Wall heat transfer $-\theta'(0)$ of the thermal Falkner–Skan boundary layer

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Numbers

$\beta$

branch

$\mathrm{Pr}$

normalisation

$-\theta'(0)$

upper

1/8

wedge:

1.531448476030393233432213071563277727546889588331286709531255420905402714919718964313985972179296383e-1

upper

1/8

Hartree:

2.165795204877789878814698921306422983531011479387168760272259933152579492700156180563235587371246749e-1

upper

1/6

wedge:

1.715320973686532118984193878091370134802936330581196614885068057793043161500174835347636737766092635e-1

upper

1/6

Hartree:

2.425830184810516621804181566925032868753665687045865793535223651924037220269848024938260984588521422e-1

upper

1/4

wedge:

2.004909317106613135970121861302098826953827469785516202466201972899009181691184852185608346345926762e-1

upper

1/4

Hartree:

2.835369947580352636285148604715651970991965103373016589210169721224067625792565733486627932352666737e-1

upper

1/3

wedge:

2.233819065812624867199134075800669910424305438243046009553515138617355351817950120495623111757786242e-1

upper

1/3

Hartree:

3.159097218759811471384968355473920768425414594250063253993831874472120032076207017080959600471471680e-1

upper

1/2

wedge:

2.592935574336310252214395786617604690093478469330670318835674427101761424524820761605064321232691160e-1

upper

1/2

Hartree:

3.666964655586080532367879507310714083743247268047154132350671145135128891874007453338110421802113458e-1

upper

2/3

wedge:

2.876226021185022671865443108250480127498546762367936661562508559518755615250437183430861049902765448e-1

upper

2/3

Hartree:

4.067597847610264061568298502930545699227385439145022509300171584738820682465980819414725497817608877e-1

upper

wedge:

3.320573362151962989371800620105829665470935614126798181004475640198724174018064405070490731855146369e-1

upper

Hartree:

4.695999883610133045093336534475884587213340784502918309470179692512837860512064536254971722517555790e-1

upper

3/2

wedge:

3.824528403373075200058724892599949689997863125755461298850405332607891347794882468547250692890426413e-1

upper

3/2

Hartree:

5.408699937731322104868834089619276069449639596449638219943508417699612264107099593033893544515743636e-1

upper

wedge:

4.223081722951666457763276133482075563846841919709458851782795078770621351942677210380167951043920120e-1

upper

Hartree:

5.972339447608184256587496990966868147103254688941211262083904635075871982749894685622489232735527513e-1

upper

wedge:

4.850479710142787014427874656546244809560218708752646427636740914061778842749023211283463812594029269e-1

upper

Hartree:

6.859614190099448316448714108754933132132847113928999149813592653770775639093995694003519645277613480e-1

upper

wedge:

5.347873090726434246385374953541464551116923101547058086181101453756825621213414665376979190575730468e-1

upper

Hartree:

7.563034654755444714909263633117304407935724021506870543351543138858493186380085298380717247155992216e-1

upper

wedge:

6.132578546354764000163300137604280537072146460333029395644985464102282959313663649791427120456307209e-1

upper

Hartree:

8.672775752573187604372729239888953737918186265958855766451540154487892298836865753212791679061778403e-1

upper

wedge:

6.755811925315489192905789573715058805997707278098376199191612681029347315989145532717907719078857882e-1

upper

Hartree:

9.554160849623055925242414902443540315965769949274036201389194246912507320455905839263816830282255607e-1

-1/6

upper

1/8

wedge:

1.332802402847473978343465929978123591489900114713604457055411030266716450480515261597178690303298433e-1

-1/6

upper

1/8

Hartree:

1.961832017336007120780442812533547962841985578338087895794154972875266113970216941281011288574469602e-1

-1/6

upper

1/6

wedge:

1.478767713712368412345885880839222527523439393314489593851653676833031121287244704501409142076820690e-1

-1/6

upper

1/6

Hartree:

2.176687137392332711366360089983076255913396400274255843085614125503161756405067387754527282690544694e-1

-1/6

upper

1/4

wedge:

1.704623921636292958477218736384475343764768664670426552309405735398878847306560959132274005902932719e-1

-1/6

upper

1/4

Hartree:

2.509138473818953071209078346118076561863859935643511232709617030329603829673664830210331055116821164e-1

-1/6

upper

1/3

wedge:

1.880180042207219392535605889688677931950803049145709862349385921790712071241588584834295014505243494e-1

