Posts tagged with fluid mechanics

Just playing with z² / z² + 2z + 2

$g(z)=\frac{z^2}{z^2+2z+2}$

on WolframAlpha. That’s Wikipedia’s example of a function with two poles (= two singularities = two infinities). Notice how “boring” line-only pictures are compared to the the 3-D ℂ→>ℝ picture of the mapping (the one with the poles=holes). That’s why mathematicians say ℂ uncovers more of “what’s really going on”.

As opposed to normal differentiability, ℂ-differentiability of a function implies:

• infinite descent into derivatives is possible (no chain of C¹ ⊂ C² ⊂ C³ ... Cω like usual)

• nice Green’s-theorem type shortcuts make many, many ways of doing something equivalent. (So you can take a complicated real-world situation and validly do easy computations to understand it, because a squibbledy path computes the same as a straight path.)


Pretty interesting to just change things around and see how the parts work.

• The roots of the denominator are 1+i and 1−i (of course the conjugate of a root is always a root since i and −i are indistinguishable)
• you can see how the denominator twists
• a fraction in ℂ space maps lines to circles, because lines and circles are turned inside out (they are just flips of each other: see also projective geometry)
• if you change the z^2/ to a z/ or a 1/ you can see that.
• then the Wikipedia picture shows the poles (infinities)

Complex ℂ→ℂ maps can be split into four parts: the input “real”⊎”imaginary”, and the output “real"⊎"imaginary”. Of course splitting them up like that hides the holistic truth of what’s going on, which comes from the perspective of a “twisted” plane where the elements z are mod z • exp(i • arg z).

ℂ→ℂ mappings mess with my head…and I like it.

## Ocean Currents

Just some pictures of ocean currents.

I love looking at oceanographic maps because the water-covered parts of the globe are so often de-emphasised—I guess because most of us humans live on land—even though most of the interesting things happening on Earth involve water.

Oceanographic maps are like looking at the world inside-out—perhaps as it should be seen, not from our perspective but from the majority perspective.

Maths concepts:

A great lecture on this topic was given by Stephen C Stearns and recorded & displayed by the people at AcademicEarth.org.

West coasts are really different the world over because, duh, the Earth is spinning. (Eastward.) That throws stuff up on the shores of the west (e.g. Nitrogen on one of the Chilean coasts which ends up becoming the world’s major source of guano) You also get a different clockwise/anticlockwise circulation of the ocean currents in the Northern and Southern hemispheres because that eastward force pushes differently below -vs- above the equator. Pretty simple logic and it makes things be the way they are.

If you want to see one of the key differences between Eastern and Western thought, look at the classical art of the two cultures. The West is deeply concerned with static form … the naturalistic reproduction of shape. The Eastern tradition … is like the apotheosis of a sketcher’s technique…. It is, in short, an art that embraces change….

It’s not easy to paint something that is never still. Yet [a] traditional Chinese artist .. capture[s] the fundamental forms of motion. This is nowhere more clear than in the ways in which these two traditions … depict … flowing water….

We like to think that the calculus of Newton and Leibniz gave us a tool to handle the science of change; but for a problem like turbulence, calculus … provides a formalism … we can stare at this equation and realize that we can’t solve it, and in the end we are forced to go back, like … Jean Leray … and gaze instead at the real thing…[the Seine].

Like so many other pattern-forming processes, convection is a non-equilibrium phenomenon…. heat flow need not itself involve motion of the bulk fluid: if the temperature differences are only slight or gradual, heat can be redistributed by conduction….

Convection is … brought about by the fact that a warmer fluid is generally less dense than a cooler one…. One can watch convection currents carry dust aloft above radiators in a heated room—the dust traces out the otherwise invisible motions of the air….

For a heating rate just sufficient to start convection the [Bénard] cells are generally sausage-like [oblong square] rolls…. Neighbouring roll cells circulate in opposite directions…Clearly, the symmetry of the fluid is broken…

As in the case of Turing patterns or viscous fingering, a particular pattern with a particular size has been selected; yet, a moment before its appearance, there was nothing in the system to give any clue of its imminent arrival or its scale.

In 1916 Lord Rayleigh tried to understand what triggered the sudden appearance of this convection pattern. It does not arise as soon as there is a [temperature] gradient…. Rather, a certain threshold in temperature difference has to be reached…and this threshold depends on the composition and the depth of the fluid….

The Rayleigh number is basically a measure of the balance between the forces that promote convection (the buoyancy of the fluid…) and those that oppose it (the frictional forces that arise from the fluid’s viscosity, and the thermal diffusivity…). … Only when the driving force (the temperature gradient) becomes big enough to overcoe …. resistance do the convection cells appear. … one can map out the generic behaviour of convecting fluids as a function of Rayleigh number, without having to worry about whether the fluid is water, oil or glycerine.. For what it is worth, the critical Rayleigh number for the start of convection is 1708.

## Boundaries and Flows

Gauß’ divergence theorem states that, unless matter is created or destroyed, the density within a region of space V can change only by flowing through its boundary ∂V. Therefore

$\large \dpi{150} \bg_white \int_{\partial V} \mathbf{flow}\ d \, \text{surface area} \quad = \quad \int_V \nabla \mathbf{flow}\ d \, \text{volume}$

i.e., you can measure the changes in an entire region by simply measuring what passes in and out of the boundaries of the region.

"Stuff passing through a boundary " could be:

• tigers through a conservation zone (2-D)
• sodium ions through a biological cell (3-D)

• magnetic flux through a toroidal fusion chamber

• water through a reservoir (but you’d have to measure evaporation, rain, dew/condensation, and ground seepage in order to get all of ∂V)

• in the other direction, you could measure water upstream and downstream in a river (no tributaries in between) and infer the net amount of water that was drunk, evaporated, or seeped

• probability mass through a set of possibilities

• particulate pollution through "greater Los Angeles"

• ¿ notes through a symphonic orchestra ?
• chromium(VI) through a human body
• smoke or steam through an industrial cooling tower or smokestack

• imports and exports through an economy

• goods or cash through a limited liability company

Said in words, the observation that you can measure change within an entire region by just measuring all of its boundaries sounds obvious, even trivial. Said symbolically, Gauß’ discovery amounts to a nifty tradeoff between boundaries  and gradients . (The gradient  is the net amount of a flow: flow in direction 1 plus flow in orthogonal direction 2 plus flow in mutually orthogonal direction 3 plus…) It also amounts to a connection between 2-D and 3-D.



Because of Cartan-style differential geometry, we know that the connection is much more general: 1-D shapes bound 2-D shapes, 77-D shapes bound 78-D shapes, and so on.

Nice one, Fred.

(Source: ocw.mit.edu)

## Mathematical Biofluiddynamics

Covers two very different questions:

1. How do fluids in an animal’s body flow? Like how does blood squeeze through arteries and veins, how does air (a fluid) flow into and out of sacs during respiration, how do capillaries work, etc?
2. How do aquatic animals swim themselves along with their tails, fins, hydrodynamic bodies, etc?