Pressure Flow Reading Page PHYS-FL-01
Pressure Flow
Bernoulli's Principle and Energy Conservation in Fluids
The Ideal Fluid
To simplify fluid dynamics, we use the Ideal Fluid model. In AP Physics, fluids are assumed to be:
Incompressible
Non-viscous
Laminar Flow
Steady Velocity
Conservation Law
"Energy is conserved along a streamline. A gain in speed requires a loss in pressure or potential energy."
The Bernoulli Equation
\( P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \)
STATIC PRESSURE Work done by fluid pressure
DYNAMIC PRESSURE Kinetic Energy density
HYDROSTATIC Potential Energy density
The Venturi Effect: Velocity vs. Pressure
LOW v HIGH P HIGH v LOW P
Fig 1.1: Velocity-Pressure Tradeoff
The Continuity Equation (\(A_1 v_1 = A_2 v_2\)) ensures mass is conserved. When the pipe narrows, the fluid must accelerate.
Acceleration requires a net force, provided by higher pressure behind pushing into lower pressure ahead.
FLUID DYNAMICS // BERNOULLI PAGE 1 OF 2
Real World Applications
Aerodynamic Lift
Airfoils force air to travel faster over the top than the bottom, creating a pressure difference.
Top: High Velocity \(\rightarrow\) Low PBottom: Low Velocity \(\rightarrow\) High P
LOW P HIGH P
Concept Check
1. The Attic Effect: Tornadoes and Roofs
Why does a high-speed wind blowing over a stationary house cause the roof to blow off rather than collapse?
2. Energy Balance
In a vertical pipe where speed is constant, how does pressure change with height? Justify with Bernoulli's equation.
UNIT: FLUID DYNAMICS // AP PHYSICS 1 PAGE 2 OF 2
Flow Mastery Slides Module 04 // Fluid Dynamics
Flow Mastery
Continuity, Bernoulli, Venturi, & Torricelli
1. The Continuity Equation
Based on the Conservation of Mass . In an incompressible fluid, the volume entering a pipe must equal the volume leaving it.
Mass Conservation
\( A_1 v_1 = A_2 v_2 \)
Decreasing Area \(\rightarrow\) Increasing Velocity
LARGE A // LOW v SMALL A // HIGH v
Fig 01: Flow Velocity Balancing
2. Bernoulli's Theorem
Conservation of Energy
\[ P + \frac{1}{2}\rho v^2 + \rho gh = C \]
Static Pressure
Energy density from internal fluid pressure.
Kinetic energy
Energy density due to fluid motion (\(v\)).
Potential energy
Energy density due to elevation (\(h\)).
The Core Logic
Trade-Off Rule
If kinetic energy increases, pressure or potential energy must decrease to maintain balance.
Ideal Flow
Assumes zero viscosity, constant density, and steady flow.
Application Alpha
The Venturi Effect
The Velocity Trade
As fluid enters a narrow throat, the Continuity Equation forces it to speed up. Bernoulli's Theorem then dictates that the static pressure must drop to compensate for the higher kinetic energy.
Wider Section Low v // High P
Narrow Section High v // Low P
HIGH P LOW P
Dynamic Pressure Gradient Visualization
3. Torricelli's Law
The Leaking Tank Paradox
POINT 1: Surface (P atm) POINT 2: Hole (P atm) h (Liquid Head)
Efflux Velocity
\[ v = \sqrt{2gh} \]
The velocity of fluid exiting a tank depends only on the height of the fluid column above the opening and gravity.
Mathematically identical to a solid object in free fall!
Connecting the Dots
How Bernoulli creates Torricelli
Step 1
Both Point 1 (surface) and Point 2 (hole) are open to air \(\rightarrow\) \(P_1 = P_2 = P_{\text{atm}}\) .
Step 2
The tank area is massive compared to the hole \(\rightarrow\) \(v_1 \approx 0\) . Surface KE is negligible.
\[ \rho gh_1 = \frac{1}{2}\rho v_2^2 + \rho gh_2 \]
\[ v_2 = \sqrt{2g\Delta h} \]
Fluid Mastery Summary
Continuity
Area shrinks? Flow speeds up. Volume flow rate remains constant. Mass is conserved.
Bernoulli
Total energy density (P, KE, PE) is constant. Flow is a trade-off between energy states.
