Cross-diffusion-driven gravitational instability in a Hele-Shaw cell saturated with a ternary solution

Min Chan Kim, Kwang Ho Song

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The effect of cross diffusion on the onset of the gravitational instabilities in a Hele-Shaw cell saturated with a ternary solution is analyzed. Based on the linear stability theory, new stability equations are derived in the global domain and then transformed into the similar domain. These stability equations are solved by employing various methods such as an initial value problem approach and quasi-steady state approximations (QSSA's). Through the initial growth rate analysis without the QSSA, we prove that initially the system is unconditionally stable. However, the QSSA in the global domain showed that the system can be initially unstable for a certain condition. Based on the QSSA in the similar domain (QSSAζ), we obtain the critical time for the onset of instability motion. As expected, the higher |δ21β| makes the system more unstable, i.e., accelerates the onset of instability motion; here δ21 and β represent the normalized cross diffusion coefficient and the ratio of densification coefficients, respectively. Based on the linear analysis, fully nonlinear analyses are also conducted by using the Fourier spectral method. The present nonlinear analyses show that the double-diffusive and diffusive-layer convection-type of instabilities are possible for the positive and negative δ21β-values, respectively. From the present nonlinear analysis, the system having δ22 > 1 prefers the instabilities with a larger wavelength than the system having δ22 <1. Here δ22 is the normalized normal diffusion coefficient of component B.

Original languageEnglish
Article number084103
JournalPhysics of Fluids
Volume28
Issue number8
DOIs
Publication statusPublished - 2016 Aug 1

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gravitational instability
quasi-steady states
cells
approximation
Nonlinear analysis
diffusion coefficient
spectral methods
densification
Initial value problems
boundary value problems
Densification
convection
Wavelength
coefficients
wavelengths

ASJC Scopus subject areas

  • Condensed Matter Physics

Cite this

Cross-diffusion-driven gravitational instability in a Hele-Shaw cell saturated with a ternary solution. / Kim, Min Chan; Song, Kwang Ho.

In: Physics of Fluids, Vol. 28, No. 8, 084103, 01.08.2016.

Research output: Contribution to journalArticle

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