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July  2020, 25(7): 2775-2791. doi: 10.3934/dcdsb.2020032

## 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

Received  May 2019 Published  July 2020 Early access  April 2020

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.

Citation: Deniz Bozkurt, Ali Deliceoğlu, Taylan Şengül. Interior structural bifurcation of 2D symmetric incompressible flows. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2775-2791. doi: 10.3934/dcdsb.2020032
##### References:
 [1] A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Ont., 1973. [2] P. Bakker, Bifurcations in Flow Patterns., Nonlinear Topics in the Mathematical Sciences, 2, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3512-2. [3] A. Bisgaard, M. Brøns and J. N. Sørensen, Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mechanica, 187 (2006), 75-83.  doi: 10.1007/s00707-006-0367-y. [4] M. Brøns and A. V. Bisgaard, Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers, J. Fluid Mech., 568 (2006), 329-349.  doi: 10.1017/S0022112006002424. [5] M. Brøns and J. N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11 (1999), 314-324.  doi: 10.1063/1.869881. [6] M. Brøns, B. Jakobsen, K. Niss, A. V. Bisgaard and L. K. Voigt, Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers, J. Fluid Mech., 584 (2007), 23-43.  doi: 10.1017/S0022112007006234. [7] M. Brøns, L. K. Voigt and J. N. Sørensen, Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech., 428 (2001), 133-148.  doi: 10.1017/S0022112000002512. [8] C. H. Chan, M. Czubak and T. Yoneda, An ODE for boundary layer separation on a sphere and a hyperbolic space, Phys. D, 282 (2014), 34-38.  doi: 10.1016/j.physd.2014.05.004. [9] M. Dam, J. J. Rasmussen, V. Naulin and M. Brøns, Topological bifurcations in the evolution of coherent structures in a convection model, Phys. Plasmas, 24 (2017). doi: 10.1063/1.4993613. [10] P. Gaskell, M. Savage and M. Wilson, Stokes flow in a half-filled annulus between rotating coaxial cylinders, J. Fluid Mech., 337 (1997), 263-282.  doi: 10.1017/S0022112097005028. [11] M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J., 50 (2001), 159-180.  doi: 10.1512/iumj.2001.50.2183. [12] F. Gürcan and A. Deliceoğlu, Streamline topologies near nonsimple degenerate points in two-dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005), 7pp. doi: 10.1063/1.2055527. [13] F. Gürcan and A. Deliceoğlu, Saddle connections near degenerate critical points in Stokes flow within cavities, Appl. Math. Comput., 172 (2006), 1133-1144.  doi: 10.1016/j.amc.2005.03.012. [14] F. Gürcan, A. Deliceoğlu and P. Bakker, Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms, J. Fluid Mech., 539 (2005), 299-311.  doi: 10.1017/S0022112005005689. [15] F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997. [16] J. N. Hartnack, Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75.  doi: 10.1007/BF01292298. [17] M. Heil, J. Rosso, A. L. Hazel and M. Brøns, Topological fluid mechanics of the formation of the Kármán-vortex street, J. Fluid Mech., 812 (2017), 199-221.  doi: 10.1017/jfm.2016.792. [18] C.-H. Hsia, J.-G. Liu and C. Wang et al., Structural stability and bifurcation for 2-D incompressible flows with symmetry, Methods Appl. Anal., 15 (2008), 495-512.  doi: 10.4310/MAA.2008.v15.n4.a6. [19] D. D. Joseph and L. Sturges, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part Ⅱ, SIAM J. Appl. Math., 34 (1978), 7-26.  doi: 10.1137/0134002. [20] R. Liu, T. Ma, S. Wang and J. Yang, Topological phase transition Ⅴ: Interior separations and cyclone formation theory, preprint. Available from: https://hal.archives-ouvertes.fr/hal-01673496. [21] H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Anal. Real World Appl., 22 (2015), 336-341.  doi: 10.1016/j.nonrwa.2014.09.007. [22] T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126.  doi: 10.1016/S0167-2789(02)00587-0. [23] T. Ma and S. Wang, Interior structural bifurcation and separation of 2D incompressible flows, J. Math. Phys., 45 (2004), 1762-1776.  doi: 10.1063/1.1689005. [24] T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/119. [25] T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759 [26] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180.  doi: 10.1016/0040-9383(65)90018-2. [27] A. Perry, M. Chong and T. Lim, The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech., 116 (1982), 77-90.  doi: 10.1017/S0022112082000378. [28] R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237. [29] L. Tophøj, S. Møller and M. Brøns, Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions, Phys. Fluids, 18 (2006), 8pp. doi: 10.1063/1.2337660. [30] Q. Wang, H. Luo and T. Ma, Boundary layer separation of 2-D incompressible Dirichlet flows, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) 675–682. doi: 10.3934/dcdsb.2015.20.675. [31] J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152.

