### 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 |δ_{21}β| makes the system more unstable, i.e., accelerates the onset of instability motion; here δ_{21} 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 δ_{21}β-values, respectively. From the present nonlinear analysis, the system having δ_{22} > 1 prefers the instabilities with a larger wavelength than the system having δ_{22} <1. Here δ_{22} is the normalized normal diffusion coefficient of component B.

Original language | English |
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Article number | 084103 |

Journal | Physics of Fluids |

Volume | 28 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2016 Aug 1 |

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### 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.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 28, no. 8, 084103. https://doi.org/10.1063/1.4960126

}

TY - JOUR

T1 - Cross-diffusion-driven gravitational instability in a Hele-Shaw cell saturated with a ternary solution

AU - Kim, Min Chan

AU - Song, Kwang Ho

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84981508200&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84981508200&partnerID=8YFLogxK

U2 - 10.1063/1.4960126

DO - 10.1063/1.4960126

M3 - Article

AN - SCOPUS:84981508200

VL - 28

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 8

M1 - 084103

ER -