Cats2D Multiphysics > Research Topics > Computing mass transport in liquids

*All unpublished results shown here are *Copyright ©
2016–2019 Andrew Yeckel, all rights reserved

## Computing mass transport in liquids

I have collated some entries from my developments page into this article that discusses how accurate pathline integration can be used to study mass transport in liquids where small diffusion coefficients tend to make the problem very stiff.

### Comparing the convective-diffusion model to the particle convection model

I'm intrigued by the possibility of using pathline integration to study strongly convected mass transport. The mixing studies I presented elsewhere correspond to a limiting case of mass Peclet number equal to infinity. In the absence of molecular diffusion there is no mass transport between neighboring points, so each point traced by the pathline integration maintains its original concentration. There is plenty of mixing so that points of low and high concentration are brought close together, but this mixing does not extend to length scales smaller than the thickness of the striated layers formed by convective transport along the streamlines of the flow. Furthermore, points within closed recirculations do not mix with points outside.

Unless these points are actual particles with properties
distinct from the liquid, molecular diffusion will eventually
cause these layers to mix. Yet convective flows often yield
Peclet numbers greater than 10^{5} for transport of
ordinary molecules in liquid, and 10^{8} or larger for
macromolecules. We expect the limiting behavior of pathline
transport without diffusion to accurately describe mixing over
some initial period of time in such systems. Continuum mass
transport at large Peclet numbers results both in stiff time
integration requiring small time steps, and sharply defined
spatial oscillations requiring a fine mesh. As these
requirements can become prohibitive, it would be useful to
know under what circumstances it is feasible to apply pathline
integration in lieu of continuum modeling.

To study this issue I compare mixing computed by pathline integration to continuum mass transport computed from a convective-diffusion model. Mixing starts from a linear concentration distribution shown on the left below. The model flow, shown on the right, is a steady state Stokes flow driven by motion of the upper and lower surfaces in opposite directions. This flow features two co-rotating recirculations enclosed by a saddle streamline in the form of a figure eight or hourglass.

After 12.6 dimensionless time units, the continuum model with
Peclet = 10^{5} shown below on the left strongly
resembles pathline integration shown on the right. I've seeded
paths at 291,384 node positions, compared to 56,481
concentration unknowns in the continuum model. Diffusion has
reduced the difference between global maximum and minimum
concentration by 5% compared to the starting concentration
field. The pathline integration never reduces this measure
because each point retains its initial concentration, and
mixing only occurs at larger length scales by the formation of
ever thinner spiral layers.