In this post we write a general description of the set of partial differential equations that govern fluid dynamics in space and time.
Here we write the equations that describe a compressible, unsteady, and viscous fluid flow.
\[\frac{\partial \rho}{\partial t}+ \nabla\cdot\left(\rho \mathbf{u}\right) = 0\] \[\frac{ \partial \rho \mathbf{u}}{\partial t} + \nabla\cdot\left(\rho \mathbf{u}\mathbf{u}^T\right) = \nabla\cdot \overline{\overline \sigma} + \rho\mathbf{F}\] \[\frac{\partial \rho E}{\partial t} + \nabla\cdot\left(\rho\mathbf{u}E\right) = -\nabla\cdot\left(p\mathbf{u}\right) + \nabla\cdot\left(\overline{\overline\tau}\cdot\mathbf{u}\right) + \nabla\cdot\left(K \nabla T\right) + \rho \dot{q} + \rho \mathbf{F}\cdot\mathbf{u}\]where the following two relations hold
\[p = \rho r T\] \[e = e(p,T)\]The first three equations refer to the conservation of mass, momentum and energy of a fluid element moving in space and time respectively. And the two remaining equations are related to the equation of state of a perfect gas and the thermodynamics relationship that relates internal energy with pressure and temperature.
Note that we end up with a system of \(7\) partial differential equations and \(7\) dependent variables (on \(\mathbf{x} = x\mathbf{e}_1+y\mathbf{e}_2+z\mathbf{e}_3\) and \(t\) since we are using a cartesian set of coordinates) which are:
| Density: | \(\rho = \rho(\mathbf{x},t)\) |
| Velocity: | \(\mathbf{u} = \mathbf{u}(\mathbf{x},t) = u\mathbf{e_1} + v\mathbf{e_2} + w\mathbf{e_3}= u_1\mathbf{e_1} + u_2\mathbf{e_2} + u_3\mathbf{e_3}\) |
| Pressure: | \(p = p(\mathbf{x},t)\) |
| Internal energy: | \(e = e(\mathbf{x},t)\) where \(E = e + \frac{\mathbf{u}^2}{2} = e + \frac{|\mathbf{u}|^2}{2}\) |
| Temperature: | \(T= T(\mathbf{x},t)\) |
Finally, the following quantities are assumed to be known:
| \(\overline{\overline\sigma}:=\) | Stress tensor where \(\overline{\overline\sigma} = -p\overline{\overline I} + \overline{\overline\tau}\) and \(\overline{\overline I}\) is the unit tensor |
| \(\overline{\overline\tau}:=\) | Viscous stress tensor |
| \(\mathbf{F}:=\) | External volume forces where \(\mathbf{F} = F_x\mathbf{e_1} + F_y\mathbf{e_2} + F_z\mathbf{e_3}\) |
| \(K:=\) | Thermal conductivity |
| \(\dot{q}:=\) | Volumetric heat sources |
| \(r:=\) | Specific gas constant where \(r = k/M\), \(k=8314\frac{J}{Kg.mol.K}\), and \(M=\) molar mass of gas |
Compressibility shows up in the equations as an explicit dependence of density on space and time, and real effects of this might appear depending on the Mach number regimes. Unsteadiness is the time dependence of the unknowns in the equations. And viscosity refers in the equations to the viscous stress tensor \(\tau\) term.
The 3D Navier-Stokes equations follow from expressing the conservation of mass, momentum,and energy in a differential form for a fluid element that moves in space and time. Normally, you end up from this process with an integral form of the equations and finally you apply Reynolds transport theorem to express them in the differential conservative form expressed in this post.
Chapters 1 and 2 of Hirsch’s book are a superb explanation for the derivation and context of these set of equations, as well as the computational aspects and numerical schemes used nowdays to simulate fluids. This was the main source used to write this post.
NASA has a beautiful summary on physics concepts related to Aerodynamics including the equations here studied.
A classical book that covers topics on fluid dynamics is Anderson’s book.