Avoiding singularities in the numerical solution of the motion of a deformable ellipse immersed in a viscous fluid.

*(English)*Zbl 1126.76020Summary: Geological materials are largely heterogeneous and are typically comprised of approximately ellipsoidal objects immersed in a matrix with different physical properties. Methodologies for the identification of ancient regional tectonic patterns may be developed based on an understanding of the behaviour of heterogeneous materials. In this contribution, the differential equation governing the rotation of a deformable ellipse immersed in a viscous fluid is considered and is found to contain a singularity when the ellipse becomes circular in shape. This problem is avoided by reformulating the equations using the standard algebraic representation of ellipse. Thus, the equations can be numerically solved without difficulty.

##### MSC:

76D99 | Incompressible viscous fluids |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

##### Keywords:

algebraic representation of ellipse
PDF
BibTeX
XML
Cite

\textit{K. F. Mulchrone}, Math. Geol. 39, No. 7, 647--655 (2007; Zbl 1126.76020)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Hart D, Rudman AJ (1997) Least-squares fit of an ellipse to anisotropic polar data: application to azimuthal resistivity surveys in karst regions. Comput Geosci 23:189–194 |

[2] | Jeffery GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc Lond A 102:201–211 |

[3] | Lamb H (1932) Hydrodynamics. Cambridge University Press, Cambridge, 738 p · JFM 58.1298.04 |

[4] | Mulchrone KF, Walsh K (2006) The motion of a non-rigid ellipse in a general 2D deformation. J Struct Geol 28:392–407 |

[5] | Mulchrone KF, O’Sullivan F, Meere PA (2003) Finite strain estimation using the mean radial length of elliptical objects with bootstrap confidence intervals. J Struct Geol 25:529–539 |

[6] | Obdam ANM, Veling EJM (1987) Elliptical inhomogeneities in groundwater flow–an analytical description. J Hydrol 95:87–96 |

[7] | Strang G (1988) Linear algebra and its applications, 3rd edn. Harcourt Brace Jovanovich, San Diego, 520 p · Zbl 0338.15001 |

[8] | Zhao C, Hobbs BE, Ord A, Hornby P, Peng S, Liu L (2006) Theoretical and numerical analyses of pore-fluid flow patterns around and within inclined large cracks and faults. Geophys J Int 166:970–988 |

[9] | Zimmerman RW (1996) Effective conductivity of a two-dimensional medium containing elliptical inhomogeneities. Proc R Soc Lond A 452:1713–1727 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.