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Monday, July 9, 2018

Vortexes in Marbling


J. K. Benson (investigating paper marbling in the Islamic world at Leiden University) has raised the question of whether vortex shedding appears in marbling.  He found some 16th century marbling patterns at the Harvard Houghton Library which appear to have vortexes next to longer strokes.



"Vortex shedding in Water"  from "Harvard Natural Sciences Lecture Demonstrations" shows vortexes being shed from a cylinder at flow speeds in the range of marbling strokes.

My work has focused on laminar and Oseen flows https://arxiv.org/abs/1702.02106 which successfully model most common marbling techniques.

At the lowest Reynolds numbers is Stokes flow, where the passage of the stylus displaces the liquid only temporarily.  The next range of Reynolds numbers produces Oseen flow, where viscous forces dominate inertial forces.  Straight strokes of finite length result in persistent movement along the stroke and rotation to both sides of the stroke.  As the inertial forces grow relative to viscous forces, instabilities such as vortex shedding appear (Re > 90).  Much higher Reynolds numbers (> 40000) can produce turbulence.

To answer Benson's question and to better quantify the fluid dynamics parameters of marbling, Dan and Regina St.John, the Chena River Marblers, recently hosted a session where we performed experiments using their equipment and expertise.

The idea was to increase the Reynolds number of marbling strokes by increasing the stylus size and speed until instabilities such as vortexes appeared.  We increased the stylus size to 25 mm, but instabilities did not appear.  We increased the speed to the point that it created a tear in the paints, but no vortexes appeared.



Reynolds number being the characteristic length times the velocity divided by the kinematic viscosity, the only other thing to try was reducing the viscosity.  Diluting the sizing by half with water resulted in a sea change.  Instead of fluid motion stopping when the stylus stopped, it would glide for as long as 5 seconds before coming to rest, showing that inertia was in play.  Stylus strokes at speeds around 25.cm/s (which is fast for marbling) created the mushroom shapes pictured.  Although the St.Johns were able to find an example of this shape in one of their books, it is not a common marbling motif.  Looking back at the photo of the 16th century marbling, mushrooms are present.



Are these mushrooms due to flow instabilities?  No.  The mushrooms appear where the stylus was stopped.  In vortex shedding, the vortexes are shed to alternating sides of the ongoing stroke.  Even for a fast stroke, the train behind the stylus was smooth and without wiggles.

We know from the video of vortex shedding that it happens in water. Viscosity near that of water may be required in order to see it in marbling.

There is more of interest here.  The mushrooms in our marbling have smaller mushrooms inside of them.  In the photograph, I have outlined mushrooms at 3 different scales.  The mushroom in the smallest box ls less obvious than the others; perhaps because the bands of color comprising it are larger relative to its size.

Pure Oseen flow is reversible; reversing the flow at the origin returns the system to its original state.  With its sub-mushrooms, the mushroom flow does not look reversible.  Could this mushroom flow be a regime which transfers energy from larger to smaller scales, yet doesn't exhibit instability?

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