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 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 and transport properties Newtonian fluids (and a brief word about rheology) stress versus rate of strain; pressure and density variations; Reynolds number; Navier-Stokes eqns, additional body forces; interfacial tension: statics, interface deformation, gradients • Part II: Prototypical flows: pressure and shear driven flows; instabilities; oscillatory flows • Part III: Lubrication and thin film flows • Part IV: Suspension flows - sedimentation, effective viscosities, an application to biological membranes Applied Physics 298r A fluid dynamics tour 2 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 Isaac Newton • statistical mechanics 1642–1727 – properties of large numbers Continuum mechanics:: (materials viewed as continua) electrodynamics solid mechanics thermodynamics fluid mechanics Applied Physics 298r A fluid dynamics tour 3 5 April 2004 A fluid dynamicist’s view of the world* ** *** Snow avalanche Galaxies Mathematics Astrophysics Engineering aeronautical Geophysics biomedical Fluid chemical dynamicist environmental mechanical Chemistry Biology Physics * after theme of H.K. Moffatt ** http://zebu.uoregon.edu/messier.html *** Courtesy of H. Huppert Applied Physics 298r A fluid dynamics tour 4 5 April 2004 Fluid motions occur in many forms around us: Here is a short tour (Water) Big Little Waves Waves waves Ship waves Ref.: An Album of Fluid Motion, M. Van Dyke Applied Physics 298r A fluid dynamics tour 5 5 April 2004 Flow and design in sports Cycles and cycling Yacht design and the America’s 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 6 5 April 2004 Swimming (large and small) Micro-organisms: flagella, cilia Running on water Rowing Basilisk or Jesus lizard Ref: McMahon & Bonner,On Size and Speed vs. # of rowers? Life; Alexander Exploring Biomechanics T.A. McMahon, Science (1971) Applied Physics 298r A fluid dynamics tour 7 5 April 2004 Small fluid drops (surface tension is important) Water issuing from a millimeter-sized nozzle (3 images on right: different oscillation frequencies given to liquid; ref: Van Dyke, An Album of Fluid Motion) Bubble ink jet printer (Olivetti) also: deliver reagents to DNA (bio-chip) arrays Three-dimensional printing -- MIT (Prof. E. Sachs & colleagues) Hagia Sophia (‘original’ in Istanbul Turkey) *This url is no longer 5 inches working -mea Applied Physics 298r A fluid dynamics tour 8 5 April 2004 *http://www.olivision.com/powerpoint/OlivettiPrinter/sld003.htm .… and a pretty picture …. A dolphin blowing a toroidal bubble Applied Physics 298r A fluid dynamics tour 9 5 April 2004 Elementary Ideas I • A brief tour of basic elements leading through the governing partial differential equations • Physical ideas, dimensionless parameters Applied Physics 298r A fluid dynamics tour 10 5 April 2004 Elementary Ideas II Applied Physics 298r A fluid dynamics tour 11 5 April 2004 Elementary Ideas III Applied Physics 298r A fluid dynamics tour 12 5 April 2004 Elementary Ideas IV Applied Physics 298r A fluid dynamics tour 13 5 April 2004 Elementary Ideas V 4. Viscosity and Newtonian fluids Applied Physics 298r A fluid dynamics tour 14 5 April 2004 Elementary Ideas VI 5. On to the equations of motion Applied Physics 298r A fluid dynamics tour 15 5 April 2004 Elementary Ideas VII Applied Physics 298r A fluid dynamics tour 16 5 April 2004 Elementary Ideas VIII The Reynolds number Osborne Reynolds • Newton’s second law: (1842–1912) mass.acceleration = å forces Forces (pressure) Friction from surrounding acting on fluid to fluid which resists motion: cause motion viscosity (µ) High Reynolds number flow Low Reynolds number flow U L Emphasizes ratio of inertial r UL inter-relation of effects to viscous Re = size, speed, effects in the flow m viscosity Applied Physics 298r A fluid dynamics tour 17 5 April 2004 Elementary Ideas IX Applied Physics 298r A fluid dynamics tour 18 5 April 2004 Quiz 1 • Consider the rise height of a liquid on a plane. • Use dimensional arguments to show that the rise height is proportional to the capillary length. Applied Physics 298r A fluid dynamics tour 19 5 April 2004 PART II: Prototypical Flows I Steady pressure-driven flow Applied Physics 298r A fluid dynamics tour 20 5 April 2004 Prototypical Flows II Additional effects when the mean free path of the fluid is comparable to the geometric dimensions GAS FLOW IN A MICROCHANNEL: COMPRESSIBLE