Research Article Revisiting Blasius Flow by Fixed Point Method

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1 Abstract ad Applied Aalysis, Article ID , 9 pages Research Article Revisitig Blasius Flow by Fixed Poit Method Dig Xu, 1 Jiglei Xu, 2 ad Goga Xie 3 1 State Key Laboratory for Stregth ad Vibratio of Mechaical Structures, School of Aerospace, Xi a Jiaotog Uiversity, No. 28, Xiaig West Road, Xi a , Chia 2 School of Jet Propulsio, Beijig Uiversity of Aeroautics ad Astroautics, Xueyua Road, No. 37, Beijig , Chia 3 School of Mechaical Egieerig, Northwester Polytechical Uiversity, Xi a, Shaaxi , Chia Correspodece should be addressed to Jiglei Xu; xujl@buaa.edu.c Received 28 October 2013; Revised 7 December 2013; Accepted 7 December 2013; Published 12 Jauary 2014 Academic Editor: Mohamed Fathy El-Ami Copyright 2014 Dig Xu et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The well-kow Blasius flow is govered by a third-order oliear ordiary differetial equatio with two-poit boudary value. Specially, oe of the boudary coditios is asymptotically assiged o the first derivative at ifiity, which is the mai challege o hadlig this problem. Through itroducig two trasformatios ot oly for idepedet variable bur also for fuctio, the difficulty origiated from the semi-ifiite iterval ad asymptotic boudary coditio is overcome. The deduced oliear differetial equatio is subsequetly ivestigated with the fixed poit method, so the origial complex oliear equatio is replaced by a series of itegrable liear equatios. Meawhile, i order to improve the covergece ad stability of iteratio procedure, a sequece of relaxatio factors is itroduced i the framework of fixed poit method ad determied by the steepest descet seekig algorithm i a coveiet maer. 1. Itroductio The Navier-Stokes equatios are the fudametal goverig equatios of fluid flow. Usually, this set of oliear partial differetial equatios has o geeral solutio, ad aalytical solutios are very rare oly for some simple fluid flows. However, i some certai flows, the Navier-Stokes equatios may be reduced to a set of oliear ordiary differetial equatios uder a similarity trasform [1, 2]. These similarity solutios could ot oly provide some physical sigificace to the complex Navier-Stokes equatios but also act as a bechmarkig for umerical method. The well-kow Blasius flow [3 5] is possibly the simplest example amog these similarity solutios. It describes the idealized icompressible lamiar flow past a semi-ifiite flat plate at high Reyolds umbers, which is mathematically a third-order oliear two-poit boudary value problem: A f [f] =2 +f =0, (1) subject to the boudary coditios: f (0) =0, f (0) =0, lim η f (η) = 1, (2) where the prime deotes differetiatio to the variable η ad f(η) is the odimesioal stream fuctio related to the stream fuctio ψ(x, y) as follows: ψ (x, y) =f(η) ]xu. (3) η=y U /(]x) is the similarity variable, where U is the free stream velocity, ] is the kiematic viscosity coefficiet, ad x ad y are the two idepedet coordiates. The two velocity compoets are the determied: u= ψ y =U f (η), V = ψ x = 1 2 [ηf (η) f (η)] ]U x. Accordig to (1)ad(2) the solutioisdefied o a semiifiite iterval η 0, ad oe of the boudary coditios is asymptotically assiged o the first derivative of fuctio at ifiity, which are the mai challeges o solvig the (4)

