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Interior structural bifurcation of 2D symmetric incompressible flows

  • * Corresponding author: Taylan Şengül

    * Corresponding author: Taylan Şengül
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  • 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.

    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

    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 $

    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 $

    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 $

    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 $

    Figure 6.  The illustration of the dimensionless boundary value problem

    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). $

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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