Kalliroscope644

J.
FluidMech.
(2010),vol.
644,pp.
5–33.
cCambridgeUniversityPress2010doi:10.
1017/S00221120099922055Thehydrodynamicsofwater-walkingarthropodsDAVIDL.
HUANDJOHNW.
M.
BUSHDepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA(Received24May2008;revised9September2009;accepted11September2009)Wepresenttheresultsofacombinedexperimentalandtheoreticalinvestigationofthedynamicsofwater-walkinginsectsandspiders.
Usinghigh-speedvideography,wedescribetheirnumerousgaits,someanalogoustothoseoftheirterrestrialcounterparts,othersspecializedforlifeattheinterface.
Thecriticalroleoftheroughsurfaceofthesewaterwalkersinbothoatationandpropulsionisdemonstrated.
Theirwaxy,hairysurfaceensuresthattheirlegsremaininawater-repellentstate,thatthebulkoftheirlegisnotwetted,butrathercontactwiththewaterarisesexclusivelythroughindividualhairs.
Maintainingthiswater-repellentstaterequiresthatthespeedoftheirdrivinglegsdoesnotexceedacriticalwettingspeed.
Flowvisualizationrevealsthatthewakesofmostwaterwalkersarecharacterizedbyaseriesofcoherentsubsurfacevorticesshedbythedrivingstroke.
Atheoreticalframeworkisdevelopedinordertodescribethepropulsionintermsofthetransferofforcesandmomentumbetweenthecreatureanditsenvironment.
Theapplicationoftheconservationofmomentumtobiolocomotionattheinterfaceconrmsthatthepropulsionofwaterwalkersmayberationalizedintermsofthesubsurfaceowsgeneratedbytheirdrivingstroke.
Thetwoprincipalmodesofpropulsionavailabletosmallwaterwalkersareelucidated.
Atdrivinglegspeedsinexcessofthecapillarywavespeed,macroscopiccurvatureforcesaregeneratedbydeformingthemeniscus,andthesurfacebehaveseectivelyasatrampoline.
Forslowerspeeds,thedrivinglegsneednotsubstantiallydeformthesurfacebutmayinsteadsimplybrushit:theresultingcontactorviscousforcesactingontheleghairscrossingtheinterfaceservetopropelthecreatureforward.
1.
IntroductionWepresentheretheresultsofanextensiveseriesofexperimentsaimedatelucidatingthepropulsionmechanismsofwater-walkingarthropods.
Ourstudyismotivatedprincipallybyfundamentalinterest,specicallyadesiretorationalizeanumberofnature'sdesigns.
Nevertheless,owingtothescalesinvolved,thisclassofproblemsmayservetoinspireandinformthebiomimeticdesignofmicrouidicdevices.
Forexample,thedynamicinteractionbetweenwater-repellentsolidsanduidsisaproblemofconsiderableinterestinanumberofengineeringapplications,forexamplethedesignofself-cleaninganddrag-reducingsurfaces(Bush,Hu&Prakash2008;Quere2008).
Therehasbeenconsiderableworkreportedinthebiologyliteratureonwater-walkinginsects,beginningwiththatofAldrovandi(1618)andRay(1710).
ThePresentaddress:DepartmentsofMechanicalEngineeringandBiology,GeorgiaInstituteofTechnology,Atlanta,GA30318,USAEmailaddressforcorrespondence:bush@math.
mit.
edu6D.
L.
HuandJ.
W.
M.
Bushintroductionoftheconceptofsurfacetension(Plateau1873)providedthephysicalbasisforunderstandingtheweightsupportofsmallwater-walkinginsects(Brocher1910;Baudoin1955),andthestaticequilibriaofwater-walkinginsectsisnowwellunderstood(Manseld,Sepangi&Eastwood1997;Keller1998).
Thedynamicsofwater-walkingarthropodsissubtlerandrequiresconsiderationofboththesurfacechemistryandthetextureresponsibleforthecreature'swaterrepellency(Bushetal.
2008)andtheinterfacialuiddynamicsgeneratedbyitslegstroke(Bush&Hu2006).
Theformandfunctionofthesurfacelayer(or'integument')ofwater-walkingcreatureshasbeenreviewedinBushetal.
(2008),andthedynamicalsignicanceofthehairgeometryhasbeendemonstratedinPrakash&Bush(submitted).
Water-walkingarthropodsarewaterrepellentbyvirtueoftheirwaxysurfacecoatingthatincreasesthecontactanglebeyondπ/2(Holdgate1955),andasurfaceroughnessthatconsistsofadensearrayofhairs(Andersen1977;Gao&Jiang2004;Stratton,Suter&Miller2004b).
Whenadjoiningthefreesurface,thesehairstrapairpockets,precludebulkwettingofthelegandsomaintaintheleg'swaterrepellency.
Whenthelegismoving,however,themannerinwhichwaterrepellencyismaintainedandforceistransferredfromtheuidtothedrivinglegsisnotimmediatelyclear.
Weheredemonstratethemeansbywhichpropulsiveforcesaregeneratedbythedrivingstrokeofwater-repellentwater-walkingarthropods.
Ithaslongbeenknownthatsurfacetensionplaysacriticalpropulsiveroleformostwater-walkingcreatures(Dufour1833;Brocher1910;Baudoin1955).
Withtheinventionofthehigh-speedcamerain1942,greatstridesweremadetowardsdescribingtheirmotiononthewatersurface.
