A two-phase solver for complex fluids: Studies of the Weissenberg effect

Abstract

In this work a new two-phase solver is presented and described, with a particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed code is based on a combination of classical Volume-of-Fluid and Continuum Surface Force methods, along with a generic kernel-conformation tensor transformation to represent the rheological characteristics of the (multi)-fluid phases. Benchmark test problems are solved in order to assess the numerical accuracy of distinct levels of physical complexities, such as the interface representation, the influence of advection schemes, the influence of surface tension and the role of fluid rheology. In order to demonstrate the new features and capabilities of the solver in simulating of complex fluids in transient regime, we have performed a set of simulations for the problem of a rotating rod inserted into a container with a viscoelastic fluid, known as the Weissenberg or Rod-Climbing effect. Firstly, our results are compared with numerical and experimental data from the literature for low angular velocities. Secondly, we have presented results obtained for high angular velocities (high elasticity) using the Oldroyd-B model which displayed very elevated climbing heights. Furthermore, above a critical value for the angular velocity, it was observed the onset of elastic instabilities driven by the combination of elastic stresses, interfacial curvature and secondary flows, that to the authors best knowledge, were not yet reported in the literature.

The effect of viscoelasticity in an airway closure model

“…The log-conformation approach applied to the Oldroyd-B model is provided as an embedded library in Basilisk and has been validated against the results of Figueiredo et al. (2016) for a viscoelastic axisymmetric droplet impacting and spreading on a plane wall. An additional validation is reported in the appendices, comparing the results of Basilisk with the corresponding simulations carried out with the code of Izbassarov & Muradoglu (2015).…”

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

“…The VOF method is directly based on the volume conservation, as it essentially solves a transport equation of the color function, either with a geometric scheme or an algebraic scheme. A geometric VOF method was implemented in [46] to simulate various free-surface flows of viscoelastic liquids, in two dimensions. Three-dimensional simulations of viscoelastic jet buckling and filament stretching were achieved with an algebraic VOF scheme in [47].…”

An adaptive solver for viscoelastic incompressible two-phase problems applied to the study of the splashing of weakly viscoelastic droplets

“…This test case has been used by diverse authors with very different schemes [27,12]. As in the previous work of [27] the dimensionless parameters were fixed to: F r = 2.26, h = 2, De = 1, Re = 5 and β = 0.1. [27] do not report values for the outer medium; in the present work we set either µ r and ρ r to 10 −3 .…”

“…The simulation performed by [27] were made with uniform meshes ranging from ∆r = ∆r * /D = ∆z = ∆z * /D = 2.5 × 10 −2 up to ∆r = ∆z = 1.25×10 −2 . Since in [27] negligible difference between meshes are shown, For both adaptation strategies, A1 and A2, the cell widths are comprised between ∆r = ∆z = 2.03 × 10 −2 and ∆r = ∆z = 8.12 × 10 −2 . The maximum timestep has been fixed in all simulations to ∆t = U o ∆t * /D = 10 −3 .…”

“…Recently it has been shown that a streamfunction-logconformation methodology [25,26] can provide stable numerical simulations of flows with very high Weissenberg numbers. Figueiredo et al [27] have shown that the log-conformation formulation can be used together with the Continuum Surface Force method (CSF) to simulate accurately highly viscoelastic, surface tension dependent, two phase flows. Some implementations are constructed, profiting from existing CFD toolboxes, such as OpenFOAM c [28,29,30,31].…”