DTIC ADA520854: On a Class of Laplace Inverses Involving Doubly-Nested Square Roots and their Applications in Continuum Mechanics

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On a class of Laplace inverses involving doubly-nested square roots and
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MechanicsResearchCommunications37(2010)282–284
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Mechanics Research Communications
journal homepage: www.elsevier.com/locate/mechrescom
On a class of Laplace inverses involving doubly-nested square roots and their
applications in continuum mechanics
P. Puria,1, P.M. Jordanb,*
aDept.ofMathematics,UniversityofNewOrleans,NewOrleans,LA70148,USA
bCode7181,NavalResearchLaboratory,StennisSpaceCenter,MS39529,USA
a r t i c l e i n f o a b s t r a c t
Articlehistory: TheexactinversesofaclassofLaplacetransformsinvolvingdoubly-nestedsquarerootsandtwobranch
Received14January2010 pointsarepresented.FromtheseresultstheexactsolutionofStokes’firstproblem,awellknowninitial-
Availableonline4April2010 boundaryvalueproblems(IBVP)influidmechanics,foramicropolarfluidisdetermined.
PublishedbyElsevierLtd.
Keywords:
Laplacetransform
Stokes’firstproblem
Micropolarfluids
Dipolarfluids
1.Introductionandproblemformulation uðy;0Þ¼utðy;0Þ¼0 ðy>0Þ: ð1:3Þ
InIBVP(1), thevelocityvectorhas theformv= (u(y, t), 0, 0),
UsingtheLaplacetransform,we(JordanandPuri,1999)derived
wheretheincompressibilityconditionr(cid:5)v=0isidenticallysatis-
theexactsolutiontoStokes’firstproblemfor a dipolarfluid(see fied;h((cid:5))denotestheHeavisideunitstepfunction;I,c,k,andthe
Cowin (1974), Straughan (1987) and the references therein) that dynamic viscosity l are positive constants; here, m=l/q denotes
isvalidforarbitraryvaluesofthematerialparametersd(P0)and
the kinematic viscosity; U is a nonzero constant; M is taken as
0 s
l(>0). The present Note is devoted to extending this earlier work constant;andwehavesetj:=k/qforconvenience.
of ours to the case of micropolar fluids, the theory of which was
formulatedbyEringen(1966).Indoingso,wealsoextendthere-
Remark 1. The boundary conditions (BC)s assumed in Eq. (1.2),
sultsofJordanetal.(2000)onLaplaceinversesinvolvingdoubly-
whichinthepresentcontextareequivalenttoBCsof‘‘typeA”in
nestedsquarerootswithonebranchpointtoamoregeneralclass
dipolarfluidtheory(Straughan,1987),weresuggestedbyKirwan
ofsuchinversespossessingtwobranchpoints.
and Newman (1972). For more on the question of what BCs are
Tothisend,weobservethatbyreplacingthestraightchannelof
appropriateinthemicropolarcontext,seealsoEringen(1966)and
KirwanandNewman(1972),2whoconsideredtheunsteadyplane
Cowin(1974).
Couette flow of an incompressible micropolar fluid, with the half-
spacey>0,andsettingthepressuregradienttermintheirequation
ofmotiontozero,onepossiblewaytoformulationStokes’firstprob- 2.ExactsolutionviatheLaplacetransform
lemforamicropolarfluidisastheIBVP
qIuttþ2kut(cid:2)kð2mþjÞuyy(cid:2)½cþIðlþkÞ(cid:3)uyyt ingTnoorneddiumceentshioennaulmqubaenrtoitfiecso:euffi0c=ieun/Uts,,wy0e=iyn/tLro,dtu0=cet/tTh,ewfohlelorwe-
þcðmþjÞu ¼0; ðy;tÞ2ð0;1Þ(cid:4)ð0;1Þ; ð1:1Þ 0 0 0
yyyy
rcffiffiffiffiþffiffiffiffiffiIffiffiffiffiðffiffi2ffiffilffiffiffiffiffiþffiffiffiffiffikffiffiÞffiffi 2L2
uð0;tÞ¼U0hðtÞ; uð1;tÞ¼0; uyyð0;tÞ¼Ms; uyyð1;tÞ¼0 ðt>0Þ; L0¼ 2k and T0¼2mþ0j; ð2Þ
ð1:2Þ
andrecastIBVP(1)inthesimplerform
* Correspondingauthor.Tel.:[phone];fax:[phone]. hu þu (cid:2)u (cid:2)u þ‘2u ¼0; ðy;tÞ2ð0;1Þ(cid:4)ð0;1Þ; ð3:1Þ
tt t yy yyt yyyy
E-mailaddress:[email protected](P.M.Jordan).
1 Presentaddress:812RochelleAve.,Monroe,LA71201,USA. uð0;tÞ¼hðtÞ; uð1;tÞ¼0; uyyð0;tÞ¼M; uyyð1;tÞ¼0 ðt>0Þ;
2 Themassdensity,q,appearstohavebeeninadvertentlyomittedfromEqs.(7)–(9) ð3:2Þ
ofKirwanandNewman(1972);seeDevakarandIyengar(2009),whereinthefirsttwo
uðy;0Þ¼uðy;0Þ¼0 ðy>0Þ; ð3:3Þ
oftheseequationsarecorrectlystatedinEqs.(3.1)and(3.2),respectively. t
0093-6413/$-seefrontmatterPublishedbyElsevierLtd.
doi:10.1016/j.mechrescom.2010.03.006
Author's personal copy
P.Puri,P.M.Jordan/MechanicsResearchCommunications37(2010)282–284 283
8 9
whereMisthenondimensionalformofM andallprimeshavebeen
s ><1 1 Z 1=h e(cid:2)gtsin½yxðgÞ(cid:3)dg >=
(s0u,p1p/2re)sasreedgibvuetnrbeymain understood. Furthermore, ‘>0 and h2 u2ðy;tÞ¼hðtÞ>:2ð1(cid:2)e(cid:2)y=‘Þ(cid:2)2p 0 gqffiðffigffiffiffiffi(cid:2)ffiffiffiffi1ffiffiffiÞffiffi2ffiffiþffiffiffiffiffi4ffiffigffiffiffi‘ffiffi2ffiffiðffiffi1ffiffiffiffi(cid:2)ffiffiffiffihffiffigffiffiffiÞffiffi>;
‘¼L10sffikfficffiðffiðffi2ffilffiffilffiffiffiþffiþffiffiffikffiffikffiÞffiÞffiffi; h¼Ið24lkLþ20 kÞ: ð4Þ (cid:2)hðtÞ8>>>>>>>>>>>><ZZ01g(cid:7)eexðxgpp½þ½(cid:2)(cid:2)1ttð=ðgghþÞþqq11ffi½=ffi=ffiðffiffiffihffighffiffiffiffiÞffiffiÞþffiffi(cid:3)ffi(cid:3)ffiffisffisffiffiffi1ffiiffiiffinffiffin=ffiffiffiffi½ffiffih½ffiyffiyffiffiffiÞffixffixffiffiffi(cid:2)ffiffiffiffi(cid:2)ffiffi(cid:2)ffi1ffiffiffiðffiðffiffigffi(cid:3)gffiffi2ffiffiffiÞffiÞffiffi(cid:2)(cid:3)ffiffi(cid:3)ffifficffifficffiffi4ffioffioffiffiffiffigsffiffisffiffi½ffiffi‘½ffiyffiyffi2ffiffiffixffixffiðffiffiffihffiffiffiþffiffiþffigffiffiffiðffiðffiffiffigþffigffiffiffiffiÞffiÞffiffi1ffi(cid:3)ffi(cid:3)ffiffidffidffiÞffiffiffigg;; ‘6ð4hÞ(cid:2)1=2;