2 2 Abstract ad Applied Aalysis Blasiusflow.Thesolutiotothisproblemhasthefollowig asymptotic property [6, 7]: f f (0) η 2, 2 as η 0, f η+b, as η, where B is a costat ad the bechmarkig value provided by Boyd [6, 7] isb = As kow, o simple closed-form solutio to the Blasius problem is available, despite the simple form ad such a log history of it sice 1908 [3]. Much attetio has bee paid to this problem. Blasius [3] himself firstly ivestigated this problem by the perturbatio method ad obtaied a approximate solutio by matchig a power series solutio for small η to a asymptotic expasio for large η.however,this procedure may be improper because of somewhat restricted radius of covergece i the first power series [8]. Later, this problem was hadled by Beder et al. [9] with δ-expasio i a smart maer. The approximate solutios were obtaied by He [10], Liao [11, 12], ad Turkyilmazoglu [13 16] with the variatioal iteratio method, homotopy aalysis method, ad homotopy perturbatio method, respectively. Wag [17] also ivestigated this problem by the Adomia decompositio method. Meatime, there are a lot of umerical methods emergig to hadle the Blasius problem icludig, but ot limited to, shootig method, fiite differeces method, ad spectral method [18 31]. A vast bibliography of umerical methods has developed for this problem, so a full accout of them is out of the scope of this paper, ad readers are suggested to refer to the review articles [6, 7]. It is oted that the existig umerical methods usually itegrate this problem over a fiite iterval η [0,η ], although the Blasius problem is origially defied o the semi-ifiite iterval η [0,+ ).Thusthevalueofη should be chose sufficietly large to assure the accuracy of the asymptotical boudary coditio at ifiity. However, the appropriate value η could ot be determied beforehad, so usually the trial-ad-error approach is ivolved, ad some differet values should be tried to fid the appropriate η to satisfy the demaded accuracy. I order to exactly assure the boudary coditios (2) ad obtai a uiformly valid solutio o the semi-ifiite iterval η [0,+ ), two trasformatios ot oly for the idepedet variable η but also for fuctio f(η) are itroduced i this paper. The trasformed oliear differetial equatio is subsequetly ivestigated with the fixed poit method (FPM) [33], which trasforms the oliear differetial equatio ito a series of itegrable liear differetial equatios. Hece, a approximate semiaalytical solutio to the Blasius problem is fially obtaied, which is validothewholedomaiadcasatisfytheasymptotic property automatically. Meatime, i order to improve the covergece ad stability of iteratio procedure, a sequece of relaxatio factors is itroduced i the framework of FPM, which are determied by the steepest descet seekig algorithm. Thus, the accuracy of this approximate solutio could be improved step by step i a coveiet maer. (5) Res (0) λ = 1/ Figure 1: The covergece history of Res (λ=1/5) FPM (λ = 1/5) Bechmark (Boyd s) Figure 2: The covergece history of (0) (λ =1/5). 2. Revisitig the Blasius Equatio by Fixed Poit Method 2.1. Trasformatios. As metioed i Sectio 1, the mai challege o hadlig the Blasius problem origiates from the semi-ifiite iterval η [0,+ ) ad the asymptotic boudary coditio lim η f (η) = 1. I order to overcome these difficulties, two trasformatios are itroduced for idepedet variable η ad fuctio f(η),respectively, z= (λη 1) (λη + 1), g (z) = λ(f η) (1 + λη), (6)

3 Abstract ad Applied Aalysis 3 f, f ad f FPM ( = 100) Howarth η df/dη d 2 f/dη 2 Figure 3: Compariso of FPM result (λ = 1/5, = 100) with Howarth s umerical result. A f [f ] η FPM ( = 100) FPM ( = 150) FPM ( = 200) Figure 4: The residual error fuctio A f [f ]=2 1/5). +f (λ= where λ (>0) is a free parameter. 1/λ stads for the legth dimesio ad its physical meaig is related to the scale of boudary layer thickess. The ifluece of λ o the solutio will be discussed i detail i Sectio 3.2. Hece, the origial Blasius equatio becomes A g [g]=λ 2 (1 z) 3 g +[1 3λ 2 (1 z) 2 +z+2g]g =0, (7) which is beeficial to the acquiremet of the valid solutio i the whole domai The Idea of Fixed Poit Method (FPM). The fixed poit, a fudametal cocept i fuctioal aalysis [34], has bee widely adopted i studyig the existece ad uiqueess of solutios by pure mathematicias. Recetly, the fixed poit cocept has bee used to hadle oliear differetial equatios, ad the fixed poit method (FPM) has bee proposed to obtai the explicit approximate aalytical solutio to the oliear differetial equatio [33]. To outlie the idea of FPM, let us cosider the followig oliear differetial equatio: A [u] =0, B + [u] =0, (10) where A[ ] is a oliear operator ad u is a ukow fuctio. Here, B + [u] = 0 is the boudary coditio ad/or iitial coditio for u. T[ ] is a cotractive map: T [u] =u π L 1 C [A [u]], (11) where L C [ ] is a liear cotiuous bijective operator, amed as the liear characteristic operator of the oliear operator A[ ] ad L 1 C [ ] is the iverse operator of L C[ ]. π is a real ozero free parameter, amed as the relaxatio factor, which couldimprovethecovergeceadstabilityofiteratio procedure. The optimal value π is usually depedet o the problem to be solved [33]. The, a solutio sequece {u = 0, 1, 2, 3,...} ca be obtaied from the followig iteratio procedure: u +1 = T [u ]=u π +1 L 1 C [A [u ]], B + [u +1 ]=0 L C [u +1 ]=L C [u ] π +1 A [u ], B + [u +1 ]=0 = 0, 1, 2,... (12) =0, 1, 2,.... (13) If the covergece of the solutio sequece {u = 0, 1, 2, 3,...} is esured, it is clear that the limit value u is exactly the zero poit of the origial oliear operator A[ ]: with the followig boudary coditios: g ( 1) =0, g ( 1) = 1, g(1) =0, (8) 2 A [u ]=0, B + [u ]=0, (14) where the prime deotes differetiatio to the ew variable z. It is clear that the semi-ifiite iterval η [0,+ ) is mapped to the bouded iterval 1 z 1,adtheorigial asymptotic boudary coditio lim η f (η) = 1 becomes λ(f η) g (1) = lim η 1+λη = lim λ[f (η) 1] =0, (9) η λ ad u is also amed as a fixed poit of the cotractive map T[u]. I [33], oly oe relaxatio factor π is itroduced ad determied by the so-called π-curves i a heuristic maer. Here, a sequece of relaxatio factors {π = 1,2,3,...} is itroduced i (12), which will be decided accordig to the steepest descet seekig algorithm i the followig Sectio 2.3.