Itbecameclearthatwaterwalkerspossessavarietyofgaits(Andersen1976;Suteretal.
1997)thatincludewalking,rowingandgalloping.
LocomotionbywaterwalkerssuchasMicrovelia,MesoveliaandHydrometraischaracterizedbyanalternatingtripodgaitanalogoustothatemployedonlandbycockroachesandotherterrestrialhexapods(Altendorferetal.
2001).
Themostecientwaterwalkers,suchaswaterstridersandsherspiders,rowusingtheirmiddlepairoflegs(Andersen1976;Suter,Stratton&Miller2003;Stratton,Suter&Miller2004a).
Bowdan(1978)notedthatwaterstridersreverttoawalkinggaitonauidofhigherviscosity.
Fisherspidersswitchfromrowingtogallopinginordertoachievetheirpeakspeed(Suter&Wildman1999).
Certainwaterwalkerscanclimbstaticmeniscisimplybydeformingthefreesurfacequasi-statically,therebygeneratinglateralcapillaryforces(Baudoin1955;Miyamoto1955;Hu&Bush2005).
Finally,Microveliaandcertainshore-dwellingcreaturespossessanemergencyformofpropulsion:byexcretingsurfactant,theygeneratesurface-tensiongradientsthatpropelthemshortdistancesalongthewatersurface(Linsenmair&Jander1976;Schildknecht1976).
Darnhofer-Demar(1969)showedexperimentallythatwaterstridersgeneratemillimetre-scaleindentationsbystrikingthefreesurfaceandsurmisedthatitwastheassociatedcurvatureforcesthatprovidethestrider'sthrust.
Heemphasizedthepresenceofwavesinthewake,andthisfuelledworksupportingtheideathatwavedragonthedrivinglegplaysacriticalpropulsiverole(Andersen1976;Denny1993;Sun&Keller2001).
ThisinferenceconcerningthecriticalroleofwavedragledtoDenny'sparadox(Suteretal.
1997),theproposalthatinfantwaterstridersunabletogeneratewavesshouldbeincapableofself-propulsion(Denny1993,2004).
Suteretal.
(1997)performedaseriesofexperimentsinwhichthesteady-stateforcesweremeasuredonaspiderlegsuspendedinarotatingume.
Theauthorsestimatedtherelativemagnitudesofforcesonthelegowingtowavedrag,uidinertiaandcurvatureforces;bydemonstratingthepersistenceoftheinertialforcesevenintheThehydrodynamicsofwater-walkingarthropods7absenceofwaves,theauthorseectivelyresolvedDenny'sparadox.
Hu,Chan&Bush(2003)arguedthatthepropulsionofthesecreaturesismosteasilyunderstoodandtheparadoxmosteasilyresolvedbyconsideringmomentumtransferintheirwake,anargumenttobesupportedbythetheoreticaldevelopmentsof§5.
2.
Huetal.
(2003)showedexperimentallythatthemostcommonwater-walkinginsect,thewaterstrider,transfersmomentumprincipallythroughsubsurfacevortices.
Inourexperimentalstudyreportedin§4,wedemonstratethatthegenerationofvorticesbywaterwalkershasabroadgenerality;moreover,wedemonstratethatevenintheabsenceofthegenerationofpronouncedwavesorvortices,water-walkinginsectsareabletopropelthemselvesbyvirtueofthemicroscaleinteractionbetweentheirroughintegumentandthewatersurface.
In§5,wedevelopatheoreticalframeworkthatmaybeusedtointegrateandrationalizeourexperimentalobservationsofwater-walkingarthropods.
Biolocomotionisgenerallyrationalizedintermsofthetransferofforceandmomentumbetweenthecreatureandthesurroundinguid(Childress1981).
Conservationofmomentumallowsonetorationalizethehigh-Reynolds-numberpropulsionofsh(Wilga&Lauder2002)andbirds(Spedding,Rosen&Hedenstrom2003)byconsideringthemomentumtransferintheirwakes,anapproachofparticularvaluewhenthesewakesarecharacterizedbycoherentvorticalstructures.
Themotionofwaterwalkersiscomplicatedbythepresenceofthefreesurface.
ImplicitintheanalysisofHuetal.
(2003)andB¨uhler(2007)forthewaterstriderandinthatofHsieh(2004)forthebasilisklizardisthatmomentumissimilarlyconservedforpropulsionatafreesurface,aresulttobeprovenin§5.
2.
Inthispaper,wereportmicro-andmacroscaleobservationsofthedierentlocomotorystylesofwater-walkingarthropods.
In§2,wedescribeourexperimentaltechniques.
In§3,weconsiderthestateofwettingofthewaterwalker'sleginbothstaticanddynamicstates.
In§4,wecharacterizethelegstrokeandtheresultingdynamicsoftheunderlyingwater.
Weproceedin§5byconsideringthehydrodynamicsofwater-walkingarthropodsfromatheoreticalperspective,characterizingthehydrodynamicforceandmomentumtransferbetweentheinsect,theuidandthefreesurface.
Lastly,in§6wediscusstheimplicationsofourworkandsuggestdirectionsforfutureresearch.
2.
ExperimentaltechniquesSixwater-walkingand20terrestrialinsectsandspiders,commontoourarea,exhibitedarangeofwater-walkingtechniques(gure1).
FreshwaterwalkersweregatheredfromFreshPond,Massachusetts;marinewaterwalkersfromthecoastatRockport,Massachusetts;andterrestrialinsectsfromCambridge,Massachusetts.
