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  • 1. FLUID MECHANICSDepartment of Nuclear Engineering and Fluid MechanicsUniversity College of EngineeringUniversity of Basque Country (EHU/UPV)Vitoria-Gasteiz Instructor: Iñigo Errasti Arrieta
  • 2. CONTENTS LESSON 1. INTRODUCTION LESSON 2. FLUID STATICS LESSON 3. FLUID KINEMATICS LESSON 4. FLUID DYNAMICS LESSON 5. THE ENERGY EQUATION LESSON 6. APPLICATIONS OF BERNOULLI EQUATION LESSON 7. LINEAR MOMENTUM THEOREM LESSON 8. DIMENSIONAL ANALYSIS AND SIMILITUDE LESSON 9. INCOMPRESSIBLE VISCOUS FLOW LESSON 10. ENERGY LOSSES IN PIPES LESSON 11. STEADY-STATE FLOW IN PIPES LESSON 12. TRANSIENT REGIMES IN PIPES LESSON 13. FLOW THROUGH OPEN CHANNELS LESSON 14. PUMPS AND TURBINES
  • 3. LESSON 1. INTRODUCTION TO FLUID MECHANICS 1. Field of application of Fluid Mechanics 2. Brief history of Fluid Mechanics 3. Fluid as a continuum. Fluid definition 4. Dimensions and Units 5. Operators 6. Physical properties of fluids
  • 4. 1. Field of application of Fluid Mechanics “Fluid Mechanics”, definition Physical phenomena in nature Engineering Other aspects in common life Main branches: • Statics • Kinematics • Dynamics • Aerodynamics • Computational Fluid Dynamics (CFD)
  • 5. 1. Field of application of Fluid Mechanics Weather & climate Vehicles Environment
  • 6. 1. Field of application of Fluid Mechanics Physiology and medicine Sports & Recreation
  • 7. 2. Brief history of Fluid Mechanics Archimedes Mariotte, Torricelli, Pascal, Castelli Newton, Bernoulli, Euler, D’Alembert Chezy, Navier, Coriolis, Darcy Pouiseuille, Hagen, Reynolds, Stokes Froude, Francis, Pelton, Herschel Thomson, Kelvin, Rayleigh, Lamb Prandtl, von Karman, Blasius Taylor, Kolmogorov, Nikuradse
  • 8. 2. Brief history of Fluid Mechanics Archimedes Newton Leibniz Bernoulli Euler (287-212 BC) (1642-1727) (1646-1716) (1667-1748) (1707-1783) Navier Stokes Reynolds Prandtl Taylor Kolmogorov(1785-1836) (1819-1903) (1842-1912) (1875-1953) (1886-1975) (1903-1987)
  • 9. 3. Fluid as a continuum. Definition of fluid Definition of fluid Comparison to solid States of matter (liquid and gas) Modelling the fluid as a continuum
  • 10. 3. Fluid as a continuum. Definition of fluidComparison to solid Time t0 Time t1 Time t2 - Deformation of solid: F r F Ф1 r F Ф2 = Ф1 r F invariable with time τ= Solid Solid Solid A τ : Shear stress Time t0 Time t1 Time t2 F: Shear force - Deformation of fluid: r Ф2 > Ф1 r A: Contact area r Ф1 F continuous with time F F Fluid Fluid FluidFigure 1.1. Deformation of solids and fluidsFigure 1.2b. Molecules are at relatively fixed positions in a solid. Figure 1.2a. Unlike a liquid, a gas does notGroups of molecules move about each other in the liquid phase. form a free surface and it expands to fill theMolecules move around at random in the gas phase (Cengel-Cimbala) entire available space (Cengel-Cimbala)
  • 11. 3. Fluid as a continuum. Definition of fluidThe model of the continuum - Differential volume range for analysing the fluid as a continuum - Definition of the density in a fluid as a continuum Figure 1.3. Differential volume in a fluid Figure 1.4. Density calculated in function of region with varying density (from White) the differential volume (from White) δm ρ = lim δV →δV * δ 0V
  • 12. 4. Dimensions and Units Dimensions, magnitudes and units Primary dimensions Secondary dimensions SI and English systems Conversion ratios Some SI units: English units: mass: kg pound-mass (lbm) length: meter foot (ft) time: second second (s) force: newton pound-force (lbf) work: joule British thermal unit (btu)
