Jet Propulsion
Calculate the propulsive discharge force or thrust induced by an incompressible jet flow.
The velocity out of a free jet can be expressed as
v_{2}= (2 (p_{1}  p_{2}) / ρ) ^{ 1/2 } (1)
where
v_{2}= velocity out of the jet (m/s)
p_{1} = pressure before the jet (N/m^{2}, Pa)
p_{2}= ambient pressure after the jet (N/m^{2}, Pa)
ρ = density of the fluid (kg/m^{3} )
Note!  the density ρ is constant in incompressible flows and the equations are valid for liquids (like water) but not for gases (like air).
The flow volume out of a jet can be expressed as
q = A v_{2}(2)
where
q = volume flow (m^{3} /s)
A = orifice area of the jet (m^{2})
The propulsive force or thrust induced by the jet can be expressed as
F = ρ q (v_{2} v_{1} ) (3)
where
v_{1} = jet velocity (m/s)
If the jet is not moving
v_{1} = 0
and (3) can be expressed as
F = ρ q v_{2}(3b)
 or alternatively with (2)
F = ρ A v_{2}^{ 2 } (3c)
Substituting v_{2} with eq. (1) in eq. (3c)  the propulsive force or thrust induced by a jet not moving (v_{1} = 0) be expressed as
F = 2 A (p_{1}  p_{2}) (4)
Example  Propulsive Force
Water flows through a garden hose with diameter 15 mm . The water pressure just before the outlet is 4 10^{5} Pa (abs) and the atmospheric pressure is 1 10^{5} Pa (abs).
The outlet area of the hose can be calculated as
A = π ((15 mm) (0.001 m/mm) / 2)^{2}
= 1.77 10 ^{4} (m^{2})
The propulsive force can be calculated as
F = 2 ( 1.77 10^{4} m^{2}) (( 4 10^{5} Pa)  ( 1 10^{5} Pa) )
= 106 N
Related Topics

Fluid Mechanics
The study of fluids  liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
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