Thesecreatureswereraisedincaptivityinaquaria,sustainedonadietofground-dwellinginsects.
Insectlegwidthsweremeasuredusingseverallightmicroscopes(SKope3000100byBoreal,LSMPascalconfocalmicroscopebyZeissandstereomicroscopeSTEMI2000byZeiss);theirmicroscalehaircoveringswereexaminedwithascanningelectronmicroscope(XL30ESEMbyFEI).
Welmedtheinsectsusingadigitalstillcamera(SonyDSC-F707),adigitalvideocamera(SonyDCR-TRV950)andahigh-speeddigitalvideocamera(RedlakeMotionscopePCI8000).
High-speedlmsweredigitizedusingMidasmotionanalysissoftware.
Insectswerelmedat1/1000sexposuretimeat30–500f.
p.
s.
Planviewsoftheirlocomotionwerelmedinashallow8D.
L.
HuandJ.
W.
M.
Bush(a)(d)(e)(f)(b)(c)Figure1.
Thewater-walkingarthropodsexaminedinourexperimentalstudy,orderedbysize:(a)thebroad-shoulderedwaterstriderMicrovelia,(b)thewatertreaderMesovelia,(c)thespringtailAnuridamaritima,(d)thewatermeasurerHydrometrastagnorum,(e)thewaterstriderGerris,(f)thesherspiderDolomedestriton.
Scalebars,1mm.
Water+NaOHLightThymolblueCameraInsectWakeFigure2.
Theexperimentalapparatususedforowvisualization.
Amildlybasicsolutionisilluminatedfrombelowandviewedfromabove.
Themotionofanopaquetracer,ThymolBluedye,isusedtotrackthemotionoftheowsgeneratedbywater-walkinginsects.
tank(15cm*15cm*4cm)andsideviewsinaslendertank(3cm*20cm*20cm)thatconstrainedthepathoftheinsectsothatitremainedinfocus.
Flowvisualizationwasaccomplishedinaseriesofparticletrackinganddyestudiesusingtheapparatusillustratedingure2.
Particletrackingwasperformedbylightingthesubjectfromabove,placingadarkbackgroundbeneaththetankandseedingtheuidwitheitherKalliroscopeAQ-1000orPlioliteparticles(S-6B,GoodyearChemicals),groundtoagrainsizeof50–100μm.
AThymolBluetechnique(Voropayev&Afanasyev1994)wasusedtovisualizesubsurfaceows.
Theshallowtankwaslitfrombelowusingalighttable(LightTracer,Artograph)andlledtoadepthof0.
5–1cmwithaweakdilutionofsodiumhydroxide(pH9.
2).
ThymolBluepowderwassprinkledonthesurfaceinthepathoftheinsect;theinsectpropelleditselfthroughthedyeeldasthedyesanktothebottomofthetank.
TheThymolBluepromptedMarangoniconvectionrollsofwidthcomparabletothedepthofthedyelayer(Scriven&Sternling1970).
ThegraininessoftheseconvectionrollscouldThehydrodynamicsofwater-walkingarthropods9(a)20μm20μm5μmAcc.
VSpotMagn3306xDetSEWD10.
0MIT3.
01.
30kV(b)(c)Figure3.
ScanningelectronmicroscopeimagesofthehairlayeronthedrivinglegsofMesovelia(a,scalebar20μm)andthewaterstrider(b,scalebar20μm;c,scalebar5μm).
Thelegisahairybrushwhosehairsaretiltedat30tothelegsurface.
Hairsaretypically30μmlong,1–3μmthickatthebaseandtapered;theirdensityis12000–16000hairsmm2.
(c)Acloserviewofthehairsshowsthattheirtipsarebentinwardtowardsthelegs;moreover,eachhairispatternedwithgroovesofcharacteristicwidth400nmthatrunitslength.
ImagescourtesyofManuPrakash.
bereducedbystirringtheuid.
Stirringalsoallowedthedye-basedmixturetobeusedrepeatedly.
Wealsoobservedthedeectionofthefreesurfacebytheinsectleg.
Thelegwasneverobservedtopenetratethefreesurface.
Instead,awater-repellentstatewasmaintained:athinairlayerremainedtrappedinitsintegument(Bushetal.
2008).
Someinsectsarealsoabletopulluponthefreesurfacebyvirtueoftheirhydrophilicclaws(Baudoin1955;Noble-Nesbitt1963;Andersen1976),animportantadaptationformeniscusclimbing(Hu&Bush2005).
ThesignofthesurfacedeectionwasdeterminedusingalightingtechniqueadoptedfromBaudoin(1955).
Bylightingdirectlyfromabovetheinsect,thesurfacedeectioncouldbeinferredqualitativelyfromthemannerinwhichlightwasfocusedontothetankbottom.
Depressionsofthefreesurfacedefractedlightradiallyoutwardandweremarkedbydarkspotsonthetankbottom;conversely,peaksinthefreesurfaceweremarkedbybrightspots(gures11,14and16).
Theinsectswereencouragedtomoveusingavarietyoftechniques.
Whentheroomwasdark,abeamoflightproducedbyaashlightwouldattracttheinsects.
Proddingwithawire,shakingthetankorblowingontheinsectwerealsoeectiveinencouragingittomoveinapreferreddirectionor,inthecaseofMicrovelia,tosecretesurfactant.
3.
MicroscaleconsiderationsThemacroscopicowsgeneratedbythedrivingstrokeofwater-walkinginsectswillbeconsideredin§§4and5.
Werstconsiderthemicroscopicinteractionbetweenthearthropodcuticleandtheinterface,specicallyitsroleinwaterrepellency(§3.
1)andpropulsion(§3.
2).
3.
1.
WaterrepellencyWater-walkingarthropodsarecoveredwithadensehairmatthatrendersthemwater-repellent(Figure3;Bushetal.
2008).
Thebodiesofwaterwalkershavetwodistincthairlayers,namelythemacrotrichiaforwaterproongtheinsectonthewatersurfaceandtheshortermicrotrichiafortrappingairshouldtheinsectbesubmergedbyraindropsoracrashingwave(Thorpe&Crisp1947;Hinton1976).
Wenotethatinthecaseofsubmergence,theairtrappedinthemicrotrichiaservesbothasabuoyandanexternalgillthatforcertainarthropodsenablesunderwaterbreathing10D.
L.
HuandJ.
W.
M.
Bush10–4(a)(b)AirHairsWatergdLegHairLeg10–310–2BoδWeδδφσ10–110010010–110–210–3MicroveliaMesoveliaHydrometraWaterstriderFisherspider10–4Figure4.
TheroughnessWebernumberWeδ=ρU2δ/σandBondnumberBoδ=ρgδ2/σcharacterizingthewaterrepellencyofthehairlayersofvespeciesofwater-walkinginsectsandspiders.
HereUrepresentsthepeaklegspeedandδtheinter-hairspacing.
AllwaterwalkersarecharacterizedbylowWebernumber,indicatingthattheirdrivinglegsremaininaCassiestateastheypropelthemselvesonthewatersurface.
Moreover,thelowBondnumberindicatesthattheairlayersbetweenthehairsremainintactinthefaceofhydrostaticpressuresgeneratedbythedrivingstroke.
(a)Across-sectionalviewofanarrayofhairswithspacingδlyingtangenttothefreesurface.
(b)ShowsanobliqueviewofasinglehairoflengthL,widthdandangleφpiercingthefreesurface.
(Flynn&Bush,2008).
Weproceedbyinvestigatingthecharacteristicsofthehairsonthedrivinglegs.
Thesurfaceofthewater-walkingarthropodlegconsistsofamatofmacrotrichiatiltedtowardsthelegtips;thetiltangleφandspacingδvaryamongspecies(gure4a).
TheleghairsofMicroveliaareshowningure3(a)andschematicallyingure5.
ThehairsgenerallyhavealengthLof20–60μmandadiameterdof1–2μmatthebaseandtapertoapointattheirtip.
Inter-speciesvariationinthehairtiltandspacingareshownintable1.
Figure6(a)showsthatthelegofMesoveliaresemblesahairybrush.
Aviewfrombelow(gure7)showsthatthearthropodlegtrapsairwithinitsintegument,thusmaintainingawater-repellentstate.
Theseairpocketscanbeseenasasilveryenvelopearoundthelegswhentheystrikethewatersurfaceasisthecaseforthesherspider(gure8).
Weproceedbyrationalizingthemaintenanceofthisairlayerinbothstaticanddynamicsettings.
Thewaxymaterialcoveringtheintegumentofwater-walkingarthropodshasachemicalcontactangleofθe=105(Holdgate1955)andsoishydrophobic.
Becauseθe>π/2,theadditionofsurfaceroughnessincreasestheenergeticcostofwettingandsodiscourageswetting(Dussan1979;deGennes,Brochard-Wyart&Quere2003).
Theformofcontactbetweenarough,hydrophobicsolidandwaterdependsexplicitlyontheformoftheroughnessandtheuidpressure.
Withmoderateroughness,thewaterentirelywetsthesubstrate,yieldingaWenzelstate(Figure9c;Wenzel1936).
Whentheroughnessisincreasedsuciently,itisenergeticallyfavourableforairinclusionstobetrappedwithintheroughsurface,sothatthesolidiswettedonlyatThehydrodynamicsofwater-walkingarthropods11GaitSpeciesn(hairscm2)L(μm)d(μm)φ(deg.
)δ(μm)HairdensityLengthWidthTiltangleSpacingRowingStrider1.
2–1.
6*10620–401.
5–230–507Velia1*10630–401–250–60–Halobates0.
8–1.
2*10620–30120–40–Fisherspider2.
5–3.
6*105–6–13WalkingMesovelia4*10530–602–35010Hydrometra2–3*1051559015Table1.
Tarsalhairpropertiesofwater-walkinginsectsandspiders.
LeghairsoflengthL,widthdandangleφwithrespecttothelegsurfacearearrangedwithdensitynandspacingδ.
DatareprintedfromAndersen(1976,1977);spiderdatafromStrattonetal.
(2004b).
(a)b(c)(e)(f)(d)(b)c,defFigure5.
ThecontactbetweenthewatertreaderMesoveliaandthefreesurface.
(a)Mesoveliasupportsitsweightbydeformingthesurface.
Scalebar,1mm.
(b)Aschematicofasectionofahairyleg.
Scalebar,100μm.
(c,d)Furtherschematicsofindividualhairspenetratingthesurface.
Scalebar,1μm.
(e)Asinglehairpenetratingthefreesurface.
Scalebar,1μm.
(f)Thehairsarecoveredinnanogroovesthattrapairwhenthehairissubmerged.
Scalebar,0.
1μm.
theextremitiesofitsroughnesselements.