4 4 Abstract ad Applied Aalysis λ = 1/9 λ = 1/ λ = 1/3 λ = Res Res λ = 1/8 λ = 1/7 λ = 1/ λ = 1/4 λ = 1/ λ = 1/ (a) Compariso amog λ=0.4, λ=1/3, λ=1/4,adλ=1/ (b) Compariso amog λ=1/5, λ=1/6, λ=1/7, λ=1/8, λ=1/9,ad λ = 1/10 Figure 5: The ifluece of λ value o the square residual error Res. Table 1: Compariso of (0) betwee FPM (λ =1/5) ad others. Preset (FPM) (0) Fazio [31] ZhagadChe[32] Boyd[6, 7](Bechmark) The Steepest Descet Seekig Algorithm (SDS). As metioed i Sectio 2.2, the relaxatio factor {π =1,2,3,...} couldimprovethecovergeceadstabilityofiteratio procedure, ad usually the optimal value of relaxatio factor is depedet o the problem to be solved. Here, a algorithm, amed as the steepest descet seekig algorithm (SDS), is adopted to determie the optimal value of the relaxatio factor. Let Res deote the square residual error of the aforemetioed iteratio procedure i (13): Res = Res (π 1,π 2,...π ) = (A [u ]) 2 dω, =1,2,3,..., Ω (15) where Ω is the defiitio domai of the variable ad Res is a kid of global residual error ad ca evaluate the accuracy of the approximatio u. The it is suggested that the optimal value of relaxatio factor π,opt correspods to the value π such that Res obtais the miimum value mi(res ).For example, whe =1, the square residual error Res 1 (π 1 ) is afuctioofπ 1 oly ad thus the optimal value π 1,opt ca be obtaied by solvig the oliear algebraic equatio: dres 1 dπ 1 =0. (16) Whe = 2, the square residual error Res 2 (π 1,π 2 ) is depedet o π 1 ad π 2.Becausetheoptimalvalueπ 1,opt is kow from the previous step, the optimal value π 2,opt is govered by the followig oliear algebraic equatio: dres 2 dπ 2 =0. (17) Similarly, for the th-higher order, the square residual error Res actually cotais a ukow relaxatio factor

5 Abstract ad Applied Aalysis 5 Table 2: Compariso of f betwee FPM (λ =1/5)adHowarth. f η FPM = 50 = 200 = 800 Howarth [21] / / / Table 3: Compariso of f betwee FPM (λ =1/5)adHowarth. η FPM = 50 = 200 = 800 Howarth [21] / / / f

