Numerical simulation for viscoplastic fluids via finite element methods

Abstract

The design of efficient, robust and flexible numerical schemes to cope with nonlinear CFD problems has become the main nerve in the field of numerical simulation. This work has developed and analyzed the Newton-Multigrid process in the frame of monolithic approaches to solve stationary and nonstationary viscoplastic fluid problems. From the mathematical point of view, the viscoplastic problem exhibits several severe problems which might be arisen to draw the mathematical challenges. The major difficulty is the unbounded value of the viscosity which needs regularization. Several regularization techniques have been proposed to cope with this problem yet, while the accuracy is still not even close to be compared to the real model. Herein, two methods are used for the treatment of the non-differentiability, namely Bercovier-Engelman and modified bi-viscous models regularizations. To compute the solution at very small values of the regularization parameter which can be considered numerically as zero, we use the continuation technique. Other difficulties would be addressed in the circle of the nonlinearity, the solenoidal velocity field, as well as the convection dominated problem which are typically involved in the standard Navier-Stokes equation. The use of mixed higher order finite element methods for flow problems is advantageous, since one can partially avoid the addition of stabilization terms to handle for instance the lack of coercivity, the domination of the convective part as well as the incompressibility. In the case of mixed lower order finite element methods, edge oriented stabilization has been introduced to provide results in the case of the lack of coercivity and convection dominated problems. The main drawback of this stabilizer is to optimize or choose appropriately the free parameters to maintain high accuracy results from the scheme. Viscoplastic fluids are involved in many industrial applications which require numerical simulation to get a big mathematical insight and to predict the fluids behavior. The dependence of pressure on the viscoplastic constitutive law is confirmed as much as the dependence of velocity. Moreover, the behavior of the pressure is strongly related to the yield property for the unyielded regimes. In the case of a constant yield stress value together with the absence of the external densities, the field of pressure is prescribed by the null value wherever the null value of the deformation tensor is considered. Real life examples to prescribe the behavior of the viscoplastic fluids might be described in case of standard benchmarks: viscoplastic flow in channel, viscoplastic flow in a lid driven cavity and viscoplastic flow around a cylinder. In each case we confirm the experimental and theoretical results which are used to analyze viscoplastic problems for the physical behavior with respect to the unyielded regimes and the cessation of time.