Graphics Reference
In-Depth Information
We've already seen two examples of force density, body forces (such as
gravity) and pressure. These are fundamentally different however: to get
the net gravitational force on a volume of fluid you integrate the gravita-
tional body force density
ρg
over the
volume
, but to get the net pressure
force on the volume you integrate the pressure times the normal
pn
over
the
surface
of the volume. The difference is that a body force acts over
a distance to influence everything inside the volume, whereas other forces
(like pressure) can only act locally on the surface of contact. It is these
local contact forces that we need to generalize from pressure to figure out
viscosity and more exotic fluid effects.
An accepted assumption (verified to be accurate in many experiments),
called Cauchy's Hypothesis, is that we can represent the local contact forces
by a function of position and orientation only. That is, there is a vector
field called
traction
,
t
(
x, n
), a function of where in space we are measuring
it and what the normal to the contact surface there is. It has units of force
per area: to get the net contact force on a volume of fluid Ω, we integrate
the traction over its surface:
F
=
∂
Ω
t
(
x, n
)
.
I'd like to underscore here that this volume can be an arbitrary region
containing fluid (or in fact, any continuum substance); e.g., it could be a
tiny subregion in the interior of the fluid, or a grid cell, etc.
Once we accept this assumption, it can be proven that the traction
must depend linearly on the normal; that is, the traction must be the
result of multiplying some matrix by the normal. Technically speaking
this is actually a rank-two tensor, not just a matrix, but we'll gloss over
the difference for now and just call it a tensor from now on.
1
The tensor
is called the
stress tensor
, or more specifically the
Cauchy stress tensor
,
2
which we label
σ
.Thuswecanwrite
t
(
x, n
)=
σ
(
x
)
n.
1
Basically a matrix is a specific array of numbers; a rank-two tensor is a more ab-
stract linear operator that can be represented as a matrix when you pick a set of basis
vectors with which to measure it. For the purposes of this topic, you can think of them
interchangeably, as we will always use a fixed Cartesian basis where
y
is in the vertical
direction.
2
There are other stress tensors, which chiefly differ in the basis in which they are
represented. For elastic solids, where it makes sense to talk of a rest configuration to
which the object tries to return, it can be convenient to set up a stress tensor in terms of
rest-configuration coordinates, rather than the world-space coordinates in which Cauchy
stress operates.