FLOW WITH SLIP REF: ARKILIC, SCHMIDT & BREUER Applied Physics 298r A fluid dynamics tour 21 5 April 2004 Prototypical Flows III Even simple flows suffer dynamical instabilities! Ref. D. Acheson Applied Physics 298r A fluid dynamics tour 22 5 April 2004 Prototypical Flows IV Applied Physics 298r A fluid dynamics tour 23 5 April 2004 Lubrication Flows I Applied Physics 298r A fluid dynamics tour 24 5 April 2004 Lubrication Flows II Applied Physics 298r A fluid dynamics tour 25 5 April 2004 Lubrication Flows III Applied Physics 298r A fluid dynamics tour 26 5 April 2004 Quiz 2 • Consider pressure-driven flow in a rectangular channel of height h and width w with h<< w. • Find an approximate expression for the flow rate through the channel. • If the permeability is the ratio of the µu/(∆p/L), find the permeability of such a rectangular channel. Applied Physics 298r A fluid dynamics tour 27 5 April 2004 Lubrication Flows IV Applied Physics 298r A fluid dynamics tour 28 5 April 2004 Lubrication Flows V Applied Physics 298r A fluid dynamics tour 29 5 April 2004 Lubrication Flows VI Time-dependent geometries Applied Physics 298r A fluid dynamics tour 30 5 April 2004 Lubrication Flows VII Applied Physics 298r A fluid dynamics tour 31 5 April 2004 Lubrication Flows VIII Applied Physics 298r A fluid dynamics tour 32 5 April 2004 Lubrication Flows IX Applied Physics 298r A fluid dynamics tour 33 5 April 2004 Suspension Flows I Applied Physics 298r A fluid dynamics tour 34 5 April 2004 Suspension Flows II Applied Physics 298r A fluid dynamics tour 35 5 April 2004 Suspension Flows III Applied Physics 298r A fluid dynamics tour 36 5 April 2004 Suspension Flows IV Applied Physics 298r A fluid dynamics tour 37 5 April 2004 Suspension Flows IV Brownian motion and diffusion: The Stokes-Einstein equation • A typical diffusive displacement 2()2txtDt= in time τ are∪ linked by (distance)2 D τ. btkTDζ=6btkTDaπµ=25cmt 10secliqtD−∪ 21cm10secgastD−∪ Diffusion coefficient: Stokes-Einstein equation • Translation diffusion of spherical particles • Einstein: related thermal fluctuations to mean square displacement; with resistivity: ζ = force/velocity where ζ = F/U • Stokes: ζ = 6πµa µ=fluid viscosity Stokes-Einstein equation a F,U • Typical magnitudes (small molecules in water): … can also investigate other shapes, rotational diffusion Applied Physics 298r A fluid dynamics tour 38 5 April 2004 Suspension Flows VI Applied Physics 298r A fluid dynamics tour 39 5 April 2004 Suspension Flows VII Ref. Stone & Ajdari 1996 Applied Physics 298r A fluid dynamics tour 40 5 April 2004 Marangoni Flows: Surface-driven motions Applied Physics 298r A fluid dynamics tour 41 5 April 2004 More on thermally-driven flows Applied Physics 298r A fluid dynamics tour 42 5 April 2004 Gradients in surface tension: Marangoni stresses • Local value of surface tension is altered by change of temperature or surfactant concentration low high air tension Fluid dragged tension from low to liquid high tension • Contaminants typically lower surface tension • Example: alcohol and water surfactants: amphiphilic molecules Hydrocarbon tail air water Polar head group Courtesy of Professor Maria Teresa Carlo Marangoni Aristodemo, Florence, and Dr. Raffaele (1840–1925) Savino, Naples Applied Physics 298r A fluid dynamics tour 43 5 April 2004 Gradients in surface tension: Marangoni stresses • Local value of surface tension is altered by change of temperature or surfactant concentration low high air tension Fluid dragged tension from low to liquid high tension • Contaminants typically lower surface tension • Example: alcohol and water surfactants: amphiphilic molecules (soap) Hydrocarbon tail air water Polar head group Courtesy of Professor Maria Teresa Carlo Marangoni Aristodemo, Florence, and Dr. Raffaele (1840–1925) Savino, Naples Applied Physics 298r A fluid dynamics tour 44 5 April 2004 Wine tears An example of the Marangoni effect Evaporation from thin film high surface tension low surface tension Applied Physics 298r A fluid dynamics tour 45 5 April 2004 Wine tears An example of the Marangoni effect Evaporation from thin film high surface tension Marangoni stress low surface tension Applied Physics 298r A fluid dynamics tour 46 5 April 2004 In fact: an improved history • Fluid motions due to gradients in surface were first properly described by James Thomson in 1855 On certain curious motions observable at the surfaces of wine and other alcoholic liquors • James Thomson