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Holding force on a contracting nozzle 2026-10-14

For steady ideal flow through a horizontal contracting nozzle, Bernoulli equation gives inlet gauge pressure

p_{0} = ρ Q^{2} (A_{1}^{- 2} - A_{0}^{- 2}) /2

. The integral momentum equation gives the fluid's force on the nozzle as

p_{0} A_{0} - ρ Q^{2} (A_{1}^{- 1} - A_{0}^{- 1})

in the outlet direction. Simplification yields the displayed nonnegative magnitude. The external holding force has equal magnitude and the opposite direction.

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Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 2 / 8D / Solution Created 2026-09-24 Updated 2026-10-14

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Use the ideal steady incompressible-flow model, neglecting height change across the smooth nozzle. Flow conservation gives inlet and outlet speeds

u_{0} = Q / A_{0}

and

u_{1} = Q / A_{1}

. Set atmospheric pressure to zero. The Bernoulli equation between inlet and outlet gives

p_{0} = \frac{ρ}{2} (u_{1}^{2} - u_{0}^{2}) = \frac{ρ Q ^{2}}{2} (\frac{1}{A _{1}^{2}} - \frac{1}{A _{0}^{2}}) .

(1)

Choose a control volume containing the fluid in the nozzle, with the positive axis pointing toward the outlet. The integral momentum equation gives

p_{0} A_{0} + F_{wall on fluid} = ρQ (u_{1} - u_{0}) .

(2)

Thus the fluid's force on the nozzle is

p_{0} A_{0} - ρQ (u_{1} - u_{0})

, in the outlet direction. Substitute the pressure and velocities:

F_{fluid on nozzle} = ρ Q^{2} [\frac{A _{0}}{2 A _{1}^{2}} - \frac{1}{A _{1}} + \frac{1}{2 A _{0}}] = \frac{ρ Q ^{2}}{2 A _{0}} (\frac{A _{0}}{A _{1}} - 1)^{2} .

(3)

The fireman's holding force is equal and opposite. Therefore the required force points upstream and has magnitude

F = \frac{ρ Q ^{2}}{2 A _{0}} (\frac{A _{0}}{A _{1}} - 1)^{2} .

(4)

This is the holding force on a contracting nozzle; retaining the inlet pressure force is essential to its sign and value.

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