A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio.

*(English)*Zbl 1063.76077Summary: A numerical algorithm for the linear equation of state is developed for the volume-of-fluid interface-tracking code SURFER++, using the continuous surface stress formulation for the description of interfacial tension. This is applied to deformation under simple shear for a liquid drop in a much more viscous matrix liquid. We choose a Reynolds number and capillary number at which the drop settles to an ellipsoidal steady state, when there is no surfactant. The viscosity ratio is selected in a range where experiments have shown tip streaming when surfactants are added. Our calculations show that surfactant is advected by the flow and moves to the tips of the drop. There is a threshhold surfactant level, above which the drop develops pointed tips, which are due to surfactant accumulating at the ends of the drop. Fragments emitted from these tips are on the scale of the mesh size, pointing to a shortcoming of the linear equation of state, namely that it does not provide a lower bound on interfacial tension. One outcome is the possibility of an unphysical negative surface tension on the emitted drops.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76D45 | Capillarity (surface tension) for incompressible viscous fluids |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

PDF
BibTeX
XML
Cite

\textit{Y. Y. Renardy} et al., Eur. J. Mech., B, Fluids 21, No. 1, 49--59 (2002; Zbl 1063.76077)

Full Text:
DOI

##### References:

[1] | Siegel, M, Influence of surfactant on rounded and pointed bubbles in 2D Stokes flow, SIAM J. appl. math., 59, 6, 1998-2027, (1999) · Zbl 0940.76015 |

[2] | V. Cristini, Drop dynamics in viscous flow, PhD thesis, Yale University, 2000 |

[3] | C.D. Eggleton, T.M. Tsai, K.J. Stebe, Tip streaming from a drop in the presence of surfactants, ICTAM 2000, QK2, 2000 |

[4] | De Bruijn, R.A, Tipstreaming of drops in simple shear flows, Chem. eng. sci., 48, 277-284, (1993) |

[5] | Eggleton, C.D; Pawar, Y.P; Stebe, K.J, Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces, J. fluid mech., 385, 79-99, (1999) · Zbl 0959.76017 |

[6] | Y. Wang, C. Maldarelli, D.T. Papageorgiou, Increased mobility of a surfactant retarded bubble at high bulk concentrations, J. Fluid Mech. (2000) (accepted for publication) · Zbl 0971.76092 |

[7] | Scardovelli, R; Zaleski, S, Direct numerical simulation of free surface and interfacial flow, Annu. rev. fluid mech., 31, 567-604, (1999) |

[8] | Gueyffier, D; Li, J; Nadim, A; Scardovelli, R; Zaleski, S, Volume-of-fluid interface tracking and smoothed surface stress methods for three-dimensional flows, J. comp. phys., 152, 423-456, (1999) · Zbl 0954.76063 |

[9] | Lafaurie, B; Nardone, C; Scardovelli, R; Zaleski, S; Zanetti, G, Modelling merging and fragmentation in multiphase flows with SURFER, J. comp. phys., 113, 134-147, (1994) · Zbl 0809.76064 |

[10] | Li, J, Calcul d’interface affine par morceaux (piecewise linear interface calculation), C. R. acad. sci. II B, 320, 391-396, (1995) · Zbl 0826.76065 |

[11] | Li, J; Renardy, Y; Renardy, M, A numerical study of periodic disturbances on two-layer Couette flow, Phys. fluids, 10, 3056-3071, (1998) |

[12] | Li, J; Renardy, Y, Direct simulation of unsteady axisymmetric core-annular flow with high viscosity ratio, J. fluid mech., 391, 123-149, (1999) · Zbl 0973.76067 |

[13] | Li, J; Renardy, Y; Renardy, M, Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method, Phys. fluids, 12, 2, 269-282, (2000) · Zbl 1149.76454 |

[14] | Renardy, Y; Li, J, Parallelized simulations of two-fluid dispersions, SIAM news, () |

[15] | Brackbill, J.U; Kothe, D.B; Zemach, C, A continuum method for modeling surface tension, J. comp. phys., 100, 335-354, (1992) · Zbl 0775.76110 |

[16] | Rider, W.J; Kothe, D.B, Reconstructing volume tracking, J. comp. phys., 141, 112-152, (1998) · Zbl 0933.76069 |

[17] | Kothe, D.B; Williams, M.W; Puckett, E.G, Accuracy and convergence of continuum surface tension models, (), 294-305 · Zbl 0979.76014 |

[18] | Stone, H.A, Dynamics of drop deformation and breakup in viscous fluids, Annu. rev. fluid mech., 26, 65-102, (1994) · Zbl 0802.76020 |

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.