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    <title>Dimensional-Analysis on Sebastian Spicker</title>
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      <title>How Cats Drink: Inertia, Gravity, and the Froude Number at the Tip of a Tongue</title>
      <link>https://sebastianspicker.github.io/posts/how-cats-drink-froude-number/</link>
      <pubDate>Mon, 22 Jul 2024 00:00:00 +0000</pubDate>
      <guid>https://sebastianspicker.github.io/posts/how-cats-drink-froude-number/</guid>
      <description>Cats do not scoop water with their tongues — they exploit a delicate balance between inertia and gravity at the air-water interface. The tip of the tongue just touches the surface; rapid withdrawal pulls a fluid column upward; the jaw closes at exactly the moment the column peaks. Reis, Jung, Aristoff, and Stocker (Science, 2010) showed that the lapping frequency of all felids — from domestic cats to lions — is tuned so that the Froude number at the tongue tip is approximately unity.</description>
      <content:encoded><![CDATA[<p><em>I have spent a non-trivial amount of time watching our cats drink — they are
indoor-only cats, on our vet&rsquo;s advice, which gives them few distractions and
gives me ample opportunity to observe. This is not entirely voluntary. Once you have noticed that something is happening at the
water bowl that does not look right — the tongue moves too fast, the water
column is pulled upward rather than scooped, the jaw closes before the tongue
returns — you find yourself crouching beside the bowl with your phone propped
against a chair, filming at 240 frames per second and feeling that you have
perhaps chosen an unusual way to spend a Tuesday morning.</em></p>
<p><em>Pedro Reis, Sunghwan Jung, Jeffrey Aristoff, and Roman Stocker had the same
impulse, with better equipment. Their 2010 paper in Science, &ldquo;How Cats Lap:
Water Uptake by Felis catus,&rdquo; is one of the more elegant pieces of dimensional
analysis in recent biology.</em></p>
<hr>
<h2 id="how-cats-do-not-drink">How Cats Do Not Drink</h2>
<p>The simplest hypothesis — that cats curl the tongue into a spoon and scoop
water into the mouth — is false. High-speed photography shows that the cat&rsquo;s
tongue does not form a cup shape. Instead, the cat extends the tongue tip
downward toward the water surface and then rapidly retracts it. The motion is
fast — too fast for normal video — and the tongue barely contacts the surface.</p>
<p>The contrast with dogs is instructive. Dogs <em>do</em> scoop: the tongue curls
backward (not forward), forming a ladle shape that scoops water upward and
backwards into the mouth. The mechanism is vigorous and inefficient — a
significant fraction of the water misses the mouth entirely, which is why
drinking dogs produce splashing and dogs often have wet chins. The mechanism
works but is inelegant.</p>
<p>Cats produce almost no splash. The mechanism is different in kind.</p>
<hr>
<h2 id="the-physical-mechanism">The Physical Mechanism</h2>
<p>Reis et al. (2010) used high-speed photography (1000 frames per second) to
resolve the cat&rsquo;s lapping motion. Their observations:</p>
<ol>
<li>
<p>The cat extends the tongue tip downward until the <em>dorsal surface</em> (the top
side) just touches the water surface. The ventral surface (the smooth
underside) does not contact the water.</p>
</li>
<li>
<p>The cat then rapidly retracts the tongue upward. The tongue tip is moving
at roughly $v \approx 0.7\,\mathrm{m/s}$ during this retraction.</p>
</li>
<li>
<p>As the tongue tip pulls away from the surface, a column of liquid is pulled
upward by the adhesion between the liquid and the retreating tongue. The
column rises against gravity.</p>
</li>
<li>
<p>The column eventually stalls — inertia is overcome by gravity — and begins
to fall back. The cat closes its jaw at exactly the moment of maximum column
height, capturing the peak volume of water.</p>
</li>
<li>
<p>The cat then extends the tongue for the next lap.</p>
</li>
</ol>
<p>The cat closes its jaw before the tongue fully retracts. This is important:
the jaw closure captures the water column, not the water adhering to the tongue.
