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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.

SUMMARYIn many cases, boundary integral equations contain a domain integral. This can be evaluated by discretization of the domain into domain elements. Historically, this was seen as going against the spirit of boundary element methods, and several methods were developed to avoid this discretization, notably dual and multiple reciprocity methods and particular solution methods. These involved the representation of the interior function with a set of basis functions, generally of the radial type. In this study, meshless methods (dual reciprocity and particular solution) are compared to the direct domain integration methods. The domain integrals are evaluated using traditional methods and also with multipole acceleration. It is found that the direct integration always results in better accuracy, as well as smaller computation times. In addition, the multipole method further improves on the computation times, in particular where multiple evaluations of the integral are required, as when iterative solvers are used. The additional error produced by the multipole acceleration is negligible.

SUMMARYThe analysis of transient heat conduction problems in large, complex computational domains is a problem of interest in many technological applications including electronic cooling, encapsulation using functionally graded composite materials, and cryogenics. In many of these applications, the domains may be multiply connected and contain moving boundaries making it desirable to consider meshless methods of analysis. The method of fundamental solutions along with a parallel domain decomposition method is developed for the solution of three-dimensional parabolic differential equations. In the current approach, time is discretized using the generalized trapezoidal rule transforming the original parabolic partial differential equation into a sequence of non-homogeneous modified Helmholtz equations. An approximate particular solution is derived using polyharmonic splines. Interfacial conditions between subdomains are satisfied using a Schwarz Neumann-Neumann iteration scheme. Outside of the first time step where zero initial flux is assumed, the initial estimates for the interfacial flux is given from the converged solution obtained during the previous time step. This significantly reduces the number of iterations required to meet the convergence criterion. The accuracy of the method of fundamental solutions approach is demonstrated through two benchmark problems. The parallel efficiency of the domain decomposition method is evaluated by considering cases with 8, 27, and 64 subdomains.