90 research outputs found
Poisson-Nernst-Planck Systems for Narrow Tubular-like Membrane Channels
We study global dynamics of the Poisson-Nernst-Planck (PNP) system for flows
of two types of ions through a narrow tubular-like membrane channel. As the
radius of the cross-section of the three-dimensional tubular-like membrane
channel approaches zero, a one-dimensional limiting PNP system is derived. This
one-dimensional limiting system differs from previous studied one-dimensional
PNP systems in that it encodes the defining geometry of the three-dimensional
membrane channel. To justify this limiting process, we show that the global
attractors of the three-dimensional PNP systems are upper semi-continuous to
that of the limiting PNP system. We then examine the dynamics of the
one-dimensional limiting PNP system. For large Debye number, the steady-state
of the one-dimensional limiting PNP system is completed analyzed using the
geometric singular perturbation theory. For a special case, an entropy-type
Lyapunov functional is constructed to show the global, asymptotic stability of
the steady-state
A comparison of no-slip, stress-free and inviscid models of rapidly rotating fluid in a spherical shell
We investigate how the choice of either no-slip or stress-free boundary conditions affects numerical models of rapidly rotating flow in Earth's core by computing solutions of the weakly-viscous magnetostrophic equations within a spherical shell, driven by a prescribed body force. For non-axisymmetric solutions, we show that models with either choice of boundary condition have thin boundary layers of depth E^(1/2), where E is the Ekman number, and a free-stream flow that converges to the formally inviscid solution. At Earth-like values of viscosity, the boundary layer thickness is approximately 1m, for either choice of condition. In contrast, the axisymmetric flows depend crucially on the choice of boundary condition, in both their structure and magnitude (either E^(-1/2) or E^(-1)). These very large zonal flows arise from requiring viscosity to balance residual axisymmetric torques. We demonstrate that switching the mechanical boundary conditions can cause a distinct change of structure of the flow, including a sign-change close to the equator, even at asymptotically low viscosity. Thus implementation of stress-free boundary conditions, compared with no-slip conditions, may yield qualitatively different dynamics in weakly-viscous magnetostrophic models of Earth's core. We further show that convergence of the free-stream flow to its asymptotic structure requires E ≤10^(-5)
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