What is the surface tension of this kitty?

Ankur M. MehtaDepartment of Electrical Engineering

University of CaliforniaBerkeley, CA 94720

mehtank@eecs.berkeley.edu

Jenny HuDepartment of Mechanical Engineering

University of MarylandCollege Park, MD 20742jjhu@alum.mit.edu

Abstract—Cats are often observed in puddle form asthey lie on the floor. Key parameters relating to this felinefluid can be extracted from the resulting geometry. Fittingan empirical curve around the volume of a kitty-puddleallows us to calculate its surface tension. This combinedwith the contact angle of the puddle with respect to thefloor gives us the energy of adhesion of the cat-floorinterface. Analysis of a representative sample kitten ona rather ailurophobic hardwood floor with contact angleof 130o indicates a surface tension of30N/m. Thiscalculated value is over an order of magnitude greaterthan any conventional fluid.

Index Terms—Feline fluids, Kitty-puddle, Surface ten-sion, Adhesion energy, Empirical model

I. I NTRODUCTION

Puddles are formed when a liquid comes in con-tact with solid surface in a gaseous medium. Coun-teracting forces define the geometry of this sessiledrop, both at the liquid-gas interface and the liquid-solid-gas triple point. The cohesive force betweenliquid molecules draws the drop together while bodyforces such as gravity pull the molecules towards thesolid-liquid interface. Adhesion between the liquidand the solid spreads the drop out.

Known parameters can be applied to generatethe resultant puddle shape. Conversely, an empiricalmatch of a puddle’s geometry to one of a family ofsuch shapes can be used to infer the fluid parame-ters. This study seeks a numerical approximationto feline surface tensionσ and adhesion energyper unit area to a wooden floor∆W in air. Therepresentative kitty-puddle to be analyzed is shownin figure 1 [1].

Fig. 1. What is the surface tension of this kitty?

II. BACKGROUND

A. Bond Number

The Bond numberBo is a dimensionless ratioof body (gravitational) effects to cohesive (surfacetension) forces of a liquid in a gaseous medium.The gravitational forces are parameterized by thedensities of the fluidsρ along with a characteristicphysical dimensionr0, the radius of curvature atthe top of the liquid drop. The cohesive force isrepresented by the surface tension of the fluid-gasinterfaceσ [2].

Bo =gr2

0 (ρliq − ρgas)

σ. (1)

A Bond number of zero, indicating no bodyforces at all, results in a perfectly spherical liquid-gas interface. Higher Bond numbers indicate greaterinfluence of gravity (or lower surface tension), caus-ing flattening of the bubble. The specific shapeof the drop can be numerically determined and

tabulated for a range of Bond numbers [3]. Acomparison of puddles withBo = 1, 5 is shownin figure 2.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5Drop shapes of varying Bond numbers

Bo = 1Bo = 5

Fig. 2. Two puddles with different Bond numbers demonstrate thedifferent drop shapes resulting from varying body forces or surfacetension. These nondimensionalized shapes can be scaled up or downto fit an empirical drop.

A physical drop can be matched against thefamily of shapes to determine the best fit, andthereby infer the Bond number for that liquid-gasinterface. This match can be effectively determinedby a computer program [4], or more simply byinspection.

B. Contact Angle

The puddle shape defined by the Bond number isindependent of the adhesive forces between the solidand the liquid, and in fact describes a drop in whichthe two phases do not interact. The boundary of theliquid then lies tangent to the solid surface. Anyadhesion between the two will cause the liquid-solidinterface to have non-zero extent. This results in thetangent to the liquid-air interface intersecting thesolid surface. The angle between the two throughthe liquid volume is defined to be the contact angleθC . This is shown in figure 3.

The drop geometry described by the Bond num-ber is truncated by the plane of the solid to givethe final shape of the puddle; the resulting contactangle of the solid-liquid interface is defined by theadhesion energy∆W of the liquid per unit area to

Fig. 3. The contact angleθC of a liquid drop on a solid surfaceis a function of the adhesive forces between the two phases. Imagefrom [5].

the solid in the gaseous medium. They are relatedby the Young-Dupre equation [5]:

σ(1 + cos(θC)) = ∆W. (2)

Horizontal surfaces intersecting a drop at varyingcontact angles and the resulting puddle shapes areshown in figure 4.

III. K ITTY-PUDDLE FITTING

The profile of the puddle shown in 1 can now beanalyzed to characterize the feline fluid. The contactangle is measured to average130o between theadvancing and receding edges of the drop. Fittinga drop shape cut off atθCkitty

= 130o aroundthe cat, the Bond number can be approximated byBokitty = 1. The resulting theoretical puddle isoverlaid on the empirical photograph in figure 5.

