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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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

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

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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 Newton16421727

Continuum mechanics:: (materials viewed as continua) (materials viewed as continua)

electrodynamics solid mechanics thermodynamics fluid mechanics

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A fluid dynamicists 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 theAmericas 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 (originalin 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

Newtons second law:

ratio of inertialeffects to viscouseffects in the flow

Re = rmUL Emphasizesinter-relation of

size, speed,viscosity

Osborne Reynolds (18421912)

mass acceleration forces. =

High Reynolds number flowLow Reynolds number flow

Forces (pressure)acting on fluid tocause motion

Friction from surroundingfluid which resists motion:viscosity ()

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

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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: = force/velocity

Stokes: = 6a

Typical magnitudes (small molecules in water):

btkTD=6btkTDa=

can also investigate other shapes, rotational diffusion

a

25cm10secliqtD 21cm10secgastD

F,U

where = F/U

=fluid viscosity

Stokes-Einstein equation

A typical diffusive displacementin time are linked by(distance)2 Dt.

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(18401925)

Courtes

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