Theapparentcontactangleθofadropintheresultingwater-repellentor'Cassie'stateisgivenbytheCassie–Baxterrelation(Cassie&Baxter1944)cosθ=fs1+fscosθ,(3.
1)12D.
L.
HuandJ.
W.
M.
Bush(a)(b)Figure6.
Thehairylegsof(a)Mesoveliaand(b)thesherspider.
Asisevidentintheinsetof(b),thehairsmaintainanarrayofairpocketswhenpressedagainstthewatersurface.
Scalebars:(a)100μm,(b)0.
15cm.
(a)(b)Figure7.
Brighteldimagesofwater-walkingarthropodsinaCassiestate,asseenfrombelowthroughaninvertedmicroscope.
(a)AliveMicroveliastandingonthewatersurfacegroomingitswater-repellentlegs.
Scalebar400μm.
Acloserlookatthesupportingfoot(thehatchedboxina,magniedinbanditsinset)showsindividualhairpinholescorrespondingtothecontactlinesbetweenthecuticleandthewatersurface.
Scalebars,100μm.
ImagescourtesyofManuPrakash.
wherefsistheexposedareafractionofthesolidsubstrate(e.
g.
theratiooftheareaofthepillartopstototalbaseareaingure9d).
Forsucientlysmallfs,theeectofthetextureistoincreasethecontactangledramaticallyfromθtoθandsoqualifytheintegumentasbeingsuperhydrophobic(θ>150).
Gao&Jiang(2004)measuredthecontactanglesofwaterstriderintegumenttobe168andrationalizedthishighvaluebyvirtueofthenanogroovesonthehairs.
Strattonetal.
(2004b)reportedvaluesof152forsherspiders.
Contactanglesforavarietyofwater-walkingarthropodsarereportedinBushetal.
(2008).
Wenotethataconsequenceofthehydrophobicityviatexturingoftheintegumentofwater-walkingarthropodsisthatifthelegisimmersedinauidwithwhichitThehydrodynamicsofwater-walkingarthropods13(a)(b)Figure8.
High-speedimagesofthesherspider(a)gallopingand(b)leapingbydrivingitslegsagainstthefreesurface.
In(a),itslegpenetratesthefreesurface,asshownintheclose-upviewintheinset,butmaintainsaCassiestate.
Scalebars,7mm.
(a)θeθ*θ*θ*(b)(c)(d)Figure9.
Rougheningasurfacewillamplifyitswettingtendencies:(a)θedenestheequilibriumorchemicalcontactangleonaatsurace.
(b)Onaroughsurface,themicroscopiccontactangleremainsθe,buttheobservedcontactangleonthemacroscopicscale,θ,dependsexplicitlyonthesurfaceroughness.
Thetwogenericstatesofwetting,theWenzelandCassiestates,areshownschematicallyin(c)and(d).
IntheWenzelstate,theporesareimpregnatedbyuid,increasingtheuid–solidcontact.
IntheCassiestate,airpocketsaretrappedbytheoverlyinguid,reducingtheuid–solidcontact.
hasalowcontactangle(θ1,asisthecaseforbasilisklizards(Glasheen&McMahon1996a,b).
Inastaticsituation(Mc=0),(5.
5)yieldsageneralizedformoftheArchimedesprinciple.
Theforceonastaticoatingbodyisequaltotheweightoftheuiddisplaced:Mg·z=Fb·z+Fc·z=Sbρgzn·zds+σCt·zd=ρgVb+ρgVm.
(5.
6)Manseldetal.
(1997)andKeller(1998)showedthatthemagnitudesofthebuoyancyandcurvatureforcesonaoatingbodyareequaltotheweightsoftheuiddisplacedbythemeniscus,respectively,insideandoutsidethelineoftangencyC(respectivelyVbandVmingure20b).
Forlongthinbodiessuchaswater-walkinginsectlegs,theratioofbuoyancytocurvatureforcesisthusgivenbytheratioofthelegradiuswtothecapillarylengthc=√σ/(ρg)or,equivalently,thesquarerootoftheBondnumber:FbFcVbVmwc√Bo.
(5.
7)26D.
L.
HuandJ.
W.
M.
Bush(a)cΘθρσccww(b)VbVmFigure20.
Thestaticweightsupportofthewaterstrider.
(a)Theweightofwater-walkingarthropodsissupportedbycurvatureforcesassociatedwithdeformationsofthefreesurface.
ThelegofwidthwintersectsthefreesurfaceatacontactlineC.
Theinitialangleoftangencybetweenthefreesurfaceandthehorizontalisθ:theresultingmeniscusdecaysoverthecapillarylengthc.
(b)Thelegincross-section;VbandVmdenotethevolumeofuiddisplacedinsideandoutsidethecontactline,respectively.
Asshown,thehairinessofthelegeectivelyincreasesitsvolumewithoutsubstantiallyincreasingitsmass,thuscontributingtoitsbuoyancy.
ThecharacteristicBondnumbersofthewater-walkinginsectsconsideredinourstudyarelistedintable2.
Waterstriders(withtypicalweight3–10dynesandlegwidth20–80μm)have104Therelativelystockysherspider(weight102–103dynesandlegwidthupto0.
17cm)has103Finallywenotethataninsectaugmentsitsbuoyancywiththeairlayertrappedinitsintegument,whichmayincreasethevolumeofuiddisplacedby20–30%(Bushetal.
2008).
5.
1.
2.
