&=&
0.
@f}
+
+ An alternative to this is when one wants the flow to be <i>parallel</i>
+ rather than perpendicular to the boundary (in deal.II, the
+ VectorTools::compute_no_normal_flux_constraints function can do this for
+ you). This is frequently the case for problems with a free boundary
+ (e.g. at the surface of a river or lake if vertical forces of the flow are
+ not large enough to actually deform the surface), or if no significant
+ friction is exerted by the boundary on the fluid (e.g. at the interface
+ between earth mantle and earth core where two fluids meet that are
+ stratified by different densities but that both have small enough
+ viscosities to not introduce much tangential stress on each other).
+ In formulas, this means that
+ @f{eqnarray*}
+ \textbf{n}\cdoy\textbf u &=& 0,
+ \\
+ (\textbf 1-\textbf n\otimes\textbf n)
+ \left(\textbf{n}\cdot [p \textbf{1} -
+ \varepsilon(\textbf{u})] \right)
+ &=&
+ 0,
+ @f}
+ the first condition (which needs to be imposed strongly) fixing a single
+ component of the velocity, with the second (which would be enforced in the
+ weak form) fixing the remaining two components.
</ol>
Despite this wealth of possibilities, we will only use Dirichlet and