assoonlldvuOit(nni3og.an2pt)ihp,seluyfsoirinuengsngudtlththteionetgbeinemsiutpibaoslriadcloianLradypitleiaoqcnuesattrgiaoivnnes,nfothrinmeE,trqLa.n½((cid:5)s3(cid:3)f,.o3tro)m,Eaqndsdo.m(t3ha.e1inn) p >>>>>>>>>>>>:þ0Zg(cid:7)1ðegxðþgp½þ1(cid:2)=1thð=ÞghþÞq½ð1gffij=ffi½ffihþðffiffigffiÞffi1ffi(cid:3)þffisffi=ffiiffiffih1nffiffiÞ=ffihffiffi(cid:2)hffi½ffiyffiÞffi1xffi(cid:2)ffiffi(cid:3)ffi2ffi(cid:2)1ffiffi(cid:2)ðffi(cid:3)ffig2ffiffi4ffi(cid:2)Þffiffig(cid:3)ffiffic4ffi‘ffioffi2gffiffisðffi‘ffih½ffi2yffigffiðffixffihffiþffigffiþffiffi1ðffiþffigffiÞffiffi1Þffiffi(cid:3)ffiÞdffiffijffig; ‘>ð4hÞ(cid:2)1=2;
‘2½e(cid:2)r2yr2(cid:2)e(cid:2)r1yr2(cid:2)Mðe(cid:2)r2y(cid:2)e(cid:2)r1yÞ(cid:3) ð12Þ
u(cid:2)ðy;sÞ¼ qffiffiffiffiffiffiffi1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð5Þ 8 9
s s2ð1(cid:2)4h‘2Þþ2sð1(cid:2)2‘2Þþ1 ><1 Z 1=h e(cid:2)gtsin½yxðgÞ(cid:3)dg >=
wheresisthetransformparameter,abaroveraquantitydenotes u3ðy;tÞ¼hðtÞ>:2p 0 qffiðffigffiffiffiffi(cid:2)ffiffiffiffi1ffiffiffiÞffiffi2ffiffiþffiffiffiffiffi4ffiffigffiffiffi‘ffiffi2ffiffiðffiffi1ffiffiffiffi(cid:2)ffiffiffiffihffiffigffiffiffiÞffiffi>;
the image of that quantity in the Laplace transform domain (e.g., 8Z 1exp½(cid:2)tðgþ1=hÞ(cid:3)sin½yx ðgÞ(cid:3)cos½yx ðgÞ(cid:3)dg
ur(cid:2)1ð;y2T;¼soÞ1‘s:¼ivuutmLpffisffiffil½ffiþiffiuffifffiðyffi1ffiyffiffi;ffit(cid:6)ffitffihffiÞffieffiq(cid:3)ffi)ffiffi,ffitffiffiðaffiaffiffisffiffiffisnffiffiffiþffikffiffidffiffiffiffiffiffio1ffiffiffiffiffi2ffifÞffiffiffiffi2ffiffiiffiffiffinffi(cid:2)ffiffiffiffivffiffiffiffiffiffie4ffiffiffiffirffiffi‘ffiffisffiffi2ffiffiiffiffisffiffioffiffiðffiffinffihffiffiffiffiffis,ffiffiffiffiffiffiiþffiffiffitffiffiffiffiffiffiffii1ffiffisffiffiffiffiÞffiffiffiffi:convenienttorecastEð6qÞ. þhpðtÞ>>>>>>>>>>>><>>>>>>>>>>>>:Zþ00Zgg(cid:7)(cid:7)1exepx½p(cid:2)q½qt(cid:2)ðqffi½gtffi½ffiðffiðffiðffiffigþffiggffiffijffiffiffiffi½ffiþffiffiffiðþþffiffi1ffiffiffiffigffiffiffi=ffiffi1ffi11ffiffiffiþffiffihffi=ffiffiffi==ffiffiffiffiÞffiffihffiffih1hffi(cid:3)ffiffiffiffiffiÞffisÞÞ=ffiffiffiffiffiffi(cid:2)(cid:3)iffiffi(cid:2)hffiffiffinffisffiffiffiffiÞffiffii11ffiffi½ffinffiffiffiy(cid:2)ffiffiffi(cid:3)ffiffi(cid:3)ffi2xhffiffiffi2ffiffi1ffiffiffiffi(cid:2)½ffiffi(cid:2)ffi(cid:2)ffi(cid:2)ffiyffi(cid:3)ffiffiffi2ffiffiffix4ðffiffiffi4ffiffiffi(cid:2)gffiffiffigffiffigffi(cid:2)ffiffiffiÞffiffiffi‘4ffiffi‘(cid:3)ffiðffiffi2ffiffi2fficgffigffiffiffiðffiffiðoffiffiffiÞh‘ffihffiffiffiffis(cid:3)ffi2ffiffigffiffigcffiffi½ffiðffiffiffiyoffiffihffiþffiffiþffixffiffiffisgffiffiffiffiffiffi1½ffiffi1ffiþffiffiþyffiþffiffiffiÞffiffiÞffixðffiffiffi1gffiffiþffiÞÞffiffij(cid:3)ffiðdggÞ(cid:3);;dg; ‘‘6>ðð44hhÞÞ(cid:2)(cid:2)11==22;;
(5)as
ð13Þ
u(cid:2)ðy;sÞ¼u(cid:2) ðy;sÞ(cid:2)ð2‘2M(cid:2)1Þu(cid:2) ðy;sÞþu(cid:2) ðy;sÞ; ð7Þ where
1 2 3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where xðgÞ¼1uutg(cid:2)1þqffiðffigffiffiffiffi(cid:2)ffiffiffiffiffi1ffiffiffiÞffiffi2ffiffiffiþffiffiffiffiffi4ffiffihffiffiffi‘ffiffi2ffiffigffiffiffiðffiffi1ffiffiffi=ffiffihffiffiffiffi(cid:2)ffiffiffiffiffigffiffiffiÞffi; ð14Þ
e(cid:2)r2yþe(cid:2)r1y e(cid:2)r2y(cid:2)e(cid:2)r1y ‘ 2
uu(cid:2)(cid:2)1¼¼su(cid:2) : 2s ; u(cid:2)2¼2sqffiðffisffiffiffiþffiffiffiffiffi1ffiffiffiÞffiffi2ffiffiffi(cid:2)ffiffiffiffiffi4ffiffiffi‘ffi2ffiffiffisffiffiðffihffiffiffisffiffiffiþffiffiffiffiffi1ffiffiffiÞffiffi; ð8Þ x(cid:6)ðgÞ¼21‘sffi(cid:3)(cid:3)(cid:3)(cid:3)ffiðffiffigffiffiffiffiþffiffiffiffiffi1ffiffi=ffiffiffihffiffiffiÞffiffi(cid:2)ffiffiffiffiffi1ffiffiffiffi(cid:6)ffiffiffiffiffi2ffiffiffi‘ffiffiqffiffiffiffiffihffiffiffiffiffigffiffiffiffiffiffiðffiffiffiffigffiffiffiffiffiffiffiffiþffiffiffiffiffiffiffiffiffiffi1ffiffiffiffi=ffiffiffiffiffiffihffiffiffiffiffiÞffiffiffi(cid:3)(cid:3)(cid:3)(cid:3)ffiffi; ð15Þ
3 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1(cid:2)hð1þ2‘2(cid:2)2‘ ‘2þh(cid:2)1Þ
Sincetheinverseswerequiredonotappeartobegiveninanyof g(cid:7)¼ ; ð16Þ
hð4h‘2(cid:2)1Þ
teh.ge.,sDtaunfdfya,r2d0t0a4b)leasn,wdeobtusernrvteotthhaetctohmepinlevxerisnevseorsfiothnefotermrmuslain(sEeqe., andweobservethatg*>0ifandonlyif‘>1=pffi4ffiffihffiffiffi.
