Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II

Engineering hybrid nanotube wires for high-power biofuel cells

Poor electron transfer and slow mass transport of substrates are significant rate-limiting steps in electrochemical systems. It is especially true in biological media, in which the concentrations and diffusion coeffi cients of substrates are low, hindering the development of power systems for miniaturized biomedical devices. In this study, we show that the newly engineered porous microwires comprised of assembled and oriented carbon nanotubes (CNTs) overcome the limitations of small dimensions and large specific surface area. Their improved performances are shown by comparing the electroreduction of oxygen to water in saline buffer on carbon and CNT fi bres. Under air, and after several hours of operation, we show that CNT microwires exhibit more than tenfold higher performances than conventional carbon fi bres. Consequently, under physiological conditions, the maximum power density of a miniature membraneless glucose / oxygen CNT biofuel cell exceeds by far the power density obtained for the current state of art carbon fi bre biofuel cells