-1/6

upper

1/3

Hartree:

2.767550086402726601150003098117422879500250343413896125094619486141231006451818170419467395059613783e-1

-1/6

upper

1/2

wedge:

2.151288179681915513533930924390655040727909851594074809598973909727417341212841580209979628201805766e-1

-1/6

upper

1/2

Hartree:

3.166610459584735137291173885044311002820875757809584006709864931506907197335055075451659129914760403e-1

-1/6

upper

2/3

wedge:

2.362143243982433654535035006468497105419559772180326771124161583830947436602826630962145495847256949e-1

-1/6

upper

2/3

Hartree:

3.476980710477461462434276544428713438898815573558683205022010148465865683630063235113292993893508594e-1

-1/6

upper

wedge:

2.688754311421251295402114626898247208484232570496670651354914605824259619436076943947155400744660988e-1

-1/6

upper

Hartree:

3.957739184463413945952819729608147101875552313587343672255180685570607933114561963020471568685327922e-1

-1/6

upper

3/2

wedge:

3.054596862703575393154095430229204936378132831391139175672152490578358771508782162786695697746191476e-1

-1/6

upper

3/2

Hartree:

4.496244839072208681591397328711055085708438854955426522220044212499703365538492255121939137025425162e-1

-1/6

upper

wedge:

3.341365653802642058664934729251524484200492464210304340301607952383571472024576615201796188298165228e-1

-1/6

upper

Hartree:

4.918357070224355740223073701661443809195829633861433747542914603002342228002522915704147794258989663e-1

-1/6

upper

wedge:

3.789431973020843428042758484101439788754028155485418927112255350272259501988781364303465689725367924e-1

-1/6

upper

Hartree:

5.577892834156167690633084281196734026123157066268475665849964800570335642019011500266682260261407838e-1

-1/6

upper

wedge:

4.142442590267187426746235000262100323682226846220784592605214798673901200344673495286115947307263553e-1

-1/6

upper

Hartree:

6.097510393288584195614058589730220568134503821427589344299942264503972948500789833666916864612457759e-1

-1/6

upper

wedge:

4.696456458993714715909907434817316719478731322415604165823772379105687650445782989725542639345657532e-1

-1/6

upper

Hartree:

6.912996727492223641093131806643591674404876470214022253782219981990450102402831086433684045842300927e-1

-1/6

upper

wedge:

5.134547811999088093224915227587995569867718181353933906290445804818709753154992915101132711592947102e-1

-1/6

upper

Hartree:

7.557849738717136133371644072899040689381092288444490808841149370877921740631134938653383288577819726e-1

1/3

upper

1/8

wedge:

1.786012611232528438405813325855844773530291610279685490705692811016825818533460641996858697778547321e-1

1/3

upper

1/8

Hartree:

2.305732366473334782709946374920537611615051968075898292080434045956875563078496572435928972549747246e-1

1/3

upper

1/6

wedge:

2.012718715720942919537244833497855802758518048708043902396820753525705899437397467428648490796243111e-1

1/3

upper

1/6

Hartree:

2.598408688862397385847072200572730720220980990520162963394078260681637334459692128006141299612208167e-1

1/3

upper

1/4

wedge:

2.373614653918116405579620390642974330260352368185869190477730440036271147678486285317633473025650513e-1

1/3

upper

1/4

Hartree:

3.064323341646248748210268993246322842060342779098975772591479600880760708172382996435414340077897805e-1

1/3

upper

1/3

wedge:

2.661744665537674517772337297148058454456239561092518770144310356586707266588206396671000936213113603e-1

1/3

upper

1/3

Hartree:

3.436297587161281443523064463851859931367052499979060099931531065166695731551031301721229348750494723e-1

1/3

upper

1/2

wedge:

3.117863596908358809596286301464383372265476379836218106407839968729178420492867629788847798036174601e-1

1/3

upper

1/2

Hartree:

4.025144595524142771147784769311676466150032780835024146190220261460793810164744332171734415598396865e-1

1/3

upper

2/3

wedge:

3.480470802054665017659733016323182872455409563746123433757192025550642781011366625121190857488286588e-1

1/3

upper

2/3

Hartree:

4.493268484439629535908166691489594592051560994085048115722490445040967540163168369936878081104157141e-1

1/3

upper

wedge:

4.052868089493258483555678636190783818867512577615701136153948201628797543228464438136771199876412486e-1

1/3

upper

Hartree:

5.232230204994286987083163381060726767260444481329104751010856430430455475204800061584400913018276682e-1

1/3

upper

3/2

wedge:

4.705784958791028012359805603074684543639824897665834811799630456418940040716127609754425889847881278e-1

1/3

upper

3/2

Hartree:

6.075142258743669014854960692901609507931107948464699697725738783763893592489514929514891136041103224e-1

1/3

upper

wedge:

5.224002157228804613519588015418366532190539238950506514549182187043269434229938514471281300322533120e-1

1/3

upper

Hartree:

6.744157785166260011757164182962042094160310288213174421466159430577831876986334567733650772785499998e-1

1/3

upper

wedge:

6.041867995039130013415884625701562425054094404338900466507218270087425874259971136076464087838717224e-1

1/3

upper

Hartree:

7.800018041590048866827929136418500222495554652043011333960039689105194769696716854376594282321812640e-1

1/3

upper

wedge:

6.691424881676899714668145150136854155750730610166794976123543989087592101845787320682998658870072599e-1

1/3

upper

Hartree:

8.638592376377522551396778384997422227199140030721702465278686199008747304057195689532291608353854860e-1

1/3

upper

wedge:

7.717409029027153003114687293375741652143894338883309161180894645772230903036649157735367454891131425e-1

1/3

upper

Hartree:

9.963132215097638368437407028813120087059876258825228570167787852234351046438236046240154942218116264e-1

1/3

upper

wedge:

8.532908827746151225620737615183011418320829565696743575688857126125987163790510157783613435378812226e-1

1/3

upper

Hartree:

1.101593792818903192766545628870314205636821084908207026549242765220507495360362755739628776934840106e+0

1/2

upper

1/8

wedge:

1.914711682904652215141729815774795983299332125525156288125598651039646605816760696093583398671984730e-1

1/2

upper

1/8

Hartree:

2.345033313831031309740315509886251869199112107864963691787129396367605656037950751543926819080320972e-1

1/2

upper

1/6

wedge:

2.161422203471876750729354413028633976174240692805707516068010958667450709687321323969675006854177660e-1

1/2

upper

1/6

Hartree:

2.647190758614088708558166131291712048259892454844496368427336070654887631327589759431058440961266047e-1

1/2

upper

1/4

wedge:

2.555343146517362054517610472544617466295633623664836293053521077763225695945936715245195502827682254e-1

1/2

upper

1/4

Hartree:

3.129643413342785081585931528450503213445794553811376864517846484691692050810270121349584700128393999e-1

1/2

upper

1/3

wedge:

2.870722747490645275869539097519544597997152510306261147050951498670946164636274473882000238706429025e-1

1/2

upper

1/3

Hartree:

3.515902962176339513017300935283049658462204799912385107297994465583872718353802556767876136466234329e-1

1/2

upper

1/2

wedge:

3.371260417679209904871552030125947125913924602164785806619380533458731563249254103722061671251938048e-1

1/2

upper

1/2

Hartree:

4.128933906678078715546723852473642685136054934158834598511367824906009227435609045846631551165897801e-1

1/2

upper

2/3

wedge:

3.770060793971506961076808734751196131971848835397850948087685850919363577828786438130143631436532332e-1

1/2

upper

2/3

Hartree:

4.617362622251105392074991956581287969570367344017403659085007363560517490116069198159365446285171889e-1

1/2

upper

wedge:

4.400744577146972761132929615683359329560685171952229382253030234185101164411924721598215940395274329e-1

1/2

upper

Hartree:

5.389789351165102086665252303732848468884932393202753927093590074604650732198899514078487401383390900e-1

1/2

upper

3/2

wedge:

5.121329945052412483775014481838107440881434780004731084510942999308135289359185476143457278640224109e-1

1/2

upper

3/2

Hartree:

6.272322584907110738983294583995749992449450495915728242448335693862465987049289177248073062499411229e-1

1/2

upper

wedge:

5.693838026695548150000307482533393496174978575559374382466679521583056402895076339141902168375961678e-1

1/2

upper

Hartree:

6.973498921729778293841813333558950400951892736697062839003669107507450732039088413024020791580808229e-1

1/2

upper

wedge:

6.598030379471413754021002430202737296201037823937735245836057343279103905028454368282805241522082504e-1

1/2

upper

Hartree:

8.080903868543514128577241546439631921718198716212587586402794426860150340406466532596806174573792979e-1

1/2

upper

wedge:

7.316493405240012263444894837525479796112602193929858894985627145631032021061326533500691048473114706e-1

