058:0160 Chapter 1Professor Fred Stern Fall 2009 1058:160Intermediate Mechanics of FluidsClass NotesFall 2007Prepared by:Professor Fred SternTyped by:Derek Schnabel (Fall 2004)Nobuaki Sakamoto (Fall 2006)Hamid Sadat-Hosseini (Fall 2006)Maysam Mousaviraad (Fall 2006)Corrected by:Jun Shao (Fall 2004)Mani Kandasamy (Fall 2005)Tao Xing, Hyun Se Yoon (Fall 2006)Hamid Sadat-Hosseini (Fall 2007)Hamid Sadat-Hosseini (Fall 2009)058:0160 Chapter 1Professor Fred Stern Fall 2007 2Chapter 1: IntroductionDefinition of a fluid: A fluid cannot resist an applied shear stress and remain at rest, whereas a non-fluid (i.e., solid) can.Solids resist shear by static deformation up to an elastic limit of the material, after which they undergo fracture.Fluids deform continuously (undergo motion) when subjected to shear stress. Consider a fluid between two parallel plates, with the lower one fixed and the upper moving at speed U, which is an example of Couette flow (i.e, wall/shear driven flows) V = u(y) î1-D flow velocity profileu(y)xyu=0u=Uh058:0160 Chapter 1Professor Fred Stern Fall 2007 3No slip condition: Length scale of molecular mean free path (λ) << length scale of fluid motion (ℓ); therefore, macroscopically there is no relative motion or temperature between the solid andfluid in contact. Knudsen number = Kn = λ/ℓ << 1Exceptions are rarefied gases and gas/liquid contact line.Newtonian fluids:Rate of Strain:dydydtudytanyudtd.(u+uy dy)dtxydyd = tanӨ-1 uydtFluid element with sides parallel to the coordinate axes at time t=0.Fluid element deformation at time t + dtyxdyu+uydyuu dt058:0160 Chapter 1Professor Fred Stern Fall 2007 4dydu.(rate of strain = velocity gradient)For 3D flow, rate of strain is a second order symmetrictensor: 12jiijj iuux xe� ���= +� �� �� �� �= εjiDiagonal terms are elongation/contraction in x,y,z and offdiagonal terms are shear in (x,y), (x,z), and (y,z).Liquids vs. Gases:Liquids GasesClosely spaced with largeintermolecular cohesiveforcesWidely spaced with smallintermolecular cohesiveforcesRetain volume but takeshape of containerTake volume and shape ofcontainerβ << 1ρ ~ constantβ >> 1ρ = ρ(p,T)Where β = coefficient of compressibility =change involume/density with external pressurepp 11 Bulk modulus 1p pK rr b� �=- " = =�" �Recall p-v-T diagram from thermodynamics:Single phase, two phase, triple point (point at which solid,liquid, and vapor are all in equilibrium), critical point058:0160 Chapter 1Professor Fred Stern Fall 2007 5(maximum pressure at which liquid and vapor are both in equilibrium).Liquid, gases, and two-phase liquid-vapor behave as fluids.058:0160 Chapter 1Professor Fred Stern Fall 2007 6Continuum HypothesisFluids are composed of molecules in constant motion and collosion; however, in most cases, molecular motion can be disregarded and the assumption is made that the fluid behaves as a continuum, i.e., the number of molecules within the smallest region of interest (a point) are sufficient that all fluid properties are point functions (single valued at a point). For example: Consider definition of density of a fluid VMVVlimt,x*V* = limiting volume below which molecular variations may be important and above which macroscopic variations may be importantV* 10-9 mm3 for all liquids and for gases at atmospheric pressure10-9 mm3 air (at standard conditions, 20C and 1 atm) contains 3x107 molecules such that M/V = constant = x = position vector x y z= + +i j kt = time058:0160 Chapter 1Professor Fred Stern Fall 2007 7Exception: rarefied gas flowNote that typical “smallest” measurement volumes are about 10-3 – 100 mm3 >> V* and that the “scale” of macroscopic variations are very problem dependent.A point in a fluid is equivalently used to define a fluid particle or infinitesimal material element used in defining the governing differential equations of fluid dynamics.At a more advanced level, the Knudsen number is used to quantify the separation of molecular and fluid motion length scales:nKll= l = molecular length scalel =fluid motion length scale058:0160 Chapter 1Professor Fred Stern Fall 2007 8Molecular scales:Air atmosphere conditions:86 10 ml-= � = mean free path tλ =10-10 s = time between collisionsSmallest fluid motion scales: ℓ = 0.1 mm = 10-4 mVmax ~ 100 m/s incompressible flow 0.3aM �tℓ = 10-6 s Thus Kn~10-3 << 1, and ℓ scales larger than 3 order of magnitude l scales.An intermediate scale is used to define a fluid particle λ << ℓ* << ℓAnd continuum fluid properties are an average over * *3V l=V* = ℓ*3 ~ 10-9 mm 3 (i.e. ℓ* ~ 10-6 m)Previously given smallest fluid motion scales are rough estimates for incompressible flow. Estimates are VERY conservative for laminar flow since for laminar flow, l is usually taken as smallest characteristic length of the flow domain and maxV can not exceed Re restriction imposed bytransition from laminar to turbulent flow.058:0160 Chapter 1Professor Fred Stern Fall 2007 9For turbulent flow, the smallest fluid motion scales are estimated by the Kolmogorov scales, which define the length, velocity, and time scales at which viscous dissipation takes place i.e. at which turbulent kinetic energy is destroyed.( )1 43h n e= ( )1 2ht n e= ( )1 4uhne=n= kinematic viscosity; e=dissipation rateWhich can also be written:4/30Rel Ll 0 4/10Reuu ULlu000Re1 20Reht t-=Which even for large Re of interest given *lFor example:100 watt mixer in 1 kg water:2 3100 100watt kg m se = =6 210 m sn-=for water *210 lmm The smallest fluid motion scales for ship and airplane:058:0160 Chapter 1Professor Fred Stern Fall 2007 10U(m/s) L(m) Re(m)u(m/s)(s)Ship 10.3 ( 20 knots)150 1.5E09 3E-3 0.05 0.06Airplane 120 28 3.4E09 2E-6 0.49 4E-6Fluid Properties:(1) Kinematic: linear (V) angular (ω/2) velocity, rate of strain (εij), vorticity (ω), and acceleration (a).(2) Transport: viscosity (μ), thermal conductivity (k), and mass diffusivity (D).(3) Thermodynamic: pressure (p), density (ρ), temperature (T), internal energy (û), enthalpy (h = û + p/ρ), entropy (s), specific heat (Cv, Cp, γ = Cp/ Cv, etc).(4) Miscellaneous: surface tension (σ), vapor pressure(pv), etc.058:0160
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