Momentum For Steady State and Uniform Flow
Introduction: The momentum equation can be simplified for the steady state condition as it was. The unsteady term (where the time derivative) is zero.
Momentum for For Constant Pressure and Frictionless Flow: Another important sub category of simplification deals with flow under approximationof the frictionless flow and uniform pressure. This kind of situations arises when friction(forces) is small compared to kinetic momentum change. Additionally, in these situations,flow is exposed to the atmosphere and thus (almost) uniform pressure surroundingthe control volume. In this situation, the mass flow rate in and out is equal.
In situations where the velocity is provided and known (remember that density is constant) the integral can be replaced by
The average velocity is related to the velocity profile by the following integral
Equation is applicable to any velocity profile and any geometrical shape
Momentum Equation in Acceleration System:
For accelerate system; the right hand side has to include the following acceleration
Where r is the distance from the center of the frame of reference and the add force is
External Forces: First, the terms on the left hand side, or the forces, have to be discussed. The forces, excluding the external forces, are the body forces, and the surface forces as the following:
Ftotal = Fb Fs
In this book (at least in this discussion), the main body force is the gravity. The gravity acts on all the system elements. The total gravity force is
This acts through the mass center towards the center of Earth. After infinitesimal time the gravity force acting on the system is the same for control volume, hence,
The integral yields a force trough the center mass which has to be found separately, the surface forces are
Divided into two categories: one perpendicular to the surface and one with the surface direction (in the surface plain see Figure Thus, it can be written as
Where the surface “force”, Sn, is in the surface direction and ¿ are the shear stresses. The surface “force”, Sn, is made out of two components, one due to viscosity (solid body) and two consequence of the fluid pressure. Here for simplicity, only the pressure component is used which is reasonable for most situations. Thus,