Viscous Fluids - Fluid Simulation for Computer Graphics - page 110

8.4 Boundary Conditions

The two types of boundaries considered in this topic are free surfaces
and solid walls, and each has particular conditions associated with
viscosity.
In the case of a free surface, things are fairly straightforward. On the
other side of the boundary there is a vacuum, or another fluid of much
smaller density whose effect we assume is negligible. Thus there is nothing
with which to transfer momentum: there can be no traction at the free
surface. In other words, the boundary condition for the stress at the free
surface is

σn = − pn + τ n =0 .

Note that if the viscous stress τ is zero, this reduces to p =0asbefore;
however, this becomes significantly more complex when τ isn't zero.
At solid walls, things are also a little more interesting. Physically speak-
ing once we model viscosity, it turns out the velocity field must be contin-
uous everywhere: if it weren't, viscous transfer of momentum would in the
next instant make it smooth again. This results in the so-called no-slip
boundary condition:

u = u solid ,

which of course simplifies to u = 0 at stationary solids. Recall that in the
inviscid case, only the normal component of velocities had to match: here
we are forcing the tangential components to match as well.
The no-slip condition has been experimentally verified to be more ac-
curate than the inviscid no-stick condition. However, the caveat is that in
many cases something called a boundary layer develops. Loosely speaking,
a boundary layer is a thin region next to a solid where the tangential ve-
locity rapidly changes from u solid at the solid wall to u at the other side of
the layer, where u is the velocity the inviscid no-stick boundary condition
( u

n ) would have given. That is, the effect of viscous drag
at the surface of the solid is restricted to a very small region next to the
solid, and can be ignored elsewhere. When we discretize the fluid flow on
a relatively coarse grid, that boundary layer may be much thinner than a
grid cell, and thus it's no longer a good idea to implement the no-slip con-
dition numerically—we would artificially be expanding the boundary layer
to at least a grid cell thick, which would be a much worse approximation
than going back to the inviscid no-stick boundary condition. In this case,

· n = u solid ·