Venturi
High velocity through constrictions causes a measurable drop in static pressure.
Torricelli
The gravity-driven case where velocity depends only on depth. Equivalent to free fall.
System Status: Ideal Flow Active
AP Physics 1 // Unit 4
Flow Check Exit Ticket Unit 4: Fluids
Flow Check Exit Ticket
Name:
Date:
1
The Constricted Rising Pipe
An ideal fluid flows through a pipe from Point A (wide, low elevation) to Point B (narrow, high elevation).
A) According to the Continuity Equation, how does fluid velocity change at Point B? Explain.
B) Using Bernoulli's Principle, identify which forms of energy density increased and which decreased at Point B.
POINT A POINT B
2
The Efflux Challenge
A water tank is filled to height H . If the tank is moved to a planet with twice the gravitational acceleration (\(2g\)), what happens to the efflux speed? Use Torricelli's Law to justify.
3
The Venturi Paradox
"When a fluid is 'squeezed' through a narrow pipe, the pressure should increase." Refute this statement using the Work-Energy Theorem.
Self-Check:
Got it!
Almost
Help
Score
Fluid Force Worksheet Fluid Force
Bernoulli & Continuity Problem Set
Name:
Date:
Continuity \( A_1 v_1 = A_2 v_2 \)
Bernoulli Equation \( P + \frac{1}{2}\rho v^2 + \rho gh = \text{const} \)
Constants: \(\rho_w = 1000 \text{ kg/m}^3\) // \(g = 9.8 \text{ m/s}^2\)
01
Horizontal Pipe Constriction
Water flows through a pipe. At A (\(10 \text{ cm}^2\)), velocity is \(2.0 \text{ m/s}\). At B (\(5.0 \text{ cm}^2\)), the pipe narrows.
Part A: Velocity at B
Part B: Pressure at B (\(P_A = 1.5 \times 10^5 \text{ Pa}\))
FLUID DYNAMICS // PROBLEM SET PAGE 1 OF 2
02
Torricelli's Tank Leak
A large water tank is open to atmosphere. A leak develops \(5.0 \text{ meters}\) below the water surface.
Derive the efflux speed using Bernoulli's equation. Assume surface velocity is zero.
03
Qualitative: Sheet Attraction
Explain why two vertical sheets move together when air is blown rapidly between them.
FLUID DYNAMICS // PROBLEM SET PAGE 2 OF 2
Bernoulli Deep Dive Reading Page Extended Technical Guide // FLUID-01
The Bernoulli Foundation
Energy Conservation in Dynamic Fluids
1. The Ideal Fluid Constraint
Bernoulli’s Principle operates within a specific set of physical boundaries. To apply these energy conservation laws accurately, physicists use the Ideal Fluid model. For Bernoulli’s Theorem, we assume:
Incompressible
Density \((\rho)\) is uniform. The volume of a liquid parcel does not change under external pressure.
Non-Viscous
No internal "stickiness." There is zero mechanical energy loss due to friction between fluid layers.
Laminar Flow
Fluid moves in smooth, steady streamlines that never cross. There are no turbulent swirls or eddies.
Irrotational
The fluid parcel does not rotate about its center of mass as it translates through the pipe.
2. Bernoulli’s Theorem: Energy Bookkeeping
Bernoulli’s Principle is the Work-Energy Theorem applied to a moving fluid. It states that for a steady, ideal flow, the sum of all forms of energy density along a streamline is constant.
\( P + \frac{1}{2}\rho v^2 + \rho gh = \text{Constant} \)
Static Pressure Work done by internal fluid forces \(P\).
Kinetic Energy Density of motion \(\frac{1}{2}\rho v^2\).
Potential Energy Density of elevation \(\rho gh\).
Physics Insight: The Work Link
When a fluid parcel accelerates, work must be performed on it. This work is done by the pressure gradient . Fluid at the high pressure side "pushes" harder than fluid at the low pressure side, resulting in net work on the parcel.
FLUID DYNAMICS // COMPREHENSIVE GUIDE PAGE 01 OF 03
3. The Anatomy of Flow
The diagram below illustrates the General Case of Bernoulli's Principle. As the fluid moves from Point 1 to Point 2, area decreases (Continuity) and elevation increases.
