A liquid is traditionally defined as a material that adapts its shape to fit a container. Yet under certain conditions, cats seem to fit this definition.
This somewhat paradoxical
observation emerged
on the web a few years ago and joined the long list of internet memes involving
our feline friends. When I first saw this question it made me laugh, and then
think.
I decided to reformulate it
to illustrate some problems at the heart of rheology, the study of the
deformations and flows of matter. My study on the rheology of cats won the
2017 Ig Nobel Prize in
Physics.
The prizes are awarded every
year by Improbable Research, an organization devoted to science and humor. The
goal is to highlight scientific studies that first make people laugh, then
think. A ceremony is
held every year at Harvard University.
What is a liquid?
At the center of the definition of a liquid is an action: A material must be able to modify its form to fit within a container. The action must also have a characteristic duration. In rheology this is called the relaxation time. Determining if something is liquid depends on whether it's observed over a time period that's shorter or longer than the relaxation time.
If we take cats as our
example, the fact is that they can adapt their shape to their container if we
give them enough time. Cats are thus liquid if we give them the time to become
liquid.
In rheology, the state of a
material is not really a fixed property – what must be measured is the
relaxation time. What is its value and on what does it depend? For example,
does the relaxation time of a cat vary with its age? (In rheology we speak of thixotropy.)
Could the type of container
be a factor? (In rheology this is studied in "wetting" problems.) Or
does it vary with the cat's degree of stress? (One speaks of "shear
thickening" if the relaxation time increases with stress, or "shear
thinning" if the opposite is true.)
Of course, we mean stress in
the mechanical sense rather than emotional, but the two meanings may overlap in
some cases.
The 'Deborah number' and the flow of mountains
What cats show clearly is
that determining the state of a material requires comparing two time periods:
the relaxation time and the experimental time, which is the time elapsed since
the onset of deformation initiated by the container.
For instance, it may be the
time elapsed since the cat stepped into a sink. Conventionally, one divides the
relaxation time by the experimental time, and if the result is more than 1, the
material is relatively solid; if the result is lower than 1, the material is
relatively liquid.
This is referred to as
the Deborah number,
after the biblical priestess who remarked that on geological time scales
("before God") even mountains flowed. On shorter time scales one can
see glaciers progressively flowing down valleys.
Even if the relaxation time
is very large (days, years), the behavior can be that of a liquid if the
Deborah number is small (compared to 1).
Conversely, even if the
relaxation time is very small (milliseconds), the behavior can be that of a
solid if the Deborah number is large (compared to 1). This is the case if one
observes a water balloon at the instant when it's popped.
The Deborah number is an
example of dimensionless number: Since we divide one time period by another,
the ratio does not have any unit. In rheology, and in science more generally,
there are many dimensionless numbers that can be used to determine the state or
regime of a material or system.
Measuring the speed of cake batter
For liquids there is another
dimensionless number that can be used to estimate whether the flow will be
turbulent, with vortices, or whether it will calmly follow the outline of the
container (we say that the flow is laminar).
If the flow speed is V and
the container has a typical size h perpendicular to the flow, then we can
define the velocity gradient V/h. The inverse of this velocity gradient scales
as a time.
Comparing this duration and
the relaxation time produces the Reynolds number in
the case of fluids dominated by inertia (like water), or the Weissenberg number for
those dominated by elasticity (like cake batter).
If these dimensionless
numbers are large in comparison to 1, then the flow is likely to be turbulent.
If they're small in comparison to 1 the flow is likely to be laminar.
Asking the question of
whether cats were a liquid allowed me to illustrate the use of these
dimensionless numbers in rheology. I hope that it will make people laugh and
then think.
Marc-Antoine Fardin, Chercheur en rhéologie, Université Paris Diderot – USPC.