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Chapter 4: Flowing Fluids & Pressure Variation (part 2)

Chapter 4: Flowing Fluids & Pressure Variation (part 2). Review visualizations Frames of reference (part 1) Euler’s equation of motion. Quick review of visualizations. Pathline - follows path of a single “fluid particle” one particle one starting point several times

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Chapter 4: Flowing Fluids & Pressure Variation (part 2)

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  1. Chapter 4: Flowing Fluids & Pressure Variation (part 2) Review visualizations Frames of reference (part 1) Euler’s equation of motion

  2. Quick review of visualizations • Pathline - follows path of a single “fluid particle” • one particle • one starting point • several times • Streakline - connecting the dots from several particles passing the same point at different times • several particles • one starting point • one time • Streamline - tangent to the velocity • several particles • starting point irrelevant • tangent to velocity • one time • All three can be useful (depending on the flow) • Pathlines, streamlines, streaklines are the same in steady flows.

  3. Two helpful distinctions • Laminar vs. turbulent flow • Lagrangian vs. Eulerian descriptions • Lagrangian: follow particle • Eulerian: measure local velocities, etc.

  4. Quantitative description of motion • How do we quantitatively describe moton … of a solid particle?? … of a fluid particle??

  5. First pressure review (under pressure?) • How can we find pressure in a fluid: ….. in a gas?? ….. in a liquid?? Let’s start by answering this for static fluids.

  6. Euler’s equation • F=ma • Valid for inviscid, incompressible flow only!

  7. Euler’s equation • Consider the fluid-filled accelerating truck. • Where is the pressure greatest? • How can we calculate the pressure of B relative to that of A?

  8. Euler derivation, continued • Now… what about the pressure difference between B and C? Which is greater? • How can we calculate the pressure of C relative to that of B? Relative to that of A?

  9. Euler derivation, continued • Now, what do we do when g is not perpendicular to acceleration direction? • Let’s answer this for a more general case.

  10. Euler’s equation • F=ma • Valid for inviscid, incompressible flow only!

  11. Chapter 4: Flowing Fluids & Pressure Variation (part 3) Euler’s equation of motion review Bernoulli’s equation of motion Types of fluid motion (part 2) Rotational motion

  12. Bernoulli’s equation • F=ma along a streamline • Steady flow assumption required

  13. Bernoulli’s equation • Assumptions • Viscous effects are negligible • Steady flow (time-independent) • Incompressible flow • Valid along a streamline • Equation (think energy conservation)

  14. Bernoulli – a simple application ?? ?? ?? ?? (a) (b) (c)

  15. An analogous example: Holes in a soda bottle; flow from base of dam Now, how do we calaculate the velocity of the fluid as it leaves the tank? (or soda bottle)?

  16. Continuity • Q = flow rate • If v = average velocity through a cross sectional area of area A Q = vA

  17. Venturi What is the pressure at 2? Local vaporization can occur when the pressure falls below vapor pressure

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