show all references

##### References:
 [1] A. A. Andronov, Qualitative Theory of Second-Order Dynamic Systems, Halsted Press, New York-Toronto, Ont., 1973. [2] P. Bakker, Bifurcations in Flow Patterns., Nonlinear Topics in the Mathematical Sciences, 2, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3512-2. [3] A. Bisgaard, M. Brøns and J. N. Sørensen, Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mechanica, 187 (2006), 75-83.  doi: 10.1007/s00707-006-0367-y. [4] M. Brøns and A. V. Bisgaard, Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers, J. Fluid Mech., 568 (2006), 329-349.  doi: 10.1017/S0022112006002424. [5] M. Brøns and J. N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11 (1999), 314-324.  doi: 10.1063/1.869881. [6] M. Brøns, B. Jakobsen, K. Niss, A. V. Bisgaard and L. K. Voigt, Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers, J. Fluid Mech., 584 (2007), 23-43.  doi: 10.1017/S0022112007006234. [7] M. Brøns, L. K. Voigt and J. N. Sørensen, Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech., 428 (2001), 133-148.  doi: 10.1017/S0022112000002512. [8] C. H. Chan, M. Czubak and T. Yoneda, An ODE for boundary layer separation on a sphere and a hyperbolic space, Phys. D, 282 (2014), 34-38.  doi: 10.1016/j.physd.2014.05.004. [9] M. Dam, J. J. Rasmussen, V. Naulin and M. Brøns, Topological bifurcations in the evolution of coherent structures in a convection model, Phys. Plasmas, 24 (2017). doi: 10.1063/1.4993613. [10] P. Gaskell, M. Savage and M. Wilson, Stokes flow in a half-filled annulus between rotating coaxial cylinders, J. Fluid Mech., 337 (1997), 263-282.  doi: 10.1017/S0022112097005028. [11] M. Ghil, T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows, Indiana Univ. Math. J., 50 (2001), 159-180.  doi: 10.1512/iumj.2001.50.2183. [12] F. Gürcan and A. Deliceoğlu, Streamline topologies near nonsimple degenerate points in two-dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005), 7pp. doi: 10.1063/1.2055527. [13] F. Gürcan and A. Deliceoğlu, Saddle connections near degenerate critical points in Stokes flow within cavities, Appl. Math. Comput., 172 (2006), 1133-1144.  doi: 10.1016/j.amc.2005.03.012. [14] F. Gürcan, A. Deliceoğlu and P. Bakker, Streamline topologies near a non-simple degenerate critical point close to a stationary wall using normal forms, J. Fluid Mech., 539 (2005), 299-311.  doi: 10.1017/S0022112005005689. [15] F. Gürcan, Flow Bifurcations in Rectangular, Lid-Driven, Cavity Flows, Ph.D thesis, University of Leeds, 1997. [16] J. N. Hartnack, Streamline topologies near a fixed wall using normal forms, Acta Mech., 136 (1999), 55-75.  doi: 10.1007/BF01292298. [17] M. Heil, J. Rosso, A. L. Hazel and M. Brøns, Topological fluid mechanics of the formation of the Kármán-vortex street, J. Fluid Mech., 812 (2017), 199-221.  doi: 10.1017/jfm.2016.792. [18] C.-H. Hsia, J.-G. Liu and C. Wang et al., Structural stability and bifurcation for 2-D incompressible flows with symmetry, Methods Appl. Anal., 15 (2008), 495-512.  doi: 10.4310/MAA.2008.v15.n4.a6. [19] D. D. Joseph and L. Sturges, The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part Ⅱ, SIAM J. Appl. Math., 34 (1978), 7-26.  doi: 10.1137/0134002. [20] R. Liu, T. Ma, S. Wang and J. Yang, Topological phase transition Ⅴ: Interior separations and cyclone formation theory, preprint. Available from: https://hal.archives-ouvertes.fr/hal-01673496. [21] H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Anal. Real World Appl., 22 (2015), 336-341.  doi: 10.1016/j.nonrwa.2014.09.007. [22] T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126.  doi: 10.1016/S0167-2789(02)00587-0. [23] T. Ma and S. Wang, Interior structural bifurcation and separation of 2D incompressible flows, J. Math. Phys., 45 (2004), 1762-1776.  doi: 10.1063/1.1689005. [24] T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, Mathematical Surveys and Monographs, 119, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/119. [25] T. Ma and S. Wang, Topological phase transitions Ⅳ: Dynamic theory of boundary-layer separations. preprint, https://hal.archives-ouvertes.fr/hal-01672759 [26] M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 2 (1963), 179-180.  doi: 10.1016/0040-9383(65)90018-2. [27] A. Perry, M. Chong and T. Lim, The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech., 116 (1982), 77-90.  doi: 10.1017/S0022112082000378. [28] R. C. T. Smith, The bending of a semi-infinite strip, Australian J. Sci. Res. Ser. A, 5 (1952), 227-237. [29] L. Tophøj, S. Møller and M. Brøns, Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions, Phys. Fluids, 18 (2006), 8pp. doi: 10.1063/1.2337660. [30] Q. Wang, H. Luo and T. Ma, Boundary layer separation of 2-D incompressible Dirichlet flows, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) 675–682. doi: 10.3934/dcdsb.2015.20.675. [31] J. Yang and R. Liu, A new fluid dynamical model coupling heat with application to interior separations, preprint, arXiv: 1606.07152.
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
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) $t = t_0 - \epsilon$
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) $t = t_0 - \epsilon$
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) $t = t_{0}+\epsilon$
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) $t = t_{0}+\epsilon$
The illustration of the dimensionless boundary value problem
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) $(A, S) = (0.16, -1).$
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
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta + 2 \lambda^2> 0$ ${\mathbf x}_{0}(\epsilon)$ 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
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
 $\epsilon<0$ $\epsilon>0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center
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
 $\epsilon<0$ $\epsilon>0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center
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
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta> 0$ ${\mathbf x}_{0}(\epsilon)$ 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
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
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta + k \lambda^2> 0$ ${\mathbf x}_{0}(\epsilon)$ 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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