1/2

upper

Hartree:

8.960837774638088238925657578156699686073106144898140458839162872971008804420574928478457906928018316e-1

1/2

upper

wedge:

8.451654953880505129883623003316815483710715276652354471675585596858846734457952376436568710735059139e-1

1/2

upper

Hartree:

1.035112105953646573141405042731552448477741106178156251387030915479968925648620572393197094469220750e+0

1/2

upper

wedge:

9.354095046391864887558309808822995579553444485299971794581170313117316841040807372343527333094412595e-1

1/2

upper

Hartree:

1.145637993457790476155693989740959549217531367812622021928111187789821476844872164195866050257343892e+0

2/3

upper

1/8

wedge:

2.057204859326668789486993357973919213465467069402926658588673230295019239705797030807487421483963070e-1

2/3

upper

1/8

Hartree:

2.375455558620916857371950854755512378024324801117603096699671613991797507234700116994441185841153295e-1

2/3

upper

1/6

wedge:

2.325334569063422266293312942481632391982728392434642611346221989899586935190608867932743490370022994e-1

2/3

upper

1/6

Hartree:

2.685065078809418537806474664082238234408145198878091656511494083386627862917083275190471152167076685e-1

2/3

upper

1/4

wedge:

2.754464174584361312139111646714047849434771725863483550429845497420262288845807083710356156688750715e-1

2/3

upper

1/4

Hartree:

3.180581265338922683350660092687536820282303524768432973781132192264111546117918264817727349482132200e-1

2/3

upper

1/3

wedge:

3.098794568888598900177195700128344849147708992456867221637941753780671402926521430126790031186163586e-1

2/3

upper

1/3

Hartree:

3.578179757022365790465413516492245653006071088465483243798348300236343676315617055060339334311920250e-1

2/3

upper

1/2

wedge:

3.646398707914634626984751174654969472910641687470922309748200530971475095260669659048658635177313952e-1

2/3

upper

1/2

Hartree:

4.210498551174435747239691285733201294784333793582477090167435900481826465262942163387704423258684485e-1

2/3

upper

2/3

wedge:

4.083476262602590618731264665206164603153994691789011369412381171042939607304627716587762047352402885e-1

2/3

upper

2/3

Hartree:

4.715192238886105283944699853871845680510006517150538285699164840652062848627233046185479733875869461e-1

2/3

upper

wedge:

4.775724404091095214616304722750840932138351700943244527554500395064836269980967023641612764520767925e-1

2/3

upper

Hartree:

5.514531540554917827748218350981895580384358881233162479826793476976143070882070847254489747648427602e-1

2/3

upper

3/2

wedge:

5.567720858284550569998214440742558347401929922484073863906781309354918052090099897124461663981871602e-1

2/3

upper

3/2

Hartree:

6.429050272606558948916628706571676649851659957663690738833506432997188155151337476698000780146926029e-1

2/3

upper

wedge:

6.197499891980275628925178859513433310338510063847867777704392370977308966354354390230299422986981433e-1

2/3

upper

Hartree:

7.156256461874977464232445893282963737190152204394948950050224913647384092759151764269645529949141287e-1

2/3

upper

wedge:

7.192740698485166155478512097366956366889612297773472386769963444172422132203929093416127383032347869e-1

2/3

upper

Hartree:

8.305461556963175052483066426269284216970348849735987859996778009658534612451014050535923173815638001e-1

2/3

upper

wedge:

7.983873765489265972698694722624024412354241656026758397324309863091095392069115325632050925455662903e-1

2/3

upper

Hartree:

9.218983335362437584951153078616969687094640385772492304022894562043834060116038231865044938879563062e-1

2/3

upper

wedge:

9.234175175509304006339108114329562890038685999374337049123095654454382315138492266414397465760231611e-1

2/3

upper

Hartree:

1.066270704664891281466003333178116239610832247738435534539182776057219171099652168845727552344566533e+0

2/3

upper

wedge:

1.022830465624606412131211388133802048125039882874024080467478978878515375590649143578544138756477549e+0

2/3

upper

Hartree:

1.181062889327433548140788741135223037997356337209903909735374534784310875244609553371473667214928967e+0

upper

1/8

wedge:

2.420541084421814676661731304272651178823124321811575610358506036726596539160615203298230590005296471e-1

upper

1/8

Hartree:

2.420541084421814676661731304272651178823124321811575610358506036726596539160615203298230590005296471e-1

upper

1/6

wedge:

2.741395731482791636131972982973287617376958410805016661849946450640775400112709238869073110408450348e-1

upper

1/6

Hartree:

2.741395731482791636131972982973287617376958410805016661849946450640775400112709238869073110408450348e-1

upper

1/4

wedge:

3.256746831324081526518126804851810657871766431218458299540027549065119389556453407774486993581015241e-1

upper

1/4

Hartree:

3.256746831324081526518126804851810657871766431218458299540027549065119389556453407774486993581015241e-1

upper

1/3

wedge:

3.671667642410391012067646106537805389034528237499849982844870976542375322742948527006251738359143759e-1

upper

1/3

Hartree:

3.671667642410391012067646106537805389034528237499849982844870976542375322742948527006251738359143759e-1

upper

1/2

wedge:

4.333635308522682964404738909350111163136315583680169684584671491350167355398107412254191963458254866e-1

upper

1/2

Hartree:

4.333635308522682964404738909350111163136315583680169684584671491350167355398107412254191963458254866e-1

upper

2/3

wedge:

4.863476866761530869946446287507213621125400233460363094240087625768173656281050164972440807108091424e-1

upper

2/3

Hartree:

4.863476866761530869946446287507213621125400233460363094240087625768173656281050164972440807108091424e-1

upper

wedge:

5.704652524990626870921516289432265767791103217300387095013157319380590732142759383719993565703730316e-1

upper

Hartree:

5.704652524990626870921516289432265767791103217300387095013157319380590732142759383719993565703730316e-1

upper

3/2

wedge:

6.669165540072669857584917982112539856880926441389762444480640777635703925860196253847908982303063996e-1

upper

3/2

Hartree:

6.669165540072669857584917982112539856880926441389762444480640777635703925860196253847908982303063996e-1

upper

wedge:

7.437213333338799709872226384150367452455222855317411680068291705330722755491848420755831865765026302e-1

upper

Hartree:

7.437213333338799709872226384150367452455222855317411680068291705330722755491848420755831865765026302e-1

upper

wedge:

8.652204659295467648879846733922756625819061422491209340838866051784934091841669162018130290816316465e-1

upper

Hartree:

8.652204659295467648879846733922756625819061422491209340838866051784934091841669162018130290816316465e-1

upper

wedge:

9.618710649422889164619584166582532984276442312135322357000779584734515119103226090172050873976276968e-1

upper

Hartree:

9.618710649422889164619584166582532984276442312135322357000779584734515119103226090172050873976276968e-1

upper

wedge:

1.114687656454669354406371988197295467763744651470075940832506780611111636905389390649862736317621394e+0

upper

Hartree:

1.114687656454669354406371988197295467763744651470075940832506780611111636905389390649862736317621394e+0

upper

wedge:

1.236228921967116355365800692416260728507179045759405426063272546474198205789984918762712488825826423e+0

upper

Hartree:

1.236228921967116355365800692416260728507179045759405426063272546474198205789984918762712488825826423e+0

Definition

For a Falkner-Skan similarity solution $f$ [2], this table gives $-\theta'(0)$ for the constant-property, uniform-wall-temperature thermal boundary layer [1], with $\theta=(T-T_\infty)/(T_w-T_\infty)$. The wedge and Hartree entries are the same heat transfer in two similarity coordinates.

Parameters

$\beta$

— Falkner-Skan pressure-gradient parameter ($\beta_s<\beta<2$, where $\beta_s$ is the separation value)

branch

— solution branch

$\mathrm{Pr}$

— Prandtl number ($\mathrm{Pr}>0$)

normalisation

— normalisation of the wall-normal coordinate

Formulas

(1)

In the wedge normalisation, with $m=\beta/(2-\beta)$, the uniform-wall temperature problem without viscous dissipation is $\theta''+\frac{m+1}{2}\mathrm{Pr}\,f\theta'=0$, $\theta(0)=1$ and $\theta(\infty)=0$.

(2)

In Hartree variables, $-\theta_H'(0)=\left(\int_0^\infty \exp\left(-\mathrm{Pr}\int_0^\eta f_H(s)\,ds\right)\,d\eta\right)^{-1}$.

(3)

With $m=\beta/(2-\beta)$, $-\theta'_{\mathrm{wedge}}(0)= \sqrt{(m+1)/2}\,[-\theta'_{\mathrm{Hartree}}(0)]$.

