One of our strays discovered, sometime in her first winter indoors — they are strictly indoor cats now, on our vet’s recommendation — that she could fit into a salad bowl. Not sit beside it, not rest her head on its rim: fit into it, curled into a precise sphere with her tail tucked under her chin and her ears folded flat, filling the bowl as liquid fills a container. The bowl has a diameter of 22 centimetres. I did not find this as surprising as perhaps I should have: there is a quantity in materials science that determines, rigorously, whether a given material in a given situation should be classified as a solid or a liquid. For a cat in a bowl, this quantity is comfortably below one.

The material is a liquid. The material is also a cat.

The Definition of a Fluid

The intuitive distinction between solids and liquids is that solids hold their shape and liquids conform to their container. But this distinction is one of timescale, not of material identity.

A classic demonstration: place a ball of silly putty on a table. Over the course of an hour, it flows slowly outward, taking the shape of the table surface — clearly a liquid. Strike it sharply with a hammer and it shatters — clearly a solid. The material has not changed. The timescale of the interaction has.

The same principle applies to glass (contrary to popular myth, medieval window glass is not thicker at the bottom because it has flowed — the variation is from the manufacturing process, and the relaxation time of soda-lime glass at room temperature is of order $10^{23}$ years — but at elevated temperatures near the glass transition, silicate glass flows readily). It applies to mantle rock, which is solid on the scale of earthquake waves and liquid on the scale of continental drift. It applies to pitch, to ice sheets, to asphalt on a hot day.

The formal tool for capturing this is the Deborah number.

The Deborah Number

The Deborah number was introduced by Marcus Reiner in 1964, in a short note in Physics Today (Reiner 1964). It is defined as:

$$\mathrm{De} = \frac{\tau}{T},$$

where $\tau$ is the relaxation time of the material — roughly, the characteristic time over which it can rearrange its internal structure and relieve stress — and $T$ is the observation time or the timescale of the imposed deformation.

  • $\mathrm{De} \ll 1$: The material relaxes quickly relative to the timescale of observation. Internal stresses are continuously relieved. The material behaves as a fluid.
  • $\mathrm{De} \gg 1$: The material relaxes slowly relative to the observation timescale. Internal stresses persist. The material behaves as a solid.
  • $\mathrm{De} \sim 1$: The material is in a viscoelastic regime — partly fluid, partly solid, exhibiting time-dependent behaviour that is neither.

The name comes from the prophetess Deborah, who sang in Judges 5:5: “The mountains flowed before the Lord.” At the timescale of a divine perspective, mountains are liquid. At the timescale of a human lifetime, they are not. Reiner’s point was that the solid-liquid distinction is not a property of the material but of the relationship between the material’s internal dynamics and the observer’s timescale.

For Newtonian fluids (water, air at ordinary conditions), $\tau \to 0$ and $\mathrm{De} \to 0$ for any finite observation time — they are always liquid. For a perfectly elastic solid (an ideal spring), $\tau \to \infty$ and $\mathrm{De} \to \infty$ for any finite observation time — always solid. Real materials lie between these extremes.

The Maxwell Viscoelastic Model

The simplest model of a material with a finite relaxation time is the Maxwell element: a spring (elastic, spring constant $G$) in series with a dashpot (viscous, viscosity $\eta$). Under a step stress $\sigma_0$ applied at time $t = 0$, the strain evolves as:

$$\epsilon(t) = \frac{\sigma_0}{G} + \frac{\sigma_0}{\eta}\,t,$$

where $\tau = \eta / G$ is the Maxwell relaxation time. The first term is the instantaneous elastic deformation of the spring; the second is the linear viscous creep of the dashpot. For $t \ll \tau$, the elastic strain dominates and the material behaves as a solid; for $t \gg \tau$, the viscous flow dominates and the material behaves as a liquid. The material “decides” whether to be solid or liquid depending on the ratio of $\tau$ to the duration of the applied stress — which is precisely the Deborah number.

The creep compliance $J(t) = \epsilon(t)/\sigma_0 = t/\eta + 1/G$ grows linearly with time for $t \gg \tau$, confirming liquid behaviour on long timescales. The relaxation modulus $G(t) = \sigma(t)/\epsilon_0 = G e^{-t/\tau}$ decays exponentially to zero, confirming that the material cannot sustain a permanent stress — again, liquid behaviour on long timescales.