(7)aregivenby
3.Special/limitingcasesandotherapplications
1 Z rþiT
u ðy;tÞ¼ lim estu(cid:2) ðy;sÞds ðt>0Þ; ð9Þ
1;2;3 2piT!1 r(cid:2)iT 1;2;3 Remark 2. The steady-state solution, u1(y), is defined as
u (y):=lim u(y,t) and is given by u (y):=1(cid:2)M‘2[1(cid:2)
whereu (y,t)=0fort<0andthearbitraryrealnumberr>0lies 1 t?1 1
1,2,3 exp((cid:2)y/‘)], which is identical in form to that of the dipolar fluid
to the right of all singularities. The first step in evaluating these
case(seeJordanandPuri,1999,Eq.(4.3)).
integralsisdeterminingthesingularitiesoftheirintegrands.Omit-
tingthedetails,itcanbeshownthattheonlynon-removablesingu- Remark 3. If we take h=1(cid:2)‘2, implying ‘22(cid:4)1;1(cid:5), then u(cid:2)
laritiesofu(cid:2)1;2 consistofasimplepoleats=0andbranchpointsat simplifytosuchanextentthattheycanbeeasily2invertedusi1n;2g;3,
s={(cid:2)1/h,0}whilethoseofu(cid:2)3consistofonlythetwobranchpoints. e.g., the tables of Laplace inverses given in (Erdélyi, 1954, Chap.
Thus,byCauchy’stheoremwehave V).Thus,forthisspecialcaseuðy;tÞ¼Uðy;tÞ,whereUisdefinedas
21piICestu(cid:2)1;2;3ðy;sÞds¼0; ð10Þ Uðy;tÞ:¼hðtÞ(erfc(cid:6)2pyffitffi(cid:7)(cid:2)a2‘2eat(cid:8)eypffiaffierfc(cid:6)2pyffitffiþpffiaffiffitffiffi(cid:7)
wtehrcelroeckthweisBerodmirewcitcihonc.ontourC(seeFig.1)istraversedinthecoun- þe(cid:2)ypffiaffierfc(cid:6) pyffiffi(cid:2)pffiaffiffitffiffi(cid:7)(cid:2)ecypffiaffiþffiffiffibffierfc(cid:6) cpyffiffiþqffiðffiaffiffiffiffiþffiffiffiffiffibffiffiÞffiffitffiffi(cid:7)
2 t 2 t
Followingstandardtechniquesofcontourintegration(see,e.g.,
pffiffiffiffiffiffi (cid:6) cy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(cid:7)(cid:9)
Duffy,2004),itcanbeestablishedthat,forallh2 (0,1/2)and‘>0, (cid:2)e(cid:2)cy aþberfc pffiffi(cid:2) ðaþbÞt
2 t
u1ðy;tÞ¼hðtÞ(12ð1þe(cid:2)y=‘Þ(cid:2)21pZ01=he(cid:2)gtsin½ygxðgÞ(cid:3)dg) (cid:2)M‘2(cid:8)erfc(cid:6)2pyffitffi(cid:7)(cid:2)12eat(cid:8)eypffiaffierfc(cid:6)2pyffitffiþpffiaffiffitffiffi(cid:7)
hðtÞ8>>>>>>>><ZZ01g(cid:7)eexxpp½½(cid:2)(cid:2)ttððggþþ11==hhÞÞ(cid:3)(cid:3)ssiginnþ½½yyx1x=þhððggÞÞ(cid:3)(cid:3)ccooss½½yyxx(cid:2)ððggÞÞ(cid:3)(cid:3)ddgg; ‘6ð4hÞ(cid:2)1=2; þe(cid:2)ypffiaffierfc(cid:6)2pyffitffi(cid:2)pffiaffiffitffiffi(cid:7)(cid:2)ecypffiaffiþffiffiffibffierfc(cid:6)2cpyffitffiþqffiðffiaffiffiffiffiþffiffiffiffiffibffiffiÞffiffitffiffi(cid:7)
(cid:2) p >>>>>>>>:þ0Zg(cid:7)1exp½(cid:2)tðgþ1=hÞ(cid:3)gsiþng½1þy=xþh1þ=hðgÞ(cid:3)cosh½(cid:2)yx(cid:2)ðgÞ(cid:3);dg; ‘>ð4hÞð(cid:2)11=12;Þ þ(cid:2)ee(cid:2)(cid:2)ccyyppffiaffibffiffiþeffiffiffibrffiefcr(cid:6)fc2(cid:6)cp2ycffitpffiy(cid:2)ffitffi(cid:2)pffibqffiffitffiffi(cid:7)ffiðffiaffiffi(cid:9)ffiffiþ(cid:9)ffiffiffi(cid:10)ffiffibffi:ffiÞffiffitffiffi(cid:7)(cid:9)(cid:2)12(cid:8)ecypffibffierfc(cid:6)2cpyffitffiþpffibffiffiðtffiffi1(cid:7)7Þ
Author's personal copy
284 P.Puri,P.M.Jordan/MechanicsResearchCommunications37(2010)282–284
Fig.1. Bromwichcontour,C,usedtoevaluatetheintegralsinEq.(10).Here,(cid:2)andearesmallpositivequantities.
