We simulate the behaviour of suspensions of large-particle, non-Brownian,
neutrally-buoyant spheres in a Newtonian liquid with a Galerkin,
finite element, Navier–Stokes
solver into which is incorporated a continuum constitutive relationship
described by
Phillips et al. (1992). This constitutive description couples
a Newtonian
stress/shear-rate relationship (where the local viscosity
of the suspension is dependent on the
local volume fraction of solids) with a shear-induced migration model of
the suspended
particles. The two-dimensional and three-dimensional (axisymmetric) model
is benchmarked with a variety of single-phase and two-phase analytic solutions
and
experimental results. We describe new experimental results using nuclear
magnetic
resonance imaging to determine non-invasively the evolution of the solids-concentration
profiles of initially well-mixed suspensions as they separate when subjected
to slow
flow between counter-rotating eccentric cylinders and to piston-driven
flow in a pipe.
We show good qualitative and quantitative agreement of the numerical predictions
and the experimental measurements. These flows result in complex final
distributions
of the solids, causing rheological behaviour that cannot be accurately
described with
typical single-phase constitutive equations.
A method is presented for tracing rays through a medium discretized as finite-element volumes. The ray-trajectory equations are cast into the local element coordinate frame, and the full finite-element interpolation is used to determine instantaneous index gradient for the ray-path integral equation. The finite-element methodology is also used to interpolate local surface deformations and the surface normal vector for computing the refraction angle when launching rays into the volume, and again when rays exit the medium. The procedure is applied to a finite-element model of an optic with a severe refractive-index gradient, and the results are compared to the closed-form gradient ray-path integral approach.
Finite element discretization of fully-coupled, incompressible flow problems with the classic mixed velocity-pressure interpolation produces matrix systems that render the best and most robust iterative solvers and preconditioners ineffective. The indefinite nature of the discretized continuity equation is the root cause and is one reason for the advancement of pressure penalty formulations, least-squares pressure stabilization techniques, and pressure projection methods. These alternatives have served as admirable expedients and have enabled routine use of iterative matrix solution techniques; but all remain plagued by exceedingly slow convergence in the corresponding nonlinear problem, lack of robustness, or limited range of accuracy. The purpose of this paper is to revisit matrix systems produced by this old mixed velocity-pressure formulation with two approaches: (1) deploying well-established tools consisting of matrix system reordering, GM-RES, and ILU preconditioning on modern architectures with substantial distributed or shared memory, and (2) tuning the preconditioner by managing the condition number using knowledge of the physical causes leading to the large condition number. Results obtained thus far using these simple techniques are very encouraging when measured against the reliability (not efficiency) of a direct matrix solver. Here we demonstrate routine solution for an incompressible flow problem using the Galerkin finite element method, Newton-Raphson iteration, and the robust and accurate LBB element. We also critique via an historical survey the limitations of pressure-stabilization strategies and all other commonly used alternatives to the mixed formulation for acceleration of iterative solver convergence. The performance of the new iterative solver approaches on other classes of problems, including fluid-structural interaction, multi-mode viscoelasticity, and free surface flow is also demonstrated.
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