LateralpropulsionThehorizontalcomponentoftheforcebalanceonawaterwalkerisgivenbythexcomponentof(5.
4):M˙U·x/(σw)=WeAdSbφtn·xdS+Sbn·xdS+BoSbzn·xdS+Ct·xd.
(5.
8)Forlargecreatures(We1andBo1),surfacetensionisnegligible,and(5.
8)showsthatsuchwater-walkingcreaturesmaypropelthemselvesusinginertialandhydrostaticpressures.
Wenotethatthehydrostaticpressurescanonlyproducealateralforceonabodywithacavitythatisnotfore–aftsymmetric.
Forexample,thebasilisklizardgeneratesthrustusinghydrostaticpressurebygeneratinganaircavitywithitsfeetandpressingagainstthecavity'sbackwall(Glasheen&McMahon1996a).
Intheparameterregimeofmostwater-walkingarthropods,(We,Bo)1,surfacetensiondominatesbothinertialandhydrostaticforces,sothatM˙U·x/(σw)=Ct·xd.
(5.
9)Thedominantlateralpropulsiveforcecomesfromthecurvatureforcethatmaybegeneratedbythedrivingleg.
Theinterfacethusrespondsroughlylikeatrampoline.
Thehydrodynamicsofwater-walkingarthropods27Thecurvaturepressuregeneratedbythestrokeofalegofwidthwisσ/w;therefore,thenetcurvatureforceactingonthedrivinglegsisFσ=Aσ/w,whereAistheareaofthedrivinglegadjoiningthedistortedinterface.
Wenotethatsuchaforcecanonlybeeectivelygeneratedifthespeedofthelegstrikeexceedsthecapillarywavespeed;otherwise,theinterfacewillrespondinaquasi-staticmanner,assumingafore–aftsymmetricformincapableofpropellingthecreatureforwardwithcurvatureforces.
Atsuchlowspeeds,thecreaturemustrelyonthebrushingtechnique,inwhichthepropulsiveforcehasitsoriginsinthemicroscopiccontactforcesorviscousstressesactingonthewettedhairtipsonthedrivingleg.
Byconsideringthelegstrikeonamacroscopicscale,wehavededucedanestimateforthecurvatureforceFσgeneratedbythedrivingstroke.
In(3.
2),wededucedanestimateforthebrushingforceFbrush=nAFcontactassociatedwiththemicrostructureonthedrivinglegs.
TherelativemagnitudesofthesetwoforcesactingonalegareawithhairdensitynandwidthwaregivenbyFbrushFσ=nσLAcosθAσ/w=nwLcosθ.
(5.
10)Themagnitudeofthisratiodependsonthemagnitudeofthecontactanglehysteresis,butitisnoteworthythatthebrushingforceresultingfromthemicroscaleinteractionbetweenintegumentandinterfacemaybecomparabletothecurvatureforces.
Forthephysicalvariableslistedintable2,thisratioassumesvaluesbetween0.
1and1,suggestingthedominanceofmacroscopiccurvaturepressuresinthepropulsiveforce.
However,situationsariseinwhichwater-walkingarthropodsaretoosmall,slowandweaktogeneratesubstantialsurfacedistortionsandpropulsivecurvatureforces.
Insuchsituations,thecreaturesarestillabletomovebybrushingthefreesurface.
Weproceedbyconsideringthehypotheticalcaseofacreaturejumpingonasoaplm.
Notethatthisisbutatheoreticalabstractionowingtothefactthatsoapdestroysthewaterrepellencyofinsectlegs,causingthemtopunctureandbreakthelm.
Intherelevant(We,Bo1)limit,theforcebalanceonthecreature,(5.
1),assumestheformM˙U·z=Cσt·zd+Mg.
(5.
11)Theonlyforcesactingonthecreaturearethosethatareduetosurfacetensionandgravity.
Thecreatureisstatic(˙U=0)whenitsweightisbalancedbythecurvatureforceassociatedwithdeformationofthefreesurface.
Alateralpropulsiveforceispossibleonlyifthemeniscusisdistortedasymmetrically;however,suchadistortionrequiresthatthelegstrikesthelmataspeedexceedingthecapillarywavespeed.
Onasoaplmofthicknessh4μm,thiswavespeedisapproximately√2σ/ρh5ms1,wellbeyondthepeaklegspeedofanywaterwalker.
Consequently,water-walkingarthropodscouldnotpropelthemselvesviacurvatureforcesonasoaplm:theabsenceofinertiaoftheunderlyinguidprecludestheirprinciplemodeofpropulsion.
Wenotethatbrushingforcesmightyetallowmotiononasoaplm,astheydoforslowwaterwalkersatthewatersurface.
Whilethisisbutatheoreticalabstraction,itunderscoresthesubtleinterplayofwettingproperties,wavesandmomentumtransferthatliesattheheartofthisstyleofbiolocomotion.
5.
2.
MomentumtransferHavingconsideredthehydrodynamicforceonanobjectstrikingthefreesurface,wehereconsiderthetransferofmomentumbytheobjectintothecontrolvolume28D.
L.
HuandJ.
W.
M.
BushtAirWater–ngnznnxρσCVSSbFigure21.
Abodystrikingthefreesurface.
ThecontrolvolumeVisboundedoutsidebySandinsidebythebodysurfaceSb.
ThebodyintersectsthefreesurfaceatacontactlineC;ndenotestheoutwardunitnormaltotheuidboundbySb.
illustratedingure21.
