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.

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.

"Mathematical Marbling" to appear in IEEE Computer Graphics and Applications

A paper on marbling, which I wrote with Shufang Lu, Xiaogang Jin, Hanli Zhao, and Xiaoyang Mao has been accepted for publication:

Lu, S.; Jaffer, A.; Jin, X.; Zhao, H.; Mao, X.; ,
"Mathematical Marbling,"
Computer Graphics and Applications, IEEE , vol.PP, no.99, pp.1, 0
doi: 10.1109/MCG.2011.51
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5887299&isnumber=5185484

Marbling the Torus

I have cleaned up the math on Marbling the Torus and changed my scripts to oversample the images, resulting in less graininess.

Modeling Ink Drops

Added to http://people.csail.mit.edu/jaffer/Marbling/Dropping-Ink.

... given a point P and a new ink drop of radius r centered at C, move the point radially from C to:

C + (P − C) · sqrt

(

1 +

r2 ||P − C||2

)

One correspondent complains that, because the divergence of this transform is not zero, it can't be incompressible.

Divergence is defined for a continuously differentiable vector field. But this transform is not continuously differentiable around C; thus its divergence isn't well-defined. The common definition of incompressible is in terms of the divergence of a vector field. For a vector-field where the divergence isn't well-defined, the definition is silent.

However, I can show that this transform preserves the area of all neighborhoods not containing C. Consider the annulus centered on C having inner radius sqrt(a/π) and outer radius sqrt((a+b)/π). Its area is b.

If ink is injected at C forming a new circular region centered on C having area e, the annulus having area b will expand to have an inner radius of sqrt((a+e)/π) and an outer radius of sqrt((a+b+e)/π). The area of the expanded annulus is still b. Note that the expanded annulus is thinner than the original.

Because this transform is radially symmetric, any annulus slice with sides forming angles with C of θ and η will map to the expanded annulus slice between the angles of θ and η. Because the annulus and slices can be made arbitrarily small, all neighborhoods not containing C map to regions having the same area.