  • 13. 4. UnitsPrefixes, dimensions and unity conversion factors Table 1.1. Prefixes Table 1.2. Units in the SI and US system Table 1.3. Unity conversion factors
  • 14. 5. Operators• Gradient of a scalar function f: r r ∂f ( x, y, z ) r ∂f ( x, y, z ) r ∂f ( x, y, z ) r gradf ( x, y, z ) = ∇f = i+ j+ k ∂x ∂y ∂z r• Divergence of a vector function f : r r r ∂f x ( x, y , z ) ∂f y ( x, y, z ) ∂f z ( x, y, z ) divf ( x, y, z ) = ∇f = + + ∂x ∂y ∂z r • Curl of a vector function f : r r r i j k r r r r ∂ ∂ ∂ cu rlf ( x, y, z ) = ∇xf = ∂x ∂y ∂z f x ( x, y , z ) f y ( x, y , z ) f z ( x, y , z )
  • 15. 6. Physical properties of fluids 1. Density, specific weight and specific gravity 2. Pressure 3. Ideal gas equation of state 4. Compressibility 5. Viscosity 6. Vapour pressure. Saturation pressure. Cavitation 7. Surface tension and capillary effect
  • 16. 6.1. Density and specific weight Density (definition, dimensions, units) Specific weight (definition, dimensions, units) Relative density or specific gravity (definition)
  • 17. 6.1. Density, specific weight and specific gravity• Density (definition) m dm ρ= ρ= dV V dV dm Mass per unit volume (kg/m3) V Figure 1.5a. Concept of density of a fluid 1• Specific volume (definition) Vs = ρ • Specific weight (definition) γ = ρg ρ SG = • Specific gravity or relative density (definition) ρH O 2
  • 18. 6.1. Density, specific weight and specific gravity Figure 1.5b. Approximate physical properties of common liquids at atmospheric pressure
  • 19. 6.2. Pressure Definition (2 forms) Dimensions. Units Pressure level with different references
  • 20. 6.2. Pressure. Units• Definition (2 forms) Atmosphere Ratio of normal force (Fn) to area at a point Fn A Fn P= AFigure 1.6. Pressure in plane A r dA ΔFn dFn r P = lim = dFn ΔA→0 ΔA dA 1 Pa= 1 Nw m-2 A 1 baria = 1 dyne cm-2 V 1 atm = 760 mmHg=1.013 x 105 Pa = 10.33 mwcFigure 1.7. General concept of pressure = 2116 lbf ft-2 1 psi = 6895 Pa
  • 21. 6.2. Pressure • Pressure references Absolute pressure (pabs): pressure relative to absolute zero, absolute vacuum, p = 0 Pa. pabs > patm ; pgauge > 0 (overpressure) Gauge pressure (pgauge): pressure relative to the local atmospheric pressure. pabs = patm ; pgauge = 0 pabs = p gage + patm pabs < patm ; pgauge < 0 • Example 1: (vacuum, suction) A gage pressure of 50 kPa recorded in a location where the atmospheric pressure is 100 kPa is pabs = 0 expressed as either p= 50 kPa gage or p=150 kPa abs • Example 2: Figure 1.8. Pressure references
  • 22. 6.3. Ideal gas equation of state • Definition of the equation of state • Equation of state for an ideal gas pV = nRT R = 8.314 J / Kmol K Universal gas constant • Other forms of the equation of state ⎡R⎤ p = ρR * T R* = ⎢ ⎥ ⎣M ⎦ (J/kg K) M: molecular weight (Kg/Kmol) of the gas p = γR T ⎡ R ⎤ R = ⎢ ⎥ ⎣ Mg ⎦ (m/K)
  • 23. 6.4. Compressibility and elasticity Parameters: dp E =− • Bulk modulus of elasticity E: (dV / V ) 1 E= (dV / V ) α • Compressibility α: α=− dp F A p=F/A ; V Increase in F (p + dp) ; (V + dV) V Negative value (decrease) Figure 1.9. Decrease in volume by an increase in pressure
  • 24. 6.4. Compressibility and elasticity Table 1.4 Values of bulk modulus of elasticity for some liquids Liquid E (GPa) Water 2,07 Ethanol 1,21 Benzene 1,03 Carbon tetrachloride 1,10 Mercury 26,20 Ideal gases: E = kp Newtons formula for speed of sound c2 = dp E = (Newton – Laplace equation): dρ ρ
  • 25. 6.5. Viscosity 1. Viscosity: Dynamic and kinematic 2. Newton’s law of viscosity 3. Rheological diagram 4. Dependence on pressure and temperature 5. Viscometers
