Linear Momentum

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The linear momentum equation is a simple yet effective equation that relates the forces acting on a fluid to its velocity and mass flow rate. It can be used in situations where Bernoulli's equation cannot since Bernoulli's equation is essentially conservation of energy and so does not take into account external forces, while linear momentum does.

Contents

Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 245 - 259.

Derivation

[1]The linear momentum equation is essentially Newton's Second Law rewritten. Newton's Second Law states that:

NewtonsSecondLawLinearMomentum.png

Where:

m = mass (kg)
v = velocity (m/s)

Since the density and velocity may change over the fluid volume, it can be rewritten as:

NewtonsSecondLawLinearMomentum2.png

Where:

V = volume (m³)

That is, the sum of all the forces acting on a fluid equals to the time rate of change of the fluids density, volume and velocity. [2]Under steady flow (constant density) and a controlled volume, the equation collapses to:

LinearMomentum.png

Where:

β = momentum flux correction factor which is usually given

It is important to note that both F and v are vectors and should therefore be treated as such in the mathematics here.

Flow with One Inlet and One Outlet

[3]If a system only has one inlet and one outlet and under steady flow where the mass flow rate is the same at both inlet and outlet, then the equation boils down to a very simple:

LinearMomentumOneInletandOutlet.png

References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).

  1. Textbook p.245
  2. Textbook p.249
  3. Textbook p. 250
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