Abstract
A system of two integral equations is presented to describe the system of 3D axisymmetric inviscid stagnation flows related to NavierStokes 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.
MSC: 34B18.
Keywords:
NavierStokes equations; 3D flows; similarity solutions; integral systems; existence results1 Introduction
The following system of two differential equations arising in the boundary layer problems in fluid mechanics
with boundary conditions
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 [3] presented a numerical study for the case which can be applied to the stagnation region of an ellipsoid. Davey [2] investigated numerically the stagnation region near a saddle point (). The twodimensional cases, or and , and the special cases of the FalknerSkan equation were solved by Hiemenz [4] and by Homann [5], respectively. Regarding the FalknerSkan problems, further analytical study can be found in [610]. Also, one may refer to recent review of similarity solutions of the NavierStokes equations [11].
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.
Let
and
Lemma 2.1Ifis a solution of (1.1)(1.3), then.
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 .
Differentiating (1.2) with η, we have
then
a contradiction. Hence, is decreasing on and then .
This completes the proof. □
Theorem 2.1Ifis a solution of (1.1)(1.2), then
has a solution, wheredenotes the Green function forwithanddefined by
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., .
Using the chain rule to , we obtain and by the inverse function theorem, we have
Integrating the last equality from 0 to t implies
Let
Then . By , we know that y is continuous from the left at 1 and then .
Differentiating with t, we have
Differentiating with t and utilizing , we have
Hence,
Substituting g, , , and f into (1.2) implies
Integrating (2.5) from t to 1, we have
Substituting f, , , and g into (1.1) implies
Therefore,
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).
Let
It is easy to verify
We define some functions
By computation, , , there exists such that for and .
In order to study the existence of solutions of (2.2)(2.3) in Q for , we denote the norm of the Banach space by
Let and be a natural number, we define
Notation
and
Let , we define an operator F as follows:
where
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.
Then the following assertions hold:
(ii) and, whereis a total variation ofyon.
(iii) If, thenis increasing onand thenfor.
Proof We shall use the basic fact: let and () be local minimum (maximum), then (≤0).
(i) If there exists such that , by , we know that there exists such that . Differentiating (3.3) with t twice, we have
If there exists such that , let , by and , we may assume such that . This implies , i.e., , and . By (3.4) and , we know
then
a contradiction. Hence, (i) holds.
(ii) Let such that and . If , we prove that is increasing on and decreasing on .
Since and , then . Let . If there exist with such that , let such that , then by (i). From , and (3.4), we know
a contradiction.
If there exist with such that , let such that , then by (i). Analogously, we know easily
a contradiction. Hence,
(iii) Let . By (i) and , we know and then is increasing on and then for . Hence, (iii) holds. □
Lemma 3.3[12]
LetEbe a Banach space, Dbe a bounded open set of E and, is compact. Iffor anyand, thenFhas a fixed point in.
Lemma 3.4Let, thenFhas a fixed pointin, i.e., there existssuch that
hold.
Proof Let
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 .
Since and for , this, together with for and , implies
By (3.3), we have
Noticing that and for , we obtain for . This, together with (3.7) and Lemma 3.2(ii), implies
a contradiction.
By Lemmas 3.1 and 3.3, F has a fixed point in . □
Lemma 3.5Letbe in Lemma 3.4, then
Proof By Lemma 3.3(i), we know . By (3.5), we have
(i) For , we know for , i.e., is decreasing in , by , for . By for and (3.5), we have
And then for . Obviously, for . This, together with the decrease in , implies
It is easy to verify for . And then
The last two inequalities imply that is bounded on .
(ii) By (3.8),
we know that is bounded on for any . □
Lemma 3.6Letbe in Lemma 3.4, then
(ii) is bounded and equicontinuous infor any.
Proof
(i) Lemma 3.2(i) and (iii) imply the desired results.
(ii) For , let such that . Since on , by Lemma 3.2(ii), , we obtain .
Differentiating (3.6) with t twice, we have . Integrating this equality from 0 to , we have
Noticing that and for and Lemma 3.2(ii), we know
i.e., is bounded on . Let (where ), we know
This implies that is equicontinuous on . □
Theorem 3.1There existssuch that
hold, where
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 ArzelaAscoli 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 ArzelaAscoli 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 .
Since
converges to and converges to for , by the Lebesgue dominated theorem (the dominated function , , we have that satisfies (3.11) and .
Fix and choose such that , then
Noticing that and converges to for , by the Lebesgue dominated theorem (the dominated function on ), we have
Differentiating the last equality twice, we know
By (i), we know and and then . This, together with (2.4), implies that satisfies (3.12). Clearly, . □
Theorem 3.2For, the system (2.2)(2.3) has at least a solutioninQ.
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, .
By (3.11), we have
we know
Integrating the last inequality from to 1 and utilizing , we have
Hence, (3.13) holds.
i.e.,
Hence,
This, together with (3.13), implies
i.e.,
In fact, if there exists such that , by , there exists such that for and .
From
By (3.11), we have
i.e., , . Integrating this inequality from to δ, we have
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 4.1For, the system (1.1)(1.3) has at least a solution.
Proof Let , by Theorem 3.2, the system (2.2)(2.3) has at least a solution in Q. By and (2.2), we know
Let
Then is strictly increasing on and
Let be the inverse function to , we define the function
Then
and
From (4.1), we have
Differentiating (4.2) with respect to η, we have
Differentiating (4.3) with respect to η, we have
Differentiating (2.2) with respect to t, we have
By setting and utilizing and (4.3), we have
By (4.3), (4.4), (4.5) and (4.6), we have
By (4.1), we have . Differentiating with respect to η, we have
Differentiating (2.3) with t twice and combining (4.5) and (4.6), we obtain
This completes the proof. □
Remark 4.1 For , by Theorem 1 [2], (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 .
In fact, if (1.1)(1.3) has a solution for some , by Theorem 2.1, then (1.1)(1.3) has a solution in . Noticing that
we know
a contradiction.
This research uses integrals of equations to investigate the existence of solutions of the 3D axisymmetric inviscid stagnation flows related to NavierStokes equations and supplies a gap of analytical study in this field.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
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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