Nicole Sharp Applied Mathematics Undergraduate Seminar Texas AampM University 2 October 2013 2 O C T 2 0 1 3 N S H A R P What is a fluid A fluid is a substance that deforms continuously ID: 310838
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Slide1
The Beauty of the Flow
Nicole Sharp
Applied Mathematics Undergraduate Seminar
Texas A&M University
2 October 2013Slide2
2 O C T 2 0 1 3
N. S H A R P
What is a fluid?
A fluid is a substance that
deforms continuously
under an application of
shear stress
.
A M U S E
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
GelsSlide3
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N. S H A R P
A droplet falling into a pool
A M U S E
Video credit:
S.
Trainoff
and N. PhillipsSlide4
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N. S H A R P
A droplet falling into a pool
A M U S E
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.Slide5
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A falling viscous stream
A M U S E
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:Slide6
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N. S H A R P
A falling viscous stream
A M U S E
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.Slide7
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Highly viscous flow
A M U S E
Video credit:
U. Penn General Motors LabSlide8
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Highly viscous flow
A M U S E
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.Slide9
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N. S H A R P
Instability in fluids
A M U S E
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.Slide10
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N. S H A R P
Instability in fluids
A M U S E
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.Slide11
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N. S H A R P
Fluid-object interaction: vortex shedding
A M U S E
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.Slide12
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N. S H A R P
Fluid-object interaction: vortex shedding
A M U S E
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.Slide13
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N. S H A R P
Fluid-object interaction: flutter
A M U S E
Video source:
B.
PatheSlide14
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Fluid-object interaction: flutter
A M U S E
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.Slide15
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So where’s the math?
A M U S E Slide16
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So where’s the math?
A M U S E
Virtually all fluid motion is described by the same three sets of equations.Slide17
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So where’s the math?
A M U S E
Virtually all fluid motion is described by the same three sets of equations.
Conservation of mass (a.k.a. continuity):
Conservation of momentum (a.k.a.
Navier
-Stokes equation):
Conservation of energy:Slide18
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N. S H A R P
Where can you find more fluid dynamics?
A M U S E
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. Slide19
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N. S H A R P
A M U S E
nicole.sharp@gmail.com
For more fluid dynamics:
http://fuckyeahfluiddynamics.tumblr.com
For a copy of these slides:
http://
tinyurl.com/nss-slides
Nicole
SharpSlide20
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N. S H A R P
For more information on…
A M U S E
Coalescing droplets:
more high-speed videos
Bouncing emulsions:
D.
Terwange
et al.
Hydrodynamic quantum analogs:
Y.
Couder et al.; J. Bush et al.
Plasma:
applications
; electrohydrodynamics; magnetohydrodynamics
Granular 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 instabilities
Mixing:
turbulence
;
Rayleigh-Taylor instabilitiesSlide21
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N. S H A R P
For more information on…
A M U S E
Fluid instabilities:
examples
;
Rayleigh-Taylor
;
Plateau-Rayleigh
;
Saffman
-Taylor; Richtmyer-Meshkov; Kelvin-Helmholtz
Vortex shedding:
examples; wakes;
von Karman vortex street
Flapping: 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
The math:
continuity
;
Navier
-Stokes
;
energy conservation
The
Millenium
Prize:
Navier
-Stokes existence and smoothness Slide22
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N. S H A R P
Just one more video…
A M U S E
Video credit:
B. Tomlinson