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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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  • 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

  • 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 Newton16421727

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

    electrodynamics solid mechanics thermodynamics fluid mechanics

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

    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

    ** ***

  • 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

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

    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

  • 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

  • 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 (originalin Istanbul Turkey)

    5 inches

    Three-dimensional printing -- MIT(Prof. E. Sachs & colleagues)

    *This url is no longerworking -mea

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

    . and a pretty picture .

    A dolphin blowing a toroidal bubble

  • 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

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

    Elementary Ideas II

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

    Elementary Ideas III

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

    Elementary Ideas IV

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

    Elementary Ideas V

    4. Viscosity and Newtonian fluids

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

    Elementary Ideas VI

    5. On to the equations of motion

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

    Elementary Ideas VII

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

    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

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

    Elementary Ideas IX

  • 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.

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

    PART II: Prototypical Flows I

    Steady pressure-driven flow

  • 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

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

    Prototypical Flows III

    Even simple flows suffer dynamical instabilities!

    Ref. D. Acheson

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

    Prototypical Flows IV

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

    Lubrication Flows I

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

    Lubrication Flows II

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

    Lubrication Flows III

  • 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

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

    Lubrication Flows IV

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

    Lubrication Flows V

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

    Lubrication Flows VI

    Time-dependent geometries

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

    Lubrication Flows VII

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

    Lubrication Flows VIII

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

    Lubrication Flows IX

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

    Suspension Flows I

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

    Suspension Flows II

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

    Suspension Flows III

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

    Suspension Flows IV

  • 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: = 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=

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

    Suspension Flows VI

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

    Suspension Flows VII

    Ref. Stone & Ajdari 1996

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

    Marangoni Flows:Surface-driven motions

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

    More on thermally-driven flows

  • Applied Physics 298r A fluid dynamics tour 43 5 April 2004

    Gradients in surface tension:Marangoni stresses

    Carlo Marangoni(18401925)

    Courtes