Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner’s equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped ‘mathematical’ plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.


    Access

    Download


    Export, share and cite



    Title :

    Geodesic Loewner paths with varying boundary conditions


    Contributors:
    McDonald, R (author)

    Publication date :

    2020-10-01


    Remarks:

    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 476 (2242) , Article 20200466. (2020)


    Type of media :

    Article (Journal)


    Type of material :

    Electronic Resource


    Language :

    English



    Classification :

    DDC:    629



    Fast approximate geodesic paths on triangle mesh

    Tang, J. / Wu, G. S. / Zhang, F. Y. et al. | British Library Online Contents | 2007


    Combination of Piecewise-Geodesic Paths for Interactive Segmentation

    Mille, J. / Bougleux, S. b. / Cohen, L. D. | British Library Online Contents | 2015


    Tracking Meteorological Structures Through Curve Matching using Geodesic Paths

    Cohen, I. / Herlin, I. / IEEE; Computer Society | British Library Conference Proceedings | 1998



    Processing of Textured Surfaces Represented as Surfel Sets: Representation, Compression and Geodesic Paths

    Darom, T. / Ruggeri, M. R. / Saupe, D. et al. | British Library Conference Proceedings | 2005