Theoretical background
Fluid mechanics is governed by a set of equations called the Navier-Stokes equations , which are the mass conservation, momentum conservation as well as the energy conservation equations. These will be briefly discussed below but references such as the publications of [Versteeg et Malalasekera]1 or [Ferziger et Peric]2 present them in more detail and depth.
Mass conservation
$$ \underbrace{\frac{\partial \rho}{\partial t} }_\mathrm{transient} + \underbrace{\vec{\nabla} \cdot (\rho \vec{v})}_\mathrm{advection} = 0 $$
where \(\rho\) is the fluid density and \(\vec{v}\) the velocity vector.
This equation stipulates that if a fixed volume is considered within a studied domain, the mass of fluid entering the volume is equal to the mass of the outgoing fluid.
Momentum conservation
$$ \underbrace{\frac{\partial \rho \vec{v}}{\partial t}}_\mathrm{transient}+\underbrace{ \vec{\nabla}\cdot(\rho\vec{v}\otimes\vec{v})}_\mathrm{advection} = \underbrace{- \vec{\nabla} p + \vec{\nabla} \cdot [\tau]}_\mathrm{diffusion} + \underbrace{\rho \vec{g}}_\mathrm{source} $$
where \(p\) is the fluid pressure,\(\vec{g}\) the gravity field and \([\tau]\) the tensor of viscous stresses.
This equation, derived from Newton’s second law , governs the fluid movement. It establishes that the amount of movement is equal to the sum of the surface and volume forces acting on the fluid.
Energy conservation
$$ \underbrace{\frac{\partial \rho h_{\text{tot}}}{\partial t} - \frac{\partial p}{\partial t}}_\mathrm{transient} + \underbrace{ \vec{\nabla} \cdot (\rho \vec{v} h_\mathrm{tot})}_\mathrm{advection} = \underbrace{\vec{\nabla} \cdot (\lambda \vec{\nabla} T) + W_\mu}_\mathrm{diffusion} + \underbrace{S_{\text{h}}}_\mathrm{source} $$
where \(h_{\text{tot}}\) is the enthalpy of the fluid, \(\lambda\) the thermal conductivity and \(W_\mu\) the thermal dissipation resulting from viscous stresses.
This equation, derived from the first principle of thermodynamics , ensures energy conservation within a fixed volume in space.
These three equations serve as a basis for contemporary fluid mechanics studies. Different versions of these equations can be used depending on the assumptions considered ( incompressible fluids , Boussinesq approximation , etc.). Bernouilli’s theorem , which is also commonly used, can be seen as an application of the momentum conservation to an incompressible, inviscid and stationary fluid. This can then be expressed with the following equation:
$$ \frac{p}{\rho}+\frac{v^2}{2}+g z = \mathrm{constant} $$