The drag of a body moving transversely in a confined stratified fluid

The slow motion of a body through a stratified fluid bounded laterally by insulating walls is studied for both large and small Peclet number. The Taylor column and its associated boundary and shear layers are very different from the analogous problem in a rotating fluid. In particular, the large Peclet number problem is non-linear and exhibits mixing of statically unstable fluid layers, and hence the drag is order one; whereas the small Peclet number flow is everywhere stable, and the drag is of the order of the Peclet number

Multiscale Mixing Efficiencies for Steady Sources

Multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales. We consider scalars maintained by temporally steady but spatially inhomogeneous sources, stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence. The mixing efficiencies are rigorously bounded in terms of the Peclet number and specific quantitative features of the source. Scaling exponents for the bounds at high Peclet number depend on the spectrum of length scales in the source, indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories.Comment: 4 pages, 1 figure. RevTex4 format with psfrag macros. Final versio

Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

Enhanced tracer transport by the spiral defect chaos state of a convecting fluid

To understand how spatiotemporal chaos may modify material transport, we use direct numerical simulations of the three-dimensional Boussinesq equations and of an advection-diffusion equation to study the transport of a passive tracer by the spiral defect chaos state of a convecting fluid. The simulations show that the transport is diffusive and is enhanced by the spatiotemporal chaos. The enhancement in tracer diffusivity follows two regimes. For large Peclet numbers (that is, small molecular diffusivities of the tracer), we find that the enhancement is proportional to the Peclet number. For small Peclet numbers, the enhancement is proportional to the square root of the Peclet number. We explain the presence of these two regimes in terms of how the local transport depends on the local wave numbers of the convection rolls. For large Peclet numbers, we further find that defects cause the tracer diffusivity to be enhanced locally in the direction orthogonal to the local wave vector but suppressed in the direction of the local wave vector.Comment: 11 pages, 12 figure

Stationary convection-diffusion between two co-axial cylinders

In this note, we examine the high Peclet number limit of the stationary extended Graetz problem for which two families of real and imaginary eigenvalues are associated, respectively, with a downstream convective relaxation and the upstream diffusive establishment. The asymptotic behavior of both families of eigenvalues is studied, in the limit of large Peclet number and thin wall, which bring to the fore a single parameter dependence, previously mentioned in the literature from numerical investigations [M.A. Cotton, J.D. Jackson, in: R.W. Lewis, K. Morgan (Eds.), Numerical Methods in Thermal Problems, vol. IV, Pineridge Press, Swansea, 1985, pp. 504–515]. The fully developed region is specifically studied thanks to the first eigenvalue dependence on the Peclet number, on the thermal conductivity coefficients and on the diameter ratio of the cylinders. The effective transport between the fluid and the solid is investigated through the evaluation of the fully developed Nusselt number and experimental measurements

Diffusion effects in rotating rotary kilns

A novel approach to the modeling of mass transfer in rotary kilns has been described (Heydenrych et al, 2001). It considers the mass transfer to occur by the inclusion of gas in the interparticle voids in between the particles that move concentrically with the kiln. By doing so, the rate of mass transfer was found to be dependent on bed fill and the ratio of reaction rate constant to angular velocity (k/ ). The model was found to be valid at slow to medium fast reactions. For fast reactions it under-predicted mass transfer. Therefore in this paper, the model will be extended to include diffusion effects. An additional dimensionless number is necessary then to describe the system. This can either be a Peclet number (R2/De) or a Thiele modulus (kR2/De)1/2. The solution of the 2-dimensional partial differential equations that describe the extended model gives a handle on the effect of scale-up in rotary kilns. For industrial-scale kilns, the Peclet number is large, which means that diffusion within the lower (passive) layer of the bed is unimportant for slower rates. With high reaction rates, iso-concentration lines are closely stacked near the surface of the bed, implying that it is important to model the active layer rather than the bed as a whole in these circumstances. However, the stiff differential equations are not easily solved then, and other methods of solution are advisable

Separation of suspended particles in microfluidic systems by directional-locking in periodic fields

We investigate the transport and separation of overdamped particles under the action of a uniform external force in a two-dimensional periodic energy landscape. Exact results are obtained for the deterministic transport in a square lattice of parabolic, repulsive centers that correspond to a piecewise-continuous linear-force model. The trajectories are periodic and commensurate with the obstacle lattice and exhibit phase-locking behavior in that the particle moves at the same average migration angle for a range of orientation of the external force. The migration angle as a function of the orientation of the external force has a Devil's staircase structure. The first transition in the migration angle was analyzed in terms of a Poincare map, showing that it corresponds to a tangent bifurcation. Numerical results show that the limiting behavior for impenetrable obstacles is equivalent to the high Peclet number limit in the case of transport of particles in a periodic pattern of solid obstacles. Finally, we show how separation occurs in these systems depending on the properties of the particles