The tongue is the mechanism that <em>creates</em> the column; the jaw captures it.</p>
<hr>
<h2 id="dimensional-analysis-the-froude-number">Dimensional Analysis: The Froude Number</h2>
<p>The relevant competition is between <strong>inertia</strong> (which drives the column
upward) and <strong>gravity</strong> (which pulls it back down). Surface tension plays a
role in stabilising the column but is not the primary factor governing the
column height.</p>
<p>The balance between inertia and gravity for a fluid column moving at speed
$v$ and of characteristic length scale $L$ (here, the diameter of the tongue
tip, $L \approx 5\,\mathrm{mm}$ for a domestic cat) is captured by the
<strong>Froude number</strong>:</p>
$$\mathrm{Fr} = \frac{v}{\sqrt{gL}},$$<p>where $g = 9.81\,\mathrm{m/s}^2$ is gravitational acceleration.</p>
<p>When $\mathrm{Fr} \ll 1$: gravity dominates, inertia is insufficient to pull a
significant column of water upward. Very slow tongue motion would lift almost
no water.</p>
<p>When $\mathrm{Fr} \gg 1$: inertia dominates, the column rises far above the
surface but the jaw must be closed quickly before the large amount of water
falls back. Very fast tongue motion wastes water and requires rapid jaw closure.</p>
<p>The optimal lapping frequency — maximising captured volume per lap — occurs
near $\mathrm{Fr} \approx 1$, where inertial and gravitational forces are
comparable and the column height is matched to the jaw closure dynamics.</p>
<h3 id="checking-the-numbers-for-a-domestic-cat">Checking the Numbers for a Domestic Cat</h3>
<p>For a domestic cat:</p>
<ul>
<li>Tongue tip diameter: $L \approx 5\,\mathrm{mm} = 5 \times 10^{-3}\,\mathrm{m}$</li>
<li>Characteristic tongue tip speed: $v \approx 0.7\,\mathrm{m/s}$</li>
</ul>
$$\mathrm{Fr} = \frac{0.7}{\sqrt{9.81 \times 5 \times 10^{-3}}}
= \frac{0.7}{\sqrt{0.049}} = \frac{0.7}{0.22} \approx 3.2.$$<p>Reis et al. found Fr of order unity — inertial and gravitational forces
comparable — confirming that the lapping speed is tuned to the inertia-gravity
balance. (The exact numerical value depends on the choice of characteristic
length scale; using the tongue tip diameter as above gives Fr in the range
1–3, squarely in the regime where neither force dominates.)</p>
<hr>
<h2 id="scaling-across-felids">Scaling Across Felids</h2>
<p>The Froude number prediction yields a scaling law for lapping frequency across
felid species of different sizes. If all felids lap at $\mathrm{Fr} \approx 1$,
then the characteristic speed scales as $v \sim \sqrt{gL}$, and the lapping
frequency scales as:</p>
$$f = \frac{v}{d} \sim \frac{\sqrt{gL}}{d},$$<p>where $d$ is the distance the tongue travels per lap (roughly proportional to
tongue length, which scales with body size). Since $L \sim d$ scales with body
size, we get:</p>
$$f \sim \frac{\sqrt{g \cdot d}}{d} = \sqrt{\frac{g}{d}} \propto d^{-1/2}.$$<p>Larger cats have longer tongues and lap more slowly. The prediction is that
lapping frequency scales as the square root of inverse tongue length — or,
equivalently, as the inverse square root of body mass (since linear dimensions
scale as mass$^{1/3}$):</p>
$$f \propto m^{-1/6}.$$<p>Reis et al. tested this against high-speed footage of large felids. A domestic
cat laps at approximately $4\,\mathrm{Hz}$; a lion laps at approximately
$1.2\,\mathrm{Hz}$; a tiger at roughly $1\,\mathrm{Hz}$. The scaling is
consistent with $f \propto m^{-1/6}$ across three orders of magnitude in
body mass.</p>
<p>The table below shows the predicted versus observed scaling:</p>
<table>
  <thead>
      <tr>
          <th>Species</th>
          <th>Body mass (kg)</th>
          <th>Predicted $f$ relative to cat</th>
          <th>Predicted $f$ (Hz)</th>
          <th>Observed $f$ (Hz)</th>
      </tr>
  </thead>
  <tbody>
      <tr>
          <td>Domestic cat</td>
          <td>4</td>
          <td>1.0</td>
          <td>4.0</td>
          <td>~4.0</td>
      </tr>
      <tr>
          <td>Jaguar</td>
          <td>80</td>
          <td>$\left(\frac{4}{80}\right)^{1/6} \approx 0.61$</td>
          <td>2.4</td>
          <td>~2.0</td>
      </tr>
      <tr>
          <td>Lion</td>
          <td>200</td>
          <td>$\left(\frac{4}{200}\right)^{1/6} \approx 0.52$</td>
          <td>2.1</td>
          <td>~1.5</td>
      </tr>
      <tr>
          <td>Tiger</td>
          <td>220</td>
          <td>$\left(\frac{4}{220}\right)^{1/6} \approx 0.51$</td>
          <td>2.1</td>
          <td>~1.0</td>
      </tr>
  </tbody>
</table>
<p>The $m^{-1/6}$ scaling captures the correct trend — larger cats lap more
slowly — though the predicted frequencies for the largest cats somewhat
overestimate the observed values. The discrepancy may reflect the limitations
of the simple allometric assumption (that all linear dimensions scale as
$m^{1/3}$) and the fact that tongue geometry does not scale isometrically
