Research Article An Upper Bound on the Critical Value β Involved in the Blasius Problem

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1 Hindawi Publihing Corporaion Journal of Inequaliie and Applicaion Volume 2010, Aricle ID , 6 page doi: /2010/ Reearch Aricle An Upper Bound on he Criical Value Involved in he Blaiu Problem G. C. Yang 1, 2 1 School of Mahemaic and Saiic, Lanzhou Univeriy, Lanzhou, Ganu , China 2 College of Mahemaic, Chengdu Univeriy of Informaion Technology, Chengdu , China Correpondence hould be addreed o G. C. Yang, cuiyang@yahoo.com.cn Received 21 February 2010; Revied 29 April 2010; Acceped 6 May 2010 Academic Edior: Michel C. Chipo Copyrigh q 2010 G. C. Yang. Thi i an open acce aricle diribued under he Creaive Common Aribuion Licene, which permi unrericed ue, diribuion, and reproducion in any medium, provided he original work i properly cied. Uilizing he Schauder fixed poin heorem o udy exience on poiive oluion of an inegral equaion, we obain an upper bound of he criical value involved in he Blaiu problem, in paricular, < 18733/ Previou reul only preened a lower bound 1/2 and numerical inveigaion Inroducion The following hird-order nonlinear differenial equaion ariing in he boundary-layer problem f ( η ) f ( η ) f ( η ) 0 on 0, 1.1 ubjec o he boundary condiion f 0 0, f 0, f 1, 1.2 called he Blaiu problem 1, ha been ued o decribe he eady wo-dimenional flow of a lighly vicou incompreible fluid pa a fla plae, where η i he imilariy boundarylayer ordinae, f η i he imilariy ream funcion, and f η and f η are he velociy and he hear re funcion, repecively. Problem alo arie in he udy of he mixed convecion in porou media 2. The mixed convecion parameer i given by 1 ε, wihε R a /P e where R a i he

2 2 Journal of Inequaliie and Applicaion Rayleigh number and P e he Pécle number. The cae of < 0 correpond o a fla plae moving a eady peed oppoie o ha of a uniform mainream 3. The boundary value problem ha been widely udied analyically. Weyl 4 proved ha ha one and only one oluion for 0; Coppel 5 udied he cae of >0; he cae of 0 <<1 6 and >1 7 were alo inveigaed, repecively. Alo, ee 8. Blaiu problem i a pecial cae of he Falkner-Skan equaion, for 0; we may refer o 9 13 for ome recen reul on he Falkner-Skan equaion. Very recenly, Brighi e al. 14 ummarized hiorical udy on he Blaiu problem and analyzed he cae <0 in deail, in which he hape and he number of oluion were deermined. We may refer o 14 and he reference herein for more recen reul. However, up o oday, we know only ha here exi a criical value 1/2, 0 uch ha ha a lea a oluion for, no oluion for < 15. Numerical reul howed ha An open queion i wha i exacly? To our knowledge, here i lile udy on i. In hi paper, we will udy he open queion menioned above by udying he exience on poiive oluion of an inegral equaion and preen an upper bound of, in paricular, < 18733/ An Upper Bound of By he baic fac in 14, we know eaily ha if f i a oluion of , hen f > 0for η 0,. In hi cae, he mo powerful mehod i he o-called Crocco ranformaion ee 14, 15, which coni of chooing f a independen variable and expreing z f a a funcion of. Differeniaing z f f he variable i omied for impliciy, weobain z f f f ff ; hence z f f.differeniaing once again, we obain z f f f. Then become he Crocco equaion 14 d 2 z d 2 z, <1 2.1 wih he boundary condiion z ( ) 0, z Inegraing 2.1 from o, we have z z d on [, 1 ). 2.3 Inegraing hi equaliy from o 1, we obain he following inegral equaion ha i equivalen o : z 1 z z d for [, 1 ). 2.4

3 Journal of Inequaliie and Applicaion 3 Le g 1/ for 1/2, 0, hen g > 0for 1/2, 0. By direc compuaion ( g 1 ) ( < 0, g ) > Hence here exi 1/5, 18733/10 5 uch ha g 0andg > 0for, 0. We hall prove ha ha a lea a oluion for, 0. Le, 0 and C, 1 be he Banach pace of coninuou funcion on, 1 wih he norm z max{ z :, 1 } and S : C, 1 C, 1 wih Sz max{z,c }, where c c 1 for, 1 and c 3/3 g ( ) 4 ( 1 ). 2.6 Clearly, Sz c for z C, 1 and 0 <c 3/12. Noaion. One ha Az Sz d, Bz d for < Sz We conider he following inegral equaion of he form z Az 1 Bz for < Lemma 2.1. The inegral equaion 2.8 ha a oluion z C, 1. Proof. Le C {z C, 1 : z 2M} wih M 1 /d. We define an operaor T on C by eing Az 1 Bz if [, 1 ), Tz 0 if Since Az d Sz 0 Sz d 1 0 c 1 c 1 2 2c ln 1 c lim 1 1 ln 1 0, for 0, 1, for 0, 1, 2.10

4 4 Journal of Inequaliie and Applicaion we know ha lim 1 Tz 0 and hen T map C ino C, 1. We how ha T i coninuou and compac from C ino C. Le z n C, z C, and lim n z n z 0. Since 1 1 for 1, we have Tz n Tz Az n Az 1 Bz n Bz Sz n Sz d ( ) 1 Sz n Sz 1 d 2 Sz n Sz d Since 1 lim n Sz n Sz for [, 1 ) 2.12 and Sz c, he Lebegue dominaed convergence heorem, he dominaed funcion F 1/c for, 1 implie ha Tz n Tz 0, ha i, T i coninuou. By d Tz /d /Sz d, we have d Tz d Sz d d for < Noicing ha d d 1 d 1 M<, 2.14 we have d Tz /d d M. Thi, ogeher wih he abolue coninuiy of he Lebegue inegral, implie ha T C {Tz : z C} i equiconinuou. On he oher hand, Tz 1 Sz Sz d 1 2M I follow from he Schauder fixed poin heorem ha here exi z C uch ha 2.8 hold.

