| Citation: |
Ali Deliceoğlu, Deniz Bozkurt. STRUCTURAL BIFURCATION OF DIVERGENCE-FREE VECTOR FIELDS NEAR NON-SIMPLE DEGENERATE POINTS WITH SYMMETRY[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 718-738. doi: 10.11948/2156-907X.20180151
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STRUCTURAL BIFURCATION OF DIVERGENCE-FREE VECTOR FIELDS NEAR NON-SIMPLE DEGENERATE POINTS WITH SYMMETRY
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Abstract
In this study, topological features of an incompressible two-dimensional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergence-free vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the x-axis. Moreover, we also show that bifurcation of the flow structure near a non-simple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and -2 in a double lid driven cavity and in two dimensional peristaltic flow. -
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References
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Ali Deliceoğlu, Deniz Bozkurt. STRUCTURAL BIFURCATION OF DIVERGENCE-FREE VECTOR FIELDS NEAR NON-SIMPLE DEGENERATE POINTS WITH SYMMETRY[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 718-738. doi: 10.11948/2156-907X.20180151
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Figure 1. Index of
$\bf u^0$ given by (3.5) about a point$x_{0}$ with symmetry about an axis. (a) A degenerate saddle point with$ind(u^0, x_0)=-2$ for the case where$n=2$ and$\alpha \beta < 0$ ; (b) A degenerate point on the x-axis with$ind(u^0, x_0)=0$ for the case where$n$ is even and$n\geq 2$ ,$\alpha \beta >0$ ; (c) A degenerate saddle point with$ind(u^0, x_0)=-2$ for the case where$n$ is even and$n\geq 4$ ,$\alpha \beta < 0$ ; (d-e) A degenerate cusp point on the x-axis with$ind(u^0, x_0)=-1$ for the case where$n$ is odd and$n\geq 3$ . -
Figure 2. Index of non-simple degenerate singular points of
$\bf u^0$ given by (3.15) with double symmetry. (a) A degenerate saddle point with$ind(u^0, x_0)=-2$ for the case$k=1$ and$\alpha \beta < 0$ ; (b) A degenerate point on the x-axis with$ind(u^0, x_0)=0$ for the case$\alpha \beta >0$ ; (c) A degenerate saddle point with$ind(u^0, x_0)=-2$ for the case$k>1$ and$\alpha \beta < 0$ . -
Figure 3. Index of non-simple degenerate singular points of
$\bf u^0$ given by (3.22) with double symmetry. (a) A degenerate saddle point with$ind(u^0, x_0)=-2$ for the case$n=1$ and$\alpha \beta < 0$ ; (b) A degenerate point on the x-axis with$ind(u^0, x_0)=0$ for the case$\alpha \beta >0$ ; (c) A degenerate saddle point with$ind(u^0, x_0)=-2$ for the case$n>1$ and$\alpha \beta < 0$ . - Figure 4. A description of the unfolding of codimension-one singularities for the case index 0 with symmetry about an axis. A similar diagram was also found by Gürcan and Deliceoğlu [15].
- Figure 5. A description of the unfolding of codimension-one singularities for the case index -2 with symmetry about an axis. A similar diagram was also found by Gürcan and Deliceoğlu [15].
- Figure 6. Schematic illustration of the unfolding of codimension-one singularities for Theorem 4.3.
- Figure 7. Schematic illustration of the unfolding of codimension-one singularities for the case index -1 with symmetry about an axis.
- Figure 8. The boundary value problem for the double lid-driven cavity with two solid walls.
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Figure 9. Flow structure development found by Gürcan and Deliceoğlu ([15]) in the double lid driven cavity as
$A$ increases from$A=2.95$ to$3.45$ for$S=1$ ; (a)$A=2.95$ , (b)$A=3.225$ , (c)$A=3.45$ . -
Figure 10. Streamline patters of peristaltic flow in moving frame for different values of the flow rate
$q$ with fixed amplitude ratio$\phi=0.6$ , reproduced from Ref. [25]. -
Figure 11. Illustration of flow structure in a sectorial cavity with lids moving in the same direction. (a)
$(A, S)=(1.65, 1)$ , (b)$(A, S)=(1.641, 1)$ , (c)$(A, S)=(1.62, 1)$ . -
Figure 12. Instantaneous streamlines for
$D=0.01$ ,$Ha=100$ and$Re_\omega=1$ : (a)$\phi = -927 \pi /1000 $ and (b)$\phi = -752\ pi/1000 $ , reproduced from Ref. [4]