OuranalysisbuildsupontheframeworkdevelopedbyChildress(1981)todescribeswimmersandierstoincorporatetheinuenceoftheinterfaceonpropulsion.
Fromrstprinciples,wemaywritetheconservationofmomentumintheuidastVρudV=S(ρuu·n+pn)dS+SbtappdS+VρgdV,(5.
12)wheretappisthelocalstressappliedbytheobjecttotheuidandSbisthewettedbodysurface.
Newton'sthirdlawrequiresthatSbtappdS=SbT·ndS+Cσtd.
(5.
13)Thesumof(5.
1),(5.
12)and(5.
13)yieldsM˙U+tVρudV=S(ρuu·n+pn)dS+Mgzρg(Vm+Vb)z,(5.
14)whereVm+Vbisthenetuidvolumedisplacedbytheobject(gure20).
WenotethatthisreducestothegeneralizedformoftheArchimedesprinciple(5.
6)inthestaticcase.
Italsoindicatesthatcreaturestooheavytooatatthesurfacecanonlyremaintherebygeneratingaverticaluxofuid.
Onemaythusrationalizetheverticalcomponentofthevorticesshedbythedrivingstrokeofthebasilisklizard(Hsieh2003,2004).
Conversely,water-walkingarthropodsrelyonsurfacetensionforweightsupportbutrequirethehorizontaltransferofmomentumfortheirlateralpropulsion.
Equation(5.
14)isthestatementofconservationofmomentumforanobjectatafreesurface:thecreationofhorizontalmomentumintheuidbythedrivingstrokeofawaterwalkerindicatesalateralforceonthecreature.
Notethatsurfacetensiondoesnotarisein(5.
14):whileitcontributesaforcetoboththebodyandthecontrolvolume,thesecontributionsareequalandopposite.
Ofcourse,theabsenceofsurfacetensioninthemomentumbalancemayalsobesimplyunderstoodonthegroundsthatThehydrodynamicsofwater-walkingarthropods29theinterfacecarriesnomass.
Asforswimmingsh(Wilga&Lauder2002)andyingbirds(Speddingetal.
2003),onemayrationalizethepropulsionofwater-walkingcreaturesbyconsideringthemomentumtransferredintheirwake(Dickinson2003;Dabiri2005).
6.
DiscussionThedynamicsofwater-walkingarthropodswaspreviouslydescribed(Denny1993;Vogel1994)bytreatingtheirlegsassmoothcylinders:macroscopiccurvatureforcesweregeneratedastheinterfacerespondedasatrampolinetothedrivingstroke.
Wehavedemonstratedherethatpropulsionattheinterfaceisinsteadarich,multiscaleprobleminwhichthemicron-scaleroughnessoftheintegumentplaysacriticalrole.
Overtherangeofhydrostaticanddynamicpressuresexperiencedbywaterwalkers,theirintegumentmaintainsaCassiestate.
Theintegumentcoupleswiththeunderlyinguidonthemicroscalethroughcombinedviscousandcontactforces,therelativemagnitudesofwhichdependonthegeometryoftheintegument.
Thenotionofcontactforcesisusefulforrationalizingseveralpreviouslyenigmaticbehavioursofwaterwalkers.
Largerwaterwalkerscanstrikethesurfaceataspeedexceedingthecapillarywavespeed,thusgeneratinganasymmetricmeniscusandtheconcomitantmacroscopiccurvatureforces.
Conversely,smaller,weakerwaterwalkersgeneratenegligiblesurfacedeformationduringtheirstrokeandsorelyprincipallyonmicroscalecontactandviscousforcesgeneratedbythelegbrushingthefreesurface.
WehavedemonstratedtheutilityoftheStrouhalnumber,acommonmeasureofdynamiceciencyinothermodesofbiolocomotion,forcharacterizingpropulsiononthewatersurface.
TheStrouhalnumberistheratioofacreature'speakappendagespeedtoitsbodyspeed.
Byvirtueoftheirsolidcontactwiththeground,terrestrialcreatureshaveStrouhalnumbersnear1.
EcientswimmersandierstendtohaveStrouhalnumbersof0.
2–0.
4,asshowninthestudiesofTaylor,Nudds&Thomas(2003)andAlexander(2003).
Forwaterwalkers,forwhich0.
14,thrustisgeneratedbydrivingagainstwater,whiledragisexperiencedinair,atleastduringtheirairbornephase.
Thus,fromaStrouhalnumberperspective,rowingonwaterisofcomparableeciencytoswimmingandying.
Inourexperimentalinvestigation,wehavereportedtheprevalenceofcoherentvorticesinthewakeofwater-walkingarthropods.
Thepropulsionmechanismofwater-walkinginsectsthushasfeaturesincommonwithswimmersandiersinthateachdrivingstrokegeneratesavortex.
Wedevelopedatheoreticalframeworkthatdescribesthetransferofforceandmomentumbetweenthewaterwalkeranditsenvironmentonboththemicroscopicandthemacroscopicscale.
Ourdevelopmentsmakeitclearthatmomentumisconservedbetweentheuidandthecreatureacrosstheinterface,anassumptionmadeimplicitlybyHuetal.
(2003)andB¨uhler(2007)intheirstudiesofthewaterstriderandbyHsieh(2004)inherstudyofthebasilisklizard.
ThevorticalwakesreportedbyHuetal.
(2003)andHsieh(2004)andmorecomprehensivelyheremaythusbeusedtosimplyrationalizethepropulsionofwater-walkingcreatures.
