Andreas N. Alexandrou

Performance of the finite volume method in solving regularised Bingham flows: inertia effects in the lid-driven cavity flow

Abstract We extend our recent work on the creeping flow of a Bingham fluid in a lid-driven cavity, to the study of inertial effects, using a finite volume method and the Papanastasiou regularisation of the Bingham constitutive model (Papanastasiou, 1987) [7]. The finite volume method used belongs to a very popular class of methods for solving Newtonian flow problems, which use the SIMPLE algorithm to solve the discretised set of equations, and have matured over the years. By regularising the Bingham constitutive equation it is easy to extend such a solver to Bingham flows since all that this requires is to modify the viscosity function. This is a tempting approach, since it requires minimum programming effort and makes available all the existing features of the mature finite volume solver. On the other hand, regularisation introduces a parameter which controls the error in addition to the grid spacing, and makes it difficult to locate the yield surfaces. Furthermore, the equations become stiffer and more difficult to solve, while the discontinuity at the yield surfaces causes large truncation errors. The present work attempts to investigate the strengths and weaknesses of such a method by applying it to the lid-driven cavity problem for a range of Bingham and Reynolds numbers (up to 100 and 5000 respectively). By employing techniques such as multigrid, local grid refinement, and an extrapolation procedure to reduce the effect of the regularisation parameter on the calculation of the yield surfaces (Liu et al., 2002) [55], satisfactory results are obtained, although the weaknesses of the method become more noticeable as the Bingham number is increased.

Solution of the square lid-driven cavity flow of a Bingham plastic using the finite volume method

We investigate the performance of the finite volume method in solving viscoplastic flows. The creeping square lid-driven cavity flow of a Bingham plastic is chosen as the test case and the constitutive equation is regularised as proposed by Papanastasiou [J. Rheol. 31 (1987) 385–404]. It is shown that the convergence rate of the standard SIMPLE pressure-correction algorithm, which is used to solve the algebraic equation system that is produced by the finite volume discretisation, severely deteriorates as the Bingham number increases, with a corresponding increase in the non-linearity of the equations. It is shown that using the SIMPLE algorithm in a multigrid context dramatically improves convergence, although the multigrid convergence rates are much worse than for Newtonian flows. The numerical results obtained for Bingham numbers as high as 1000 compare favourably with reported results of other methods.

Squeeze Flow of Semi-Solid Slurries

A standard method used to determine material properties of semi-solid slurries is the squeeze flow experiment in which a fixed amount of material is squeezed under constant force or velocity. The relation between the force and the displacement provides information about the rheology of the slurry. The thixotropy and the time response of the sample however is rarely, if ever, taken into consideration. In this work we study how thixotropy affects the flow characteristics and consequently the predicted material properties. We show that depending on the method of compression and the thixotropic constants the flow can be significantly different. Therefore the predicted material constants can vary and hence cannot be unique.

Cessation of the lid-driven cavity flow of Newtonian and Bingham fluids

We provide benchmark results for a transient variant of the lid-driven cavity problem, where the lid motion is suddenly stopped and the flow is left to decay under the action of viscosity. Results include Newtonian as well as Bingham flows, the latter having finite cessation times, for Reynolds numbers Re ∈ [1, 1000] and Bingham numbers Bn ∈ [0, 10]. The finite-volume method and Papanastasiou regularisation were employed. A combination of Re and Bn, the effective Reynolds number, is shown to convey more information about the flow than either Re or Bn alone. A time scale which characterises the flow independently of the geometry and flow parameters is proposed.

Rotating Thin Films in Bell Sprayers and Spin Coating

Steady axisymmetric thin film flows on a rotating cone of Newtonian and non‐Newtonian liquids are analyzed by means of the axisymmetric boundary layer with swirl equations, Galerkin finite‐element discretization with free surface parameterization, and Newton iteration, which permits the simultaneous evaluation of the boundary layer thickness with the primary unknowns which are the three velocity components. The significance of the inlet boundary conditions on the accuracy of the finite‐element predictions is discussed in detail. It is shown that natural inlet boundary conditions compatible with the local dynamics are superior to essential boundary conditions which may give rise to artificial finite‐element solutions. The analysis accounts for inertia, viscous, and surface tension effects. Both centrifugal and Coriolis accelerations are included. Results are presented and compared for Newtonian, shear‐thinning and shear‐thickening, Bingham plastic, and second‐order viscoelastic liquids at different operati...

Thixotropic flow past a cylinder

Abstract We study the flow of a thixotropic fluid around a cylinder. The rheology of the fluid is described by means of a structural viscoplastic model based on the Bingham constitutive equation, regularised using the Papanastasiou regularisation. The yield stress is assumed to vary linearly with the structural parameter, which varies from zero (completely broken structure) to one (fully developed skeleton structure), following a first-order rate equation which accounts for material structure break-down and build-up. The results were obtained numerically using the Finite Element Method. Simulations were performed for a moderate Reynolds number of 45, so that flow recirculation is observed behind the cylinder, but vortex shedding does not occur. The effects of the Bingham number and of the thixotropy parameters are studied. The results show that the viscous character of the flow can be controlled within certain limits through these parameters, despite the fact that the Reynolds number is fixed.

On the Modeling of Semisolid Suspensions

Due to the distinct rheology of semisolid slurries the process has well established advantages over competing near-net-shape manufacturing technologies. Despite the obvious advantages of the process its adoption by the casting industry, however, has been slow. This is primarily due to lack of confidence of how these slurries flow in die cavities. The added cost associated with the specially prepared slurry has also affected the process commercial success. Nevertheless despite these problems attention to the semi-solid metal process has indeed increased over the years. The main focus of this review is the modeling of semisolid slurries. The objective here is to present an overview of relevant aspects of modeling by focusing on the physics of the slurry and by stressing consistent mathematical and analysis methods to determine the material constants.

Semi-Solid Metal Processing: "Unlimited" Flow Velocity without Turbulence in Thin Cast Sections

The objective of this presentation is to show and explain why semisolid slurries can fill thin sections at seemingly unlimited flow velocity; the suitability of SSM parts with very thin sections is a characteristic of the process that is often overlooked by the industry. This fact provides a unique opportunity for new advanced applications of the process, not possible by any of the existing metal-mold casting methods.

Determining True Material Constants of Semisolid Slurries from Rotational Rheometer Data

In this work we revisit the issue of obtaining true material constants for semisolid slurries. Therefore, we consider the circular Couette flow of Herschel-Bulkley fluids. We first show how true constants can be obtained using an iterative procedure from experimental data to theory and vice versa. The validity of the assumption that the rate-of-strain distributions across the gap share a common point is also investigated. It is demonstrated that this is true only for fully-yielded Bingham plastics. In other cases, e.g., for partially-yielded Bingham plastics or fully-yielded Herschel-Bulkley materials, the common point for the fully-yielded Bingham case provides a good approximation for determining the material constants only if the gap is sufficiently small. It can thus be used to simplify the iterative procedure in determining the material constants.

Parameter Estimation for Semi-Solid Aluminum Alloys Using Transient Experiments

A rotating vane-cup rheometer is used to determine the rheological properties of semi-solid slurries, and a procedure is established for characterizing the rheology with emphasis given to the proper and self-consistent evaluation of the material constants.