ALBERT

Notes: The steady distributions of temperature and concentration in a reactive fluid adjacent to a catalytic wall are investigated. It is shown that for the mass action kinetics considered, the number of steady solutions is either one or three. Several simple sufficient criteria, as well as a slightly more complicated necessary and sufficient criterion, are given for the uniqueness of the steady solution. If the Lewis number Le (ratio of molecular and thermal diffusivities) is at least one, then a steady state that is unique is always stable with respect to small disturbances. For Le≥1, the high and low temperature steady states are stable when the number of steady states is three. When Le〈1, even a unique steady state can be unstable, and when there are three steady states, one, two, or three of them can be unstable. The Le≥1 and Le〈1 cases are further distinguished by the fact that oscillatory instability can occur for Le〈1 but not for Le≥1.

Notes: The onset of thermal convection due to heating from below in a system consisting of a fluid layer overlying a porous layer with anisotropic permeability and thermal diffusivity is studied. Flow in the porous medium is assumed to be governed by Darcy's law; the Beavers–Joseph condition is applied at the interface between the two layers. The linear perturbation equations are solved numerically. It is found that the effects of anisotropy on the onset of thermal convection are most profound for small values of the depth ratio ζ (ratio of fluid layer thickness to porous layer thickness), since in that case, the onset of convection corresponds to significant motion in both layers. For fixed values of the vertical permeability in the porous medium, decreasing the value of ξ (ratio of horizontal to vertical permeability) leads to stabilization of the superposed layer configuration because of increased resistance to motion in the porous medium. For larger values of ζ, the onset of motion is increasingly confined to the fluid layer, with the transport of heat through the porous layer occurring primarily by conduction. Accordingly, the influence of ξ on the stability characteristics for larger ζ is less significant than the effects of an anisotropic thermal conductivity.

Notes: Several techniques have been proposed for determining two- or three-dimensional velocity fields from measurements of one passive scalar. It is shown that measurements of one scalar and knowledge of the equation governing its transport determine a velocity field, only up to an additive vector field locally perpendicular to the gradient of the scalar field but otherwise arbitrary. Three previously proposed procedures for selecting a unique velocity field from among the uncountable infinity consistent with the scalar transport data and equation are then discussed, and it is shown that a recent "iterative inversion'' procedure for "solution'' of a singular linear equation system (obtained using only measurements of one scalar and the equation governing its transport) cannot converge as claimed. A method for determining the correct n-dimensional (n=2 or 3) divergence-free velocity field from measurements of n−1 passive or reactive scalars is then developed. Finally, it is shown how the velocity field in an n-dimensional compressible flow can be determined from measurements of density and n−1 passive or reactive scalars. © 1995 American Institute of Physics.

Notes: Based on a closed surface of triangles fitted to atomic coordinates determined crystallographically, Brune and Kim [Proc. Natl. Acad. Sci. USA 90, 3835–3839 (1993)] proposed a boundary-element Stokes-flow technique for ab initio computation of a translational diffusion coefficient and the rotational diffusion tensor Dr of globular proteins. They applied their approach to atomic coordinates for a tetragonal structure of hen egg-white lysozyme, and reported that computed values of a translational diffusion coefficient and Dr=tr(Dr)/3 agreed well with experiment. After establishing the identity between the infinite-dilution tracer diffusion coefficient of the protein macroion (D+ for lysozyme cation) and the "translational diffusion coefficient" computed by Brune and Kim, we adopt a somewhat different computational approach and show how convergence of D+ and Dr for tetragonal lysozyme depends on two computational parameters characterizing the fidelity of the geometric approximation to the protein surface and two others characterizing the accuracy of the Stokes-flow computations. We then compute D+ and Dr for lysozyme using atomic coordinates for the triclinic crystal structure, three structures determined by nuclear magnetic resonance spectroscopy in the liquid phase (presumably corresponding more closely to in vivo structures), the solvated tetragonal structure (with 108 water molecules) considered by Brune and Kim, and a "dry" version of the same structure. These computations show that D+ and Dr computed for all of the dry crystal structures are in excellent agreement with those for the liquid-phase conformations. Values of D+ and Dr computed for the solvated structure are lower, consistent with the larger volume and area of the corresponding polyhedral surface. We also show that several choices of the origin of the force system [discussed by Brenner, J. Colloid Interface Sci. 23, 407–436 (1967)] give rise to nearly identical translational diffusion coefficients. Finally, we show how to estimate the thickness of the "solvation shell" contributing to the hydrodynamic resistance of the protein cation, and use the binary Nernst–Hartley equation to then estimate the effective cation charge at the two pH values at which the binary diffusion coefficient has been accurately measured in recent interferometric experiments. © 2002 American Institute of Physics.

Notes: The onset of convection in a triply diffusive, incompressible motionless Newtonian fluid layer of infinite horizontal extent bounded by rigid parallel walls is studied by means of a linear stability analysis. The qualitative features of most of the results carry over from the case of stress-free boundaries studied earlier by Pearlstein, Harris, and Terrones [J. Fluid Mech. 202, 443 (1989)]. One striking feature that does not carry over from the stress-free case, however, is the possibility of quasiperiodic bifurcation from the steady motionless state via two pairs of complex conjugate temporal eigenvalues with incommensurable imaginary parts crossing into the right half-plane at the same combination of Rayleigh numbers. The impossibility of this type of quasiperiodic bifurcation from the rest state in the rigid case is discussed in terms of the topology of the disconnected neutral curves. The numerical method employed is suitable for the computation of disconnected oscillatory neutral curves in other stability problems for which a low-order exact dispersion relation does not exist.

Notes: The effect of the viscosity–temperature relation, μ(T), on the onset of convection in a horizontal fluid layer is discussed. Linear analysis shows that, for aqueous glycerol solutions, the critical Rayleigh numbers (Racrit) obtained using the Arrhenius approximation, μA(T), are in excellent agreement with those employing the actual μ(T) data. The results for the exponential approximation μe(T) differ to an extent that depends on the glycerol mass fraction and the temperature difference (ΔT) between the plates. The error associated with use of μe(T), while small, is of the same order as, or larger than, the uncertainty in careful experiments. For the broad class of liquids for which μA(T) is an excellent approximation to μ, we have assessed the errors in Racrit associated with the widely used μe(T). As ΔT and the viscosity contrast increase, the values of Racrit deviate increasingly from those predicted using μA(T). Also, the relative error in Racrit is much smaller than the maximum relative error in the coefficients [involving μ(T) and its derivatives with respect to T] of the linear disturbance equations.

Notes: Two extensions of the classical proper orthogonal decomposition (POD) methodology exposited by Lumley for turbulent flows are presented. First, the extended POD (EPOD) can deal with flows lacking statistical stationarity by summing the velocity correlations of an ensemble of time series. Thus it reduces to the classical POD for a statistically stationary flow, and to the "snapshot'' of Sirovich for data from a single instant in each realization of an ensemble. Second, the EPOD can deal with flows having one or more length scales exceeding the spatial range over which simultaneous measurements can be made, by generating composite modes using nonsimultaneous, spatially overlapping data. Both new capabilities have been exercised by applying the EPOD to the streamwise velocity component measured by a cross-stream hot-wire rake in an anharmonically forced turbulent plane mixing layer that is not statistically stationary, and is statistically homogeneous in only one direction. It is discovered that as few as three one-dimensional modes are capable of capturing as much as 78% of the "action'' in data from ten wires in a cross-stream rake.