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. Non-Newtonian: Non-linear relation i) Shearing thinning (τ, μapparent ) e.g. Latex paint, suspension ii) Shearing thickening (τ, μapp ) e.g. water-corn 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 mid-plane (y = 0) τ = 0 lb/ft

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