Abstract. In recent work, Li et al. (Comm. Math. Sci., 7:81-107, 2009) developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages.Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in . However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the L 2 and L ∞ norms for selected test problems.

A one-dimensional multi-phase flow model for thermomagnetically pumped ferrofluid with heat transfer is proposed. The thermodynamic model is a combination of a simplified particle model and thermodynamic equations of state for the base fluid. The magnetization model is based on statistical mechanics, taking into account non-uniform particle size distributions. An implementation of the proposed model is validated against experiments from the literature, and found to give good predictions for the thermomagnetic pumping performance. However, the results reveal a very large sensitivity to uncertainties in heat transfer coefficient predictions.

The level-set method is a popular interface tracking method in two-phase flow simulations. An often-cited reason for using it is that the method naturally handles topological changes in the interface, e.g. merging drops, due to the implicit formulation. It is also said that the interface curvature and normal vectors are easily calculated. This last point is not, however, the case in the moments during a topological change, as several authors have already pointed out. Various methods have been employed to circumvent the problem. In this paper, we present a new such method which retains the implicit level-set representation of the surface and handles general interface configurations. It is demonstrated that the method extends easily to 3D. The method is validated on static interface configurations, and then applied to two-phase flow simulations where the method outperforms the standard method and the results agree well with experiments.

This article addresses the use of the level-set method for capturing the interface between two fluids. One of the advantages of the level-set method is that the curvature and the normal vector of the interface can be readily calculated from the level-set function. However, in cases where the level-set method is used to capture topological changes, the standard discretization techniques for the curvature and the normal vector do not work properly. This is because they are affected by the discontinuities of the signed-distance function half-way between two interfaces. This article addresses the calculation of normal vectors and curvatures with the level-set method for such cases. It presents a discretization scheme based on the geometry-aware curvature discretization by Macklin and Lowengrub [1]. As the present scheme is independent of the ghost-fluid method, it becomes more generally applicable, and it can be implemented into an existing level-set code more easily than Macklin and Lowengrub's scheme [1]. The present scheme is compared with the second-order central-difference scheme and with Macklin and Lowengrub's scheme [1], first for a case with no flow, then for a case where two drops collide in a 2D shear flow, and finally for a case where two drops collide in an axisymmetric flow. In the latter two cases, the Navier-Stokes equations for incompressible two-phase flow are solved. The article also gives a comparison of the calculation of normal vectors with the direction difference scheme presented by Macklin and Lowengrub in [2] and with the present dis- * Corresponding author Email address: karl.y.lervag@ntnu.no (Karl Yngve Lervåg) URL: http://folk.ntnu.no/lervag (Karl Yngve Lervåg)Preprint submitted to Computers and Fluids April 9, 2013 cretization scheme. The results show that the present discretization scheme yields more robust calculations of the curvature than the second-order central difference scheme in areas where topological changes are imminent. The present scheme compares well to Macklin and Lowengrub's method [1]. The results also demonstrate that the direction difference scheme [2] is not always sufficient to accurately calculate the normal vectors.