Fluid Mechanics/Control Volume Analysis

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Control Volume Analysis[edit]

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[edit]

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,


  \dot m = \int_{CS} \rho
  \left(
    \mathbf{V}\cdot\mathbf{n}
  \right) dA +\frac{d}{dt}\int_{CV}\rho dV

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:


  \int_{CS} \rho
  \left(
    \mathbf{V}\cdot\mathbf{n}
  \right) dA =-\frac{d}{dt}\int_{CV}\rho dV

For steady flow, the time derivative of the quantity is zero, we have


  \int_{CS} \rho
  \left(
    \mathbf{V}\cdot\mathbf{n}
  \right) dA =0

And for incompressible flow, we have


  \int_{CS}
  \left(
    \mathbf{V}\cdot\mathbf{n}
  \right) dA = 0

If we consider flow through a tube, we have, for steady flow,


  \rho_1A_1V_1 = \rho_2A_2V_2

and for incompressible steady flow, A1V1 = A2V2.

Conservation of Momentum[edit]

Law of conservation of momentum as applied to a control volume states that


  \sum F = \frac{d}{dt}
  \left(
    \int_{CV} \!\mathbf{V} \mathbf{\rho} \,dV
  \right) + \int_{CS} \mathbf{V}\mathbf{\rho}
  \left(
    \mathbf{V}\cdot\mathbf{n}
  \right)dA

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[edit]

The law of Conservation of Energy in fluid mechanics is a specific application of the First Law of Thermodynamics.


  \frac{d\mathbf{Q}}{dt} + \frac{d\mathbf{W}}{dt} = \frac{d}{dt}
  \left(
    \int_{CV} e\mathbf{\rho} \,dV
  \right) + \int_{CS} e\mathbf{\rho}
  \left(
    \mathbf{V}\cdot\mathbf{n}
  \right)dA

where e is the energy per unit mass.

Conservation Equations of Mass, Momentum and Energy[edit]

Equation of Continuity[edit]

  • A differential mass balance relating density change to velocity.

  \frac{\partial\rho}{\partial t} + \nabla\cdot
  \left(
    \rho \mathbf{V}
  \right) = 0


  • For incompressible fluids the equation of continuity reduces to:
 
  \rho 
  \left(
    \nabla \cdot 
    \mathbf{V}
  \right) = 0
since

  \frac{\partial\rho}{\partial t} = 0
for all incompressible fluids


Euler's Equation[edit]

  • applies conservation of momentum to inviscid, incompressible flow.

  \rho \mathbf{g} - \nabla p = \rho\frac{d\mathbf{V}}{dt}

Stokes' Equation[edit]

  • applies conservation of momentum in creeping flow limit (low Reynold's Number)

  \nabla p = \mu \nabla^2 \mathbf{V}