Halvor Lund

RELAXATION TWO-PHASE FLOW MODELS AND THE SUBCHARACTERISTIC CONDITION

The subcharacteristic condition for hyperbolic relaxation systems states that wave velocities of an equilibrium system cannot exceed the corresponding wave velocities of its relaxation system. This condition is central to the stability of hyperbolic relaxation systems, and is expected to hold for most such models describing natural phenomena. In this paper, we study a hierarchy of two-phase flow models. We consider relaxation with respect to volume transfer, heat transfer and mass transfer. We formally verify that our relaxation processes are consistent with the first and second laws of thermodynamics, and present analytical expressions for the wave velocities for each model in the hierarchy. Through an appropriate choice of variables, we prove directly by sums-of-squares that for all relaxation processes considered, the subcharacteristic condition holds for any thermodynamically stable equation of state.

A Hierarchy of Relaxation Models for Two-Phase Flow

A hierarchy of relaxation two-phase flow models is considered, formulated as hyperbolic relaxation systems with source terms. The relaxation terms cause volume, heat, and mass transfer due to differences in pressure, temperature, and chemical potential, respectively, between the two phases. The subcharacteristic condition is a concept closely related to the stability of such relaxation systems. It states that the wave speeds of an equilibrium system never can exceed the speeds of the corresponding relaxation system. The work of Fl\aatten and Lund [Math. Models Methods Appl. Sci., 21 (2011), pp. 2379--2407] is extended, with analytical expressions for the wave velocities in each model in the mentioned hierarchy. The subcharacteristic condition is explicitly shown to be satisfied using sums of squares, subject only to physically fundamental assumptions.

Effects of City-Size Heterogeneity on Epidemic Spreading in a Metapopulation: A Reaction-Diffusion Approach

We review and introduce a generalized reaction-diffusion approach to epidemic spreading in a metapopulation modeled as a complex network. The metapopulation consists of susceptible and infected individuals that are grouped in subpopulations symbolizing cities and villages that are coupled by human travel in a transportation network. By analytic methods and numerical simulations we calculate the fraction of infected people in the metapopulation in the long time limit, as well as the relevant parameters characterizing the epidemic threshold that separates an epidemic from a non-epidemic phase. Within this model, we investigate the effect of a heterogeneous network topology and a heterogeneous subpopulation size distribution. Such a system is suited for epidemic modeling where small villages and big cities exist simultaneously in the metapopulation. We find that the heterogeneous conditions cause the epidemic threshold to be a non-trivial function of the reaction rates (local parameters), the network’s topology (global parameters) and the cross-over population size that separates “village dynamics” from “city dynamics”.

A multi-phase ferrofluid flow model with equation of state for thermomagnetic pumping and heat transfer

Abstract 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.

Splitting methods for relaxation two-phase flow models

A model for two-phase pipeline flow is presented, with evaporation and condensation modelled using a relaxation source term based on statistical rate theory. The model is solved numerically using a Godunov splitting scheme, making it possible to solve the hyperbolic fluid-mechanic equation system and the relaxation term separately. The hyperbolic equation system is solved using the multi-stage (MUSTA) finite volume scheme. The stiff relaxation term is solved using two approaches: one based on the Backward Euler method, and one using a time-asymptotic scheme. The results from these two methods are presented and compared for a CO2 pipeline depressurisation case.