Implicit Methods for Simulating Low Reynolds Number Free Surface Flows: Improvements on MAC-Type Methods

  • José A. Cuminato
  • Cassio M. Oishi
  • Rafael A. Figueiredo
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 1)


This paper is concerned with describing the main improvements introduced to the MAC (Marker-And-Cell) method for the numerical simulation of low Reynolds number free surface flows, namely: a stable implicit treatment of the pressure boundary condition for projection methods, a semi-implicit method based on the Crank–Nicolson (C–N) discretization of the momentum equations, a more accurate method for moving the massless particles representing the free surface and a viscoelastic model based on the Pom-Pom constitutive law, are discussed. Low Reynolds number free surface flows appear in a number of important industrial processes in the oil, food, cosmetic and medical industries and their simulation present a challenge for explicit MAC-type methods due to their parabolic time step constraint. The simulation of moving boundary problems presents a number of difficulties for a numerical method. For the semi-implicit (C–N) MAC method the main difficulty appears in applying the projection method to uncouple velocity and pressure, this is in addition to other difficulties of correctly imposing the boundary conditions on the free surface and the free surface representation itself.


Navier–Stokes equations Viscoelastic fluid flows  Free surface MAC scheme Implicit strategy Jet buckling Extrudate swell MAC method review Non-Newtonian fluids 



The authors would like to acknowledge the financial support of FAPESP (projects nos. 2013/07375-0, 2011/09194-7, 2009/15892-9) and CNPq (projects nos. 305447/2010-6 , 473589/2013-3).


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Copyright information

© Springer Japan 2014

Authors and Affiliations

  • José A. Cuminato
    • 1
  • Cassio M. Oishi
    • 2
  • Rafael A. Figueiredo
    • 1
  1. 1.Instituto de Ciências Matemáticas e Computação (ICMC)Universidade de São PauloSão CarlosBrazil
  2. 2.Faculdade de Ciência e Tecnologia (FCT)Universidade Estadual Paulista Julio de Mesquita FilhoPresidente PrudenteBrazil

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