Review for exam1_2007 - Reviews for Exam1 Fall 2007 Chapter...

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Unformatted text preview: Reviews for Exam1 Fall 2007 Chapter 1 INTRODUCTION AND BASIC CONCEPTS 1. Fluids and no-slip condition • • Fluid: a substance that deforms continuously when subjected to shear stresses No-slip condition: no relative motion between fluid and boundary 2. Basic units Dimension ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ Velocity Acceleration Force Pressure Density Internal energy SI unit m⁄s m⁄s N (Kg m⁄s ) Pa (N⁄m ) Kg⁄m ⁄Kg (N m⁄kg) J 3. Weight and mass • W (N) = Kg • W (lbf) = • • • 1 N = 1 Kg × 1 m/s2 1 lbf = 1 slug × 1 ft/s2 1 slug = 32.2 lbm (weighs 32.2 lb under standard gravity) , where slug = 9.81 m/s2 , where = 32.2 ft/s2 4. Properties involving mass or weight of fluid • • Specific weight Specific gravity = (N/m3) =⁄ 5. Viscosity • Newtonian fluid: o Shear stress (N/m2) o o o • Coefficient of viscosity (Ns/m2) = ⁄ Kinematic viscosity (m2/s) Non-Newtonian fluid: Ex) Couette flow , BG unit ft⁄s ft⁄s lbf lbf⁄ft slug⁄ft BTU⁄lbm Reviews for Exam1 Fall 2007 6. 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 fluid velocity, the process is called cavitation. Cavitation number 1⁄2 • 0 implies cavitation 7. Surface tension • • • • Surface tension force = line force with direction normal to the cut = surface tension [N/m] = length of cut through the interface Chapter 2 PRESSURE AND FLUID STATICS 1. Absolute pressure, Gage pressure, and Vacuum • , • , = gage pressure = vacuum pressure Reviews for Exam1 Fall 2007 2. Pressure variation with elevation • For a static fluid, pressure varies only with elevation and is constant in horizontal , 0, • 0, If the density of fluid is constant, o = constant (piezometric pressure) = constant (piezometric head) o o 0 gage, : increase linearly with depth, decrease linearly with height 3. Pressure measurements (Manometry) 1) U-tube manometer • Δ ℓ Δ ℓ • Δ gage ℓ 2) Differential U-tube manometer ℓ ℓ Δ Δ • • • ℓ ⁄ o o ℓ ℓ ⁄ Δ ℓ ⁄ If fluid is a gass : If fluid is liquid & pipe horizontal ℓ Δ 1Δ Δ ℓ: planes. Reviews for Exam1 Fall 2007 4. Hydrostatic forces on plane surfaces 1) Horizontal surfaces • • Line of action is through centroid of , i.e., , , 2) Inclined surfaces • o o • • sin : pressure at centeroid of : 1st moment of area Magnitude of resultant hydrostatic force on plane surface is product of pressure at centeroid of area and area of surface Center of pressure o o : moment of inertia with respect to horizontal centeroidal axis For plane surfaces with symmetry about an axis normal to 0-0, 0 and Reviews for Exam1 Fall 2007 5. Hydrostatic forces on curved surfaces • ( ̂ : projection of onto plane to -direction) • ( ̂ : projection of onto plane to -direction) • V = weight of fluid above surface 6. Buoyancy • • • V Fluid weight equivalent to body volume V Line of action is through centeroid of V = center of buoyancy 7. Stability 1) Immersed bodies • • • • 0 and ∑ 0. Static equilibrium requires: ∑ ∑ 0 requires and the body is neutrally stable If is above : stable (righting moment when heeled) If is above : unstable (heeling moment when heeled) Reviews for Exam1 Fall 2007 2) Floating bodies • The center of buoyancy generally shifts when the body is rotated • Metacenter M: The point of intersection of the lines of action of the buoyant force before and after heel • GM • • o GM: metacentric height o = moment of inertia of waterplane area about centerplane axis GM > 0: stable (M is above G) GM < 0: unstable (G is above M) CG 8. Fluids in rigid-body motion • If no relative motion between fluid particles • For rigid body translation: ̂ ̂ o 0, 0, • • • • • • increase in decrease in 0, decrease in , decrease in 0 and | | , increase in 0 and | | constant ⇒ but slower than • • ̂ = unit vector in direction normal of Ω̂ For rigid body rotation: Ω̂ o Ω o o Ω 0 constant or Ω constant ( : curves of constant pressure ( Ω) : pressure at (r,z)=(0,0)) Reviews for Exam1 Fall 2007 Chapter 3 BERNOULLI EQUATION 1. Flow patterns • • • Stream line: a line that is everywhere tangent to the velocity vector at a given instant Pathline: the actual path traveled by a given fluid particle Streakline: the locus of particles which have earlier passed through a particular point 2. Streamline coordinates • Velocity : V , • , Acceleration: o = local in ̂ direction o = local in o direction = convective due to spatial gradient of V o = convective due to curvature : centrifugal acceleration o : the radius of curvature of the streamline 3. Bernoulli equation • Euler equation: • Along streamline 2 or 2 • Across streamline • Assumptions o Inviscid flow o Steady flow o Incompressible flow o Flow along a streamline 2 Reviews for Exam1 Fall 2007 4. Applications of Bernoulli equation 1) Stagnation tube • , 0, , • 2 2) Pitot tube • 0, • 2 from manometer or pressure gage 3) Simplified continuity equation • Volume flow rate: • Mass flow rate: • Conservation of mass: • For incompressible flow ( =constant): or 4) Flow rate measurement • If the flow is horizontal ( • If velocity profiles are uniform at sections (1) and (2), • Flow rate is, Ex) Venturi meter , steady, inviscid, and incompressible, ⁄ Reviews for Exam1 Fall 2007 Chapter 4 FLUIDS KINEMATICS 1. Velocity and description Methods • Lagrangian: keep track of individual fluids particles ̂ • ̂ Eulerian: focus attention on a fixed point in space , ̂ ̂ 2. Acceleration and material derivatives • Lagrangian: ̂ • ̂ Eulerian: ̂ ̂ where, ̂ • • • ̂ gradient operator = local or temporal acceleration. Velocity changes with respect to time at a given point. = convective acceleration. Spatial gradients of velocity Material (substantial) derivative Reviews for Exam1 Fall 2007 3. Flow classification • • • • • • • • One-, Two-, and Three-dimensional flow Steady vs. Unsteady flow Incompressible and Compressible flow Viscous and Inciscid flow Rotational vs. Irrotational flow Laminar vs. Trubulent viscous flow Internal vs. External flow Separated vs. Unseparated flow 4. Reynolds Transport Theorem (RTT) Special Cases: • Non-deforming CV moving at constant velocity: • Fixed CV: • Steady flow: • Uniform flow across discrete CS (steady or unsteady): 0 ∑ , 5. Continuity equation 0 Simplifications: • • Steady flow: = constant over discrete 0 ∑ (flow sections): • Incompressible fluid ( = constant): • Steady One-dimensional flow in a conduit: ∑ const (conservation of volume) 0, 0, for = ...
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