Categories: Fluid dynamics, Fluid mechanics, Physics.

Navier-Stokes equations

While the Euler equations govern ideal “dry” fluids, the Navier-Stokes equations govern nonideal “wet” fluids, i.e. fluids with nonzero viscosity.

Incompressible fluid

First of all, we can reuse the incompressibility condition for ideal fluids, without modifications:

\[\begin{aligned} \boxed{ \nabla \cdot \va{v} = 0 } \end{aligned}\]

Furthermore, from the derivation of the Euler equations, we know that Newton’s second law can be written as follows, for an infinitesimal particle of the fluid:

\[\begin{aligned} \rho \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{f^*} \end{aligned}\]

\(\mathrm{D}/\mathrm{D}t\) is the material derivative, \(\rho\) is the density, and \(\va{f^*}\) is the effective force density, expressed in terms of an external body force \(\va{f}\) (e.g. gravity) and the Cauchy stress tensor \(\hat{\sigma}\):

\[\begin{aligned} \va{f^*} = \va{f} + \nabla \cdot \hat{\sigma}^\top \end{aligned}\]

From the definition of viscosity, the stress tensor’s elements are like so for a Newtonian fluid:

\[\begin{aligned} \sigma_{ij} = - p \delta_{ij} + \eta (\nabla_i v_j + \nabla_j v_i) \end{aligned}\]

Where \(\eta\) is the dynamic viscosity. Inserting this, we calculate \(\nabla \cdot \hat{\sigma}^\top\) in index notation:

\[\begin{aligned} \big( \nabla \cdot \hat{\sigma}^\top \big)_i = \sum_{j} \nabla_j \sigma_{ij} &= \sum_{j} \Big( \!-\! \delta_{ij} \nabla_j p + \eta (\nabla_i \nabla_j v_j + \nabla_j^2 v_i) \Big) \\ &= - \nabla_i p + \eta \nabla_i \sum_{j} \nabla_j v_j + \eta \sum_{j} \nabla_j^2 v_i \end{aligned}\]

Thanks to incompressibility \(\nabla \cdot \va{v} = 0\), the middle term vanishes, leaving us with:

\[\begin{aligned} \va{f^*} = \va{f} - \nabla p + \eta \nabla^2 \va{v} \end{aligned}\]

We assume that the only body force is gravity \(\va{f} = \rho \va{g}\). Newton’s second law then becomes:

\[\begin{aligned} \rho \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \rho \va{g} - \nabla p + \eta \nabla^2 \va{v} \end{aligned}\]

Dividing by \(\rho\), and replacing \(\eta\) with the kinematic viscosity \(\nu = \eta/\rho\), yields the main equation:

\[\begin{aligned} \boxed{ \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v} } \end{aligned}\]

Finally, we can optionally allow incompressible fluids with an inhomogeneous “lumpy” density \(\rho\), by demanding conservation of mass, just like for the Euler equations:

\[\begin{aligned} \boxed{ \frac{\mathrm{D} \rho}{\mathrm{D} t} = 0 } \end{aligned}\]

Putting it all together, the Navier-Stokes equations for an incompressible fluid are given by:

\[\begin{aligned} \boxed{ \frac{\mathrm{D} \va{v}}{\mathrm{D} t} = \va{g} - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v} \qquad \nabla \cdot \va{v} = 0 \qquad \frac{\mathrm{D} \rho}{\mathrm{D} t} = 0 } \end{aligned}\]

Due to the definition of viscosity \(\nu\) as the molecular “stickiness”, we have boundary conditions for the velocity field \(\va{v}\): at any interface, \(\va{v}\) must be continuous. Likewise, Newton’s third law demands that the normal component of stress \(\hat{\sigma} \cdot \vu{n}\) is continuous there.

References

  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.

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