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The Beauty of the Flow
Nicole Sharp
Applied Mathematics Undergraduate SeminarTexas A&M University2 October 2013
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What is a fluid?A fluid is a substance that deforms continuously
under an application of shear stress.
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shear stress, τ
Image credits: W. van Hoeve et al.; A. Lindholdt et al.; NASA/ESA; P. Lovine; G. Scott
Liquids Gases
Plasmas Granular Materials Gels
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A droplet falling into a pool
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Video credit: S. Trainoff and N. Phillips
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A droplet falling into a pool
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Image credits: A. Labuda and J. Belina; D. Terwagne et al.; Y. Couder et al.; D. Harris and J. Bush
The procession of progressively smaller drops merging with the pool
is called the coalescence cascade.
The cascade can be delayed almost indefinitely by vibrating the pool,
which bounces the droplets.
Using vibration to mix bouncing drops of different immiscible fluids.
Clustered arrays of bouncing droplets. Bouncing droplets as quantum mechanical analogs.
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A falling viscous stream
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Image/video credits: Smarter Every Day; S. Morris et al.
When viscous fluids like honey fall, they tend to coil depending on
factors like height, jet diameter, viscosity, and mass flow rate.
If we instead pour the fluid onto a moving belt, we get even stranger behavior:
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A falling viscous stream
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Image credit: S. Chui-Webster and J. Lister
Each shift in behavior is called a bifurcation and appears due to nonlinearity in the governing equations. Eventually, this leads to chaos.
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Highly viscous flow
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Video credit: U. Penn General Motors Lab
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Highly viscous flow
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Image credit: U. New Mexico Physics Dept.; T. Congor
In extremely viscous (laminar) flows, only molecular diffusion and momentum
diffusion govern how the fluid moves.
Molecular diffusion is random but slow. Momentum diffusion is exactly reversible,
allowing one to unmix the fluids.
Most flows are turbulent and their motion is generated by momentum convection which is irreversible.
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Instability in fluids
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Image/video credits: V. Zecevic; J. Fontane et al.; M. Stuart
The Kelvin-Helmholtz instability occurs between fluid layers moving
at different velocities.
It can be observed through numerical simulation as well as
laboratory demonstration.
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Instability in fluids
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Image credits: G. Hart; NASA/JPL/U. of Arizona; NASA/Voyager 1
The Kelvin-Helmholtz instability is observed in nature as well at many
different scales.
Kelvin-Helmholtz clouds on Earth and on Jupiter
Lava coils on the surface of Mars.
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Fluid-object interaction: vortex shedding
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Video/image credits: M. Soltys; D. Burbank; MODIS Aqua
Blunt objects in a flow shed alternating periodic vortices to
create von Karman vortex streets.
Vortex street from islands off Baja California.Vortex streets formed by volcanic islands.
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Fluid-object interaction: vortex shedding
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Image credits: J. Buchholz and A. Smits; T. Schnipper et al.; M. Shelley and J. Zhang
Similarly complicated wake structures are made by flapping objects.
Dye visualization of the wake of a pitching plate.
Wakes of flapping foils in flowing soap films.
Wakes of flexible flapping flags.
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Fluid-object interaction: flutter
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Video source: B. Pathe
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Fluid-object interaction: flutter
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Video/image credits: Wikimedia; NASA; A. Varma
Sometimes an object’s structural dynamics and its aerodynamics get
into a potentially destructive feedback loop known as flutter.
Tacoma Narrows Bridge in flutter (circa 1940).
Piper PA-30 Twin Comanche with tail in flutter.
Male hummingbirds use flutter in their tail feathers during dives as part of their mating calls.
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So where’s the math?
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So where’s the math?
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Virtually all fluid motion is described by the same three sets of equations.
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So where’s the math?
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Virtually all fluid motion is described by the same three sets of equations.
Conservation of mass (a.k.a. continuity):
0
ut
Conservation of momentum (a.k.a. Navier-Stokes equation):
fτuuu
pt
Conservation of energy:
Tkppt
phh
t
h
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Where can you find more fluid dynamics?
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• Math• Chemistry• Physics/astrophysics• Atmospheric science• Geology• Every engineering department
What math should you study?
• Calculus• Differential and partial differential equations• Fourier transforms• Linear algebra• Perturbation theory• Nonlinear dynamics and chaos• Mathematical modeling
Image credits: F. Oefner; D. Quinn et al.
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nicole.sharp@gmail.com
For more fluid dynamics: http://fuckyeahfluiddynamics.tumblr.com
For a copy of these slides: http://tinyurl.com/nss-slides
Nicole Sharp
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For more information on…
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Coalescing droplets: more high-speed videosBouncing emulsions: D. Terwange et al. Hydrodynamic quantum analogs: Y. Couder et al.; J. Bush et al.
Plasma: applications; electrohydrodynamics; magnetohydrodynamicsGranular flows: applications; examples; similarities to traditional fluids
Coiling fluids: more examples; Kaye effect; lavaChaos in fluids: turbulence; blowing in a straw; vibrating networks
Viscous flow: Stokes flow; laminar flow; Saffman-Taylor instabilitiesMixing: turbulence; Rayleigh-Taylor instabilities
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For more information on…
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Fluid instabilities: examples; Rayleigh-Taylor; Plateau-Rayleigh; Saffman-Taylor; Richtmyer-Meshkov; Kelvin-Helmholtz
Vortex shedding: examples; wakes; von Karman vortex streetFlapping: examples; flapping flightFlow visualization: examples; smoke; dye; oil-flow; schlieren
Aeroelastic flutter: examples; use in hummingbirdsTacoma Narrows Bridge collapse: Minute Physics explains;
Billah and Scalan
fτuuu
pt
0
ut The math: continuity; Navier-Stokes; energy conservation
The Millenium Prize: Navier-Stokes existence and smoothness
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Just one more video…
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Video credit: B. Tomlinson
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