Fluid Dynamics

Warmup

• Question: Which states of matter are considered fluids?

Chapter 8 Section 1 Objectives

• Review density

• Units of density

Extensive & Intensive Quantities

• Quantity Definitions:
    - Mass:
      - Definition: Amount of matter.
      - Unit: Kilograms (kg).
      - Measuring Devices: Balance.
      - Equation: N/A
    - Volume:
      - Definition: Space occupied.
      - Unit: Cubic meters (m³).
      - Measuring Devices: Graduated Cylinder.
      - Equation: $V = l imes w imes h$
    - Density:
      - Definition: Matter per unit space.
      - Unit: Grams per Cubic Centimeter (g/cm³) or kg/m³.
      - Measuring Devices: Calculated.
      - Equation: $<br>ho = rac{m}{v}$
    - Weight:
      - Definition: Force of gravity on an object.
      - Unit: Newtons (N).
      - Measuring Devices: Scale.
      - Equation: $F_g = mg$

States of Matter

• Mass, Volume, Density, and Shape:
    - Solid:
      - Mass: Constant
      - Volume: Constant
      - Density: Constant
      - Shape: Constant
    - Liquid:
      - Mass: Constant
      - Volume: Constant
      - Density: Variable
      - Shape: Variable
    - Gas:
      - Mass: Constant
      - Volume: Variable
      - Density: Variable
      - Shape: Variable

Fluid Physics

• Two Cases:
    - Fluid Statics: Fluids "at rest"
    - Fluid Dynamics: Fluids in motion

Finding Density from Slope

• Data:
    | Mass (kg) | Volume (x10⁻⁴ m³) |
    |--------------|-------------------|
    | 1.2 | |
    | 1.7 | |
    | 2.0 | |
    | 2.9 | |
    | 3.3 | |
    | 4.1 | |
    | 4.7 | |
    | 5.4 | |
    | 5.8 | |
    | 7.7 | |
  

• Graph: Mass (Kg) vs. Volume (x10⁻⁴ m³)

Chapter 8 Section 2 Objectives

• What is pressure?

• How to measure pressure?

Pressure – Force per Area

• Units of Pressure:
    - N/m²
    - Pascal (Pa)

• Atmospheric Pressure:
    - $100,000 ext{Pa}$

• Pressure Equation:
    - $P = rac{F}{A}$

Determining the Weight of a Car Tire

• Tire Pressures:
    | Position | Pressure (PSI) | Area (in²) |
    |-------------------|----------------|-------------|
    | Driver Front | 43 | 24 |
    | Passenger Front | 44 | 24 |
    | Driver Rear | 44 | 34 |
    | Passenger Rear | 45 | 34 |

• Force Calculation:
    - $F = P imes A$

• Calculated Forces:
    - Driver Front: 1032 lbs
    - Passenger Front: 1056 lbs
    - Driver Rear: 1496 lbs
    - Passenger Rear: 1530 lbs

• Sum of Force:
    - Total: 5114 lbs

Pressure Comparison in Footwear

• Question: Does a student apply a larger pressure to the ground when wearing running shoes or soccer cleats?
    - When wearing running shoes due to larger bottom area.
    - When wearing running shoes due to smaller bottom area.
    - When wearing soccer cleats due to larger bottom area.
    - When wearing soccer cleats due to smaller bottom area.

Pressure PhET

• Interactive simulation for understanding pressure concepts.

Absolute Pressure Equation

• Variables:
    - $P$: absolute pressure (Pa)
    - $P_o$: atmospheric pressure (Pa)
    - $<br>ho$: fluid density (kg/m³)
    - $g$: gravitational field strength (N/kg)
    - $h$: fluid depth (m)

• Gauge Pressure:
    - $<br>ho g h$

• Absolute Pressure Equation:
    - $P = P_o + <br>ho g h$

Practice Problems

• Problem 1: A valve closes off the end of a full pipe of water.
    - Calculate absolute pressure at point A. Given: density of water = 1000 kg/m³.

• Problem 2: How far underwater must a diver go to experience a gauge pressure equal to atmospheric pressure?

Chapter 8 Section 3 – Fluid Statics & Archimedes' Principle

Graphs Related to Fluid Force

• Graphs include:
    - Buoyant Force vs. Object Mass
    - Buoyant Force vs. Fluid Density
    - Buoyant Force vs. Volume of Displaced Fluid
    - Buoyant Force vs. Gravitational Field Strength

Archimedes’ Principle: Buoyant Force

• Variables:
    - $F_b$: buoyant force (N)
    - $<br>ho$: density of the fluid (kg/m³)
    - $V$: volume of displaced fluid (m³)
    - $g$: gravitational field strength (N/kg)

• Buoyant Force Equation:
    - $F_b = <br>ho V g$

Buoyant Force Comparison

• Comparison:
    - Object B has a larger buoyant force due to displacing a larger volume of water.

High Buoyant Force Scenario

• Example:
    - If both objects have the same volume, the object in honey experiences higher buoyant force because density of honey (1420 kg/m³) is higher than water (1000 kg/m³).

Buoyant Force of Different Materials

• Observation: Buoyant force is based on fluid density, not the density of the objects. So, if they are in the same fluid, buoyant force is identical.

Buoyant Force and Depth

• Principle: Depth does not affect buoyant force if the volume remains constant.

Buoyant Force and Orientation

• Observation: Orientation does not impact buoyant force as it does not change the object's volume.

Difference in Pressure at Two Depths

• Method: To determine difference in pressure, calculate absolute pressure at each depth and find the difference.

Pressure on Sides of a Cube

• Equations:
    - $ext{Pressure} = <br>ho gh + P_o$
    - $ext{Pressure} = rac{ ext{Force}}{ ext{Area}}$
    - $ext{Force} = ext{Pressure} imes ext{Area}$

Four Cube Comparison

• Key Point: In physics, floating means that the upward buoyant force equals the gravitational force.

Relationship Between Density and Volume

• Variables:
    - $P_{ ext{cube}}$
    - $P_{ ext{water}}$
    - $V_{ ext{water}}$
    - $ext{Water Displaced}$
    - $V_{ ext{cube}}$

Chapter 8 Section 4 – Bernoulli's Principle

Flow Rate of Water

• Question: What happens to the flow rate of water between point 1 and 2?

Conservation of Mass and Fluid Flow Rate

• Key Principles:
    - The mass of a system is conserved.
    - Water is considered incompressible (Ideal Fluid).
    - The volume of the liquid system is conserved.
    - The volume of water passing a point per second is constant, known as the Fluid Flow Rate $Q$.

• Units: $Q ext{(m}^3/ ext{s)}$

Kinetic Energy of Water in Pipeline

• Question: Where in the pipe does the kinetic energy of the water increase?

Bernoulli’s Equation

• Variables:
    - $P_1$: pressure at height 1
    - $v_1$: velocity at height 1
    - $y_1$: height 1
    - $P_2$: pressure at height 2
    - $v_2$: velocity at height 2
    - $y_2$: height 2

Continuity Equation for Fluid Flow

• Relationship:
    - $A_1 v_1 = A_2 v_2$

Torricelli’s Theorem

• Statement: A fluid under pressure ejected from a container will have the same exit velocity as if it were dropped from the height of the fluid within the container.

• Equation:
    - $rac{v^2}{2} + h + rac{<br>ho}{ ext{constant}}$

• Variables:
    - $V$: velocity of the liquid
    - $g$: acceleration due to gravity
    - $h$: height of the liquid over the reference point
    - $P$: atmospheric pressure at the top of the container
    - $<br>ho$: density of the fluid