### Abstract

We analyse the cross-sectional evolution of an englacial meltwater conduit that contracts due to inward creep of the surrounding ice and expands due to melting. Making use of theoretical methods from free-boundary problems in Stokes flow and Hele-Shaw squeeze flow we construct an exact solution to the coupled problem of external viscous creep and internal heating, in which we adopt a Newtonian approximation for ice flow and an idealized uniform heat source in the conduit. This problem provides an interesting variant on standard free-boundary problems, coupling different internal and external problems through the kinematic condition at the interface. The boundary in the exact solution takes the form of an ellipse that may contract or expand (depending on the magnitudes of effective pressure and heating rate) around fixed focal points. Linear stability analysis reveals that without the melting this solution is unstable to perturbations in the shape. Melting can stabilize the interface unless the aspect ratio is too small; in that case, instabilities grow largest at the thin ends of the ellipse. The predictions are corroborated with numerical solutions using boundary integral techniques. Finally, a number of extensions to the idealized model are considered, showing that a contracting circular conduit is unstable to all modes of perturbation if melting occurs at a uniform rate around the boundary, or if the ice is modelled as a shear-thinning fluid.

Original language | English |
---|---|

Article number | 083101 |

Journal | Physics of Fluids |

Volume | 26 |

Issue number | 8 |

DOIs | |

Publication status | Published - 12 Aug 2014 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*26*(8), [083101 ]. https://doi.org/10.1063/1.4892389

**Free-boundary models of a meltwater conduit.** / Dallaston, Michael C.; Hewitt, Ian J.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 26, no. 8, 083101 . https://doi.org/10.1063/1.4892389

}

TY - JOUR

T1 - Free-boundary models of a meltwater conduit

AU - Dallaston, Michael C.

AU - Hewitt, Ian J.

PY - 2014/8/12

Y1 - 2014/8/12

N2 - We analyse the cross-sectional evolution of an englacial meltwater conduit that contracts due to inward creep of the surrounding ice and expands due to melting. Making use of theoretical methods from free-boundary problems in Stokes flow and Hele-Shaw squeeze flow we construct an exact solution to the coupled problem of external viscous creep and internal heating, in which we adopt a Newtonian approximation for ice flow and an idealized uniform heat source in the conduit. This problem provides an interesting variant on standard free-boundary problems, coupling different internal and external problems through the kinematic condition at the interface. The boundary in the exact solution takes the form of an ellipse that may contract or expand (depending on the magnitudes of effective pressure and heating rate) around fixed focal points. Linear stability analysis reveals that without the melting this solution is unstable to perturbations in the shape. Melting can stabilize the interface unless the aspect ratio is too small; in that case, instabilities grow largest at the thin ends of the ellipse. The predictions are corroborated with numerical solutions using boundary integral techniques. Finally, a number of extensions to the idealized model are considered, showing that a contracting circular conduit is unstable to all modes of perturbation if melting occurs at a uniform rate around the boundary, or if the ice is modelled as a shear-thinning fluid.

AB - We analyse the cross-sectional evolution of an englacial meltwater conduit that contracts due to inward creep of the surrounding ice and expands due to melting. Making use of theoretical methods from free-boundary problems in Stokes flow and Hele-Shaw squeeze flow we construct an exact solution to the coupled problem of external viscous creep and internal heating, in which we adopt a Newtonian approximation for ice flow and an idealized uniform heat source in the conduit. This problem provides an interesting variant on standard free-boundary problems, coupling different internal and external problems through the kinematic condition at the interface. The boundary in the exact solution takes the form of an ellipse that may contract or expand (depending on the magnitudes of effective pressure and heating rate) around fixed focal points. Linear stability analysis reveals that without the melting this solution is unstable to perturbations in the shape. Melting can stabilize the interface unless the aspect ratio is too small; in that case, instabilities grow largest at the thin ends of the ellipse. The predictions are corroborated with numerical solutions using boundary integral techniques. Finally, a number of extensions to the idealized model are considered, showing that a contracting circular conduit is unstable to all modes of perturbation if melting occurs at a uniform rate around the boundary, or if the ice is modelled as a shear-thinning fluid.

UR - http://www.scopus.com/inward/record.url?scp=84929121009&partnerID=8YFLogxK

U2 - 10.1063/1.4892389

DO - 10.1063/1.4892389

M3 - Article

VL - 26

JO - Physics of Fluid

JF - Physics of Fluid

SN - 1070-6631

IS - 8

M1 - 083101

ER -