058:160 Intermediate Mechanics of Fluids Class Notes Fall 2006 Prepared by: Professor Fred Stern Typed 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 1 Professor Fred Stern Fall 2006 1 Chapter 1: Introduction Definition 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) y V = u(y) î u=U 1-D flow velocity profile h u(y) xu=0058:0160 Chapter 1 Professor Fred Stern Fall 2006 2 No 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 and fluid in contact. Knudsen number = Kn = λ/ℓ << 1 Exceptions are rarefied gases and gas/liquid contact line. Newtonian fluids: Rate of Strain: (u+uy dy)dt x y dy dӨ = tan-1 uydt y u+uydy dy uxu dt Fluid element with sides parallel to the coordinate axes at time t=0. Fluid element deformation at time t + dt058:0160 Chapter 1 Professor Fred Stern Fall 2006 3 dydydtudy=θtan yudtd==.θθ dyduµθµτ==. (rate of strain = velocity gradient) For 3D flow, rate of strain is a second order symmetric tensor: 12jiijjiuuxxε⎛⎞∂∂=+⎜⎟⎜⎟∂∂⎝⎠= εji Diagonal terms are elongation/contraction in x,y,z and off diagonal terms are shear in (x,y), (x,z), and (y,z). Liquids vs. Gases: Liquids Gases Closely spaced with large intermolecular cohesive forces Widely spaced with small intermolecular cohesive forces Retain volume but take shape of container Take volume and shape of container β << 1 ρ ~ constant β >> 1 ρ = ρ(p,T) Where β = coefficient of compressibility =change in volume/density with external pressure pp ∂∂=∂∂∀∀−=ρρβ11058:0160 Chapter 1 Professor Fred Stern Fall 2006 4 Bulk modulus 1ppKρρβ∂∂=−∀ = =∂∀ ∂ 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 point (maximum pressure at which liquid and vapor are both in equilibrium). Liquid, gases, and two-phase liquid-vapor behave as fluids.058:0160 Chapter 1 Professor Fred Stern Fall 2006 5 Continuum Hypothesis Fluids 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 ρ of a fluid x = position vector xyz=++ijkt = time()VMVVlimt,x*δδδ→δ=ρ δV* = limiting volume below which molecular variations may be important and above which macroscopic variations may be important δV* ≈ 10-9 mm3 for all liquids and for gases at atmospheric pressure 10-9 mm3 air (at standard conditions, 20°C and 1 atm) contains 3x107 molecules such that δM/δV = constant = ρ058:0160 Chapter 1 Professor Fred Stern Fall 2006 6 Exception: rarefied gas flow Note that typical “smallest” measurement volumes are about 10-3 – 100 mm3 >> δV* 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: nKlλ= λ= molecular length scale l =fluid motion length scale058:0160 Chapter 1 Professor Fred Stern Fall 2006 7 sm Molecular scales: Air atmosphere conditions: 8610mλ−=× = mean free path 1010tλ−= = time between collisions Smallest fluid motion scales: 40.1 10lmm−=∼ max100V∼ ms incompressible flow 0.3aM ≤ 610tsλ−=Thus and scales larger than 3 order of magnitude 310 1nK−∼lλ scales. An intermediate scale is used to define a fluid particle *llλAnd continuum fluid properties are an average over **Vl3= *910Vm−∼3m m i.e. *3 610 10lmm−−=∼ Previsouly given smallest fluid motion scales are rough estimates for incompressible flow. Estimates are VERY conservative for laminar flow since for laminar flow, is usually taken as smallest characteristic length of the flow l058:0160 Chapter 1 Professor Fred Stern Fall 2006 8 domain and can not exceed Re restriction imposed by transition from laminar to turbulent flow. maxV 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. ()143ηνε= ()12ητνε= ()14uηνε= ν= kinematic viscosity; ε=dissipation rate Which can also be written: 340Relη−∼ 0l∼ L140Reuuη−= 000ReulULνν= ∼ 120Reηττ−= Which even for large Re of interest given *lηFor example: 100 watt mixer in 1 kg water: 23100 100watt kgmsε== 6210 msν−=for water 2*10 mm lη−= 058:0160 Chapter 1 Professor Fred Stern Fall 2006 9 Fluid 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
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