Fluid Dynamics: Energy & Momentum

Fluid dynamics considers the forces causing fluid motion. The two key principles are Conservation of Energy and Conservation of Momentum.

Bernoulli's Equation

Bernoulli's equation relates pressure, velocity, and elevation for an ideal, incompressible fluid in steady flow. It states that the total mechanical energy remains constant along a streamline.

$\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2$

• Pressure Head: $P/\gamma$
• Velocity Head: $V^2/2g$
• Elevation Head: $z$
• Total Head ($H$): Sum of the three components.

Energy Line (EL) and Hydraulic Grade Line (HGL)

• EL: Plot of total head ($H$).
• HGL: Plot of piezometric head ($P/\gamma + z$). The HGL is always below the EL by the velocity head.

Impulse-Momentum Equation

Derived from Newton's Second Law ($F=ma$). It states that the sum of external forces acting on a fluid control volume equals the rate of change of momentum.

$\sum \vec{F} = \rho Q (\vec{V}_{out} - \vec{V}_{in})$

Common applications include:

• Force exerted by a jet on a plate.
• Force on a pipe bend.

Solved Problems