1.2. THE MOMENTUM EQUATION 3
1.2 The Momentum Equation
The first differential equation (1.1), which is actually three in one wrapped up as a vector equation, is called the
“momentum equation”. This really is good old Newton’s equation
~
F = m~a in disguise. It tells us how the fluid
accelerates due to the forces acting on it. We’ll try to break this down before moving onto the second differential
equation (1.2), which is called the “incompressibility condition”.
Let’s first imagine we were simulating a fluid using a particle system (later in the course we will actually use this as
a practical method, but for now let’s just use it as a thought experiment). Each particle would represent a little blob
of fluid. It would have a mass m, a volume V , and a velocity ~u. To integrate the system forward in time all we would
need is to figure out what the forces acting on each particle are:
~
F = m~a then tells us how the particle accelerates,
from which we get its motion. We’ll write the acceleration of the particle in slightly odd notation (which we’ll later
relate to the momentum equation above):
~a ≡
D~u
Dt
(1.3)
The big D derivative notation is called the material derivative. Newton’s law is now:
m
D~u
Dt
=
~
F (1.4)
So what are the forces acting on the particle? The simplest is of course gravity: m~g. The rest of the fluid (the other
particles) also exerts forces though.
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, though. For example, if the pressure is equal in every direction
there’s going to be net force of zero. What really matters is the imbalance of higher pressure on one side of the
particle than the other, 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.
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. In the appendices again we will
rigorously derive this, 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, 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 which 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 coefficient” (dynamic
means we’re getting a force out of it; the kinematic viscosity from before is used to get an acceleration instead),
which is denoted with the Greek letter µ. I’ll note here that for fluids with variable viscosity, this term ends up being
a little more complicated, but won’t be handled in these notes.