfacial amplitude. Thus, we examine the eﬀect of horizontal quasi-periodic vibration, with two incommensurate frequencies, on the stability of the interface. The numerical study shows the existence of two types of instability: the Kelvin-Helmholtz instability and the quasi-periodic resonances. The numerical results show also that an increase of the frequency ratio has a distabilizing eﬀect on the Kelvin-Helmholtz instability and curves converge towards those of the periodic case.

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Effect of Horizontal Vibration on the Interfacial Instability in a Horizontal Hele-Shaw Cell

MATEC Web of Conferences ◽

10.1051/matecconf/20120106002 ◽

2012 ◽

Vol 1 ◽

pp. 06002 ◽

Cited By ~ 1

Author(s):

J. Bouchgl ◽

S. Aniss ◽

M. Souhar ◽

O. Caballina

Keyword(s):

Interfacial Instability ◽

Horizontal Vibration ◽

Hele Shaw Cell

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Control of the Interfacial Instabilities in a circular Hele-Shaw cell oscillating with a periodic angular velocity

MATEC Web of Conferences ◽

10.1051/matecconf/201928607014 ◽

2019 ◽

Vol 286 ◽

pp. 07014

Author(s):

J. Bouchgl ◽

M. Souhar

Keyword(s):

Stability Analysis ◽

Angular Velocity ◽

Immiscible Fluids ◽

Interfacial Instability ◽

Time Dependent ◽

Interfacial Instabilities ◽

Periodic Rotation ◽

The Stability ◽

Time Periodic ◽

Hele Shaw Cell

The stability of an interface of two viscous immiscible fluids of diﬀerent densities and confined in a Hele-Shaw cell which is oscillating with periodic angular velocityis investigated. A linear stability analysis of the viscous and time-dependent basic flows, generated by a periodic rotation, leads to a time periodic oscillator describing the evolution of the interface amplitude. In this study, we examine mainly the eﬀect of the frequency of the periodic rotation on the interfacial instability that occurs at the interface.

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Viscous fingering in Hele-Shaw cells

Journal of Fluid Mechanics ◽

10.1017/s0022112086001088 ◽

1986 ◽

Vol 173 ◽

pp. 73-94 ◽

Cited By ~ 205

Author(s):

P. G. Saffman

Keyword(s):

Boundary Conditions ◽

Viscous Fluids ◽

Parallel Plates ◽

Narrow Gap ◽

Fractal Structures ◽

The Status ◽

Interfacial Motion ◽

Selection Mechanisms ◽

Theoretical Results ◽

Hele Shaw Cell

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

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Quasi-Periodic Oscillations in Dwarf Novae

International Astronomical Union Colloquium ◽

10.1017/s0252921100073310 ◽

1979 ◽

Vol 46 ◽

pp. 77-88

Author(s):

Edward L. Robinson

Keyword(s):

Binary Systems ◽

Outer Edge ◽

Light Curves ◽

Close Binary ◽

Bright Spot ◽

Dwarf Novae ◽

Periodic Oscillations ◽

High Frequencies ◽

Short Period ◽

Typical Time

Three distinct kinds of rapid variations have been detected in the light curves of dwarf novae: rapid flickering, short period coherent oscillations, and quasi-periodic oscillations. The rapid flickering is seen in the light curves of most, if not all, dwarf novae, and is especially apparent during minimum light between eruptions. The flickering has a typical time scale of a few minutes or less and a typical amplitude of about .1 mag. The flickering is completely random and unpredictable; the power spectrum of flickering shows only a slow decrease from low to high frequencies. The observations of U Gem by Warner and Nather (1971) showed conclusively that most of the flickering is produced by variations in the luminosity of the bright spot near the outer edge of the accretion disk around the white dwarf in these close binary systems.

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