Computational modelling of non-Newtonian fluids based on the stabilised finite element method.

This work considers numerical modelling of non-Newtonian fluid flow. The motivation for this is the need for simulating industrial processes that involve non- Newtonian fluids. The non-Newtonian fluids that are included belong to the group of generalised Newtonian fluids. Generalised Newtonian fluids have a non-linear dependence between the shear stain rate and the shear stress, implying a non constant viscosity. Depending on the type of non-linearity generalised Newtonian fluids can be divided into three groups; shear-thinning, shear-thickening and visco-plastic fluids In this work all three groups are considered. The numerical modelling is performed by using a semi-discrete finite element method. The spatial domain is discretized with finite elements while the time domain is discretized with a discrete time stepping scheme. The finite element method that is used belongs to the group of stabilised finite element methods. Two types of stabilisation are employed. The first is the streamline upwind Petrov-Galerkin method (SUPG) stabilisation that is used to prevent the occurrence of spurious node- to-node oscillations that appear in the presence of dominant advective terms. The second type of stabilisation is pressure stabilisation This stabilisation is necessary to remove pressure oscillations and allow for greater flexibility when choosing interpolation functions. Thus the main unknown variables, velocity and pressure are discretized by linear equal order interpolation functions which offers substantial implementation advantages. The time stepping scheme is a single step method with a generalised midpoint rule. The scheme includes a parameter that can be used to obtain a variety of time stepping schemes from backward Euler to trapezoidal rule. The highly non linear system of equations obtained after discretization is solved via the Newton-Raphson solution procedure. After the finite element formulation was discretized, consistently linearised and implemented into a finite element program, several tests were performed to verify the implementation, to determine the accuracy and to prove suitability of this type numerical modelling for industrial applications. Several large scale problems from industrial practice have finally been solved to illustrate capabilities of the methodology