Control Volume Analysis
In this notes sheet...
As seen in thermodynamics, there is a difference between systems and control volumes. The former are used for a fixed mass of fluid, constantly moving, the latter are used for fluid flow through a defined boundary.
We hardly ever model a fixed mass of fluid, and as such we always use control volume analysis in fluid dynamics.
Mass Flow Rate
Over a set time δt, a distance δx is travelled by any given fluid particles. Therefore, the swept volume over time δt is Aδx. This volume has a mass, ρAδx.
The mass flow rate is the derivative of this with respect to time:
This can be re-written in terms of normal velocity, u:
Mass flux is another description of flow, given as the mass flow rate per unit area:
The velocity must be normal to the area.
If the velocity is not normal to the area, δA, we need to find the normal component as the scalar product of the velocity and normal vectors:
Writing δA as a vector:
This is the most important equation for mass flow rate. It will crop up in the derivations of everything that follows in this notes sheet.
Through a control volume, where there is inflow and outflow across the control surface (CS):
Outflow gives a positive value, inflow gives a negative value.
Volume Integrals
If certain properties through the control volume, such as density, are not constant, then we need to treat the total volume as an infinite number of infinitesimally small volumes, each with mass:
When density is constant:
Reynolds Transport Theorem
The Reynolds transport theorem is used to model the conservation of any given extensive property N. This could be any property: we will use it for mass, energy, and momentum.
In the Reynolds transport theorem, the specific form of the property is used, η: