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058:0160 Chapter 1Professor Fred Stern Fall 2006 0058:160Intermediate Mechanics of FluidsClass NotesFall 2006Prepared 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)058:0160 Chapter 1Professor Fred Stern Fall 2006 1Chapter 1: IntroductionDefinition of a fluid: A fluid cannot resist any 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 2006 2No 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:dydydtudytanyudtd.(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 2006 3dydu.(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 2006 4(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 2006 5Continuum 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 importantV*  10-9 mm3 for all liquids and for gases at atmospheric pressure10-9 mm3 air (at standard conditions, 20C 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 2006 6Exception: 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 equantionally 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 equantify the separation of molecular and fluid motion length scales:nKll= l = molecular length scalel =fluid motion length scale058:0160 Chapter 1Professor Fred Stern Fall 2006 7Molecular scales:Air atmosphere conditions:86 10 ml-= � = mean free path1010t sl-= = time between collisionsSmallest fluid motion scales: 40.1 10l mm m-=: max100V m s: incompressible flow 0.3aM � 610t sl-=Thus 310 1nK-: = and l scales larger than 3 order of magnitude l scales.An intermediate scale is used to define a fluid particle *l ll = =And continuum fluid properties are an average over * *3V l=* 9 310V mm-: i.e. * 3 610 10l mm m- -=:Previsouly 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 flow058:0160 Chapter 1Professor Fred Stern Fall 2006 8domain and maxV can not exceed Re restriction imposed bytransition from laminar to turbulent flow.For turbulent flow, the smallest fluid motion scales are estimated by the Kolmogorov scales, which define the length, velocity, and time scales at which viscos 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:3 40Relh-: 0l L: 1 40Reu uh-= 0 00Reu lULn n= :1 20Reht t-=Which even for large Re of interest given *lh ?For example:100 watt mixer in 1 kg water:2 3100 100watt kg m se = =6 210 m sn-=for water 2 *10 mm lh-= ?058:0160 Chapter 1Professor Fred Stern Fall 2006 9Fluid 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.(1) Kinematic Properties:Kinematics refers to the description of the flow pattern without consideration of forces and moments, whereas dynamics refers to descriptions of F and M.Lagrangian vs. Eulerian


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