DATUM (h=0) POINT 1 Area A1 (Wide) Velocity v1 (Slow) Pressure P1 (High) h1 POINT 2 Area A2 (Narrow) Velocity v2 (Fast) Pressure P2 (Low) h2
Energy Balance
Potential (\(\rho gh\)) ↑
Kinetic (\(\frac{1}{2}\rho v^2\)) ↑
Static Pressure (\(P\)) ↓
Potential and Kinetic gains are compensated by Static Pressure loss.
4. The Continuity Constraint
The change in velocity \(v\) seen above is governed by the Equation of Continuity . In an incompressible fluid, the volume flow rate \(Q = Av\) must be constant.
Fluid Force Answer Key Answer Key
Fluid Force Worksheet // Instructor Guide
SOLUTIONS
1 The Constricted Pipe
Part A: Velocity Analysis
\( A_1 v_1 = A_2 v_2 \)
\( v_2 = 4.0 \text{ m/s} \)
Part B: Bernoulli Application
\( P_A + \frac{1}{2}\rho v_A^2 = P_B + \frac{1}{2}\rho v_B^2 \)
\( P_B = 1.44 \times 10^5 \text{ Pa} \)
2 Water Tank Efflux
At surface (1): \( P_{atm}, v \approx 0, h=5.0 \). At hole (2): \( P_{atm}, v=?, h=0 \).
\( v_2 \approx 9.9 \text{ m/s} \)
3 Qualitative Guide
Core Principle: Kinetic Energy Density vs. Static Pressure.
Air between sheets has higher speed \(\rightarrow\) higher kinetic energy density. According to Bernoulli, the static pressure between the sheets must be lower than the stationary atmospheric pressure outside. This pressure gradient pushes the sheets together.
UNIT: FLUID DYNAMICS SOLUTIONS KEY // AP PHYSICS 1
Torricelli Efflux Lab Activity Lab Protocol // PHYS-FLUID-LB
Torricelli’s Efflux Lab
Validation of Bernoulli’s Theorem
Objectives
Link fluid height (\( h \)) to efflux velocity (\( v \)).
Use kinematics to find \( v \) from range (\( x \)).
Identify real-world resistive forces (viscosity, air drag).
Materials
• Large 2L Bottle
• Precision Hole Punch
• Meter Stick & Level
• Stopwatch / Tape
Theoretical Background
From Torricelli’s Law , theoretical speed \( v \) is:
\( v = \sqrt{2gh} \)
Experimental velocity is solved via range \( x \) and vertical drop height \( y \):
\( v_{expt} = x \cdot \sqrt{\frac{g}{2y}} \)
Experimental Setup
h y RANGE (x)
FLUID DYNAMICS // LAB INVESTIGATION PAGE 01 OF 02
1. Procedure & Data Collection
Trial Depth \( h \) (m) Range \( x \) (m) Expt. \( v \) (m/s) Theory \( v \) (m/s) 1 2 3 4 5
2. Analysis: Construct a sketch of \( x^2 \) vs. \( h \).
DEPTH (h)
RANGE SQ (x²)
Lab Exit Ticket
Q1. If the hole diameter was doubled, would the efflux velocity change? Justify using Torricelli's Law.
Q2. Identify one specific resistive force that caused your experimental velocity to be lower than theory.
FLUID DYNAMICS // LAB INVESTIGATION PAGE 02 OF 02
Bernoulli Visual Guide Reference Chart Ref Doc: B-PRIN-VISUAL-01
Bernoulli's Principle
Visual Dynamics & Energy Conservation
SYSTEM: IDEAL FLUID
FLOW: LAMINAR / STEADY
The General Bernoulli Equation
\( P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2 \)
STATIC PRESSURE Internal energy density provided by the fluid's pressure.
DYNAMIC PRESSURE Kinetic energy density due to the fluid's motion.
HYDROSTATIC Potential energy density due to height in gravity.
Visual Mapping
Area A1 Pressure P1 Velocity v1 DATUM LEVEL (h=0) Area A2 Pressure P2 Velocity v2 HEIGHT INCREASE PIPE GEOMETRY A2 is SMALLER
Trade-off 1: Velocity vs Pressure
In horizontal flow, as velocity increases (narrower sections), dynamic pressure increases . Total energy balance requires a corresponding drop in static pressure .
Trade-off 2: Elevation vs Pressure
As a fluid rises, potential energy density increases . To maintain balance, either velocity must decrease (uphill slowdown) or static pressure must drop.
