### Abstract

This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [Lowengrub and Truskinovsky, Proc. R. Soc. London, Ser. A 454, 2617 (1998)]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper [paper II, Phys. Fluids 14, 514 (2002)], we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [Rowlinson and Widom, Molecular Theory of Capillarity (Clarendon, Oxford, 1979)] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in paper II.

Original language | English (US) |
---|---|

Pages (from-to) | 492-513 |

Number of pages | 22 |

Journal | Physics of Fluids |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2002 |

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### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes
- Computational Mechanics
- Mechanics of Materials
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*14*(2), 492-513. https://doi.org/10.1063/1.1425843

**Modeling pinchoff and reconnection in a hele-shaw cell. I. The models and their calibration.** / Lee, Hyeong G.; Lowengrub, J. S.; Goodman, Jonathan.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 14, no. 2, pp. 492-513. https://doi.org/10.1063/1.1425843

}

TY - JOUR

T1 - Modeling pinchoff and reconnection in a hele-shaw cell. I. The models and their calibration

AU - Lee, Hyeong G.

AU - Lowengrub, J. S.

AU - Goodman, Jonathan

PY - 2002/2

Y1 - 2002/2

N2 - This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [Lowengrub and Truskinovsky, Proc. R. Soc. London, Ser. A 454, 2617 (1998)]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper [paper II, Phys. Fluids 14, 514 (2002)], we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [Rowlinson and Widom, Molecular Theory of Capillarity (Clarendon, Oxford, 1979)] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in paper II.

AB - This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [Lowengrub and Truskinovsky, Proc. R. Soc. London, Ser. A 454, 2617 (1998)]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper [paper II, Phys. Fluids 14, 514 (2002)], we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [Rowlinson and Widom, Molecular Theory of Capillarity (Clarendon, Oxford, 1979)] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in paper II.

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UR - http://www.scopus.com/inward/citedby.url?scp=0036469249&partnerID=8YFLogxK

U2 - 10.1063/1.1425843

DO - 10.1063/1.1425843

M3 - Article

VL - 14

SP - 492

EP - 513

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 2

ER -