Here,erfc((cid:5))denotesthecomplementaryerrorfunctionandthe References
positive constants a, b, and c are defined as a= (2‘2(cid:2)1)(cid:2)1, b=
pffiffiffiffiffiffiffiffiffiffiffiffiffi
(1(cid:2)‘2)(cid:2)1,andc¼‘(cid:2)1 1(cid:2)‘2. Christov,C.I.,Jordan,P.M.,2009.Commenton‘‘StokesfirstproblemforanOldroyd-
Bfluidinaporoushalfspace”[Phys.Fluids17,023101(2005)].PhysicsofFluids
21,069101.
Remark 4. Setting ‘=l2 and letting h?0 reduces u to the l1=1 Cowin,S.C.,1974.Thetheoryofpolarfluids.In:Yih,C.-S.(Ed.),AdvancesinApplied
special case of (Jordan and Puri, 1999, Eq. (3.4)), i.e., the Mechanics,vol.14.AcademicPress,NewYork,pp.279–347.
corresponding solution for the case of a dipolar3 fluid. On the Devakar, M., Iyengar, T.K.V., 2009. Stokes first problem for a micropolar fluid
throughstate–spaceapproach.AppliedMathematicalModelling33,924–936.
other hand, letting ‘?0 reduces u to the solution of Stokes’ first Duffy, D.G., 2004. Transform Methods for Solving Partial Differential Equations,
problemforan Oldroyd-B liquid;see pp.354–355 ofDuffy(2004), seconded.Chapman&Hall/CRC,BocaRaton,FL.
andalsoChristovandJordan(2009). Erdélyi,A.(Ed.),1954.TablesofIntegralTransforms,vol.I.McGraw–Hill,NewYork.
Eringen, A.C., 1966. Theory of micropolar fluids. Journal of Mathematics and
Mechanics16,1–18.
Guenther, R.B., Lee, J.W., 1996. Partial Differential Equations of Mathematical
Remark 5. It is noteworthy that Eq. (1.1), which is identical in
PhysicsandIntegralEquations.Dover,Mineola,NY.pp.217–218.
formtotheequationofmotionforthisprobleminvolvingamicro- Jordan, P.M., Puri, P., 1999. Exact solutions for the flow of a dipolar fluid on a
structure-ladenfluidofthetypedescribedbythetheoryof Kline suddenlyacceleratedflatplate.ActaMechanica137,183–194.
Jordan,P.M.,Puri, P.,Boros, G.,2000. Anewclass ofLaplace inversesandtheir
and Allen (1970), also arises in the study of incompressible flow
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ThisworkwascarriedoutwhileP.M.J. helda NASAGSRPFel- MathematicalMethodsintheAppliedSciences9,35–45.
lowship(NGT-13-52706).Thepresentmanuscriptwascompleted
while this author was supported by ONR/NRL funding (PE
061153N).
3 Inthecontextofthepresentproblem,dipolarfluidtheoryandGreen–Naghdi
theoryyield,allowingfordifferencesinthecoefficients,thesameequationofmotion;
see,e.g.,QuintanillaandStraughan(2005)andthereferencestherein.