Fluid Mechanics/Control Volume Analysis
Control Volume Analysis
A fluid dynamic system can be analyzed using a control volume, which is an imaginary surface enclosing a volume of interest. The control volume can be fixed or moving, and it can be rigid or deformable. Thus, we will have to write the most general case of the laws of mechanics to deal with control volumes.
Conservation of Mass
The first equation we can write is the conservation of mass over time. Consider a system where mass flow is given by dm/dt, where m is the mass of the system. The term on the left denotes the rate of change of the mass of the system. The first term on the right describes the amount of flow across the control surface. The second term on the right refer to the condition within the control volume. We have,
However, by definition of a system the mass of which is a constant; thus the left-hand side of the above equation equals to zero and it could be rewritten as:
For steady flow, the time derivative of the quantity is zero, we have
And for incompressible flow, we have
If we consider flow through a tube, we have, for steady flow,
and for incompressible steady flow, A1V1 = A2V2.
Conservation of Momentum
Law of conservation of momentum as applied to a control volume states that
where V is the velocity vector and n is the unit vector normal to the control surface at that point.
The sum of the forces represents the sum of forces that act on the entirety of the fluid volume (body forces) and the forces that act only upon the bounding surface of a fluid (surface forces). Body forces include the gravitational force.
Conservation of Energy
The law of Conservation of Energy in fluid mechanics is a specific application of the First Law of Thermodynamics.
where e is the energy per unit mass.
Conservation Equations of Mass, Momentum and Energy
Equation of Continuity
- A differential mass balance relating density change to velocity.
- For incompressible fluids the equation of continuity reduces to:
- for all incompressible fluids
- applies conservation of momentum to inviscid, incompressible flow.
- applies conservation of momentum in creeping flow limit (low Reynold's Number)