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1 57:020 Fluid Mechanics Class Notes Fall 2014 Prepared by: Professor Fred Stern Typed by: Stephanie Schrader (Fall 1999) Corrected by: Jun Shao (Fall 2003, Fall 2005) Corrected by: Jun Shao, Tao Xing (Fall 2006) Corrected by: Hyunse Yoon (Fall 2007 Fall 2014)

2 1 CHAPTER 1: INTRODUCTION AND BASIC CONCEPTS Fluids and the no-slip condition Fluid mechanics is the science and technology of fluids either at rest (fluid statics) or in motion (fluid dynamics) and their effects on boundaries such as solid surfaces or interfaces with other fluids. Definition of a fluid: A substance that deforms continuously when subjected to a shear stress Consider a fluid between two parallel plates, which is subjected to a shear stress due to the impulsive motion of the upper plate Fluid Element t=0 u=u u=0 No slip condition: no relative motion between fluid and boundary, i.e., fluid in contact with lower plate is stationary, whereas fluid in contact with upper plate moves at speed U. θ τ τ t= t Fluid deforms, i.e., undergoes rate of strain θ due to shear stress τ

3 2 Newtonian fluid: τ Solid γ τ θ = rate of strain τ = μθ µ = coefficient of viscosity Such behavior is different from solids, which resist shear by static deformation (up to elastic limit of material) Elastic solid: τ γ = strain τ = G γ τ G = shear modulus t=0 t= t Both liquids and gases behave as fluids Liquids: Closely spaced molecules with large intermolecular forces Retain volume and take shape of container container liquid Gases: Widely spaced molecules with small intermolecular forces Take volume and shape of container gas

4 3 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). Liquids, gases, and two-phase liquid-vapor behave as fluids.

5 4 Continuum Hypothesis In this course, 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 ρ, lim δm = δv δv * δ V ( x t) x = position vector = xi+ yj+ zk t = time δv * = limiting volume below which molecular variations may be important and above which macroscopic variations may be important δv * 10-9 mm 3 for all liquids and for gases at atmospheric pressure 10-9 mm 3 air (at standard conditions, 20 C and 1 atm) contains 3x10 7 molecules such that δm/δv = constant = ρ Note that typical smallest measurement volumes are about mm 3 >> δv * and that the scale of macroscopic variations are very problem dependent

6 5 Exception: rarefied gas flow δ * defines a point in the fluid, i.e., a fluid particle or infinitesimal material element used for deriving governing differential equations of fluid dynamics and at which all fluid properties are point functions: δ * >> molecular scales λ = mean free path = 6x10-8 m t λ = s = time between collisions δ * << fluid motions scales = m 3, i.e., l* = 10-6 m For laminar flow: l max smallest geometry scales of the flow U max < U transition to turbulent flow For turbulent flow: l max and U max determined by Kolmogorov scales at which viscous dissipation takes place, which for typical ship/airplane, η m (ship)/ m (airplane) u η 0.05 m/s (ship)/1.64 m/s (airplane) t η s (ship)/ s (airplane)

7 6 Properties of Fluids Fluids are characterized by their properties such as viscosity µ and density ρ, which we have already discussed with reference to definition of shear stress τ = μθ and the continuum hypothesis. (1) Kinematic: Linear (V) and angular (ω 2) velocity, rate of strain (ε ij ), 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 (u ), enthalpy (h = u + p ρ), specific heat (C v, C p, γ = C p C v, etc.) (4) Miscellaneous: Surface tension (σ), vapor pressure (p v ), etc.

8 7 Properties can be both dimensional (i.e., expressed in either SI or BG units) or non-dimensional: Figure B.1 Dynamic (absolute) viscosity of common fluids as a function of temperature. Figure B.2 Kinematic viscosity of common fluids (at atmospheric pressure) as a function of temperature. Table B.1 Physical Properties of Water (BG Units) Table B.2 Physical Properties of Water (SI Units) Table B.3 Physical Properties of Air at Standard Atmospheric Pressure (BG Units) Table B.4 Physical Properties of Air at Standard Atmospheric Pressure (SI Units) Table 1.5 Approximate Physical Properties of Some Common Liquids (BG Units) Table 1.6 Approximate Physical Properties of Some Common Liquids (SI Units) Table 1.7 Approximate Physical Properties of Some Common Gases at Standard Atmospheric Pressure (BG Units) Table 1.8 Approximate Physical Properties of Some Common Gases at Standard Atmospheric Pressure (SI Units)

9 8 Basic Units System International and British Gravitational Systems Primary Units SI BG Mass M kg slug=32.2lbm Length L m ft Time t s s Temperature T C ( K) F ( R) Temperature Conversion: K = C R = F K and R are absolute scales, i.e., 0 at absolute zero. Freezing point of water is at 0 C and 32 F. Secondary (derived) units Dimension SI BG velocity V L/t m/s ft/s acceleration a L/t 2 m/s 2 ft/s 2 force F ML/t 2 N (kg m/s 2 ) lbf pressure p F/L 2 Pa (N/m 2 ) lbf/ft 2 density ρ M/L 3 kg/m 3 slug/ft 3 internal energy u FL/M J/kg (N m/kg) BTU/lbm Table 1.3 Conversion Factors from BG and EE Units to SI Units. Table 1.4 Conversion Factors from SI Units to BG and EE Units.

