The Equations of Fluids - Fluid Simulation for Computer Graphics - page 13

Things are a little more complicated if the solid itself is moving too. In
general, we need the normal component of the fluid velocity to match the
normal component of the solid's velocity, so that the relative velocity has
zero normal component:

u · n = u solid · n.

In both these equations, n is of course the normal to the solid boundary.
This is sometimes called the no-stick condition, since we're only restricting
the normal component of velocity, allowing the fluid to freely slip past
in the tangential direction. This is an important point to remember: the
tangential velocity of the fluid might have no relation at all to the tangential
velocity of the solid.
So that's what the velocity does: how about the pressure at a solid
wall? We again go back to the idea that pressure is “whatever it takes to
make the fluid incompressible.” We'll add to that, “and enforce the solid
wall boundary conditions.” The p/ρ term in the momentum equation

applies even on the boundary, so for the pressure to control u

∇

· n at a solid

wall, obviously that's saying something about n ,otherwiseknownas

the normal derivative of pressure: ∂p/∂n . We'll wait until we get into
numerically handling the boundary conditions before we get more precise.
That's all there is to a solid wall boundary for an inviscid fluid. If
we do have viscosity, life gets a little more complicated. In that case, the
stickiness of the solid might have an influence on the tangential component
of the fluid's velocity. The simplest case is the no-slip boundary condition,
wherewesimplysay

∇ p ·

u =0 ,

or if the solid is moving,

u = u solid .

Again, we'll avoid a discussion of exact details until we get into numerical
implementation.
As a side note, sometimes the solid wall actually is a vent or a drain
that fluid can move through: in that case, we obviously want u n to be

different from the wall velocity-rather to be the velocity at which fluid is
being pumped in or out of the simulation at that point.
The other boundary condition that we're interested in is at a free sur-
face. This is where we stop modeling the fluid. For example, if we simulate
water splashing around, then the water surfaces that are not in contact
with a solid wall are free surfaces. In this case there really is another fluid,

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