6 6 Abstract ad Applied Aalysis Table 4: Compariso of betwee FPM (λ =1/5)adHowarth. η FPM = 50 = 200 = 800 Howarth [21] e e e e e e e e e e e e 9 / e e e 17 / e e e 18 / (0) (0) Bechmark (Boyd's) λ = 1/3 λ = 1/4 λ = 1/5 Bechmark (Boyd's) λ = 1/5 λ = 1/6 λ = 1/7 λ = 1/8 (a) Compariso amog λ=1/3, λ=1/4,adλ=1/5 (b) Compariso amog λ=1/5, λ=1/6, λ=1/7,adλ=1/8 Figure 6: The ifluece of λ value o the covergece of (0). π oly, so the optimal value π,opt is determied by the followig oliear algebraic equatio: dres =0. (18) dπ The ame of the steepest descet seekig algorithm just comes from the aforemetioed approach; that is, every optimal value π,opt is sought to miimize the correspodig square residual error Res. Accordig to this approach, oly oe oliear algebraic equatio should be solved i every iteratio step, ad the elemets of the sequece {π = 1,2,3,...} are obtaied sequetially ad separately Iteratio Procedure. Now, for (7), let us choose the liear characteristic operator: L C [g] = d3 g dz 3 =g, (19) ad costruct a iteratio procedure as follows:

7 Abstract ad Applied Aalysis 7 L C [g +1 ] = L C [g ] π +1 A g [g ], g +1 ( 1) =0, g +1 ( 1) = 1 2, g +1 (1) =0, =0,1,2... (20) g +1 =g π +1 {λ 2 (1 z) 3 g +[1 3λ2 (1 z) 2 +z+2g ]g }, g +1 ( 1) =0, g +1 ( 1) = 1 2, g +1 (1) =0, =0,1,2... (21) The iitial guess g 0 is coveietly chose as g 0 = (z2 1), (22) 4 which satisfies the followig equatio: L C [g 0 ]=0, g 0 ( 1) =0, g 0 ( 1) = 1 2, g 0 (1) =0. 3. Result ad Discussio (23) 3.1. Results as λ=1/5. Before the acquiremet of approximate solutio accordig to the iteratio procedure (21), the free parameter λ should be determied. It is foud that the iteratio procedure coverges rapidly whe the value of 1/λ takes the scale of boudary layer thickess. Here, the value of λ is firstly set to λ=1/5ad the ifluece of λ valueothe solutiowill be studied i Sectio 3.2. I order to demostrate FPM, the procedure to obtai the first-order approximatio g 1 (z) is give here i detail. Firstly, the goverig equatio for g 1 (z) is deduced accordig to (21): g 1 = π ( z + 19z2 ), g 1 ( 1) =0, g 1 ( 1) = 1 2, g 1 (1) =0. (24) The the first order approximatio g 1 (z) takes the followig form: g 1 (z) = 73π π z+( π )z2 19π z3 31π z4 19π z5. (25) It is clear that the first-order approximatio g 1 (z) i (25) is depedet o the relaxatio factor π 1,whoseoptimal value could be determied by the steepest descet seekig algorithm (SDS) as metioed i Sectio 2.3. Here, the square residual error Res of the origial equatio (1) is itroduced: + Res = 0 (A f [f ]) 2 +1 dη = 1 (1 z) 2 (A 8λ g [g ]) 2 dz. (26) The,thesquareresidualerroroff 1 (η) is as follows: Res 1 (π 1 ) = π π π π 4 1, (27) ad the optimal value π 1,opt ad the miimum of Res 1 are π 1,opt = , mi (Res 1 ) = (28) Hece, the first-order approximatio g 1 (z) is fially determied: g 1 (z) = z z z z z 5. (29) For the higher-order approximatio g (z),theprocedure is similar, ad a explicit semiaalytical solutio could be deduced by the symbolic computatio software, such as MAXIMA, MAPLE ad MATHEMATICA. I cosideratio of the trasformatio (6), the correspodig approximate solutio f (η) to the origial Blasius equatios (1)ad(2)is f (η) = g (z) (η+ 1 (λη 1) )+η, z= λ (λη + 1). (30) The covergece history of the square residual error Res is illustrated i Figure 1, which clearly shows that Res is gradually reduced durig the iteratio procedure, so the accuracy of the approximate solutio could be improved step by step to ay possibility. The secod derivative (0) is a measure of the shear stress o the plate ad plays a critical role i the Blasius problem [4, 5]. The relatioship betwee (0) ad g ( 1) ca be deduced as follows: (0) =4λg ( 1). (31) The covergece history of (0) is displayed i Figure 2, which shows that the differece betwee the approximatio (0) ad Boyd s [6, 7] bechmarkig result f (0) = decreases durig the iteratio procedure. Meawhile, the compariso betwee the preset result