10 9 Weight and Mass F = ma Newton s second law (valid for both solids and fluids) Weight = force on object due to gravity W = mg g = 9.81 m/s 2 = 32.2 ft/s 2 SI: W (N) = m (kg) 9.81 m/s 2 BG: W (lbf) = m(slug) 32.2ft/ s 2 EE: W (lbf) = ( ) g c = m lbm 32.2 ft/s 2 g c lbm ft 32.2 = 2 s lbf 1 N = 1 kg 1 m/s 2 1 lbf = 1 slug 1 ft/s 2 lbm 32.2 slug, i.e., 1 slug = 32.2 lbm

11 10 System; Extensive and Intensive Properties System = fixed amount of matter = mass m Therefore, by definition d( m) = 0 dt Properties are further distinguished as being either extensive or intensive. Extensive properties: Intensive properties: depend on total mass of system, e.g., m and W independent of amount of mass of system, e.g., p (force/area) and ρ (mass/volume)

12 11 Properties Involving the Mass or Weight of the Fluid Specific Weight, γ = gravitational force (i.e., weight) per unit volume V = W/V = mg/v = ρg N/m 3 (Note that specific properties are extensive properties per unit mass or volume) Mass Density ρ = mass per unit volume = m/v kg/m 3 Specific Gravity S = ratio of γ liquid to γ water at standard = γ/γ water, 4 C dimensionless (or γ air at standard conditions for gases) T = 4 C γ water, 4 C = 9810 N/m 3 for T = 4 C and atmospheric pressure

13 12 Variation in Density gases: ρ = ρ (gas, T, p) equation of state (p-v-t) = p/rt ideal gas R = R (gas) e.g. R (air) = N m/kg K ρ (air) = kg/m 3 at Standard Atmosphere (T = 15 C and p = kpa) liquids: ρ constant Water Note: For a change in temperature from 0 to 100 C, density changes about 29% for air while only about 4% for water. Liquid and temperature Density (kg/m 3 ) Density (slugs/ft 3 ) Water 20 o C (68 o F) Ethyl alcohol 20 o C (68 o F) Glycerine 20 o C (68 o F) 1, Kerosene 20 o C (68 o F) Mercury 20 o C (68 o F) 13, Sea water 10 o C at 3.3% salinity 1, SAE 10W 38 o C(100 o F) SAE 10W-30 8 o C(100 o F) SAE o C(100 o F)

14 13 For greater accuracy can also use p-v-t diagram ρ = ρ (liquid, T, p) T p ρ ρ Properties Involving the Flow of Heat For flows involving heat transfer such as gas dynamics additional thermodynamic properties are important, e.g. specific heats c p and c v J/kg K specific internal energy û J/kg specific enthalpy h = û + p/ρ J/kg

15 14 Viscosity Recall definition of a fluid (substance that deforms continuously when subjected to a shear stress) and Newtonian fluid shear / rate-of-strain relationship: τ = μθ. Reconsider flow between fixed and moving parallel plates (Couette flow) y δuδt=distance fluid particle travels in time δt h u(y)=velocity profile U = y h δy δf at t δf=fluid element δf at δt δθ u=u u=0 Newtonian fluid: τ = μθ = δθ μ δt δuδt δuδt tan δθ = or δθ = for small δθ δy δy therefore and δu δ θ = i.e., δy du τ = μ dy du θ = = velocity gradient dy

16 15 Exact solution for Couette flow is a linear velocity profile where U u( y) = y Note: u(0) = 0 and u(h) = U h i.e., satisfies no-slip U τ = μ = constant boundary condition h U/h = velocity gradient = rate of strain µ = coefficient of viscosity = proportionality constant for Newtonian fluid µ = τ du dy 2 N m = m m s m 2 Ns = 2 m µ ν = = = kinematic viscosity ρ s

17 16 µ = µ(fluid;t,p) = µ(gas/liquid;t) gas and liquid µ p, but small µ gas: µ T liquid: µ T Due to structural differences, more molecular activity for gases, decreased cohesive forces for liquids