Ourformulationprovidesanintegrativeviewofallformsofwalkingonwater,includingboththeinertia-basedpropulsionoflargewaterwalkersandthesurface-tension-basedpropulsionofwater-walkingarthropods.
Moreover,ourstudyinformstheresolutionofDenny'sparadox(Suteretal.
1997).
Denny'sparadox(Denny1993,2004)wasgeneralizedbySuteretal.
(1997)tothefollowingform:'smallorslow-movingsurfacedwellingarthropodsshouldnotbeabletopropelthemselveshorizontally'.
Wenowseethattheparadoxrestedon30D.
L.
HuandJ.
W.
M.
Bushtwoawedassumptions.
First,waterstrider'smotionwasassumedtorelyonthegenerationofcapillarywaves,sincethepropulsiveforcewasthoughttobethatassociatedwithwavedragonthedrivingleg.
Second,inordertogeneratecapillarywaves,itwasassumedthatthestriderlegspeedmustexceedtheminimumwavespeed,cm=(4gσ/ρ)1/2≈23cms1.
Wenotethatthissecondassumptionisstrictlytrueonlyforsteadymotions(Lighthill1978)andsoisnotstrictlyapplicabletothepropulsivedrivingstrokeofthewaterstrider;indeed,bodiesmovingatunsteadyspeedsuConsiderationofthemicroscaleforcesactingontheintegumentofthedrivinglegsmakesitclearthatwater-walkingarthropodsneedn'trelyonwavedrag,inertialforcesormacroscopiccurvatureforces:contactandviscousforcesarisingfromtheinteractionofthecuticleandtheinterfaceareavailableforpropulsionatanylegspeed.
Denny'sparadoxcanalternativelyberesolvedbynotingthatwater-walkingcreaturestransfermomentumtotheunderlyinguid(Huetal.
2003).
TheprecisepartitioningofmomentumbetweenwavesandvorticesfollowingtheimpulsivestrokewasconsideredexperimentallybyHuetal.
(2003)andtheoreticallybyB¨uhler(2007).
Huetal.
(2003)madearoughestimatethatthemomentumpartitionbetweenwavesandvorticeswasapproximately1–10.
B¨uhler(2007)deducedthatanimpulsiveforcingatafreesurfacegeneratesmomentuminwavesandvorticesofrelativemagnitudes1/3and2/3respectively.
Hsieh(2003,2004)showedthatwater-runningbasilisklizardsalsogeneratewavesandvortices.
Theprecisepartitioningofmomentumtransferinwavesandvorticesinthewakeofvariouswaterwalkersisleftasasubjectforfutureconsideration.
Wehavedemonstratedthatmostwater-walkingarthropodshaveattheirdisposaltwopropulsiveforces.
Iftheystrikethesurfaceataspeedinexcessofthecapillarywavespeed,23cms1,theycanproduceameniscuswhosefore–aftasymmetryresultsinacurvatureforcethatpropelsthemforward.
Suchisthecaseformostadultwater-walkingarthropods,forwhichtheinterfaceserveseectivelyasatrampoline.
Conversely,ifthepeakspeedofthedrivinglegissubstantiallylessthan23cms1,theinterfacerespondsquasi-statically;consequently,itsfore–aftsymmetryismaintained,andnolateralpropulsiveforceresults.
Anumberofsmallandinfantwaterwalkersthususeatechniquethathasnotpreviouslybeendiscussed:bybrushingtheirlegsacrossarelativelyunperturbedsurface,theygenerateacombinationofviscousstressesandcontactforcesontheirwettedintegumentthatservetopropelthemforward.
Wenotethatthelatterbrushingtechnique,inecientthoughitiswhencomparedwithpropulsionviacurvatureforces,operatesatalllegspeeds.
Finally,itisnoteworthythatwehaveassumedthroughoutthisstudythattheinsectintegumentiseectivelyrigid.
Aninterestingavenueforfutureresearchistheroleoftheintegument'selasticityonthewaterrepellencyandthedynamicsofthisclassofcreatures.
Prakash&Bush(submitted)havedemonstratedthatthecuticleofwaterstrider'sintegumentisunidirectionalbyvirtueofitselasticity:contactforcesactingonmovingdropletsaregreatestformotionperpendiculartothelegandsmallestformotiontowardsthelegtip.
Thisobservationraisesanumberofinterestingdynamicalquestions,includingtheroleoftheintegumentindetachingfromthefreesurface.
Thisclassofproblemsiscurrentlyunderconsiderationandislikelytoinformthedesignofbiomimeticwater-walkingdevices(Huetal.
2003,2007;Suhretal.
2005;Thehydrodynamicsofwater-walkingarthropods31Floydetal.
2006;Song,Suhr&Sitti2006;Yuetal.
2007)andsyntheticunidirectionalsuperhydrophobicsurfaces(Prakash&Bush,submitted).
Videoimagesofmanyofourexperimentscanbefoundathttp://www.
me.
gatech.
edu/hu/orontheMultimediaFluidMechanicsCD-ROM(Bush&Hu2004).
TheauthorsthankManuPrakashformanyvaluablediscussionsandhiscontributionstotheguresandLucyMendelandBrianChanfortheirassistancewithillustrations.
TheauthorsgratefullyacknowledgethenancialsupportoftheNSF:J.
B.
throughgrantCTS-0624830andCareerGrantCTS-0130465andD.
H.
throughaMathematicalSciencesPostdoctoralResearchFellowship.
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