The kitten doesn’t fit completely inside the pre-dicted shape; the posterior of the animal protrudesbeyond the expected boundary. This is a manifes-tation of the non-ideality of the surface on whichthe kitty-puddle rests. Due to the macro- and mi-croscopic irregularities in surface texture, there area range of stable interfacial states, and a dynamicanalysis of the formation of the drop would benecessary to account for the deviation from the idealdrop shape [6]. However, because the bulk of thefluid fits the predicted curve quite well, we canignore the slight perturbation in our considerationof bulk kitten fluid parameters.

A. Physical kitten parameters

According to [7], the average weight of a Moggycross breed small cat is 3.4kg. Modeling the cat as

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5Drops with Bo=1 and varying contact angles

θC

=50o

θC

=90o

θC

=130o

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

θC

=50o

θC

=90o

θC

=130o

Fig. 4. Varying surface adhesion energy causes the base of a dropto spread out along a surface. For a given bond number, this has theeffect of truncating the base of the drop shape at a given contactangle.

Fig. 5. Cat in a bubble!

a cylinder of radius 6cm and length 35cm yields adensity

ρkitty =3.4kg

π(.06m)2(.35m)= 859kg/m3.

B. Surface tension of a kitten

Equation (1) for the Bond number can be invertedto calculate the surface tension of the kitten:

σkitty =gr2

0(ρkitty − ρair)

Bo. (3)

Fitting in parameters described above gives

σkitty =9.81m

s2 (0.06m)2(859 kgm3 − 1.2 kg

m3 )

1= 30N/m.

C. Adhesion energy of a kitten on a hardwood floor

The Young-Dupre equation (2) can now be usedto calculate the adhesion energy per unit area of thecat to a hardwood floor in an air medium. Giventhe observed contact angle and calculated surfacetension,

∆W = σkitty(1 + cos(θCkitty)

= 10.7J/m2.

The estimated (from literature), measured (fromthe image), and calculated parameters for the rep-resentative feline sample are summarized in tableI.

TABLE IFELINE FLUID PARAMETER VALUES

Parameter Description Type Value Unitsr0 Radius of a kitten E 6 cm

g Acceleration (gravity) E 9.81 m/s2

ρkitty Density of a kitten C 859 kg/m3

ρair Density of air E 1.2 kg/m3

Bokitty Bond number M 1σkitty Surface tension C 30 N/m

θCkitty Contact angle M 130o

∆W Adhesion Energy C 10.7 J/m2

IV. CONCLUSION

This work features the first foray into the field offauna fluids, focusing on felines.

The calculated surface tension of the kitten is overtwo orders of magnitude greater than that of waterat room temperatureσ = 0.0728N/m, and stillnotably higher than mercuryσ = .4254N/m [8].This indicates great promise for the use of kittensas a new class of heat transfer fluid.

The high contact angle of the kitten on a hard-wood floor is indicative of a strongly ailurophobicsurface. In fact, the rather low interfacial adhesionenergy is less than the energy required to lift the catjust 2cm. This explains the ease at which nappingkitty-puddles can be picked off of the floor.

Fig. 6. Further examples of fluid puddles from throughout the animalkingdom. Top row: bunny [9], hedgehog [10]. Bottom row: penguin[11], baby [12].

Kittens aren’t the only animals that form dropson solid surfaces. It would be interesting to conductfurther research into establishing fluid properties forother animals to compare with kittens and other con-ventional liquids. Some examples of animal puddlesare shown in figure 6.

REFERENCES

[1] “Cute overload :): Kitten puddle,” http://www.cuteoverload.com/2006/03/kittenpuddle.html.

[2] V. P. Carey,Liquid-Vapor Phase-Change Phenomena: An In-troduction To The Thermophysics Of vaporization and conden-sation in heat transfer equipment: An Introduction ... in HeatTransfer Equipment (Ser). Taylor & Francis, 1992.

[3] F. Bashforth and J. C. Adams,An attempt to test the theoriesof capillary action by comparing the theoretical and measuredforms of drops of fluid. With an explanation of the method ofintegration employed in constucting the tables which give thetheoretical forms of such drops. Cambridge University Press,1883.

[4] J. N. Butler and B. H. Bloom, “A curve-fitting method forcalculating interfacial tension from the shape of a sessile drop,”Surface Science, vol. 4, no. 1, pp. 1–17, Jan 1966.

[5] “Contact angle,” http://en.wikipedia.org/wiki/Contactangle.[6] “Wetting,” http://en.wikipedia.org/wiki/Wetting.[7] “Cat weight guide,” http://www.petsgetslim.co.uk/site-media/

images/modules/advice/Catweights.pdf.[8] “Surface tension values of some common test liquids for surface

energy analysis,” http://www.surface-tension.de/.[9] “Domobunnrigato,” http://www.cuteoverload.com/2007/04/

domobunnigato.html.[10] “The question: How can we help hedgehogs on bon-

fire night?” http://www.guardian.co.uk/environment/2007/nov/05/conservation.ethicall%iving.

[11] “Baby penguin lying down,” http://www.flickr.com/photos/kathleen-andersen/33444699/.

[12] “some tips for photographing newborns!” http://proudtointroduce.com/blog/?p=469.