Questions: Control Volume Momentum Equation: Forces from Flow

Question 1

A horizontal water jet (mass flow rate ṁ, velocity V) strikes a stationary flat vertical plate and deflects at 90°. What is the force the water exerts on the plate in the original jet direction?

• A. Zero — the water is deflected, so momentum is redirected, not destroyed
• B. ṁV — the incoming x-momentum is entirely transferred to the plate since no x-momentum leaves the control volume
• C. 2ṁV — the momentum reverses direction, so the change is twice the incoming value
• D. ṁV/2 — only half the momentum is transferred because some is lost to turbulence

Answer: (B)

The jet enters with x-momentum flux ṁV and leaves perpendicular to the jet (zero x-momentum flux out). The x-momentum equation gives: F_plate_on_fluid = 0 − ṁV = −ṁV (the plate pushes back on the fluid). By Newton's third law, the fluid pushes the plate in the +x direction with force ṁV. Option C (2ṁV) would apply if the jet *reversed* direction (like bouncing off a wall) rather than deflecting sideways.

Question 2

An engineer needs to calculate the force on a 90° pipe elbow. Which approach does the momentum equation make possible?

• A. Integrate the pressure and shear stress distributions across the entire interior surface of the elbow
• B. Apply the momentum flux equation using only the conditions at the inlet and outlet of a control volume enclosing the elbow
• C. Solve the Navier-Stokes equations numerically for the full velocity field inside the elbow
• D. Use Bernoulli's equation along a streamline from inlet to outlet and differentiate

Answer: (B)

The control volume momentum method requires only the inlet and outlet mass flow rates and velocity vectors — the internal flow is treated as a black box. You draw a control surface around the elbow, apply ΣF = Σ(ṁ_out v_out) − Σ(ṁ_in v_in) in each direction, include pressure forces at the cut surfaces, and solve for the reaction force. This is vastly simpler than options A, C, or D, all of which require knowledge of internal flow details that are unnecessary for the net force calculation.

Question 3

The control volume momentum equation requires knowing the velocity distribution across the inlet and outlet cross-sections in order to compute momentum flux accurately.

• T. True
• F. False

Answer: False

For uniform (or one-dimensional) flow at inlets and outlets — the standard engineering assumption — momentum flux is simply ṁ·V, where ṁ is the mass flow rate and V is the average velocity. No velocity profile information is needed. In more precise analyses with non-uniform velocity profiles, a momentum correction factor β is introduced, but for the dominant applications (jet forces, pipe bends, nozzle thrust) the one-dimensional assumption is standard and sufficiently accurate.

Question 4

A rocket in space ejects exhaust gases at high velocity from its nozzle. The thrust force on the rocket equals the mass flow rate of exhaust multiplied by the exit velocity (neglecting pressure terms). This result follows directly from the momentum equation applied to a control volume.

• T. True
• F. False

Answer: True

Taking a control volume fixed to the rocket, exhaust exits at mass flow rate ṁ and velocity V_e relative to the rocket. There is no inlet (the rocket isn't scooping in mass). The net momentum flux out of the control volume is ṁ·V_e (in the exhaust direction), and Newton's second law requires an equal and opposite force on the rocket. This is rocket thrust: T = ṁ·V_e + (p_e − p_atm)·A_e. The control volume approach gives the exact same result as the rocket equation without needing to model the complex internal combustion or nozzle flow.

Question 5

What is the key conceptual advantage of the control volume momentum approach over analyzing forces through internal flow details, and in what type of problem is this advantage most valuable?

Model answer: The control volume approach requires only boundary conditions — mass flow rates and velocities at inlets and outlets — not the internal flow field. Forces are calculated from the difference in momentum flux across the control surface. This is most valuable when internal flow is complex (turbulent, three-dimensional, or involving phase change) but inlet and outlet conditions are measurable: jet impact forces, pipe elbow reactions, nozzle thrust, turbine blade forces.

The approach trades spatial detail for thermodynamic accounting. Engineers rarely care *how* momentum changes inside a device — they care what forces act on the device. By treating the interior as a black box and applying Newton's second law to net fluxes at the boundary, the method bypasses the need for detailed flow solutions. This is the same philosophy as using control volumes for mass and energy: the integral form of conservation laws is far more tractable than the differential form for engineering calculations.