Numerical simulation for viscoplastic fluids via finite element methods
Creators: name: El-Borhamy, Mohamed
Publisher: name: Technische Universität Dortmund
Year: 2012
Subjects: subject: finite element method
subject: multigrid method
subject: Newton method
subject: time stepping schemes
subject: viscoplastic fluids
subject: 510

description: 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.