Momentum equation and its applications

Derivation

Newton's second law for a system: the resultant external force equals the time rate of change of linear momentum of the system.

$$\sum \vec F = \frac{d(m\vec V)}{dt}\bigg|_{system}$$

Using the Reynolds transport theorem to convert from the system to a control volume (CV):

$$\sum \vec F = \frac{\partial}{\partial t}\int_{CV}\rho\vec V\,dV + \int_{CS}\rho\vec V(\vec V\cdot\hat n)\,dA$$

For steady flow the first term is zero. For a CV with one inlet (1) and one outlet (2), with uniform velocity over each area:

$$\sum \vec F = \dot m(\vec V_2 - \vec V_1) = \rho Q(\vec V_2 - \vec V_1)$$

In components:

$$\sum F_x = \rho Q(V_{2x} - V_{1x}), \qquad \sum F_y = \rho Q(V_{2y} - V_{1y})$$

$\sum \vec F$ includes the pressure forces on the control surface, body forces (weight) and the force $\vec R$ exerted by the solid boundary on the fluid.

Assumptions: steady flow, uniform velocity at the inlet and outlet sections, a fixed control volume, and an inertial frame.

Applications

1. Pipe bend: take the CV as the fluid inside the bend. The forces are the end pressure forces $p_1A_1$ and $p_2A_2$ and the reaction $R$ from the bend. The change in the direction and speed of the flow gives the force on the bend, which is used to design anchor blocks and thrust blocks.

2. Jet on a vane: the vane changes the direction of the jet. For a smooth vane deflecting the jet through $\theta$, the force in the jet direction is $F = \rho a V^2(1-\cos\theta)$ for a fixed vane and $F = \rho a (V-u)^2(1-\cos\theta)$ for a single vane moving at speed $u$, found from the change of momentum using the relative velocities. This is used in impulse turbines (Pelton wheel).

3. Jet propulsion: a fluid is accelerated backward and the reaction thrust pushes the body forward. For ships with water jets and for aircraft engines, the thrust is $F = \dot m(V_{jet} - V_{body})$, where $V_{jet}$ is the jet velocity relative to the body. For a rocket, $F = \dot m V_e + (p_e - p_a)A_e$.

Other applications: force on a nozzle, thrust on a sluice gate, and force in a hydraulic jump.