Part II Fundamentals of Fluid Mechanics By Munson, Young, and Okiishi


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1 Part II Fundamentals of Fluid Mechanics By Munson, Young, and Okiishi WHAT we will learn I. Characterization of Fluids  What is the fluid? (Physical properties of Fluid) II. Behavior of fluids  Fluid Statics: Properties of a fluid at rest (Physics of the pressure in fluids)  Fluid Dynamics: Behavior of a moving fluid Fluid kinetics and kinematics (Bernoulli Equation & Control volume analysis) Basic things of Fluids (Properties of Fluids) 1. How is a fluid different from a solid? Molecular spacing: Solid < Liquid < Gases Cohesive forces between molecules: Solid (Not easily deformed) > Liquid (Easily deformed, but not easily compressed) > Gases (Easily deformed and compressed) Fluid = Liquid + Gases A substance that deforms continuously when acted on by a shearing stress* of any magnitude * Shearing stress: Tangential force per unit area acting on the surface
2 . Heaviness of a fluid Density of a fluid, ρ : Mass per unit volume m ρ = (kg/m 3 ): Depending on pressure and temperature* V * Ideal gas law: p = ρrt where p (T): Absolute pressure (Temp.) R: Gas constant, 87.0 m /s K Specific Weight, γ : Weight (force) per unit volume γ = ρg (N/m 3 ) Specific Gravity, SG: Ratio of ρ of a fluid to ρ of water at 4 o C SG ρ = (Unitless) ρ o H 4 C 3. Compressibility of a fluid Bulk Modulus (Compressibility of fluid, when the pressure changes) Defined as E dp dp = v dv / = V dρ / ρ [lb/in or N/m ]  Large E v Hard to compress : because p (dp > 0), V (dv <0) Usually E v of liquid: Very large, (incompressible) w.r.t. gases
3 4. Fluidity of a fluid [Viscosity, μ i.e. flowing feature of a fluid] Consider a situation shown F A: Shearing stress ( F T = τa) (A: Area of upper plate) δ a : Displacement of top plate δβ : Rotation angle of line AB u(y): Fluid velocity at height y b y u U δa B B δβ A F T Step 1. Application of force F T (or Shearing stressτ)  Upper plate: Moving due to a shearing stress τ [Velocity = U] = Fluid velocity in contact with upper plate = u(b)  Bottom plate: no movement [Velocity = 0] = Fluid velocity in contact with bottom plate = u(0) Step. Deformation of Fluid If fluid velocity between two plates Vary linearly U i.e. u = u(y) = y b du U = b Special case!! For a short time period δ t, line AB rotates by an small angle δβ or, δa Uδt tan δβ δβ = = b b δβ U lim = = & γ : Shearing strain, (Function of F T ) δt 0 δt b Then, U du τ ( = F T ) & γ = A b = or du τ = μ
4 Viscosity μ : Absolute (or namic) viscosity [lb s/ft or N s/m ]  How easily (or fast) a fluid flows (deforms) due toτ  Large μ Difficult to flow  Depends on the temperature and type of a fluid* * Type of a fluid du 1. Newtonian Fluid: Linear relation between τ and. NonNewtonian: Nonlinear relation i) Shearing thinning (τ, μapparent ) e.g. Latex paint, suspension ii) Shearing thickening (τ, μapp ) e.g. watercorn starch, iii) Bingham plastic: e.g. Mayonnaise Newtonian Fluid c.f. Kinematic viscosity, μ ν = [ft /s or m /s] ρ
5 5. Speed of Sound in a fluid Propagation of Sound Wave Propagation of Disturbances (Oscillations) of fluid molecules Changes of p and ρ of the fluid due to acoustic vibration Speed of sound or Acoustic velocity, c dp c = = dρ E v ρ (since dp E v = ) dρ / ρ 6. Vapor pressure Evaporation: Escape of molecules from liquids to the atmosphere Equilibrium state of Evaporation in the closed container : Number of molecules leaving the liquid surface = No. of molecules entering the liquid surface Vapor pressure: Pressure on the liquid surface exerted by the vapors  Property of a fluid (V. P. of gasoline > V.P. of water)  Function of Temperature (T, Vapor Pressure )  High vapor pressure Easy to be vaporized (Volatility) Boiling (Formation of vapor bubble within a fluid) condition  When environmental (container) pressure = Vapor pressure e.g. Vapor pressure of water at 100 o C = 14.7 Psi (Standard atmospheric pressure)
6 7. Cohesivity of a fluid (Surface Tension, σ ) Surface tention, σ = Cohesive force ( Intermolecular attraction) Length along the boudary of surface  Property of a fluid (Especially at the boundary)  Molecules inside a fluid: No net attraction (Balanced cohesive force by surrounding molecules)  Molecules at the surface: Nonzero attraction toward the interior (Unbalanced force due to lack of outside molecules = Source of Tention) How can this unbalanced force be compensated? Surface Tensile force along the surface Number of molecular attraction per unit length (Intensity) : Surface tension, σ σ ( Length ) = ( Force) [ σ ] = N / m
7 Ex. 1 Spherical droplet cut in half Question: What is the inside pressure of a fluid drop? P i Let s cut the drop in half, then, Force due to σ [(σ ) (Length) = πrσ ] = Force due to the pressure difference [( Δ p) (Area)= Δ pπr ] σ R i.e. πrσ = ΔpπR or Δpπ R Δ p = p i p e σ = R > 0 Ex. Capillary action of liquid Q: Why do a liquid rise in a capillary tube? Strong (or Weak) molecular attraction between the wall and liquid Rise (Fall) of a liquid At the equilibrium, Vertical force due to surface tension ( πr σ cosθ ) = Weight of a liquid column ( mg = ρ Vg = γπr h) σ cosθ h = (Radius of tube R, then, h ) γr
8 Ex. 3 (Viscosity) The velocity distribution for the flow of a Newtonian fluid between to wide, parallel plates shown is give by the equation, 3V y u = 1 h where V is the mean velocity. The fluid has a viscosity μ of 0.04 lb s/ft. When V = ft/s and h = 0. in, determine (a) the shearing stress (τ ) acting on the bottom wall, and (b) the shearing stress acting on a plane parallel to the walls and passing through the centerline (midplane). Sol) Shearing stress: From the given equation, du τ = μ where μ = 0.04 lb s/ft du = d 3V 1 y h 3Vy = h 3μV The shearing stress as a function of height, τ = h y (a) Along the bottom wall (y =  h) 3 V 3(0.04lb s / ft )( ft / s) τ = μ = =14.4 lb/ft h 0.( in) (0.083 ft / in) (b) Along the midplane (y = 0) τ = 0 lb/ft
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