Cats2D Multiphysics > Research Topics > Computing mass transport in liquids

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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 10⁵ for transport of ordinary molecules in liquid, and 10⁸ 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 = 10⁵ 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.

 

The next results show 31.4 time units of mixing. The maximum concentration difference has fallen by 10% in the continuum model. The spiral layers in concentration are somewhat less distinct in the continuum model. This is roughly the point at which I judge the models to start deviating significantly from each other.

The difference in models is much starker after 62.8 time units, shown next. The pathline integration exhibits a large number of spiral layers, but the continuum model shows only a small hint of them. The maximum concentration has fallen by 30%.

The dimensionless time is based on a characteristic time t* = L_o/U_o, where L_o is the cavity width and U_o is the velocity of the moving boundary. This is the transit time of the boundary, the time taken to move fluid from one side of the cavity to the other. In one of these transit times, fluid will move a quarter of a trip around the cavity, forming a quarter of a spiral layer of high and low concentration fluid. The number of spiral layers of high and low concentration created in time t is N = t/2t*. The rainbow color map used here makes it difficult to discern the actual number of layers, however, because the map introduces spurious color bands, mainly green and yellow, that do not represent maxima or mimima (ridges or valleys) in the concentration map. The situation is clearer when the warm-cold color scale is used, shown here after 12.6 and 31.4 time units of mixing.