Lecture Notes: Fluid and Gas Flow Equations

Key Equations Discussed

• Continuity Equation
• Euler Equation

Continuity Equation

Overview

• Focuses on the conservation of mass in fluid flow.
• Relevant for flow through channels.
• Involves a streamtube formed by streamlines.

Equation Details

• Assumptions: Steady flow, ρ (density) and V (velocity) are mean values over the area.
• Variables:
  • Station 1: Velocity ( v_1 ), Area ( A_1 )
  • Station 2: Velocity ( v_2 ), Area ( A_2 )
• Formula:
  • Mass flow per second: ( \rho A V )
  • Continuity equation: ( \rho_1 A_1 V_1 = \rho_2 A_2 V_2 )
  • For incompressible flow: ( A V = \text{constant} )

Key Point

• The equation states that the mass per second entering a streamtube is equal to the mass per second exiting.

Euler Equation

Overview

• Based on conservation of momentum.
• Applies Newton’s second law to a flowing gas.

Derivation

• Consider a volume of air following a streamline with dimensions ( dx, dy, dz ).
• Forces involved: pressure force, friction force, gravity force (gravity and friction neglected for the derivation).

Calculation

• Pressure Force:

  • At left side: Pressure ( P )
  • At opposite side: ( P + \Delta P )
  • ( \Delta P = \frac{dP}{dx} dx )
  • Resulting force in x-direction: (-\frac{dP}{dx} dx \cdot dy \cdot dz)

• Mass and Acceleration:

  • Mass ( m = \rho \cdot dx \cdot dy \cdot dz )
  • Acceleration: ( a = \frac{dV}{dx} \cdot V )

Final Equation

• Combining terms: ( dP = -\rho V dV )
• Application: Applies to compressible flow due to the presence of ρ.

Historical Context

• Derived by Leonhard Euler, a Swiss mathematician.

Notes for Next Lecture

• Discussion on Bernoulli family and further exploration on Euler's contributions.