B. J. Bayly

Current-voltage relations for electrochemical thin films

The DC response of an electrochemical thin film, such as the separator in a microbattery, is analyzed by solving the Poisson--Nernst--Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler--Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (cur...

Density variations in weakly compressible flows

Density variations in real fluids are related to both pressure and entropy variations, even in the incompressible limit. The behavior of density variations depends crucially on the relative sizes of the pressure, temperature, and entropy fluctuations. It is shown how this arises from a formal asymptotic expansion of the compressible Navier–Stokes equations about a uniform state. Direct numerical simulations of the full compressible equations verify the consistency of different asymptotic regimes. In the case of turbulent flow, the Kolmogorov–Obukhov theory allows one to predict inertial range scalings for the density fluctuation power spectra in various situations. The results may be relevant to observations of the interstellar medium.

Three-dimensional stability of elliptical vortex columns in external strain flows

The Kirchhoff-Kida family of elliptical vortex columns in flows with uniform strain and rotation displays a rich variety of dynamical behaviours, even in a purely two-dimensional setting. In this paper, we address the stability of these columns with respect to three-dimensional perturbations via the geometrical optics method. In the case when the external strain is equal to zero, the analysis reduces to the stability of a steady elliptical vortex in a rotating frame. When the external strain is non-zero, the stability analysis reduces to the theory of a Schrödinger equation with quasi-periodic potential. We present stability results for a variety of different Kirchhoff-Kida flows. The vortex columns are typically unstable except when the interior vorticity is approximately the negative of the background vorticity, so that the flow in the inertial frame is nearly a potential flow.

Laminar–turbulent transition of wall‐jet flows subjected to blowing and suction

A new family of solutions describing the mean flow of an incompressible two‐dimensional wall jet subjected to boundary suction or blowing was found, in which the vertical velocity at the wall varies in a power law with downstream distance. Linear stability calculations indicate that two unstable modes, an inviscid inflectional point instability and a viscous instability, may coexist in the flow. It is demonstrated theoretically that the relative importance of each mode can be controlled by subjecting the wall jet to small amounts of blowing or suction. Experimental results, in the absence of suction and blowing, support theoretical findings about the coexistence of the two instability modes.

Unsteady dynamo effects at large magnetic reynolds number

Abstract In this paper we show that the adjoint formulation of the Stretch-Fold-Shear (SFS) fast dynamo model has smooth eigenfunctions at zero diffusivity. We thus compute fast dynamo action with smooth fields. We note that the adjoint problem is equivalent to use of a vector potential in reversed flow. We compare this behavior with analogous results for general pulsed-flow models. Some calculations using pulsed Beltrami waves are described.

Heat transfer from a cylinder in a time‐dependent cross flow at low Peclet number

The heat transfer from a constant‐temperature cylinder in a uniform, time‐dependent cross flow at low Peclet number is considered. The time dependence is allowed to be strong, so that the velocity fluctuations may be comparable to, or larger than, the mean flow. The first nontrivial term in the Oseen approximation is calculated using matched‐expansion theory, and its physical significance is discussed. As an illustration, the time‐dependent heat transfer is calculated for a steady cross flow with large sinusoidal perturbations.

Analogy between hyperscale transport and cellular automaton fluid dynamics

It is argued that the dynamics of a very large scale (hyperscale) flow superposed on the stationary small‐scale flow maintained by a force f(x) is analogous to the cellular automaton hydrodynamics on a lattice having the same spatial symmetry as the force f.

Scalar dynamo models

Abstract The equation (δt + u·∇)C = R(x, t)C + k∇ 2 C, is a scalar analogue of the magnetic induction equation. If the velocity field u(x, t) and the ‘stretching’ function R(x, t) are explicitly given, then we have the analogue of the dynamo problem. The scalar problem displays many of the same features as the vector kinematic dynamo problem. The fastest growing modes have growth rates that approach a finite limit as K→0 while the eigenfunctions develop more and more complex structure at smaller and smaller length scales. Some insight is provided by an analysis which finds a lower bound on the growth rate that is asymptotically independent of the diffusivity.

Independent degrees of freedom of dynamical systems

Numerical evidence is presented demonstrating the possibility of obtaining reduced equations of motion for dynamical systems in terms of a small number of variables. Some implications of these results are discussed.

Secondary Instabilities, Coherent Structures and Turbulence

In this paper, we review recent progress on several problems of transition and turbulence. First, we explore the role of secondary instabilities in transition to turbulence in both wall bounded and free shear flows. It is shown how the competition between secondary instabilities and classical inviscid inflectional instabilities is important in determining the evolution of free shear flows. An outline of a general theory of inviscid instability is given. Then, we explore recent ideas on the force-free nature of coherent flow structures in turbulence. The role of viscosity in generating small-scale features of turbulence is discussed for both the Taylor-Green vortex and for two-dimensional turbulence. Finally, we survey recent ideas on the application of renormalization group methods to turbulence transport models. These methods yield fundamental relationships between various types of turbulent flow quantities and should be useful for the development of transport models in complex geometries with complicated physics, like chemical reactions and buoyant heat transfer.