In this project you will apply your knowledge of linear and angular momentum
to study the aggregation of small droplets of ice to form large grains of
snow.
As snow crystals form in clouds they start falling through the cloud. Due to
air resistance, larger particles fall faster than smaller particles. A large
particle will therefore overtake smaller particles. When a smaller particle is
overtaken, it will stick to the larger particle, adding further to the size.
This process forms aggregate snowflakes, which is one of the most common types
of snowflakes. \({ }^{3}\) This mechanism is often called differential
sedimentation, and is a process important for pattern formation in many
natural systems, and it is also a process important for many industrial
processes. An example of a complex aggregate formed by a related aggregation
process called Diffusion Limited Aggregation in Fig. \(16.35\) shows the complex
geometries typically found in aggregate grains.
In this project, we will study the aggregation process in detail. We will
study an approximately spherical grain of ice of mass \(M\) and radius \(R\),
hitting and sticking to an identical grain of ice.
First, let us address why large particles fall faster than small particles.
The mass of an ice grain of radius \(R\) and mass density \(\rho_{m}\) is
$$
M=\rho_{m} \frac{4 \pi}{3} R^{3}
$$
We will assume that air resistance can be modelled using the approximation:
$$
\mathbf{F}_{v}=-k_{v} \mathbf{v}
$$
where
$$
k_{v} \simeq 20.4 R \eta
$$
is a constant depending on the viscocity \(\eta\) of the fluid.
(a) Find the forces acting on an ice grain with radius \(R\), and write down
Newton's second law of motion for the grain.
(b) Show that the acceleration of the grain is
$$
\mathbf{a}=\mathbf{g}-\frac{20.4 \eta}{\rho_{m} \frac{4 \pi}{3} R^{2}}
\mathbf{v}
$$
where \(\mathbf{g}=-g \mathbf{j}\) and \(g\) is the acceleration of gravity. Can
you now explain why larger grains fall faster than smaller grains?
We will now study a collision between two identical ice grains. One grain is
at rest relative to the reference system and the other grain has a velocity
\(v_{0}\) downwards. When the two grains collide, they stick together at the
point of contact, and remain stuck together. We call this combination of two
grains a compound grain.
(c) The moment of inertia of one ice grain around its center is
\(I_{c}=\frac{2}{5} M R^{2}\). Show that the moment of inertia, \(I\), around the
center of mass for a compound grain consisting of two grains sticking together
is \(I=(14 / 5) M R^{2}\)
First, we consider a linear collision where the upper grain hits the lower
grain directly in the center, as illustrated in Fig. 16.36a. We assume the
collision to be instantaneous, so you can ignore the effect of air resistance
and gravity during the collision.
(d) What is the velocity, \(\mathbf{v}_{1}\), of the center of mass the compound
grain after the collision?
(e) What is the angular velocity, \(\omega_{1}\), around the center of mass of
the compound grain after the collision?
Let us now consider the more general case illustrated in Fig. 16.36b. When the
two grains touch, the line between the centers of the two grains forms the
angle \(\theta\) with the horizontal. The upper grain still has the initial
velocity \(v_{0}\) downwards before the collision, and the lower grain is at
rest.
(f) What is the velocity, \(\mathbf{v}_{1}\), of the center of mass of the
compound grain after the collision?
(g) What is the angular velocity, \(\omega_{1}\), around the center of mass of
the compound grain after the collision?
(h) What is the loss of energy in the collision?
Let us now address the motion of the compound grain after the collision.
Initially, it is rotating with the angular velocity \(\omega_{1}\).
(i) If we ignore air resistance, find \(\omega(t)\) as a function of time for
the subsequent motion.
In the following we will not ignore air resistance, but rather develop a
simplified model for the air resistance. In order to determine the force
acting on the compound object due to air resistance, we either need to perform
experiments on such objects, or we can use numerical simulations of the fluid
flow around the object to determine the forces.
Here, we will use a strong simplification: We assume that we may consider the
compound object to consist of two separate spheres. The force on each of the
spheres due to air resistance is described by \((16.176)\), where the
corresponding velocity, \(v\), in (16.176) is the velocity of the center of the
sphere, and the force acts in the center of the sphere.
The compound object has velocity \(\mathbf{v}_{\mathrm{cm}}\) and angular
velocity \(\omega\).
(j) Argue that the velocities, \(\mathbf{v}_{A}\) and \(\mathbf{v}_{B}\), of each
of the ice grains \(A\) and \(B\) are \(\mathbf{v}_{A}=\mathbf{v}_{c m}+\omega
\times \mathbf{r}\) and \(\mathbf{v}_{B}=\mathbf{v}_{c m}-\omega \times
\mathbf{r}\), where \(\mathbf{r}\) describes the position of grain \(A\) relative
to the center of mass of the compound grain.
(k) Show that the net force on the center of mass of the compound object is
\(\sum \mathbf{F}=\) \(2 M \mathbf{g}-2 k_{v} \mathbf{v}_{c m}\), where
\(\mathbf{g}=-g \mathbf{j}\) and \(g\) is the acceleration of gravity.
(l) Show that the torque around the center of mass of the compound object due
to air resistance is \(\tau=-2 k_{v} \omega R^{2}\) (Hint: Use Lagrange's
formula).
(m) Show that the angular acceleration \(\alpha\) of the compound object around
its center of mass can be written as \(\alpha=d \omega / d t=-\left(1 /
t_{0}\right) \omega\), and find the characteristic time \(t_{0}\).
(n) Describe (with words) the motion of the compound object.
(o) Sketch the time development of the velocity \(v_{c m}\) and the angular
velocity \(\omega\) of the compound object, and discuss how the behavior would
change if you changed the radius, \(R\), of the grains.
(p) How would our argument change if we instead studied large particles, where
the air resistance force depends on the square of the velocity?
Final comment: Notice that the result above for the net force on the compound
grain indicates that small and large grains have the same acceleration, which
is not consistent with our initial result. This is due to our (incorrect)
simplification of adding the air resistance force for each of the grains
together to get the air resistance force for the compound grain. For a real
ice crystal formed by aggregation, the dependence of the air resistance on the
size of the compound grain is more complicated, and will also depend on the
complex geometry attained by a compound grain after a few hundred collisions
with smaller grains.
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