Fluid Dynamics: Physical ideas, the Navier-Stokes equations, and
Transcript of Fluid Dynamics: Physical ideas, the Navier-Stokes equations, and
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Fluid Dynamics: Physical ideas, theNavier-Stokes equations, and
applications to lubrication flowsand complex fluids
Howard A. StoneDivision of Engineering &
Applied SciencesHarvard University
A presentation for AP298rMonday, 5 April 2004
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Outline
• Part I: elementary ideas– A role for mechanical ideas– Brief picture tour: small to large lengths scales; fast and slow flows; gases and liquids
• Continuum hypothesis: material andtransport properties Newtonian fluids (and a brief word about rheology)
stress versus rate of strain; pressure and densityvariations;
Reynolds number; Navier-Stokes eqns, additionalbody forces; interfacial tension: statics, interfacedeformation, gradients
• Part II: Prototypical flows: pressure and sheardriven flows; instabilities; oscillatory flows
• Part III: Lubrication and thin film flows• Part IV: Suspension flows - sedimentation,
effective viscosities, an application tobiological membranes
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From atoms to atmospheres:mechanics in the physical sciences
• classical mechanics– particle and rigid body dynamics
• celestial mechanics– motion of stars, planets,
comets, ...• quantum mechanics
– atoms and clusters of atoms• statistical mechanics
– properties of large numbers
Isaac Newton1642–1727
Continuum mechanics:: (materials viewed as continua) (materials viewed as continua)
electrodynamics solid mechanics thermodynamics fluid mechanics
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A fluid dynamicist’s view ofthe world*
* after theme of H.K. Moffatt ** http://zebu.uoregon.edu/messier.html*** Courtesy of H. Huppert
Fluiddynamicist
Mathematics
BiologyChemistry Physics
Engineering
Geophysics
Astrophysics
aeronauticalbiomedicalchemicalenvironmentalmechanical
Snow avalancheGalaxies
** ***
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Fluid motions occur in manyforms around us:
BigWaves
Little waves
Ship waves
(Water)Waves
Ref.: An Album of Fluid Motion,M. Van Dyke
Here is a short tour
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Flow and design in sports
Cycles and cycling Yacht design and theAmerica’s Cup(importance of the keel)
http://www.sgi.com/features/2000/jan/cup/Rebecca Twig, Winning Jan. 1996
Bicycling Feb. 1996
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Micro-organisms:flagella, cilia
Swimming (large and small)
Rowing
Speed vs. # of rowers?T.A. McMahon, Science (1971)
Running on water
Basilisk or Jesus lizard
Ref: McMahon & Bonner,On Size andLife; Alexander Exploring Biomechanics
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Small fluid drops(surface tension is important)
Water issuing from amillimeter-sized nozzle(3 images on right: different oscillationfrequencies given to liquid; ref: VanDyke, An Album of Fluid Motion)
Bubble ink jet printer(Olivetti)
also: deliverreagents to DNA(bio-chip) arrays
*http://www.olivision.com/powerpoint/OlivettiPrinter/sld003.htm
Hagia Sophia (‘original’in Istanbul Turkey)
5 inches
Three-dimensional printing -- MIT(Prof. E. Sachs & colleagues)
*This url is no longerworking -mea
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.… and a pretty picture ….
A dolphin blowing a toroidal bubble
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Elementary Ideas I
• A brief tour of basic elementsleading through the governingpartial differential equations
• Physical ideas, dimensionlessparameters
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Elementary Ideas II
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Elementary Ideas III
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Elementary Ideas IV
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Elementary Ideas V
4. Viscosity and Newtonian fluids
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Elementary Ideas VI
5. On to the equations of motion
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Elementary Ideas VII
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Elementary Ideas VIII
• Newton’s second law:
ratio of inertialeffects to viscouseffects in the flow
Re = rmUL Emphasizes
inter-relation ofsize, speed,viscosity
Osborne Reynolds (1842–1912)
mass acceleration forces. =å
High Reynolds number flowLow Reynolds number flow
Forces (pressure)acting on fluid tocause motion
Friction from surroundingfluid which resists motion:viscosity (µ)
UL
The Reynolds number
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Elementary Ideas IX
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Quiz 1
• Consider the rise height of aliquid on a plane.
