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Created by: Siva Harikumar

Definition Edit

The Momentum equation is a statement of Newton's second law  and relates the sum of the forces acting on a body to its acceleration, or to the rate of change of momentum in the direction of the resultant external force. Newton's second law is frequently written as

F =ma

where F is the force exerted on a body of mass m and a is the acceleration. However a more general form of the equation is


F=mV

where mV is the momentum of a body of mass m [1].

It is sometimes referred to as linear momentum to distinguish it from the related subject of angular momentum. Linear momentum is a vector quantity, since it has a direction as well as a magnitude. Angular momentum is a pseudovector quantity because it gains an additional sign flip under an improper rotation.


Derivation and Description Edit

The linear momentum of a system of particles is the vector sum of the momenta of all the individual objects in the system:

Linear momentum

where P is the total momentum of the particle system.


Now let us consider the forces acting on the control volume defined within a region of fluid flow.

We can visualize these forces as two types.
Stream tube in two and three dimensions

Stream tube in two and three dimensions

Body forces Edit

These forces stem from "action at a distance", such as gravitational and electromagnetic forces that may be exerted on the fluid inside the control volume [[[2]]].

Body force3



Surface forces Edit

These forces stem from two sources: pressure and shear stress distributions over the surface.


Surfaceforce2


where the integral of the pressure force pdS is the summation of the elemental forces over the entire control surface. The negative sign indicates that the force is in the direction opposite of dS, the elemental area. The 2nd term on the right hand side denote the total viscous force exerted on the control surface.


Change in momentum Edit

The change in momentum of the fluid would be fluctuating in time simply due to the time variations in the density and velocity. In addition the fluctuations, the elemental fluid mass has a momentum since it is moving along with the fluid. Reynold's transport theorem is used to formulate the momentum equation.


The first term on the right hand side indicates the net flow of momentum out of the control volume through S and the second term denotes the time rate of change of momentum inside the control volume due to unsteady flow fluctuations. Note that it is a vector equation. We assumed that the mass flows out of the control volume and hence the positive sign before the first term. 


The change in momentum and the total force equations are combined to give the momentum equation.
Momentum3


It is the integral formulation of the Newton's second law applied to viscous fluid flows. The momentum equation for a viscous flow are also called the Navier-Stokes equations.

In a general isothermal flow the primary unknowns include the pressure, p, and the components of the velocity vector, u, which are functions of the spatial coordinates
and time [[[3]]]. Other flow variables, such as the nine components of the stress tensor, the residence time, the streamlines, etc., can be evaluated a posteriori once the primary unknowns have been calculated. It is a standard mathematical rule that, in order to determine a number of unknowns, equal number of equations that contain these unknowns must be solved. Therefore, the five equations –continuity, the three momentum components and the energy equation–can be solved for the five unknowns.

Application of the Momentum Equation to an Aircraft Engine Edit

Consider the jet engine as shown in the figure [[[4]]]. The force produced by the engine is conventionally called thrust, T. Every force outside the engine is drag, D

.
Appn to aircraft engine2

Control volume for application of momentum equation to an aircraft engine

 Therefore, gathering only those terms that relate to the fluid that passes through the engine, we have:

Thrust

where subscripts e and o denote exhaust and initial conditions respectively.


The thrust is largely composed of the net change in momentum of the air entering and leaving the engine, with a typically small adjustment for the differences in the pressure between the inlet and the exit.


ReferencesEdit

  1. Anderson, John David. Fundamentals of Aerodynamics. New York: McGraw-Hill, 2001.
  2. Hill, Philip & Peterson, Carl. Mechanics and Thermodynamics of Propulsion: Addison-Wesley, 1992.
  3. Papanastasiou, Georgiou & Alexandrou. Viscous Fluid Flow: CRC Press, 2000.
  4. web.mit.edu/.../propulsion/notes/node78.html