3.7 Momentum of Fluid Mechanics

According to Newton’s Second Law of Motion, the momentum can define as 

                             

 

Fluid contain in tube ABCD move to A’B’C’D’ after time changing, dt and since momentum = mass × velocity

AA’B’B = ρ₁A₁v₁ dt× v₁

 = ρ₁A₁v₁² dt

CC’D’D = ρ₂A₂v₂dt× v₂

= ρ₂A₂v₂² dt

Momentum changing in time = ρ₂A₂v₂² dt - ρ₁A₁v₁² dt

By dividing dt,

Momentum rate = ρ₂A₂v₂² - ρ₁A₁v₁²

In any steady flow, ρ₁A₁v₁ = ρ₂A₂v₂ = m, thus, momentum rate is

• Momentum rate = m(v₂ - v₁)

Therefore, Force, F is given by

•  Force, F = m(v₂ - v₁)

For any steady flow in three dimension equations can be express as:

The application of Momentum Equations

a)    Structure Open to Atmosphere

b)    Force of Pressure Conduits and Bend

c)     Force of A Stationary Vane or Blade

d)    Force of Nozzle 

Example

A water jet strikes on a vane at 150⁰. If water flows and velocity are 0.68 kg/s and  24 m/s respectively, calculate the resultant force at stationary vane ?

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