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

Abstract

We propose an adaptive numerical solver for the study of viscoelastic 2D two-phase flows using the volume-of-fluid method. The scheme uses the robust log conformation tensor technique of Fattal & Kupferman [1,2] combined with the time-split scheme proposed by Hao & Pan [3]. The use of this time-split scheme has been proven to increase the stability of the numerical computation of two-phase flows. We show that the adaptive computational technique can be used to simulate viscoelastic flows efficiently. The solver is coded using the opensource libraries provided by the Basilisk [4] platform. In particular, the method is implemented for Oldroyd-B type viscoelastic fluids and related models (FENE-P and FENE-CR). The numerical scheme is then used to study the splashing of weakly viscoelastic drops. The solvers and tests of this work are freely available on the Basilisk [4] web site [5]. arXiv:1807.00103v1 [physics.flu-dyn] 30 Jun 2018as the Smoothed-Particle-Hydrodynamics (SPH) method, can also be found in the literature on computational rheology [11,12].The Finite-Element method applied to viscoelastic flows goes back to the pioneering work of [13,14,15]. Successful implementations of viscoelastic fluids using FE have recently been conducted [16,17] and is the basis of commercial codes as Polyflow R . The FE implementation of the log conformation schemes done by Hulsen et al. [18] follows right after the original scheme is published. the FE implementation of the log conform performed by Hao & Pan [3] is particularly relevant for the present work since we use the time-split scheme proposed in that work.Most of the numerical simulations loose convergence and destabilize when the relaxation parameter, or its dimensionless counterpart, the Weissenberg number, is increased above a threshold value. This behaviour, known as the High-Weissenberg number problem (HWNP), has been a severe hindrance for computational rheology. Fortunately, a major relief of the HWNP problem has been provided by Fattal & Kupferman [1,2]. These authors proposed formulating the equations in terms of the logarithm of the conformation tensor. Interestingly, this log-conformation (kernel) formulation guarantees the positive definiteness of the conformation tensor during the entire simulation. The success of this kernel method has been immediate, and is substituting, in practice, the classic approach in computational rheology. The log conformation kernel has been implemented within the FD method [1,2], the FE method [18,3] and the FV method [19]. In the same spirit [20] proposed using the square root of the conformation tensor to preserve the positive definiteness. Although less extended than the log conformation kernel, the square root conformation kernel has been used recently to analyze the lid cavity problem [21,10]. Although other conformation kernels are possible [22,9], these seem to be the most accurate.The FV is, at present, the method of reference in CFD (included commercial codes). Several reasons support its popularity. Remarkably, the metho...

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