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SUMMARY:Growth of thin fingers in Laplacian and Poisson fields - Robb McDo
 nald (University College London)
DTSTART:20191031T143000Z
DTEND:20191031T153000Z
UID:TALK133522@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:(i) The Laplacian growth of thin two-dimensional protrusions i
 n the form of either straight needles or curved fingers satisfying Loewner
 &#39\;s equation is studied using the Schwarz-Christoffel (SC) map. Partic
 ular use is made of Driscoll&#39\;s numerical procedure\, the SC Toolbox\,
  for computing the SC map from a half-plane to a slit half-plane\, where t
 he slits represent the needles or fingers. Since the SC map applies only t
 o polygonal regions\, in the Loewner case\, the growth of curved fingers i
 s approximated by an increasing number of short straight line segments. Th
 e growth rate of the fingers is given by a fixed power of the harmonic mea
 sure at the finger or needle tips and so includes the possibility of &lsqu
 o\;screening&rsquo\; as they interact with themselves and with boundaries.
  The method is illustrated by examples of needle and finger growth  in  ha
 lf-plane and channel geometries. Bifurcating fingers are also studied and 
 application to branching stream networks discussed.  (ii) Solutions are fo
 und for the growth of infinitesimally thin\, two-dimensional fingers gover
 ned by Poisson&#39\;s equation in a long strip. The analytical results det
 ermine the asymptotic paths selected by the fingers which compare well wit
 h the recent numerical results of Cohen and Rothman (2017) for the case of
  two and three fingers. The generalisation of the method to an arbitrary n
 umber of fingers is presented and further results for four finger evolutio
 n given. The relation to the analogous problem of finger growth in a Lapla
 cian field is also discussed.
LOCATION:Seminar Room 1\, Newton Institute
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