Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

We propose a two fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier-Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker-Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

On discretization in time in simulations of particulate flows

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases

hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated

Modeling Inner Proton Belt Variability at Energies 1 to 10 MeV Using BAS-PRO

Abstract: Geomagnetically trapped protons forming Earth's proton radiation belt pose a hazard to orbiting spacecraft. In particular, solar cell degradation is caused by non‐ionising collisions with protons at energies of several megaelectron volts (MeV), which can shorten mission lifespan. Dynamic enhancements in trapped proton flux following solar energetic particle events have been observed to last several months, and there is a strong need for physics‐based modeling to predict the impact on spacecraft. However, modeling proton belt variability at this energy is challenging because radial diffusion coefficients are not well constrained. We address this by using the British Antarctic Survey proton belt model BAS‐PRO to perform 3D simulations of the proton belt in the region 1.15 ≤ L ≤ 2 from 2014 to 2018. The model is driven by measurements from the Radiation Belt Storm Probes Ion Composition Experiment and Magnetic Electron Ion Spectrometer instruments carried by the Van Allen Probe satellites. To investigate sensitivity, simulations are repeated for three different sets of proton radial diffusion coefficients D LL taken from previous literature. Comparing the time evolution of each result, we find that solar cycle variability can drive up to a ∼75% increase in 7.5 MeV flux at L = 1.3 over four years due to the increased importance of collisional loss at low energies. We also show how the anisotropy of proton pitch angle distributions varies with L and energy, depending on D LL . However we find that phase space density can vary by three orders of magnitude at L = 1.4 and μ = 20 MeV/G due to uncertainty in D LL , highlighting the need to better constrain proton D LL at low energy

Modeling Field Line Curvature Scattering Loss of 1–10 MeV Protons During Geomagnetic Storms

The proton radiation belt contains high fluxes of adiabatically trapped protons varying in energy from ∼one to hundreds of megaelectron volts (MeV). At large radial distances, magnetospheric field lines become stretched on the nightside of Earth and exhibit a small radius of curvature RC near the equator. This leads protons to undergo field line curvature (FLC) scattering, whereby changes to the first adiabatic invariant accumulate as field strength becomes nonuniform across a gyroorbit. The outer boundary of the proton belt at a given energy corresponds to the range of magnetic L shell over which this transition to nonadiabatic motion takes place, and is sensitive to the occurrence of geomagnetic storms. In this work, we first find expressions for nightside equatorial RC and field strength Be as functions of Dst and L* to fit the TS04 field model. We then apply the Tu et al. (2014, https://doi.org/10.1002/2014ja019864) condition for nonadiabatic onset to solve the outer boundary L*, and refine our expression for RC to achieve agreement with Van Allen Probes observations of 1–50 MeV proton flux over the 2014–2018 era. Finally, we implement this nonadiabatic onset condition into the British Antarctic Survey proton belt model (BAS-PRO) to solve the temporal evolution of proton fluxes at L ≤ 4. Compared with observations, BAS-PRO reproduces storm losses due to FLC scattering, but there is a discrepancy in mid-2017 that suggests a ∼5 MeV proton source not accounted for. Our work sheds light on outer zone proton belt variability at 1–10 MeV and demonstrates a useful tool for real-time forecasting