Oseen Flow in Ink Marbling

Mathematical modeling of ink marbling has long been a fascination of mine.  My Ink Marbling web pages have presented emulations of a number of marbling techniques.  But the raking techniques modeled were either paths across the whole tank or circular paths.

Pictorial ink marbling designs are created using short strokes, where a stylus is inserted into the tank; moved a short distance; then extracted.  There seems to be no way to adapt the line or circle draws to short strokes with endpoints.

Having bought a copy of Boundary-layer theory (Hermann Schlichting et. al.) for my convection project, I started reading from the beginning.  It didn't take long until I found a description of Oseen flow on page 115 (chapter IV, very slow motion).  Its streamline figure looked very promising.  After further research I have written: Oseen Flow in Ink Marbling arXiv:1702.02106 [physics.flu-dyn].

Unlike a cylinder in a 2-dimensional flow, the velocity field induced by an infinitesimally thin stylus can be exactly solved in closed form.  In this sense, marbling is the purest form of Oseen flow.

The partial differential equations solved include conservation of mass (divergence=0), but not conservation of momentum (Navier-Stokes).  It's not clear how much momentum is imparted by the stylus, or how that imparting momentum changes with time.

Liquid marbling is sensitive to the speed of a stylus moving through the tank.  At low speeds the induced flow is laminar; at high speeds the flow becomes turbulent.  Both are used by marblers, but only the laminar flow is possible to solve in closed form.

A Convection Surprise

Years ago I heard advice to experimental physicists to continue refining their experiments, even after they yield a hoped-for result.

Last week I realized that if I switched the wind-tunnel fan to the lower speed setting, it might be able to run at speeds lower than 100r/min.  I tried it and it worked!  This updated plot has points at Re values less than 10000 and they show significantly less convection than expected.  As in an earlier post, there would be many possible explanations for finding too much convection, but not for finding too little.

At these low wind speeds the fan speed wanders through a +/-10% range and the convection measurements are averaged over time.  I have rewritten my program to compute the convection over smaller intervals than the whole trial and included both the whole trial and smaller runs in the graph.  My program also averages measurements through a trapezoidal window, which reduces the variation between the smaller intervals within an experimental trial.

Here is a link to a pdf of plots splitting the measurements in 2 through 8 pieces.  Each trial's dots are distributed vertically, reflecting the noisy temperature readings.

The four lowest trials cluster near the asymptote for turbulent convection from a smooth surface, even though the rough surface asymptote would convect more heat.  In the "Measured over time split into 4 intervals" chart it is seen that the plate spends most of its time over the range between the two asymptotes.

If the switching between asymptotes were correlated with fan speed variations, we would expect to see the dots for each trial on a slanted line; but they are roughly vertical.  The transition between laminar and turbulent forced convective modes remains an unsolved problem for theory.  Modelling the transition between these two turbulent forced convective modes may be as intractable.