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Fluid Dynamics: Physical ideas, the Navier-Stokes equations, and applications to lubrication flows and complex fluids Howard A. Stone Division of Engineering & Applied Sciences Harvard University A presentation for AP298r Monday, 5 April 2004
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Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

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Page 1: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

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

Page 2: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 2 5 April 2004

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

Page 3: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 3 5 April 2004

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

Page 4: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 4 5 April 2004

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

** ***

Page 5: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 5 5 April 2004

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

Page 6: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 6 5 April 2004

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

Page 7: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 7 5 April 2004

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

Page 8: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 8 5 April 2004

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

Page 9: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 9 5 April 2004

.… and a pretty picture ….

A dolphin blowing a toroidal bubble

Page 10: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 10 5 April 2004

Elementary Ideas I

• A brief tour of basic elementsleading through the governingpartial differential equations

• Physical ideas, dimensionlessparameters

Page 11: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 11 5 April 2004

Elementary Ideas II

Page 12: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 12 5 April 2004

Elementary Ideas III

Page 13: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 13 5 April 2004

Elementary Ideas IV

Page 14: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 14 5 April 2004

Elementary Ideas V

4. Viscosity and Newtonian fluids

Page 15: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 15 5 April 2004

Elementary Ideas VI

5. On to the equations of motion

Page 16: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 16 5 April 2004

Elementary Ideas VII

Page 17: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 17 5 April 2004

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

Page 18: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 18 5 April 2004

Elementary Ideas IX

Page 19: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 19 5 April 2004

Quiz 1

• Consider the rise height of aliquid on a plane.

• Use dimensional arguments toshow that the rise height isproportional to the capillarylength.

Page 20: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 20 5 April 2004

PART II: Prototypical Flows I

Steady pressure-driven flow

Page 21: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 21 5 April 2004

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

Page 22: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 22 5 April 2004

Prototypical Flows III

Even simple flows suffer dynamical instabilities!

Ref. D. Acheson

Page 23: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 23 5 April 2004

Prototypical Flows IV

Page 24: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 24 5 April 2004

Lubrication Flows I

Page 25: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 25 5 April 2004

Lubrication Flows II

Page 26: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 26 5 April 2004

Lubrication Flows III

Page 27: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 27 5 April 2004

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.

Page 28: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 28 5 April 2004

Lubrication Flows IV

Page 29: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 29 5 April 2004

Lubrication Flows V

Page 30: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 30 5 April 2004

Lubrication Flows VI

Time-dependent geometries

Page 31: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 31 5 April 2004

Lubrication Flows VII

Page 32: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 32 5 April 2004

Lubrication Flows VIII

Page 33: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 33 5 April 2004

Lubrication Flows IX

Page 34: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 34 5 April 2004

Suspension Flows I

Page 35: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 35 5 April 2004

Suspension Flows II

Page 36: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 36 5 April 2004

Suspension Flows III

Page 37: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 37 5 April 2004

Suspension Flows IV

Page 38: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 38 5 April 2004

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=∪

Page 39: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 39 5 April 2004

Suspension Flows VI

Page 40: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 40 5 April 2004

Suspension Flows VII

Ref. Stone & Ajdari 1996

Page 41: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 41 5 April 2004

Marangoni Flows:Surface-driven motions

Page 42: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 42 5 April 2004

More on thermally-driven flows

Page 43: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 43 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

water

air

Polar head group

Hydrocarbon tail

Page 44: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

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

Page 45: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 45 5 April 2004

Wine tears

An example ofthe Marangoni effect

Evaporation from thin film

high surface tension

low surface tension

Page 46: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 46 5 April 2004

Wine tears

An example ofthe Marangoni effect

Evaporation from thin film

high surface tension

low surface tensionMarangoni stress

Page 47: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 47 5 April 2004

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)

Page 48: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 48 5 April 2004

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.

Page 49: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 49 5 April 2004

Page 50: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 50 5 April 2004

Page 51: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 51 5 April 2004

Page 52: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 52 5 April 2004

Page 53: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 53 5 April 2004

Page 54: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 54 5 April 2004

Page 55: Viscosity and Continuum Mechanics ... - Harvard · PDF fileNavier-Stokes equations, and applications to lubrication flows ... elementary ideas ... electrodynamics solid mechanics thermodynamics

Applied Physics 298r A fluid dynamics tour 55 5 April 2004