Bernard Meulenbroek

The multiscale nature of streamers

Streamers are a generic mode of electric breakdown of large gas volumes. They play a role in the initial stages of sparks and lightning, in technical corona reactors and in high altitude sprite discharges above thunderclouds. Streamers are characterized by a self-generated field enhancement at the head of the growing discharge channel. We briefly review recent streamer experiments and sprite observations. Then we sketch our recent work on computations of growing and branching streamers, we discuss concepts and solutions of analytical model reductions and we review different branching concepts and outline a hierarchy of model reductions.

Multiple scales in streamer discharges, with an emphasis on moving boundary approximations

Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

An empirical theory for gravitationally unstable flow in porous media

In this paper, we follow a similar procedure as proposed by Koval (SPE J 3(2):145–154, 1963) to analytically model CO2 transfer between the overriding carbon dioxide layer and the brine layer below it. We show that a very thin diffusive layer on top separates the interface from a gravitationally unstable convective flow layer below it. Flow in the gravitationally unstable layer is described by the theory of Koval, a theory that is widely used and which describes miscible displacement as a pseudo two-phase flow problem. The pseudo two-phase flow problem provides the average concentration of CO2 in the brine as a function of distance. We find that downstream of the diffusive layer, the solution of the convective part of the model, is a rarefaction solution that starts at the saturation corresponding to the highest value of the fractional-flow function. The model uses two free parameters, viz., a dilution factor and a gravity fingering index. A comparison of the Koval model with the horizontally averaged concentrations obtained from 2-D numerical simulations provides a correlation for the two parameters with the Rayleigh number. The obtained scaling relations can be used in numerical simulators to account for the density-driven natural convection, which cannot be currently captured because the grid cells are typically orders of magnitude larger than the wavelength of the initial fingers. The method can be applied both for storage of greenhouse gases in aquifers and for EOR processes using carbon dioxide or other solvents.

Laplacian Instability of Planar Streamer Ionization Fronts-An Example of Pulled Front Analysis

Streamer ionization fronts are pulled fronts that propagate into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long-time attractor out of a continuous family. A stability analysis for perturbations in the transverse direction has to take these features into account. In this paper we show how to apply the Evans function in a weighted space for this stability analysis. Zeros of the Evans function indicate the intersection of the stable and unstable manifolds; they are used to determine the eigenvalues. Within this Evans function framework, we define a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem. We also derive dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with an analysis of the Evans function leading to analytical expressions for the dispersion relation in the limit of small and large wave numbers. The paper concludes with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front.

Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling

We study the shape stability of disks moving in an external Laplacian field in two dimensions. The problem is motivated by the motion of ionization fronts in streamer-type electric breakdown. It is mathematically equivalent to the motion of a small bubble in a Hele–Shaw cell with a regularization of kinetic undercooling type, namely, a mixed Dirichlet–Neumann boundary condition for the Laplacian field on the moving boundary. Using conformal mapping techniques, linear stability analysis of the uniformly translating disk is recast into a single PDE which is exactly solvable for certain values of the regularization parameter. We concentrate on the physically most interesting exactly solvable and nontrivial case. We show that the circular solutions are linearly stable against smooth initial perturbations. In the transformation of the PDE to its normal hyperbolic form, a semigroup of automorphisms of the unit disk plays a central role. It mediates the convection of perturbations to the back of the circle where...

Construction and test of a moving boundary model for negative streamer discharges

Starting from the minimal model for the electrically interacting densities of electrons and ions in negative streamer discharges, we derive a moving boundary approximation for the ionization fronts. Solutions of the moving boundary model have already been discussed, but the derivation of the model was postponed to the present paper. The key ingredient of the model is the boundary condition on the moving front. It is found to be of kinetic undercooling type, and the relation to other moving boundary models is discussed. Furthermore, the model is compared to two-dimensional simulations of the underlying density model. The results suggest that our moving boundary approximation adequately represents the essential dynamics of negative streamer fronts.

Attendance and attainment in a Calculus course

In this paper the relationship between attendance and attainment in a standard calculus course is investigated. Calculus could in principle be studied without attending lectures due to the wealth of material available (in hardcopy and online). However, in this study we will show that the pass rate of students attending classes regularly (>75% of the classes) is much higher than the pass rate of students attending fewer classes. We use a logistic model to investigate whether this correlation is significant. We will argue why we believe that this correlation between attendance and attainment is causal, i.e. why it is necessary for most students to attend classes in order to (improve their chances to) pass the exam.

Conditions for upscalability of bioclogging in pore network models

In this work, we model the biofilm growth at the microscale using a rectangular pore network model in 2D and a cubic network in 3D. For the 2D network, we study the effects of bioclogging on porosity and permeability when we change parameters like the number of nodes in the network, the network size, and the concentration of nutrients at the inlet. We use a 3D cubic network to study the influence of the number of nodes in the z direction on the biofilm growth and on upscalability. We show that the biofilm can grow uniformly or heterogeneously through the network. Using these results, we determine the conditions for upscalability of bioclogging for rectangular and cubic networks. If there is uniform biofilm growth, there is a unique relation between permeability and porosity, K ∼ ϕ2, this relation does not depend on the volume of the network, therefore the system is upscalable. However, if there is preferential biofilm growth, the porosity-permeability relation is not uniquely defined, hence upscalability is not possible. The Damköhler number is used to determine when upscalability is possible. If the Damköhler number is less than 101, the biofilm grows uniformly and therefore the system is upscalable. However, if the Damköhler number is greater than 103, the biofilm growth exhibits a deviation from uniform biofilm growth and heterogeneous growth is observed, therefore upscalability is not possible. There is a transition from uniform growth to preferential growth if the Damköhler number is between 101 and 103.

Erratum to Laplacian Instability of Planar Streamer Ionization Fronts—An Example of Pulled Front Analysis

Streamer ionization fronts are pulled fronts propagating into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long time attractor out of a continuous family. A transverse stability analysis has to take these features into account. In this paper we introduce a framework for this transverse stability analysis, involving stable and unstable manifolds in a weighted space. Within this framework, a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem is defined and dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front are derived. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with analytical expressions for the dispersion relation in the limit of small and large wave numbers and with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front.