Interior structural bifurcation of 2D symmetric incompressible flows

Deniz Bozkurt; Ali Deliceoğlu; Taylan Şengül

doi:10.3934/dcdsb.2020032

Article Contents

2020, Volume 25, Issue 7: 2775-2791. Doi: 10.3934/dcdsb.2020032

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Interior structural bifurcation of 2D symmetric incompressible flows

1.
Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
2.
Department of Mathematics, Marmara University, 34722 Istanbul, Turkey

^* Corresponding author: Taylan Şengül
^* Corresponding author: Taylan Şengül

Received: May 2019

Early access: April 2020

Published: July 2020

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Abstract

The structural bifurcation of a 2D divergence free vector field $ \mathbf{u}(\cdot, t) $ when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{x}_0 $ of zero index has been studied by Ma and Wang [23]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when $ \mathbf{u}(\cdot, t_0) $ is anti-symmetric with respect to $ \mathbf{x}_0 $, or symmetric with respect to the axis located on $ \mathbf{x}_0 $ and normal to the unique eigendirection of the Jacobian $ D\mathbf{u}(\cdot, t_0) $, the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when $ \mathbf{u}(\cdot, t_0) $ has an interior isolated singular point $ \mathbf{x}_0 $ with index -1, 1. In particular, we show that if such a vector field with its acceleration at $ t_0 $ both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.

Keywords:

Flow structures,
structural stability,
divergence-free vector field and bifurcation.

Mathematics Subject Classification: Primary: 34D30, 34C23; Secondary: 37N10.

Citation:

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Figure 1. The topological structure of $ {\mathbf u}_0 $ near the origin, under the assumptions: (a) (S2), (b) (S7), $ k $ is even, $ \alpha \lambda < 0 $, (c) (S7), $ k $ is even, $ \alpha \lambda > 0 $, (d) (S7), $ k $ is odd

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Figure 2. Structural bifurcation diagram of the unfolding of codimension-one singularities for the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1 $ case in Theorem 3.2: (a) $ t = t_0+\epsilon $, (b) $ t = t_0 $, (c) $ t = t_0 - \epsilon $

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Figure 3. Structural bifurcation diagram of the unfolding of codimension-one singularities for the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = 1 $ case in Theorem 3.2: (a) $ t = t_0+\epsilon $, (b) $ t = t_0 $, (c) $ t = t_0 - \epsilon $

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Figure 4. Schematic illustration of the unfolding of codimension-one singularities for flows with reflectional symmetry (18) in the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1 $ case. (a) $ t = t_{0}-\epsilon $, (b) $ t = t_{0} $, (c) $ t = t_{0}+\epsilon $

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Figure 5. Schematic illustration of the unfolding of codimension-one singularities for flows with anti-symmetry (19) in the $ \operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1 $ case. (a) $ t = t_{0}-\epsilon $, (b) $ t = t_{0} $, (c) $ t = t_{0}+\epsilon $

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Figure 6. The illustration of the dimensionless boundary value problem

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Figure 7. The streamlines patterns in rectangular cavities of various aspect ratios $ A $ and various lid speed ratios $ S $. (a) $ (A, S) = (0.5, -1) $, (b) $ (A, S) = (1, -1) $, (c) $ (A, S) = (0.3, -1) $, (d) $ (A, S) = (0.16, -1). $

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Table 1. The bifurcated solutions for $ k=2 $, $ n=3 $, $ 2 \lambda \lambda_1 + \alpha \lambda_2 > 0 $

	$ \epsilon<0 $	$ \epsilon>0 $
$ \alpha \beta + 2 \lambda^2> 0 $	$ {\mathbf x}_{0}(\epsilon) $ is a saddle	$ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta + 2 \lambda^2< 0 $	$ {\mathbf x}_{0}(\epsilon) $ is a center	$ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle

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Table 2. The bifurcated solutions for $ k=2 $, $ n>3 $, $ 2 \lambda \lambda_1 + \alpha \lambda_2 > 0 $

$ \epsilon<0 $	$ \epsilon>0 $
$ {\mathbf x}_{0}(\epsilon) $ is a saddle	$ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center

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Table 3. The bifurcated solutions for $ k>2 $, $ 2k<n+1 $, $ \alpha \lambda_2 >0 $

$ \epsilon<0 $	$ \epsilon>0 $
$ {\mathbf x}_{0}(\epsilon) $ is a saddle	$ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center

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Table 4. The bifurcated solutions for $ k>2 $, $ 2k>n+1 $, $ \alpha \lambda_2>0 $

	$ \epsilon<0 $	$ \epsilon>0 $
$ \alpha \beta> 0 $	$ {\mathbf x}_{0}(\epsilon) $ is a saddle	$ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta< 0 $	$ {\mathbf x}_{0}(\epsilon) $ is a center	$ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle

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Table 5. The bifurcated solutions for $ k>2 $, $ 2k=n+1 $, $ \alpha \lambda_2 > 0 $

	$ \epsilon<0 $	$ \epsilon>0 $
$ \alpha \beta + k \lambda^2> 0 $	$ {\mathbf x}_{0}(\epsilon) $ is a saddle	$ {\mathbf x}_{\pm}(\epsilon) $ are saddles, $ {\mathbf x}_{0}(\epsilon) $ is a center
$ \alpha \beta + k \lambda^2< 0 $	$ {\mathbf x}_{0}(\epsilon) $ is a center	$ {\mathbf x}_{\pm}(\epsilon) $ are centers, $ {\mathbf x}_{0}(\epsilon) $ is a saddle

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