Existence of solutions of a system of 3D axisymmetric inviscid stagnation flows
© Yang et al.; licensee Springer 2012
Received: 7 February 2012
Accepted: 6 December 2012
Published: 28 December 2012
A system of two integral equations is presented to describe the system of 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and existence of its solutions is studied. Utilizing it, we construct analytically the similarity solutions of the 3D system. A nonexistence result is obtained. Previous study was only supported by numerical results.
KeywordsNavier-Stokes equations 3D flows similarity solutions integral systems existence results
has been used to describe the system of 3D axisymmetric inviscid stagnation flow [1, 2], which consists of three partial differential equations [2, 3], where λ is a parameter related to the external flow components.
A solution of (1.1)-(1.3) is called a similarity solution and can be used to express the solutions of the 3D system. Regarding the study of (1.1)-(1.3), Howarth  presented a numerical study for the case which can be applied to the stagnation region of an ellipsoid. Davey  investigated numerically the stagnation region near a saddle point (). The two-dimensional cases, or and , and the special cases of the Falkner-Skan equation were solved by Hiemenz  and by Homann , respectively. Regarding the Falkner-Skan problems, further analytical study can be found in [6–10]. Also, one may refer to recent review of similarity solutions of the Navier-Stokes equations .
However, up to now, there has been very little analytical study on the existence of solutions of (1.1)-(1.3).
The main aim of this paper is to study the existence of solutions of (1.1)-(1.3) analytically for the case of . The method is to present a system of two integral equations and study the existence of its solutions and then use it to construct the solutions of (1.1)-(1.3). Also, a nonexistence result is obtained.
2 A system of two integral equations related to (1.1)-(1.3)
In this section, we present a system of two integral equations to describe a system of (1.1)-(1.3) under suitable conditions, which will be utilized in Section 4.
Lemma 2.1 Ifis a solution of (1.1)-(1.3), then.
Notice that , , , and for .
If , we know and then is decreasing on , which implies that exists. Hence, by (2.1).
If , we have by (1.2). By (2.1), there exists such that and then there exists such that . Obviously, by . We prove that is decreasing on .
In fact, if there exist with such that . Let such that , then and .
a contradiction. Hence, is decreasing on and then .
This completes the proof. □
Proof Assume that . Let for be the inverse function to . It follows that is strictly increasing on and with , . Let for , by Lemma 2.1, . This implies that for and x is continuous on . By Lemma 2.1, we see that x is continuous from the left at 1. Hence, we have and , i.e., .
Then . By , we know that y is continuous from the left at 1 and then .
Notice that , , we have .
From this, we have for and .
where is defined by (2.4). Hence, is a solution of (2.2)-(2.3) in Q. □
3 Positive solutions of the system (2.2)-(2.3)
In this section, we will use the fixed point theorem to study the existence of positive solutions of the system (2.2)-(2.3).
By computation, , , there exists such that for and .
where is defined by (2.4).
It is easy to verify that , θ are continuous operators from into and , , we know the following proposition holds:
Lemma 3.1is a continuous and compact operator fromto.
and , where is a total variation of y on .
If , then is increasing on and then for .
- (i)If there exists such that , by , we know that there exists such that . Differentiating (3.3) with t twice, we have(3.4)
a contradiction. Hence, for .
Let such that and . If , we prove that is increasing on and decreasing on .
Let . By (i) and , we know and then is increasing on and then for . Hence, (iii) holds. □
Let E be a Banach space, D be a bounded open set of E and, is compact. Iffor anyand, then F has a fixed point in.
where . We prove for and with .
In fact, if there exist and μ with and such that , by Lemma 3.2(i) and (iii), we have .
And then , i.e., .
By Lemmas 3.1 and 3.3, F has a fixed point in . □
is bounded on .
is bounded on for any .
- (i)For , we know for , i.e., is decreasing in , by , for . By for and (3.5), we have(3.9)
where defined in (3.1).
- (ii)By (3.8),
we know that is bounded on for any . □
and is increasing in .
is bounded and equicontinuous in for any .
Lemma 3.2(i) and (iii) imply the desired results.
For , let such that . Since on , by Lemma 3.2(ii), , we obtain .
This implies that is equicontinuous on . □
Proof Let be in Lemma 3.4, by Lemma 3.5(ii) and (iii), we know that is bounded and equicontinuous on for any . Letting (), utilizing the diagonal principle and the Arzela-Ascoli theorem, we know that there exists a subsequence of and such that converges to for . Without loss of generality, we assume that is itself of .
By Lemma 3.6, we know that is bounded and equicontinuous on for any and then is bounded and equicontinuous on . Let (), the diagonal principle and the Arzela-Ascoli theorem imply that there exist y and in and two subsequences and with such that converges to for with and converges to for each . For the sake of convenience, we assume that and are itself of . By , we obtain and then for .
converges to and converges to for , by the Lebesgue dominated theorem (the dominated function , , we have that satisfies (3.11) and .
By (i), we know and and then . This, together with (2.4), implies that satisfies (3.12). Clearly, . □
Theorem 3.2 For, the system (2.2)-(2.3) has at least a solutionin Q.
Proof Let in Theorem 3.1. It is clear that we only prove . If , by (3.10), we obtain for and then . Next, we prove for for .
From this and , we obtain and .
By and , there exists such that . Since for , i.e., is concave down on , then for and for . Hence, .
Hence, (3.13) holds.
And then for .
Finally, we prove for .
In fact, if there exists such that , by , there exists such that for and .
and then , a contradiction.
This completes the proof. □
4 Existence of solutions of (1.1)-(1.3)
In this section, we use positive solutions obtained in Theorem 3.2 to construct the solutions of (1.1)-(1.3) in Γ.
Theorem 4.1 For, the system (1.1)-(1.3) has at least a solution.
we have .
Then for .
This completes the proof. □
Remark 4.1 For , by Theorem 1 , (1.1)-(1.3) has no solution such that with for , is a constant.
Utilizing the system (2.2)-(2.3), we know easily that (1.1)-(1.3) has no solution in Γ for .
This research uses integrals of equations to investigate the existence of solutions of the 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and supplies a gap of analytical study in this field.
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 the Scientific Research Foundation of the Education Department of Sichuan Province, China.
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