18 17 Newtonian vs. Non-Newtonian Fluids Dilatant (Shear thickening): τ du/dy Newtonian: τ du/dy Pseudo plastic (Shear thinning): τ du/dy Bingham plastic: Requires τ before becomes fluid Ex) toothpaste, mayonnaise Newtonian Fluids Non-Newtonian Fluids τ du dy τ du dy n µ = slope n > 1 (shear thickening) Slope increases with increasing τ; ex) cornstarch, quicksand n < 1 (shear thinning) Slope decreases with increasing τ; ex) blood, paint, liquid plastic

19 18 Elasticity (i.e., compressibility) Increasing/decreasing pressure corresponds to contraction/expansion of a fluid. The amount of deformation is called elasticity. dp = E v dv dp > 0 dv V < 0 V Increase pressure, decrease volume. minus sign used and by definition, m = ρv dm = ρdv + Vdρ = 0 dv = dρ V ρ Thus, E v = dp dv V = dp dρ ρ = ρ dp dρ [N m2] Liquids are in general incompressible, e.g. E v = 2.2 GN/m 2 water i.e. ΔV = 0.05%V for p = 1MN/m 2 (G=Giga=10 9 M=Mega=10 6 k=kilo=10 3 ) Gases are in general compressible, e.g. for ideal gas (i.e., p = ρrt) at T = constant (isothermal) dp = RT dρ E v = ρ RT = p

20 19 Vapor Pressure and Cavitation When the pressure of a liquid falls below the vapor pressure it evaporates, i.e., changes to a gas. If the pressure drop is due to temperature effects alone, the process is called boiling. If the pressure drop is due to fluid velocity, the process is called cavitation. Cavitation is common in regions of high velocity, i.e., low p such as on turbine blades and marine propellers. high V low p (suction side) isobars low V high p (pressure side) Cavitation number, C a = p p 1 ρv 2 v 2 streamlines around lifting surface (i.e. lines tangent to velocity vector) C a < 0 implies cavitation

21 20 Surface Tension and Capillary Effects At the interface of two immiscible fluids (e.g., a liquid and a gas), forces develop to cause the surface to behave as if it were a stretched membrane. Molecules in the interior attract each other equally, whereas molecules along the surface are subject to a net force due to the absence of neighbor molecules. The intensity of the molecular attraction per unit length along any line in the surface is call the surface tension and is designated by the Greek symbol σ. AIR F σ = surface tension force F σ Interface F σ Near surface forces are increased due to absence of neighbors such that surface is in tension σ per unit length WATER Away from interface molecular forces are equal in all directions σ air/water = N/m Fσ = σ L=line force with direction normal to the cut L=length of cut through the interface

22 21 Effects of surface tension: Contact angle: θ < 90 o, Wetting e.g., Water, θ 0 θ > 90 o, Non-wetting e.g., Mercury, θ Capillary action in small tube (See page 21) h = 4σ γd 2. Pressure difference across curved interface (See page 22) p = σ/r R = radius of curvature 3. Transformation of liquid jet into droplets 4. Binding of wetted granular material such as sand 5. Capillary waves: surface tension acts as restoring force resulting in interfacial waves called capillary waves

23 22 Capillary tube h F σ F σ Fluid attaches to solid with contact angle θ due to surface tension effect and wetty properties water reservoir d θ θ= contact angle Example: Capillary tube d = 1.6mm = m Fσ = σ L, L=length of contact line between fluid & solid (i.e., L = πd = circumference) water reservoir at 20 C, σ = N/m, γ = 9790 N/m 3 h =? ΣF z = 0 F σ,z - W = 0 σπd cosθ - ρgv = 0 θ 0 cosθ = 1 ρg = γ πd σπ d γ h 4 2 = 0 h 4 σ = = 18. mm γd 6 V πd = Δh 4 2 =Volume of fluid above reservoir

24 23 Pressure jump across curved interfaces (a) Cylindrical interface Force Balance: 2σL = 2 RL (p i p o ) p = σ/r p i > p o, i.e. pressure is larger on concave vs. convex side of interface (b) Spherical interface (Droplets) π 2Rσ = πr 2 p p = 2σ/R Bubble: π 2Rσ + π 2Rσ Two interfaces (c) General interface p = σ(r R 2-1 ) R 1,2 = principal radii of curvature = πr 2 ΔP ΔP = 4σ R

25 24 A brief history of fluid mechanics See textbook section (page 27) Fluid Mechanics and Flow Classification Hydrodynamics: flow of fluids for which density is constant such as liquids and low-speed gases. If in addition fluid properties are constant, temperature and heat transfer effects are uncoupled such that they can be treated separately. Examples: hydraulics, low-speed aerodynamics, ship hydrodynamics, liquid and low-speed gas pipe systems Gas Dynamics: flow of fluids for which density is variable such as high-speed gases. Temperature and heat transfer effects are coupled and must be treated concurrently. Examples: high-speed aerodynamics, gas turbines, high-speed gas pipe systems, upper atmosphere

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