The development and implementation of algorithms to investigate drop fragmentation under shear for viscoelastic liquids with surfactant

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The proposed research concerns the development and implementation of new algorithms for the numerical investigation of drop breakup under shear. The investigator has used thevolume-of-fluid (VOF) approach to simulate Newtonian liquids, including the effect of inertia, and insoluble surfactants at low concentration. A fixed regular grid is used. The motion of fluid interfaces is tracked by means of a color function, which represents the volume fraction for a fluid in a grid cell. At each time step of the numerical simulation, the color function is advected with the flow, and gradients of it are used to reconstruct the position of the interface and to compute the interfacial tension force. In prior work, the first and second derivatives of the color function have been approximated by finite differences. However, since the color function is actually a step function, the use of finitedifferences for derivatives in nonlinear combinations such as curvature is problematic. The conventional remedy is to use a smoothed color function. This, on the other hand, requires a length scale of smoothing which is small relative to physical scales but still large relativeto the mesh size. In practice, smoothing introduces numerical diffusion into the interfacial tension force. In drop breakup simulations, for instance, smoothing on the order of two or three times the mesh size still leads to non-convergence of satellite drop volumes with spatial mesh refinement. The investigator has recently developed the method PROST, which is aparabolic representation of the interface in the surface tension force and which requires nosmoothing. The interface is reconstructed from a least square fit of a paraboloid to the values of the color function in a given cell and its neighbors, and the curvature and normalare retrieved from this paraboloid. This leads to convergence of daughter drop volumes with mesh refinement. The objective of the proposed research is to develop algorithms with the VOF- PROST framework to simulate viscoelastic liquids with surfactants, and to implement the codeon large-scale parallel processing systems. The mathematical model allows the liquids to havememory, first and second normal stress differences, and shear-thinning. A nonlinear constitutive model is used for soluble surfactants, with advection along the interface and exchange with the bulk.The breakup of liquid drops suspended in another liquid occurs in a number of chemical processes, such as the commercial mixing of molten plastics to form new materials in the recycling industry. Upon mixing, the size and distribution of the daughter drops influence the quality of the product, and therefore it is important to be able to control these outcomes. This is difficult to predict for a commercial mixer because of the complexity of thefluid flow, resulting in a trial and error approach. The investigator's starting point for a theoretical prediction is to study the deformation of a single droplet in a well-defined flowfield. Numerical algorithms are devised to track the history of drop deformation and breakup,which is an important contribution to the fundamental knowledge base of manufacturing processes that involve mixing two immiscible liquids. The success of this work depends on thephysical modeling of the liquids to compare with available experimental data, the accuracy ofthe algorithms to solve the equations, the implementation on high-performance computing platforms, and established collaborations with engineers who perform controlled experiments. Abroader impact is that the algorithms apply to a wider class of fluid-fluid systems, such asthe processing of physiological fluids. The participation of a postdoctoral research associate and a graduate student is an essential educational component of the project.