Numerical simulation of stratified flows and droplet deformation in two-dimensional shear flow of Newtonian and viscoelastic fluids

Analysis of multi-layer fluid flow systems or, in general, flows with interfaces often leads to mathematical expressions and equations too complicated for pencil and paper hence numerical computation is almost always necessary. In this dissertation, we develop a numerical code for tracking deformable interfaces. In particular this code is a viscoelastic version of the volume of fluid algorithm developed in [11]. The code uses the piecewise linear interface calculation method to reconstruct the interface and the continuous surface force formulation to model interfacial tension forces. Our numerical algorithm is primarily designed to simulate the flow of (i) superposed fluids (herein referred to as fluid-fluid systems) and (ii) the drop in a flow problem (droplet-matrix systems) in 2D shear flows of viscoelastic fluids. However by taking the viscoelastic parameters to be zeros, we in fact can consider cases were either or both of the phases in the fluid-fluid or droplet-matrix system will be assumed viscoelastic or Newtonian. The extra stresses governing viscoelasticity will herein be treated with the Oldroyd-B constitutive equations. The part our work dealing with two-layer flows is in the same spirit as among others that of Renardy et al. [29], who investigated the Poiseuille flow counterpart. As mentioned earlier, this part can also be thought of as a natural extension of the work of Li et al., [11], to the viscoelastic regime. Our subsequent work on deformable drops is closely connected to the experimental investigation of Guido et al. [6], and the numerical works of Sheth et al. [32], Pillapakkam et al. [18], and Renardy et al. [28], all of whom considered the drop in a flow problem in various contexts. As in [11] we employ the volume of fluid scheme with a semi-implicit Stokes solver (enabling computations at low Reynolds numbers) in our numerical algorithm. In the first part, the code is validated against linear theory for the superposed shear flow of well documented fluid-fluid systems. Numerical validation in the second part will mostly be against the results of the four major works cited earlier.