8.5 Bernoulli's Equation

• Bernoulli's Equation relates pressure, velocity, and height for an incompressible fluid flowing steadily
• What is the mathematical expression for Bernoulli's Equation?

  $P +$$\frac{1}{2}\rho v^{2} +$$\rho gh =$$constant$
• Match the terms in Bernoulli's Equation with their meanings:

  P ↔️ Static pressure

  \rho ↔️ Density of the fluid

  v ↔️ Velocity of the fluid

  h ↔️ Height of the fluid
• Bernoulli's Equation assumes the fluid is incompressible
• Bernoulli's Equation applies to fluids with changing velocities over time.

  False
• Bernoulli's Equation assumes the fluid is non-viscous
• Bernoulli's Equation applies along a streamline or flow tube.
• Match the relationships in Bernoulli's Equation with their descriptions:

  Higher pressure ↔️ Lower velocity and height

  Higher velocity ↔️ Lower pressure and height

  Higher height ↔️ Lower pressure and velocity
• What is the relationship between pressure and velocity in Bernoulli's Equation?

  Higher velocity, lower pressure
• Higher pressure in a fluid implies lower velocity and height
• Bernoulli's Equation is based on the conservation of energy principle.
• Bernoulli's Equation assumes the density of the fluid remains constant
• What theorem is used to derive Bernoulli's Equation?

  Work-energy theorem
• The work done by pressure gradient forces is given by (P_1 - P_2)A \Delta x
• What is the change in kinetic energy of a fluid element in Bernoulli's Equation derivation?

  $\Delta KE =$$\frac{1}{2}\rho A \Delta x (v_{2}^{2} - v_{1}^{2})$
• In the derivation of Bernoulli's Equation, the work done is equated to the sum of kinetic and potential energy changes.
• In flow through pipes, Bernoulli's Equation relates pressure, velocity, and height
• How do aircraft wings generate lift according to Bernoulli's Equation?

  Pressure difference
• Venturi meters measure flow rate by measuring the pressure difference across a constricted section.
• The lift on aircraft wings can be calculated using the formula Lift = \frac{1}{2}\rho (v_{above}^2 - v_{below}^2)A
• What does Bernoulli's Equation relate for fluid flow in pipes?

  Pressure, velocity, and height
• Bernoulli's Equation explains how aircraft wings generate lift by creating a pressure difference due to varying airflow velocities above and below the wing
• What do Venturi meters measure using Bernoulli's Equation?

  Flow rate
• Bernoulli's Equation assumes that the fluid is incompressible, meaning its density remains constant.
• Bernoulli's Equation assumes that the fluid is non-viscous, meaning it has no internal friction
• What happens to the pressure in a fluid when its velocity increases according to Bernoulli's Equation?

  It decreases
• Steps to derive Bernoulli's Equation from the work-energy theorem

  1️⃣ Apply the work-energy theorem

  2️⃣ Express the work done by pressure gradient forces

  3️⃣ Calculate the kinetic energy change

  4️⃣ Calculate the potential energy change

  5️⃣ Combine terms to derive Bernoulli's Equation
• What fundamental principle is Bernoulli's Equation derived from?

  Conservation of energy
• Bernoulli's Equation states that the total mechanical energy of a fluid remains constant along a streamline
• What theorem is applied to derive Bernoulli's Equation?

  Work-energy theorem
• The work done on a fluid element equals its change in kinetic energy.
• What is the formula for the work done by pressure gradient forces in a fluid?

  $W =$$(P_{1} - P_{2})A \Delta x$
• In the work formula, \( P_1 \) and \( P_2 \) represent the pressures at two points.
• What is the formula for the change in kinetic energy of a fluid element?

  \Delta KE = \frac{1}{2}\rho A \Delta x (v_2^2 - v_1^2)</latex>
• In the kinetic energy change formula, \( \rho \) represents the density of the fluid.
• The change in potential energy of a fluid element depends on the acceleration due to gravity.
• What is the formula for the change in potential energy of a fluid element?

  $\Delta PE =$$\rho g A \Delta x (h_{2} - h_{1})$
• The conservation of energy principle is used to equate the work done to the change in kinetic and potential energies.
• Match the variables in Bernoulli's Equation with their meanings:

  P ↔️ Pressure

  \( \rho \) ↔️ Density

  v ↔️ Velocity

  h ↔️ Height
• What is the final form of Bernoulli's Equation after rearranging terms?

  $P_{1} +$$\frac{1}{2}\rho v_{1}^{2} +$$\rho gh_{1} =$$P_{2} +$$\frac{1}{2}\rho v_{2}^{2} +$$\rho gh_{2}$