Derivations - Fluid Simulation for Computer Graphics - page 201

Here u 3 outer-product matrix. Again, since this is true for

any arbitrary region Ω, the integrands must be equal:
∂ ( ρu )
∂t

⊗ u is the 3 ×

+ ∇· ( ρu × u )+ ∇ p = ρg.

This is the conservation law form of the momentum equation .Usingthe
product rule of differentiation, exploiting Dρ/Dt = 0 for an incompressible
fluid, and dividing through by ρ , this can readily be reduced to the form
of the momentum equation used in the rest of this topic.

B.2 The Pressure Problem as a Minimization

Here we go through a calculus of variations argument illustrated by Batty
et al. [Batty et al. 07] that the pressure problem,

Δ t
ρ ∇

∇· p = ∇· u inside Ω ,

p =0 on F, (B.1)

Δ t
ρ ∇

p · n = u · n − u solid · n on S,

is equivalent to the following energy minimization problem introduced in
Chapter 4:

2 ρ

p

2 +
S

2

Δ t
ρ ∇

1 1

min
p
p =0on F

u − −

2 ρ

u pn · Δ tu solid

Ω Ω

(B.2)
Here Ω is the fluid region, and its boundary is partitioned into S ,the
part in contact with solids, and F , the free surface. The functional being
minimized is the total work done by pressure, measured as the change in
kinetic energy of the fluid plus the work done on the solid by the fluid
pressure over the next time step, which accounts for all work done by the
pressure. As we said, in incompressible flow there is no “elastic” potential
energy, thus the pressure is fully inelastic and dissipates as much energy
as possible, hence the minimization. Also as we said before, the kinetic
energy of the intermediate velocity field is a constant, hence we can just
drop it in the minimization.
The calculus of variations is a way to solve problems such as this, min-
imizing an integral with respect to a function , as opposed to just a finite
set of variables. We'll walk through the standard method of reducing the
minimization to a PDE.