Notes on Fluid Mechanics — Continuity, Bernoulli, Dimensional Analysis

Continuity Equation

• One of the most important equations in fluid mechanics.
• Statement: The product of cross-sectional area and fluid velocity at any point along a pipe is constant.
  • $Q = A v = ext{Constant}$
• Definitions:
  • $Q$ = volumetric flow rate or discharge
  • $A$ = cross-sectional area of the pipe
  • $v$ = velocity of flow
• Assumptions of the Continuity Equation:
  • 1) The tube has a single entry and a single exit
  • 2) The fluid flowing in the tube is non-viscous (no resistance to flow) [note: transcript wording is inconsistent; typically we assume incompressible, steady flow; viscosity neglected in continuity]
  • 3) The flow is steady (velocity of a fluid particle at a fixed position does not change with time)
• Key consequence: For incompressible flow, the mass flow rate is the same at any cross-section, so $$
  ho A v = ext{constant} \