# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 1 Introduction
The marker-and-cell (MAC) approach is a method that has been used quite extensively for simulation free surface flows [14; 15; 16; 17; 18; 19; 20], in which the governing equations discretized with finite differences and the free surface is tracked while using marker particles. The method has been successfully used for the simulation, for example, of the die swell and viscoelastic jet buckling with a variety of constitutive models. The simulations are typically performed on staggered, uniform grids, which has certain limitations for complex simulation geometries and in terms of computational efficiency when high resolution is required. Furthermore, the existence of a gas phase surrounding the viscoelastic fluid is not taken into account, which may be required for some cases. The front-tracking method is another similar method [21]. The volume of fluid (VOF) method is a popular method for the simulation of Newtonian as well as viscoelastic free surface flow with finite volume discretization. VOF is a diffuse-interface method, in which the presence of two or more fluid phases is represented by their corresponding volume fractions, advected with geometric or algebraic schemes. A sharp fluid interface may then be reconstructed from the solution. A single set of momentum, continuity equations are then solved for the whole domain, and the fluid properties are locally averaged with the fluid volume fraction. Furthermore, the transport of the volume fraction may be solved with the same spatial grid as the momentum and continuity equation, which makes the method suitable for finite volume simulation of free surface flow. Some examples follow. [PERSON] et al. [22] developed a VOF-solver for viscoelastic free surface flow that is based on the open source solver OpenFOAM [23]. The simulations were stabilized by reformulating the viscoelastic constitutive equations with the log-conformation representation (LCR) and with both-sides diffusion (BSD) in the momentum equation. [PERSON] et al. [24] simulated the viscoelastic die swell effect for Oldroyd-B and Giesekus fluids with the VOF method, combined with two different schemes for the convection of the fluid volume fraction, a geometric scheme, as well as an algebraic scheme that directly solved the transport equation for the volume fraction. Furthermore, their methods were comparedto the VOF solver that is available in RheoTool [25], which is an open source toolbox for viscoelastic flow that is based on OpenFOAM. The constitutive equation was also reformulated with LCR in their study. [PERSON] et al. [26] reported a related method, using a mathematical formulation that is similar to the volume of fluid method, while solving the governing equations with a piecewise linear finite element method, contrary to finite volume discretization. In previous work, a Lagrangian-Eulerian framework for viscoelastic flow has been proposed [27; 28]. In the method, the constitutive equation is solved at Lagrangian nodes, which are convected by the flow. The momentum and continuity equations are discretized with the finite volume method on an Eulerian octree grid and then solved with SIMPLEC iterations [29]. Boundary conditions that are imposed by objects in the domain are treated using implicit immersed boundary methods [30; 31]. The contribution to the momentum equation from the viscoelastic stress is established through the unstructured interpolation of the stress tensors from the Lagrangian nodes to the cell centers of the Eulerian grid while using the radial basis function. The ability to efficiently simulate industrial applications, such as adhesive extrusion, parts assembly, and additive manufacturing, is a main motivator to develop the new method. For this purpose, it is necessary that the numerical method is accurate, computationally efficient, and robust. The ability to model viscoelastic free surface flow in complex geometry with moving objects is also necessary.