  • 26. 6.5. Viscosity Definition Physical phenomena causing viscosity: • Intermolecular cohesion: r r U2 U r 2 τ U1 r U1 Figure 1.10. Influence of intermolecular cohesion on viscosity ● Collisional exchange of momentum: r r U2 U2 τ r U1 r U1 Figure 1.11. Influence of momentum exchange on viscosity
  • 27. 6.5. Newton’s law of viscosity • Velocity profile, shear stress, deformation: τ τ y U + dU U + dU ds dθ dV dy dV dV U U μ: Dynamic viscosity τ τ U=0 τ F: Shear force Time (t) Time (t + dt) A: Contact area Figure 1.2 Deformation of a fluid element F dθ dU ● Shear stress: τ= ● Deformation rate: = dt dy A dθ dU dθ dU • Newton’s law of viscosity: τ=μ =μ F = μA = μA dt dy dt dy • Inviscid flow hypothesis (ideal fluid) μ =0 τ =0
  • 28. 6.5. Rheological diagram Newtonian fluid dθ dU μ: Dynamic viscosity τ=μ =μ Pseudo plastic fluid dt dy Dilatant fluid Ideal plastic Ideal fluid Elastic solid Ideal plastic (Bingham) Dilatant τ = k ( gradU ) , n > 1 n τ (N/m2) r 1 gradU = (τ − τ 0 ) μ Newtonian Plastic Elastic Pseudo plastic τ = k ( gradU )n , n < 1 solid Ideal fluid dθ dU −1 = (s ) dt dy Figure 1.13 Rheological diagram
  • 29. 6.5. Dynamic and kinematic viscosity. Unities Dynamic viscosity (absolute viscosity) dθ dU • Newton’s law of viscosity: τ=μ =μ dt dy SI system: : 1 Pa s = 1 Poiseuille = 1 Nw m-2s CGS system: 1 poise = 1 dyne cm-2s Kinematic viscosity: μ ν= ρ SI system: : 1 m2s-1 CGS system: 1 cm2s-1 = 1 stoke
  • 30. 6.5. Dependence on pressure and temperature Effect of temperatureμ (Pa·s) μ↓ as T↑ ν↓ decreases when T↑ - Liquid: ρ ≈ Cte (incompressible) Gas - Gas: μ ↑ con T↑ ν↑↑ increases when T↑ Liquid ρ ↓ con T↑ ( p = γR’T ) T (K) Sutherland correlation for viscosity of gases Figure 1.14 Dynamic viscosity of gases and liquids as a function of temperature Effect of pressure μ ≈ Cte μ ≈ Cte (∆p not excessive) - Liquid: ν ≈ Cte - Gas: ν↓ decreases when p↑ ρ ≈ Cte (incompressible) ρ ↑ as p↑
  • 31. 6.5. Dependence on pressure and temperature (Figures 1.15, 1.16 from White)
  • 32. 6.5. Viscometers a) b) Engler viscometer Redwood viscometer Brookfield viscometer Falling sphere viscometer c) d) Figures 1.17 Schematic of a) Engler, b) Redwood, c) Brookfield and d) falling sphere-type viscometers
  • 33. 6.6. Vapour pressure and saturation pressure Reviewing the concepts of vapour pressure and saturation pressure Figure 1.18. Phase diagram
  • 34. 6.6. Vapour pressure and saturation pressure Cavitation: Process of formation of the vapour of liquid when it is subjected to reduced pressure at constant ambient temperature Gaseous and vaporous cavitation Cavitation number: p − p sat p↓ , T p↑, T Ca = p, T boiling implosion 1 ρU 2 2 p < psat (T) p > psat (T) Figure 1.19. Schematic of the cavitation process
  • 35. 6.6. Vapour pressure and saturation pressureFigure 1.20. Consequence of cavitation damage on Figure 1.21. Consequence of cavitation damagean impeller of a pump on an impeller of a pump. DetailFigure 1.22. Consequence of cavitation damage on Figure 1.23. Consequence of cavitation damage onan impeller of a pump. Profile view an impeller of a pump. Profile view
  • 36. 6.7. Surface tension and capillary effect Surface tension Figure 1.24. Forces acting on a liquid at the surface and deep inside (Cengel-Cimbala). F σ= l Capillary rise 2σ h= cos φ ρgR Figure 1.25. Capillary rise and fall of water and mercury in a small diameter glass tube (Cengel-Cimbala). Figure 1.26. Forces acting on a liquid column that has risen in a tube.

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