across the full range of felid body sizes.</p>
<hr>
<h2 id="why-not-just-lick">Why Not Just Lick?</h2>
<p>A natural question: why not simply allow the tongue to fully submerge and
absorb water through the papillae, as the tongue already contacts water when
lapping? Several answers:</p>
<ol>
<li>
<p><strong>Papillae are not sponges.</strong> Feline papillae are hollow and scoop-shaped
(filiform papillae with hollow tips), optimised for grooming and food
manipulation, not passive absorption. Active wicking is limited.</p>
</li>
<li>
<p><strong>The cat cannot breathe with its mouth submerged.</strong> A lapping mechanism
that keeps the mouth mostly closed except for the brief jaw-closure moment
allows continuous breathing through the nose during drinking.</p>
</li>
<li>
<p><strong>Speed and efficiency.</strong> The inertial column mechanism delivers significantly
more water per jaw movement than surface tension adhesion alone. At 4 laps
per second, a domestic cat takes in roughly $0.14\,\mathrm{mL}$ per lap,
for a total of roughly $34\,\mathrm{mL/min}$ — comparable to sipping rates
in animals that use more direct intake mechanisms.</p>
</li>
</ol>
<p>The cat has converged on a hydrodynamically optimal strategy under the
constraint of keeping the oral cavity mostly sealed during the intake cycle.</p>
<hr>
<h2 id="the-robotic-tongue">The Robotic Tongue</h2>
<p>Reis et al. constructed a robotic cat tongue to verify the mechanism: a smooth
glass disc lowered to the water surface and retracted at controlled speeds.
The column height as a function of speed followed the predicted inertia-gravity
balance, confirming that the mechanism does not depend on any specifically
biological property of the tongue — it is a fluid dynamics result that applies
to any surface moving away from a water interface at the right speed.</p>
<p>The robot lapped at the same Froude number as the cat.</p>
<hr>
<h2 id="dogs-horses-and-the-comparison">Dogs, Horses, and the Comparison</h2>
<p>Dogs cup the tongue <em>caudally</em> (backwards) rather than ventrally, forming a
ladle. The mechanism is faster and delivers more water per stroke but is
messy — the ladle is formed outside the mouth, and water sloshes freely. Dogs
lap at roughly $3\,\mathrm{Hz}$ with a tongue tip speed significantly higher
than cats, producing Fr well above unity. The excess inertia is why dog
drinking generates splashing.</p>
<p>Horses, by contrast, create a near-seal with their lips and use suction —
a fundamentally different mechanism that requires no tongue projection at all.
The lapping mechanism of felids is phylogenetically specific and appears to
have evolved under selection pressure for both efficiency and noise suppression,
consistent with the ambush-predator lifestyle. A cat that splashed while
drinking would alert prey at a water source. A cat that laps near-silently
does not.</p>
<hr>
<h2 id="a-note-on-the-measurement">A Note on the Measurement</h2>
<p>Getting reliable high-speed footage of a cat drinking is harder than it sounds.
Our cats drink at different times of day, in different moods, and the presence
of a camera tripod next to the water bowl is regarded as grounds for drinking
elsewhere. Pedro Reis et al. solved this by filming their laboratory cat, Cutta
Cutta, in a controlled setting. Their footage is available online and is
genuinely beautiful: a slow-motion waterfall in miniature, rising improbably
from the tongue tip and held there by the balance between upward momentum and
downward gravity, until the jaw swings shut.</p>
<p>The physics is in the timing.</p>
<hr>
<h2 id="references">References</h2>
<ul>
<li>
<p>Reis, P.M., Jung, S., Aristoff, J.M., &amp; Stocker, R. (2010). How cats lap:
Water uptake by <em>Felis catus</em>. <em>Science</em>, 330(6008), 1231–1234.
<a href="https://doi.org/10.1126/science.1195421">https://doi.org/10.1126/science.1195421</a></p>
</li>
<li>
<p>Aristoff, J.M., Stocker, R., Jung, S., &amp; Reis, P.M. (2011). On the water
lapping of felines and the water running of lizards. <em>Communicative &amp;
Integrative Biology</em>, 4(2), 213–215.</p>
</li>
<li>
<p>Vogel, S. (1994). <em>Life in Moving Fluids: The Physical Biology of Flow</em>
(2nd ed.). Princeton University Press.</p>
</li>
</ul>
<hr>
<h2 id="changelog">Changelog</h2>
<ul>
<li><strong>2025-12-15</strong>: Updated water intake per lap from 0.04 mL to 0.14 mL (Reis et al. report ~0.14 +/- 0.04 mL per lap; the previous value was the standard deviation), and updated the intake rate accordingly (~34 mL/min). Updated the papillae location from ventral to dorsal surface. Updated the Aristoff et al. reference to the correct 2011 <em>Communicative &amp; Integrative Biology</em> article. Removed the Jung &amp; Kim (2012) PRL reference (article number 034501 resolves to a different paper).</li>
</ul>
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