, Volume 51, Issue 5, pp 1041–1055 | Cite as

Simulating the Hele-Shaw flow in the presence of various obstacles and moving particles



A generalization of the approach developed in the recent papers by the authors is presented. It aims to provide a description of the Hele-Shaw cell in the presence of multiple small obstacles/moving particles. We perform an asymptotic analysis of the dynamics of the moving boundary and the moving particles. For this, a modification of Maz’ya–Movchan–Nieves uniform asymptotic formula for the Green’s function of the mixed boundary value problem for the Laplace equation in a multiply connected domain is utilized. The paper contains extensive numerical analysis, accounting for various physical mechanisms of particle movement in the Hele-Shaw flow.


Hele-Shaw flow Point source/sink Moving obstacles Green’s function Neumann function Mixed boundary value problem Asymptotic analysis Numerical simulation 



D.P, S.R, and G.M. gratefully acknowledge the support of the European Union Seventh Framework Marie Curie Programme PARM-2 (Project Reference: PIAP-GA-2012-284544-PARM2), and M.W. acknowledges the European Union FP7 project INTERCER2 (Reference: PIAP-GA-2011-286110-INTERCER2). The authors are grateful to Dr. Michael Nieves for fruitful discussion on the asymptotic approximation of Green’s function.

Supplementary material

11012_2015_271_MOESM1_ESM.pdf (238 kb)
Supplementary material 1 (pdf 239 kb)


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • D. Peck
    • 1
  • S. V. Rogosin
    • 2
  • M. Wrobel
    • 1
  • G. Mishuris
    • 1
  1. 1.Department of MathematicsAberystwyth UniversityAberystwythUK
  2. 2.Belarusian State UniversityMinskBelarus

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