8 8 Abstract ad Applied Aalysis Table 5: The asymptotic property of f for large positive η (λ =1/5, = 800). η f η B = η lim (f η) (Bechmark) [6, 7] ad others give i [6, 7, 31, 32]istabulatediTable 1,which shows that (0) is the same as Boyd s bechmarkig result withi 7 sigificat digits whe 300 adwithi12 sigificat digits whe 800. The approximate semiaalytical solutios ad the wellkow Howarth s [32] accurate umerical result of f(η), f (η), ad (η) are compared i Figure 3 ad simultaeously tabulated i Tables 2 4, which shows that the preset result obtaied by FPM is of high accuracy. The residual error fuctio A f [f ] = 2 +f is plotted i Figure 4, which also reveals that the error of approximate solutios gradually decreases durig the iteratio procedure. Moreover, the preset approximate solutios areuiformlyvalidithewholeregio. Based o the asymptotic property of f(η) give i (5), we obtai B= lim η (f η). (32) The approximate value of B could be obtaied as follows: B f (η) η, for large η. (33) The compariso betwee the approximate value of B obtaied by FPM ( = 800, λ = 1/5) ad the bechmarkig result B = provided by Boyd [6, 7] is give i Table 5, which shows that the preset result is the same as Boyd s bechmarkig result withi 7 sigificat digits whe η 9adwithi13sigificatdigitswheη The Ifluece of λ Value o the Solutio. It is clear that 1/λ takes the legth dimesio i cosideratio of trasformatio (6). I order to ivestigate the ifluece of λ value o the solutio, some differet λ values are cosidered i the followig calculatios, ad the compariso of Res at differet λ values is displayed i Figures 5(a) ad 5(b). It is foud that all Res correspodig to differet λ values are gradually reduced durig the iteratio procedure, ad Res based o λ=1/5coverges more rapidly tha others. What is the physical meaig of λ? Letustrytofidtheaswer from the Pradtl s boudary layer theory [5]. Accordig to Pradtl s boudary layer theory, the effect ofviscosityismailycofiedtotheboudarylayersuch that η<δ, adtheouterflow(η>δ) could be cosidered as iviscid flow. From Table 3, the thickess of the boudary layerisjustaboutδ 5,whereu/U = f Now, the physical meaig of λ becomes clear. 1/λ has the same scale of boudary layer thickess δ. I cosideratio of trasformatio (6), the regio 1 z 1 is divided ito two equal parts, ad the viscous flow (η <δ 5) ad iviscid flow (η >δ 5) correspod to 1 z < 0 ad 0<z<1, respectively. Although this determiatio of λ is i a heuristic maer, it is fortuate that the solutio is quite isesitive to λ so log as 1/λ is of the same order-of-magitude as δ 5. The ifluece of λ o the covergece of (0) is give i Figures6(a) ad 6(b), which alsorevealsthatthelimitvalues of (0) with differet λ values agree well with each other. Hece, the selectio of λ is oessetial to the fial solutio. 4. Coclusio I this paper, the well-kow Blasius flow is revisited by the fixed poit method (FPM). I order to overcome the difficulties origiated from the semi-ifiite iterval ad asymptotic boudary coditio, two trasformatios are itroduced for ot oly the idepedet variable but also the depedet variable. Uder these trasformatios, all the boudary coditios are exactly assured for every order approximate solutio. I the meawhile, a free scale parameter λ is itroduced i the trasformatio, ad its physical meaig is related to the thickess of the boudary layer. Moreover, a sequece of relaxatio factors {π = 1, 2, 3,...} is itroduced to improve the covergece ad stability durig iteratio procedure, ad its elemets are obtaied i a coveiet maer by the steepest descet seekig algorithm. Fially, the compariso of the preset results with other scholars umerical results, especially with the bechmarkig results provided by Boyd, shows that FPM is a effective ad accurate approach to obtai the semiaalytical solutio to oliear problems. Nomeclature U : Freestreamvelocity,m/s u: x-compoets of the velocity, m/s V: y-compoets of the velocity, m/s ψ(x, y): Streamfuctio,m 2 /s f(η): Nodimesioal stream fuctio ]: Kiematic viscosity coefficiet, m 2 /s. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper.

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