The Physics of Marbling

Last week I spoke on The Physics of Marbling at the Form in art, toys and games workshop at the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge!

I am the one in the bright blue jacket.

The four day workshop had many fascinating presentations on a wide variety of topics.  Videos for most of the talks are available.

Mixed Convection from an Isothermal Rough Plate

In March 2017 the roughness of the aluminum plate was reduced from 3mm to 1mm.  I installed it in the wind-tunnel and started running experiments.  The forced convection measurements were nearly 30% higher than expected!

I examined nearly every aspect of the physical device and its mathematical model.  The fan-speed calibration was found to be sensitive to the distance between the test surface and the wind-tunnel wall.  Conditioning the rpm-to-speed conversion on the plate's orientation improved the earlier data taken with the plate with 3mm roughness.

When the plate is not parallel to the wind-tunnel, the forced and mixed measurements are affected.  With the 3mm roughness plate, the alignment had been controlled within a couple of millimeters over the plate's 305mm length.  The 1mm roughness plate seemed to require stricter tolerances.  Using a caliper, I am able to control the alignment to better than 1mm.

The primary cause of the measured excess was that, when the height of the posts had been reduced, the size and spacing of the posts had not been reduced.  At high wind-speeds the convection from the flat post tops was exceeding the "fully-rough" mode of convection.  The model incorporating this phenomena is developed in the "Rough to Smooth Turbulence Transition" section of my "Mixed Convection from an Isothermal Rough Plate" paper.

Writing a paper forces one to revisit all the questions and anomalies that occurred during research and experiment.  Understanding, resolving, and testing all of these issues has taken months.  I would appreciate any proof-reading or critiquing that others might provide before I submit it for publication.

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.