On the Rheology of Cats

In 2014, Marc-Antoine Fardin, a physicist at the ENS Lyon, published “On the Rheology of Cats” in the Rheology Bulletin 83(2), 16–17. The paper asked whether cats satisfy the defining rheological criterion for liquids, using the Deborah number as the test. Fardin was awarded the 2017 Ig Nobel Prize in Physics — which is awarded for research that “makes you laugh, then makes you think” — for this work.

The paper is not a joke. It is standard rheology applied to an unusual material, with appropriately hedged conclusions and correct citations to the primary literature on viscoelastic flow. The humour is in the application; the physics is serious.

Estimating the Cat’s Relaxation Time

The relaxation time $\tau$ of a cat is the time scale over which the cat’s body deforms to fill a container. This is observable. A cat placed near a suitable container — a salad bowl, a cardboard box, a bathroom sink — adopts a conformed shape on a timescale of roughly 5–30 seconds. The initial posture (stiff, alert) gives way to a relaxed conformation as the cat assesses the container and adjusts. Fardin estimated $\tau \approx 1$–$30$ seconds, with the exact value depending on the container’s attractiveness to the specific cat.

This is the material’s characteristic relaxation time. The fact that it is finite — that the cat does eventually conform to the container — is the essential observation.

Computing the Deborah Number for Various Situations

Scenario 1: Cat in a sink. A cat taking ten minutes to settle into a bathroom sink. Observation time $T = 600\,\mathrm{s}$, relaxation time $\tau \approx 5\,\mathrm{s}$.

$$\mathrm{De}_\mathrm{sink} = \frac{5}{600} \approx 0.008 \ll 1.$$

The cat is unambiguously a liquid.

Scenario 2: Cat in a cardboard box. Conformation over approximately 30 minutes, $\tau \approx 20\,\mathrm{s}$.

$$\mathrm{De}_\mathrm{box} = \frac{20}{1800} \approx 0.011 \ll 1.$$

Liquid.

Scenario 3: Cat dropping from a bookshelf. Contact time during a jump approximately $T \approx 0.05\,\mathrm{s}$, relaxation time still $\tau \approx 5\,\mathrm{s}$.

$$\mathrm{De}_\mathrm{jump} = \frac{5}{0.05} = 100 \gg 1.$$

Solid. The cat does not deform into the shape of the bookshelf during the jump; it rebounds elastically.

Scenario 4: Cat startled by a loud noise. Reaction time $T \approx 0.3\,\mathrm{s}$, $\tau \approx 5\,\mathrm{s}$.

$$\mathrm{De}_\mathrm{startle} = \frac{5}{0.3} \approx 17 \gg 1.$$

Solid. On short timescales, cats behave as elastic materials — they spring, they bounce, they do not flow.

The cat is neither permanently solid nor permanently liquid. It is a viscoelastic material whose phase classification depends on the timescale of the interaction. This is not a loose analogy; it is the definition of viscoelasticity.

Non-Newtonian Behaviour and Flow Instabilities

Fardin noted an additional complication: cat flow is not Newtonian. A Newtonian fluid has a viscosity $\eta$ that is independent of the applied shear rate $\dot\gamma$. Many real materials are shear-thinning (viscosity decreases with increasing shear rate — ketchup, blood, many polymer solutions) or shear-thickening (viscosity increases with increasing shear rate — cornstarch suspension, some dense suspensions). Cats, Fardin observed, appear to be shear-thinning: the more rapidly you attempt to move a relaxed cat from its current position, the more “liquid” (accommodating, compliant) it becomes, up to a point at which the cat transitions to solid behaviour (claws, teeth).

This is, formally, the behaviour of a yield-stress fluid: a material that behaves as a solid below a critical stress $\sigma_y$ and flows above it. The Herschel–Bulkley model describes such fluids:

$$\sigma = \sigma_y + k \dot\gamma^n, \quad \sigma > \sigma_y,$$

where $k$ is the flow consistency index and $n < 1$ for shear-thinning. The challenge of fitting $k$, $n$, and $\sigma_y$ for a specific cat is experimental, and Fardin acknowledged this was left to future work.

