8. Hydrodynamics

 
  How do you describe a flow?



Some basic truths:

General equation of motion for the flow field $\mbox{$\bf v$}(\mbox{$\bf r$},t)$ in a compressible viscous fluid: Navier-Stokes equation

$$\frac{\partial }{\partial t} \rho \mbox{$\bf v$} + \nabla \c... ...$} \right] + \nabla p - \mu \nabla \cdot \mbox{${\bf U}$} = 0$$ (8.1)

Here, $\mu$ is the viscosity, and

$$\mbox{${\bf U}$}\equiv\nabla \mbox{$\bf v$} + (\nabla \mbox{... ...{T}-\frac{2}{3}(\nabla \cdot \mbox{$\bf v$}) \mbox{${\bf I}$}$$ (8.2)

defines the Navier-Stokes tensor. (In 2 dimensions, write $1$ in place of $2/3$).

Character of Navier-Stokes PDE:

- Contains advective (hyperbolic) and diffusive (parabolic) terms
- Small viscosity: advective terms dominate $\Longrightarrow$hyperbolic
- High viscosity: diffusive terms important $\Longrightarrow$parabolic
- Stationary case ( $\partial / \partial t = 0$): $\Longrightarrow$elliptic

The NS equation results from the conservation of momentum. In addition, we have conservation of mass,

$$\frac{\partial \rho}{\partial t}+ \nabla \cdot \rho \mbox{$\bf v$}=0$$ (8.3)

and conservation of energy,

$$\frac{\partial e}{\partial t}+ \nabla \cdot \left[ (e+p) \mb... ... with}\;\;\;\; e \equiv \rho \epsilon + \frac{\rho v^{2}}{2}$$ (8.4)

where $e$ is the energy density ($\epsilon$ ... internal energy per unit mass of the fluid).

To close the set of equations some equation of state $p=p(\rho,\epsilon)$ is assumed.

The following approaches will be discussed:

• Conventional methods of solving the Navier-Stokes PDE
  
• Discretized ``Lattice gas''or ``Lattice Boltzmann''dynamics
  
• Direct Simulation Monte Carlo