The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell

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Abstract

We investigate the viscous instability of a miscible displacement process in a recti-linear geometry, when the viscosity contrast is controlled by two quantities whichdiuse at dierent rates. The analysis is applicable to displacement in a porousmedium with two dissolved species, or to displacement in a Hele-Shaw cell with twodissolved species or with one dissolved species and a thermal contrast. We carry outasymptotic analyses of the linear stability behaviour in two regimes: that of smallwavenumbers at intermediate times, and that of large times.An interesting feature of the large-time results is the existence of regimes in whichthe favoured wavenumber scales with t−1/4, as opposed to the t−3/8 scaling foundin other regimes including that of single-species ngering. We also show that theregion of parameter space in which the displacement is unstable grows with time,and that although overdamped growing perturbations are possible, these are neverthe fastest-growing perturbations so are unlikely to be observed. We also interpretour results physically in terms of the stabilising and destabilising mechanisms actingon an incipient nger.
Original languageEnglish
Pages (from-to)564-577
Number of pages14
JournalEuropean Journal of Mechanics - B/Fluids
Volume28
Issue number4
Early online date5 Feb 2009
DOIs
Publication statusPublished - Aug 2009

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Hele-Shaw
Linear Stability
Cell
cells
Perturbation
perturbation
Porous Media
Parameter Space
Viscosity
Unstable
Scaling
viscosity
scaling
geometry

Keywords

  • viscous fingering
  • instability
  • viscosity
  • Hele-Shaw cell

Cite this

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title = "The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell",
abstract = "We investigate the viscous instability of a miscible displacement process in a recti-linear geometry, when the viscosity contrast is controlled by two quantities whichdiuse at dierent rates. The analysis is applicable to displacement in a porousmedium with two dissolved species, or to displacement in a Hele-Shaw cell with twodissolved species or with one dissolved species and a thermal contrast. We carry outasymptotic analyses of the linear stability behaviour in two regimes: that of smallwavenumbers at intermediate times, and that of large times.An interesting feature of the large-time results is the existence of regimes in whichthe favoured wavenumber scales with t−1/4, as opposed to the t−3/8 scaling foundin other regimes including that of single-species ngering. We also show that theregion of parameter space in which the displacement is unstable grows with time,and that although overdamped growing perturbations are possible, these are neverthe fastest-growing perturbations so are unlikely to be observed. We also interpretour results physically in terms of the stabilising and destabilising mechanisms actingon an incipient nger.",
keywords = "viscous fingering, instability, viscosity, Hele-Shaw cell",
author = "D. Pritchard",
year = "2009",
month = "8",
doi = "10.1016/j.euromechflu.2009.01.004",
language = "English",
volume = "28",
pages = "564--577",
journal = "European Journal of Mechanics - B/Fluids",
issn = "0997-7546",
number = "4",

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TY - JOUR

T1 - The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell

AU - Pritchard, D.

PY - 2009/8

Y1 - 2009/8

N2 - We investigate the viscous instability of a miscible displacement process in a recti-linear geometry, when the viscosity contrast is controlled by two quantities whichdiuse at dierent rates. The analysis is applicable to displacement in a porousmedium with two dissolved species, or to displacement in a Hele-Shaw cell with twodissolved species or with one dissolved species and a thermal contrast. We carry outasymptotic analyses of the linear stability behaviour in two regimes: that of smallwavenumbers at intermediate times, and that of large times.An interesting feature of the large-time results is the existence of regimes in whichthe favoured wavenumber scales with t−1/4, as opposed to the t−3/8 scaling foundin other regimes including that of single-species ngering. We also show that theregion of parameter space in which the displacement is unstable grows with time,and that although overdamped growing perturbations are possible, these are neverthe fastest-growing perturbations so are unlikely to be observed. We also interpretour results physically in terms of the stabilising and destabilising mechanisms actingon an incipient nger.

AB - We investigate the viscous instability of a miscible displacement process in a recti-linear geometry, when the viscosity contrast is controlled by two quantities whichdiuse at dierent rates. The analysis is applicable to displacement in a porousmedium with two dissolved species, or to displacement in a Hele-Shaw cell with twodissolved species or with one dissolved species and a thermal contrast. We carry outasymptotic analyses of the linear stability behaviour in two regimes: that of smallwavenumbers at intermediate times, and that of large times.An interesting feature of the large-time results is the existence of regimes in whichthe favoured wavenumber scales with t−1/4, as opposed to the t−3/8 scaling foundin other regimes including that of single-species ngering. We also show that theregion of parameter space in which the displacement is unstable grows with time,and that although overdamped growing perturbations are possible, these are neverthe fastest-growing perturbations so are unlikely to be observed. We also interpretour results physically in terms of the stabilising and destabilising mechanisms actingon an incipient nger.

KW - viscous fingering

KW - instability

KW - viscosity

KW - Hele-Shaw cell

U2 - 10.1016/j.euromechflu.2009.01.004

DO - 10.1016/j.euromechflu.2009.01.004

M3 - Article

VL - 28

SP - 564

EP - 577

JO - European Journal of Mechanics - B/Fluids

JF - European Journal of Mechanics - B/Fluids

SN - 0997-7546

IS - 4

ER -