Abstract
We present a numerical study of a stabilization method for computing confined and free-surface flows of highly elastic viscoelastic fluids. In this approach, the constitutive equation based on the conformation tensor, which is used to define the viscoelastic model, is modified introducing an evolution equation for the square-root conformation tensor. Both confined and free-surface flows are considered, using two different numerical codes. A finite volume method is used for confined flows and a finite difference code developed in the context of the marker-and-cell method is used for confined and free-surface flows. The implementation of the square-root formulation was performed in both numerical schemes and discussed in terms of its ability and efficiency to compute steady and transient viscoelastic fluid flows. The numerical results show that the square-root formulation performs efficiently in the tested benchmark problems at high-Weissenberg number flows, such as the lid-driven cavity flow, the flow around a confined cylinder, the cross-slot flow and the impacting drop free surface problem.
Highlights
We present a numerical study of a stabilization method for computing confined and free-surface flows of highly elastic viscoelastic fluids
The evolutionary character of the constitutive models and the hyperbolic nature of the equations require preserving the positive definiteness of the conformation tensor [6,7], and numerical discretization errors could, eventually, lead to the loss of such positive definiteness, resulting in a loss of topological evolutionary that can trigger Hadamard instabilities [6]. This numerical breakdown, which occurs when Wi increases, is known as the High Weissenberg Number Problem (HWNP), and has been a great challenge for those working on numerical simulations of viscoelastic fluid flows
The simulations for the square-root formulation were performed at moderate elasticity, Wi = 0.6, and compared with the solution obtained in previous works using different methods, namely the results of Alves et al [42] obtained using the extra-stress tensor formulation or those results obtained with the log-conformation formulation
Summary
The governing equations for transient, incompressible and isothermal flow of viscoelastic fluids can be written in a compact and dimensionless form as follows. T is the time, u is the velocity vector, p is the pressure, g is the gravitational field, and τ and A are the extra-stress and conformation tensors, respectively
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