5 Journal of Inequaliie and Applicaion 5 Theorem 2.2. The problem ha a lea a oluion for, 0 and hen < 18733/ Proof. We fir prove ha he funcion z obained in Lemma 2.1 i a oluion of 2.4 for, 0. Clearly, we have only o prove Sz z for, 1,hai,z c for, 1. Fir of all, we prove ha here exi, 1 uch ha z >c. In fac, if z c for, 1, hen by Sz c 1 c ( 1 ) z ( ) 1 Sz 1 ( 1 2) c Thi implie ha c 2 1 /2 1 1/5 /2 2/5, which conradic c 3/12. From he relaion z Sz d, z Sz, 2.17 we know ha z i convex and increaing on, 0 and concave on 0, 1. Moreover, ince z 1 0, here exi 0, 1 uch ha z max{z :, 1 }. For, 1, we have Bz Bz z 0. Then, from 2.8 we deduce ha Az z Sz for, 1 and hence Az Az 1 for [, 1) Inegraing he la inequaliy for o 1 and uing Az 1 0, we know ha [ )] 2 Az( 2 d And hen z Az 3/3. Thi, ogeher wih c c 3/12 for 0, 1, implie ha Sz 3/3for 0, 1. Hence 1 0 Sz d 3 0 3/ Noicing ha Sz c and 1 < 0for, 0, weobain 0 0 Sz d 2. 2c 2.21 Then z ( ) 0 Sz 1 3 Sz 0 Sz d c 2.22

6 6 Journal of Inequaliie and Applicaion By direc compuaion, we have 3/6 2 /2c c 1 and hen z c. Since z i convex and increaing on, 0 and concave on 0, 1 wih z 1 0, we immediaely ge z c for, 1. Hence Sz z and z i a poiive oluion of 2.4. Since any poiive oluion of i a oluion of and i equivalen o 2.4, hence ha a lea a oluion for, 0 and we obain he deired reul < 18733/ Acknowledgmen The auhor would like o hank very much Profeor C. K. Zhong and W. T. Li in Lanzhou Univeriy, China, for heir guidance and he referee for heir valuable commen and uggeion. Thi reearch i uppored in par by he Training Fund of Sichuan Provincial Academic and Technology Leader. Reference 1 H. Blaiu, Grenzchichen in Flüigkeien mi kleiner Reibung, Zeichrif für angewande Mahemaik und Phyik, vol. 56, pp. 1 37, E. H. Aly, L. Ellio, and D. B. Ingham, Mixed convecion boundary-layer flow over a verical urface embedded in a porou medium, European Journal of Mechanic. B, vol. 22, no. 6, pp , P. D. Weidman, New oluion for laminar boundary layer wih cro flow, Zeichrif für Angewande Mahemaik und Phyik, vol. 48, no. 2, pp , H. Weyl, On he differenial equaion of he imple boundary-layer problem, Annal of Mahemaic, vol. 43, pp , W. A. Coppel, On a differenial equaion of boundary-layer heory, Philoophical Tranacion of he Royal Sociey of London. Serie A, vol. 253, pp , P. Harman, Ordinary Differenial Equaion, John Wiley & Son, New York, NY, USA, Z. Belhachmi, B. Brighi, and K. Taou, On he concave oluion of he Blaiu equaion, Aca Mahemaica Univeriai Comenianae, vol. 69, no. 2, pp , O. A. Oleinik and V. N. Samokhin, Mahemaical Model in Boundary Layer Theory, vol. 15 of Applied Mahemaic and Mahemaical Compuaion, Chapman & Hall/CRC Pre, Boca Raon, Fla, USA, J. Wang, W. Gao, and Z. Zhang, Singular nonlinear boundary value problem ariing in boundary layer heory, Journal of Mahemaical Analyi and Applicaion, vol. 233, no. 1, pp , R. P. Agarwal and D. O Regan, Singular inegral equaion ariing in Homann flow, Dynamic of Coninuou, Dicree & Impulive Syem. Serie B, vol. 9, no. 4, pp , G. C. Yang and K. Q. Lan, The velociy and hear re funcion of he Falkner-Skan equaion ariing in boundary layer heory, Journal of Mahemaical Analyi and Applicaion, vol. 328, no. 2, pp , G. C. Yang, New reul of Falkner-Skan equaion ariing in boundary layer heory, Applied Mahemaic and Compuaion, vol. 202, no. 1, pp , K. Q. Lan and G. C. Yang, Poiive oluion of he Falkner-Skan equaion ariing in he boundary layer heory, Canadian Mahemaical Bullein, vol. 51, no. 3, pp , B. Brighi, A. Fruchard, and T. Sari, On he Blaiu problem, Advance in Differenial Equaion, vol. 13, no. 5-6, pp , M. Y. Huaini and W. D. Lakin, Exience and nonuniquene of imilariy oluion of a boundarylayer problem, The Quarerly Journal of Mechanic and Applied Mahemaic, vol. 39, no. 1, pp , 1986.