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Mechanics
In this work, a numerical investigation is conducted for a drop r=0.125 of viscous liquid, suspended in another liquid of the same density and viscosity, and undergoing breakup due to shear. The computational domain is a three-dimensional box 4x0,5x1, with spatial periodicity on the sides, und no-slip conditions at the top and bottom walls. The full Navier-Stokes equations are solved, with finite element method and the volume of fluid (VOF) algorithm to track the interface [1].
Several options are available to introduce upwinding in finite element methods. Stabilized multigrid finite element methods [2] have been retained in this work for several reasons, the most important being that all the equations are then solved using the same methodology, and that velocity and pseudo-concentration are discretized in continuous spaces, simplifying the overall implementation.
The system matrix associated to equation is non-symmetric and non-definite. A standart GMRES interative solver with diagonal preconditioning has been implemented to solve. When the shear rate is increased past a critical value, the drop ruptures. For the case of equal viscosities and densities, the critical capillary number is approximately 0.41. Our computation for Ca=0.42 in a dimensionless 4x0,5x1 box. A 3D finite elements mesh of unstructured tetrahedra is generated to from the triangulated surface. The methodology is based on various a posteriori årror estimators, which indicate regions of the computational domain where the mesh must be refined, or coarsened. Among others, projection error estimators, and estimators coining from the resolution of local variational problems of the error, are the most widely used
The competition between the externally imposed shear flow and the surface tension-driven flow is clearly evident. The most noticeable initial motion is elongation of the drop, stretched by the viscous shear stress of the external flow (for times less than 30). At time t=40, formation of a waist is seen near the center of the drop, and the drop is steadily thinning. As the drop slowly lengthens, a visible neck is seen near the bulbous end. Because of this neck, the ends will eventually pinch off and the liquid thread remaining in the middle will from small satellite droplets.
For viscosity ratios larger that the critical value, drop rotation away from the extensional axis is faster, resulting in near-alignement with the flow direction and no breakup. The drop stabilizes before significant elongation takes place. Definition critical curves for several finite Reynolds and capillary numbers and asymptote (for Stokes flow) for Newtonian and non- Newtonian Fluids.
[1] Chekhonin K.A., Sukhinin P.A. The Flow non-Newtonian Fluids with Free Surface infill to axissymmetrical Volume // Mathematical Modelling, 2001. V.13, ¹1. P.89-102.
[2] Chekhonin K.A., Bulgakov V.K. Fundamentals Mixed Finite Element Method for Hydrodynamics Problems Khabarovsk: Khabarovsk Tekhnical University 1999.
Note. Abstracts are published in author's edition
Last update: 06-Jul-2012 (11:52:45)