The Deborah number and the yield stress together provide a two-parameter phase diagram for cat rheology:

  • Low stress, short timescale: solid (De ≫ 1 or σ < σ_y)
  • Low stress, long timescale: liquid (De ≪ 1)
  • High stress: yield, followed by flow

Flow Instabilities: The Rayleigh-Plateau Connection

Fardin also noted that cats confined to containers thinner than their body diameter can exhibit flow instabilities. A cat attempting to fit into a glass too narrow for its body will sometimes adopt a helical or coiled configuration — an instability reminiscent of the Rayleigh–Plateau instability of a liquid jet.

The Rayleigh–Plateau instability occurs when a cylindrical fluid jet of radius $r_0$ is subject to perturbations of wavelength $\lambda > 2\pi r_0$. Modes with wavelength longer than the cylinder’s circumference are unstable and grow, breaking the jet into droplets. The dispersion relation for growth rate $\sigma$ as a function of wavenumber $k = 2\pi/\lambda$ (for an inviscid jet) is:

$$\sigma^2 = \frac{\gamma}{\rho r_0^3}\, k r_0 \bigl(1 - k^2 r_0^2\bigr) I_1(kr_0)/I_0(kr_0),$$

where $\gamma$ is surface tension and $I_0, I_1$ are modified Bessel functions. The analogy with a cat is inexact — surface tension is not the dominant restoring force — but the qualitative instability mechanism (a long cylinder of material is unstable to perturbations whose wavelength exceeds the cylinder’s circumference) appears to apply, suggesting that very elongated cats in very narrow containers should be unstable to coiling. This is, again, left to future experimental work.

Why the Deborah Number Matters (Outside of Cat Physics)

The Deborah number is not a curiosity; it is a central dimensionless number in engineering and materials science.

Polymer processing: The flow of polymer melts through injection-moulding channels involves De in the range $10^{-2}$–$10^2$. Too high a De leads to elastic instabilities, melt fracture, and surface defects in the finished part.

Blood rheology: Blood is a non-Newtonian viscoelastic fluid. In the large arteries (low shear rate), red blood cells aggregate into rouleaux and blood behaves as a shear-thinning fluid. In the capillaries (high shear rate), rouleaux break up and individual cells deform to fit through vessels smaller than their resting diameter — liquid behaviour on short length scales.

Geophysics: The mantle is an elastic solid for seismic waves ($T \sim$ seconds, De ≫ 1) and a viscous fluid for convection ($T \sim 10^8$–$10^9$ years, De ≪ 1). The same material. Different Deborah numbers.

Glaciology: Ice is an elastic solid for rapid fracture (calving of icebergs) and a viscous fluid for glacier flow. The transition occurs at timescales of years to decades, depending on temperature and stress.

The cat is in good company.

References

  • Fardin, M.-A. (2014). On the rheology of cats. Rheology Bulletin, 83(2), 16–17.

  • Reiner, M. (1964). The Deborah number. Physics Today, 17(1), 62. https://doi.org/10.1063/1.3051374

  • Barnes, H.A., Hutton, J.F., & Walters, K. (1989). An Introduction to Rheology. Elsevier (Rheology Series, Vol. 3).

  • Bird, R.B., Armstrong, R.C., & Hassager, O. (1987). Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics (2nd ed.). Wiley-Interscience.

  • Eggers, J. (1997). Nonlinear dynamics and breakup of free-surface flows. Reviews of Modern Physics, 69(3), 865–930. https://doi.org/10.1103/RevModPhys.69.865

Changelog

  • 2025-12-15: Fixed Deborah number in summary from 0.08 to 0.008 (matching the body calculation: 5/600 = 0.00833).
  • 2025-12-15: Corrected Fardin’s institutional affiliation from “Paris Diderot University” to “ENS Lyon” — his affiliation on the 2014 Rheology Bulletin paper is Université de Lyon / ENS Lyon (CNRS UMR 5672). He moved to Paris Diderot later in 2014, after the paper was published.