### Abstract

We consider flow in a Hele-Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability - a variant of the Saffman-Taylor instability - is found in a version of the two-dimensional Darcy's law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman-Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles.

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

Pages (from-to) | 1471-1495 |

Number of pages | 25 |

Journal | Nonlinearity |

Volume | 10 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Nonlinearity*,

*10*(6), 1471-1495. https://doi.org/10.1088/0951-7715/10/6/005

**Hele-Shaw flow and pattern formation in a time-dependent gap.** / Shelley, Michael; Tian, Fei Ran; Wlodarski, Krzysztof.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 10, no. 6, pp. 1471-1495. https://doi.org/10.1088/0951-7715/10/6/005

}

TY - JOUR

T1 - Hele-Shaw flow and pattern formation in a time-dependent gap

AU - Shelley, Michael

AU - Tian, Fei Ran

AU - Wlodarski, Krzysztof

PY - 1997/11

Y1 - 1997/11

N2 - We consider flow in a Hele-Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability - a variant of the Saffman-Taylor instability - is found in a version of the two-dimensional Darcy's law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman-Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles.

AB - We consider flow in a Hele-Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability - a variant of the Saffman-Taylor instability - is found in a version of the two-dimensional Darcy's law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman-Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles.

UR - http://www.scopus.com/inward/record.url?scp=0000804616&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000804616&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/10/6/005

DO - 10.1088/0951-7715/10/6/005

M3 - Article

AN - SCOPUS:0000804616

VL - 10

SP - 1471

EP - 1495

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 6

ER -