The Lamb-Oseen Vortex and Paint Marbling

Just published The Lamb-Oseen Vortex and Paint Marbling on arXiv.

The image to the left shows the decay with time of the Lamb-Oseen vortex (starting from an impulse of circulation at the center point).

The image to the right shows the same vortex, but with exponentially increasing time.  The rotational shear propagates to larger and larger orbits while the center returns to rest.  While this animation returns to its original position, it could come to rest at any angle controlled by the magnitude of the initial circulation.

More about mathematical marbling.

Vortex marbling in Jupiter's great spot

Image from NASA's Juno spacecraft 2018-04-01

Although there are structures resembling mushrooms, the sides of the caps are vortexes, not the smaller mushrooms seen in my previous post.  There appears to be a vortex street running across the bottom half of the image.

Bubbles in Marbling

Bubbles and parted paints

The St.Johns marbling technique uses stylus speeds of only a few centimeters per second. When a stylus is moved quickly through the tank fluid, air bubbles are formed. Bubbles were formed drawing a 25 mm diameter dowel faster than (V=) 20 cm/s through the tank (which also parted the floating paints) in my previous post.

While the formation of bubbles could be due to non-Newtonian fluid behavior, it is worth examining the conditions assuming a Newtonian fluid. A 25 mm diameter dowel submerged 12.5 mm will behave more like half of a 25 mm diameter sphere than a cylinder. If the kinematic viscosity (ν=0.001 m²/s) of the liquid is 1000 times that of water, then the Reynolds number is about 5, far less than the 90 needed to spawn vortexes. Re is inversely proportional to viscosity; reducing the kinematic viscosity by a factor of 10 raises Re to 50.

A half sphere of diameter d has buoyant-pressure (restoring force divided by cross-section area) of about 81 N/m². Surface tension pressure (restoring force divided by cross-section area) of water is roughly 3.7 N/m².

Drag is the force on the object moving through the tank fluid. There must be an equal and opposite net force on the liquid. Drag D for a sphere is the product of the friction coefficient C_D, frontal area (π/4*d²), and dynamic head V²*ρ/2 (for water ρ=997). That force divided by the frontal area of the object is a pressure (suction actually).

A bubble will be formed if this suction behind the moving stylus is larger than the sum of the restoring forces at the liquid surface.

For ν=1000 mm²/s (1000 times that of water) the suction behind our 25 mm diameter dowel is 88 N/m², which exceeds the restoring pressures 81 N/m² and 3.7 N/m², and bubbles can result.

The slower motions and smaller styluses that the St.Johns usually use have Reynolds numbers much smaller than 5. Thus the marblings they create don't evidence inertial effects (versus the mushroom designs of my previous post).

"Mushroom" flow from straight strokes

Kinematic viscosities below 50 mm²/s (50 times that of water) would be needed to spawn vortexes in marbling.  At 50 mm²/s viscosity, a 25 mm cylinder would need to be moved at 20 cm/s over at least 16 cm before the first vortex was shed. 

In water, a 5 mm cylinder moving at 2 cm/s would shed vortexes 3 cm apart.   A 1 mm diameter stylus moved in a straight path at 5 cm/s would not shed vortexes.

So existing evidence of Karman (shed) vortexes is only likely to be found in marbling produced on a tank filled with water.

Vortexes in Marbling

Jake Benson (investigating paper marbling in the Islamic world at Leiden University) has raised the issue 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.

This image from the on-line Getty collection is busier, but shows some of the same features.

Autograph album of Johann Joachim Prack von Asch
Publication date 1587
In the collection of the Getty Research Institute
https://archive.org/stream/gri_33125012902959#page/n151

"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 in Newtonian fluids 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 vortex 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 and bubbles in the paints, but no vortexes appeared.  The tear indicates that the assumption that the fluid is Newtonian may not be valid; and the properties of carageenan used to make the "sizing" in the tank are complicated.  Note: my later post finds that Newtonian fluids can produce these behaviors.

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.

Fractal Scaling of Population Counts Over Time Spans

It's been 7 months since my last post.  The process of downsizing to a smaller home last winter put my projects on hold.

Although fractals have stubbornly refused to appear in my investigation of self-similar surface roughness, they have shown up at my day job as a data scientist at Digilant.

Investigating the possibility of combining weekly counts of unique user IDs, I discovered that the L^p-norm does so with surprisingly good accuracy on digital advertising datasets.  The L^p-norm implies a scaling law.  My son Martin (who also works at Digilant) noticed that the scaling law exponent is a fractal dimension.  The L^p-norm and scaling law are implied by the Pareto distribution of lifetimes in a population.  This link between the L^p-norm and fractal dimension should have application beyond counting populations.

We wrote a paper about these results at https://arxiv.org/abs/1806.06772