Conservation of momentum is a crucial principle in fluid mechanics. It builds on Newton's second law, relating forces to changes in momentum for fluid systems. This concept is essential for analyzing fluid flow and predicting behavior in various engineering applications.

The conservation of momentum equation considers forces acting on a fluid, including pressure, shear, and body forces. It balances these forces with the rate of momentum change within a control volume and the net momentum flux across its boundaries. This powerful tool helps solve real-world fluid flow problems.

Conservation of Momentum in Fluid Mechanics

Conservation of momentum equation

• Starts with Newton's second law $F = ma = m \frac{dV}{dt}$ relates net force to change in momentum
  • $F$ net force acting on the system (pressure, shear, body forces)
  • $m$ mass of the system (fluid within control volume)
  • $a$ acceleration of the system (change in velocity)
  • $V$ velocity of the system (fluid velocity)
• For a control volume, consider rate of change of momentum within control volume and net flux of momentum across control surface
  • Rate of change of momentum within control volume $\frac{\partial}{\partial t} \int_{CV} \rho V dV$ accounts for unsteady flow
  • Net flux of momentum across control surface $\int_{CS} \rho V (V \cdot n) dA$ represents momentum entering and leaving
• Combine rate of change of momentum and net flux of momentum to obtain conservation of momentum equation $F = \frac{\partial}{\partial t} \int_{CV} \rho V dV + \int_{CS} \rho V (V \cdot n) dA$
  • $\rho$ fluid density (mass per unit volume)
  • $n$ unit normal vector pointing outward from control surface (defines flow direction)

Forces on control volumes

• Pressure forces act normal to control surface
  • Calculated using pressure distribution $F_p = \int_{CS} p n dA$
    • $p$ pressure at control surface (force per unit area)
• Shear forces act tangentially to control surface caused by viscous effects and velocity gradients
  • Calculated using shear stress distribution $F_s = \int_{CS} \tau dA$
    • $\tau$ shear stress at control surface (force per unit area)
• Body forces act on entire volume of fluid within control volume (gravity, electromagnetic forces)
  • Calculated using body force per unit volume $F_b = \int_{CV} \rho f dV$
    • $f$ body force per unit mass (acceleration)

Momentum equation applications

• Steady-state problems simplify conservation of momentum equation by setting rate of change of momentum within control volume to zero $F = \int_{CS} \rho V (V \cdot n) dA$
  • Use simplified equation to solve for unknown forces or velocities (pipe flow, nozzles)
• Unsteady problems consider rate of change of momentum within control volume $F = \frac{\partial}{\partial t} \int_{CV} \rho V dV + \int_{CS} \rho V (V \cdot n) dA$
  • Solve equation for unknown forces or velocities considering time-dependent nature (valve opening/closing, surge tanks)
• Apply appropriate boundary conditions and assumptions based on specific problem
  • Uniform velocity profiles (plug flow)
  • Incompressible flow (constant density)
  • Negligible viscous effects (inviscid flow)

Momentum-flux correction factor

• Momentum-flux correction factor $\beta$ accounts for non-uniformity of velocity profile across cross-section
  • Defined as $\beta = \frac{\int_A V^2 dA}{V_{avg}^2 A}$ ratio of actual momentum flux to average momentum flux
    • $V_{avg}$ average velocity over cross-sectional area $A$
• For uniform velocity profiles, $\beta = 1$ (plug flow)
• For non-uniform velocity profiles, $\beta > 1$
  • Laminar flow in circular pipe $\beta = \frac{4}{3}$ (parabolic profile)
  • Turbulent flow in circular pipe $\beta \approx 1.02$ to $1.10$ (flatter profile)
• Incorporate momentum-flux correction factor into conservation of momentum equation $F = \frac{\partial}{\partial t} \int_{CV} \rho V dV + \int_{CS} \beta \rho V_{avg} (V_{avg} \cdot n) dA$