On the shear stress function and the critical value of the Blasius problem
© Yang et al.; licensee Springer 2012
Received: 12 August 2012
Accepted: 7 September 2012
Published: 25 September 2012
The Blasius problem has been used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past a flat plate moving at a constant speed β; and it is well known that there exists the critical value such that it has at least one solution for each and has no positive solution for . The known numerical result shows . In this paper, by the study of the integral equation equivalent to the Blasius problem, we obtain the relation between the velocity function and the shear stress functions , upper and lower bounds of and a new lower bound of . In particular, , . Regarding , previous results presented a lower bound −0.5 and an upper bound −0.18733.
has been used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past a flat plate. It also arises in the study of the mixed convection in porous media , where η is the similarity boundary layer ordinate, is the similarity stream function, and are the velocity and the shear stress functions, respectively. The case of corresponds to a flat plate moving at a steady speed opposite to that of a uniform mainstream .
Regarding the analytic study of the Blasius problem (1.1)-(1.2), Weyl  proved that (1.1)-(1.2) has one and only one solution for ; Coppel  studied the case of ; the cases of  and  were also investigated, respectively. Also, see . In 1986, Hussaini and Lakin  indicated that there exists a critical value such that (1.1)-(1.2) has at least a solution for and no solution for . A lower bound was presented with and numerical results showed . In 2008, Brighi, Fruchard and Sari  summarized historical study on the Blasius problem and analyzed the case in detail, in which the shape and the number of solutions were determined. In 2010, Yang  obtained an upper bound . The Blasius problem is a special case of the Falkner-Skan equation, for , we may refer to [12–15] for some recent results on the Falkner-Skan equation.
An open question is: What exactly is ? And what properties does the shear stress function have? To our knowledge, there is little study on it. By the study of the integral equation that is equivalent to the Blasius problem, in this paper, we present the relation between and , upper and lower bounds of and new lower bounds of . In particular, , .
2 Upper and lower bounds of
where and for .
Since , our work is restricted to the case of and begins with the following lemma.
, where , .
From , we know for . This implies , .
- (ii)Integrating (2.3) from β to 0, we have
which implies that the left inequality of (ii) holds.
And then . Hence, (ii) holds. □
Utilizing Theorem 2.1, we can obtain upper and lower bounds of z as follows.
Theorem 2.2 Let z be a solution of (2.4), then for .
Hence, for .
Next, we prove for .
Then , a contradiction.
Combining Theorems 2.1 and 2.2, we obtain
Corollary 2.1 Let z be a solution of (2.4), then . In particular, .
Proof Let , by (2.8), we have on . From , we obtain , and then . Hence, . This, together with , implies . The right hand is from Theorem 2.1(i).
Since for , hence .
From this, we have , i.e., . □
Based on Theorem 2.2, Corollary 2.1, and , we obtain the relation between the velocity function and the shear stress functions , upper and lower bounds of .
. Specially, .
3 New lower bound of
To obtain a better lower bound of , we first prove
Hence, (3.1) holds.
then there exists a unique such that , for and for .
Theorem 3.2 If , the Blasius problem (1.1)-(1.2) has no solution and then .
then , a contradiction. Hence, . □
Remark 3.1 Theorem 3.2 improves the lower bound of from −0.5 in  to −0.45.
The authors wish to thank the anonymous referees for their valuable comments. This research was supported by the National Natural Science Foundation of China (Grant No. 11171046) and Scientific Research Foundation of the Education Department of Sichuan Province, China.
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