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Wednesday, February 8, 2017

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.

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