• Use dimensional arguments toshow that the rise height isproportional to the capillarylength.
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PART II: Prototypical Flows I
Steady pressure-driven flow
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Prototypical Flows II
GAS FLOW IN A MICROCHANNEL: COMPRESSIBLEFLOW WITH SLIP
REF: ARKILIC, SCHMIDT & BREUER
Additional effects when the mean free path of thefluid is comparable to the geometric dimensions
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Prototypical Flows III
Even simple flows suffer dynamical instabilities!
Ref. D. Acheson
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Prototypical Flows IV
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Lubrication Flows I
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Lubrication Flows II
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Lubrication Flows III
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Quiz 2
• Consider pressure-driven flow ina rectangular channel of height hand width w with h<< w.
• Find an approximate expressionfor the flow rate through thechannel.
• If the permeability is the ratio ofthe µu/(∆p/L), find thepermeability of such a rectangularchannel.
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Lubrication Flows IV
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Lubrication Flows V
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Lubrication Flows VI
Time-dependent geometries
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Lubrication Flows VII
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Lubrication Flows VIII
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Lubrication Flows IX
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Suspension Flows I
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Suspension Flows II
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Suspension Flows III
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Suspension Flows IV
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Suspension Flows IV
Brownian motion and diffusion: The Stokes-Einstein equation
Diffusion coefficient: Stokes-Einstein equation
• Translation diffusion of spherical particles
• Einstein: related thermal fluctuations to mean square displacement; with resistivity: ζ = force/velocity
• Stokes: ζ = 6πµa
• Typical magnitudes (small molecules in water):
btkTDζ=6btkTDaπµ=
… can also investigate other shapes, rotational diffusion
a
25cm10secliqtD−∪ 21cm10secgastD−∪
F,U
where ζ = F/U
µ=fluid viscosity
Stokes-Einstein equation
• A typical diffusive displacementin time τ are linked by(distance)2 Dtτ.
2()2txtDt=∪
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Suspension Flows VI
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Suspension Flows VII
Ref. Stone & Ajdari 1996
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Marangoni Flows:Surface-driven motions
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More on thermally-driven flows
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Gradients in surface tension:Marangoni stresses
Carlo Marangoni(1840–1925)
Courtesy of Professor Maria TeresaAristodemo, Florence, and Dr. Raffaele
Savino, Naples
Fluid draggedfrom low tohigh tension
lowtension
hightension
liquid
air
• Local value of surface tension is altered bychange of temperature or surfactantconcentration
• Contaminants typically lower surface tension• Example: alcohol and water
surfactants: amphiphilic molecules
water
air
Polar head group
Hydrocarbon tail
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Applied Physics 298r A fluid dynamics tour 44 5 April 2004
Gradients in surface tension:Marangoni stresses
Carlo Marangoni(1840–1925)
Courtesy of Professor Maria TeresaAristodemo, Florence, and Dr. Raffaele
Savino, Naples
Fluid draggedfrom low tohigh tension
lowtension
hightension
liquid
air
• Local value of surface tension is altered bychange of temperature or surfactantconcentration
• Contaminants typically lower surface tension• Example: alcohol and water
surfactants: amphiphilic molecules (soap)
water
air
Polar head group
Hydrocarbon tail
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Wine tears
An example ofthe Marangoni effect
Evaporation from thin film
high surface tension
low surface tension
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Wine tears
An example ofthe Marangoni effect
Evaporation from thin film
high surface tension
low surface tensionMarangoni stress
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In fact: an improved history
• Fluid motions due to gradientsin surface were first properlydescribed by James Thomson in1855 On certain curious motions
observable at the surfaces of wineand other alcoholic liquors
• James Thomson was the olderbrother of William Thomson(who will appear later in thetalk)
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Conclusions
• Continuum descriptions of fluid-likesystems begin with momentumstatement involving stress (Cauchyequation)
• For Newtonian fluids the startingpoint is the Navier-Stokes equationswhich is commonly studied assumingthe density and viscosity are constant
• Common geometric configurations,including thin films, are well studiedand ammenable to analysis
• Many common features among areasof complex fluids, suspensions,lubricating films, etc.
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