(4)

At $\beta=0$ and $\mathrm{Pr}=1$, Reynolds's analogy gives $-\theta'_{\mathrm{wedge}}(0)=f''_{\mathrm{wedge}}(0)$ and $-\theta'_{\mathrm{Hartree}}(0)=f''_{\mathrm{Hartree}}(0)$.

(5)

The pressure-gradient exponent and the Falkner-Skan parameter are related by $m=\beta/(2-\beta)$.

Comments

(6)

The wedge normalisation uses $\eta=y\sqrt{u_e/(\nu x)}$ and $\psi=\sqrt{\nu x u_e}\,f(\eta)$, so that $f'''+\frac{m+1}{2}ff''+m(1-f'^2)=0$. The Hartree normalisation uses $\eta_H=\eta\sqrt{(m+1)/2}$ and $f_H=\sqrt{(m+1)/2}\,f$, giving $f_H'''+f_Hf_H''+\beta(1-f_H'^2)=0$. The two entries with the same $\beta$, branch and $\mathrm{Pr}$ are the same physical heat transfer coefficient in the two standard similarity coordinates. The thermal equation in (1) is the Pohlhausen energy equation for the prescribed Falkner-Skan velocity profile.

(7)

The six selected pressure gradients include $\beta=0,1/2,1$ and keep an adverse-gradient example alongside intermediate favourable gradients. Prandtl numbers are selected as simple ratios of momentum diffusivity to thermal diffusivity, including reciprocal pairs around $\mathrm{Pr}=1$. They are not asserted to be measurements for air, water, oils or any other named material. The exact argument sets are stated in the completeness note.

(8)

The textbook flat-plate correlation $\mathrm{Nu}_x=0.332\,\mathrm{Re}_x^{1/2}\mathrm{Pr}^{1/3}$ [3] uses the Blasius $\mathrm{Pr}=1$ value as a practical coefficient. The exact ratio $-\theta'(0)/\mathrm{Pr}^{1/3}$ is not constant as $\mathrm{Pr}$ varies.

Programs

(P1)

Python

from generate import FalknerSkan

generator = FalknerSkan()
print(generator.value({'beta': '0', 'branch': 'upper', 'normalisation': 'wedge', 'prandtl': '1'}, digits=100))

References

[1]

Marco Rosales-Vera and Alvaro Valencia, Solutions of Falkner-Skan equation with heat transfer by Fourier series, International Communications in Heat and Mass Transfer 37, 761-765, 2010. (doi)

Links

[2]

Wikipedia: Falkner-Skan boundary layer

[3]

Wikipedia: Nusselt number

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

How they were obtained:

Every listed entry is given to 100 significant decimal digits. The attached generate.py uses arbitrary-precision Taylor continuation and shooting for the upper-branch velocity profile, with integrated Taylor series for momentum thickness and thermal quadrature. The thermal integral includes a Gaussian asymptotic tail beyond the finite endpoint.

Every value is compared between two runs: 160 working decimal digits, Taylor order 140, step 1/8 and endpoint 32; and 210 digits, order 180, step 1/10 and endpoint 36. Across the three related tables, the largest observed relative difference is 4.209e-148. The momentum-integral identity and boundary residuals are checked at every pressure gradient. The Blasius case is also checked by Toepfer scaling, and the thermal value at Pr=1 is checked against the wall shear at beta=0. Uniform-flow and exact rational-solution controls test the Taylor recurrence. A separately implemented SciPy collocation solution and adaptive quadrature check all entries at ordinary floating-point precision.

Taylor truncation, finite-domain shooting and the asymptotic thermal tail are convergence-tested, not rigorously enclosed. The high-precision scaling control reuses the Taylor integrator; the separate collocation solver checks only ordinary floating-point accuracy. Arb is used internally, but step midpoints are retained. These checks support the stated heuristic rigour, not proven 100-digit error bounds. No mathematical definitions, parameter selections or entry annotations were changed.

Table is complete: no (it holds the upper branch at $\beta\in\{-1/6,0,1/3,1/2,2/3,1\}$ and $\mathrm{Pr}\in\{1/8,1/6,1/4,1/3,1/2,2/3,1,3/2,2,3,4,6,8\}$, in both Hartree and wedge normalisations (156 entries). This includes Blasius, right-angle-wedge and stagnation-flow cases, a simple adverse-gradient case and intermediate favourable gradients. The Prandtl numbers are simple dimensionless ratios below, at and above one, not measured properties of named fluids. Other arguments and the lower branch are not included.)