Graphics Reference
In-Depth Information
So what are the forces acting on the particle? The simplest is of course
gravity:
mg
. However, it gets interesting when we consider how the rest of
the fluid also exerts force: how the particle interacts with other particles
nearby.
The first of the fluid forces is pressure. High-pressure regions push on
lower-pressure regions. Note that what we really care about is the net
force on the particle: for example, if the pressure is equal in every di-
rection there's going to be a net force of zero and no acceleration due to
pressure. We only see an effect on the fluid particle when there is an im-
balance, i.e. higher pressure on one side of the particle than on the other
side, resulting in a force pointing away from the high pressure and towards
the low pressure. In the appendices we show how to rigorously derive
this, but for now let's just point out that the simplest way to measure
the imbalance in pressure at the position of the particle is simply to take
the negative gradient of pressure:
p
. (Recall from calculus that the
gradient is in the direction of “steepest ascent,” thus the negative gradi-
ent points away from high-pressure regions towards low-pressure regions.)
We'll need to integrate this over the volume of our blob of fluid to get
the pressure force. As a simple approximation, we'll just multiply by the
volume
V
. You might be asking yourself, but what is the pressure? We'll
skip over this until later, when we talk about incompressibility, but for now
you can think of it being whatever it takes to keep the fluid at constant
volume.
The other fluid force is due to viscosity. A viscous fluid tries to resist
deforming. Later we will derive this in more depth, but for now let's
intuitively develop this as a force that tries to make our particle move at
the average velocity of the nearby particles, i.e., that tries to minimize
differences in velocity between nearby bits of fluid. You may remember
from image processing, digital geometry processing, the physics of diffusion
or heat dissipation, or many other domains, that the differential operator
that measures how far a quantity is from the average around it is the
Laplacian
∇·∇
. (Now is a good time to mention that there is a quick
review of vector calculus in the appendices, including differential operators
like the Laplacian.) This will provide our viscous force then, once we've
integrated it over the volume of the blob. We'll use the
dynamic viscosity
coecient
, which is denoted with the Greek letter
η
(dynamic means we're
getting a
force
out of it; the kinematic viscosity from before is used to
get an
acceleration
instead). I'll note here that for fluids with variable
viscosity, this term ends up being a little more complicated; see Chapter 8
for more details.
−∇
