# American Institute of Mathematical Sciences

March  2015, 20(2): 675-682. doi: 10.3934/dcdsb.2015.20.675

## Boundary layer separation of 2-D incompressible Dirichlet flows

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 College of Mathematics and Software Science, Sichuan Normal University, Chengdu,Sichuan 610066, China, China

Received  March 2014 Revised  June 2014 Published  January 2015

In this paper, the solutions of Navier-Stokes equations governing 2-D incompressible flows with the Dirichlet boundary condition are analyzed. We derive a condition for boundary layer separation, and the condition is determined by initial values and external forces. More importantly, the condition can predict when and where the boundary layer separation occurs directly. In addition, we also get an algebraic equation for the separation point and the separation time. The algebraic equation can tell us where the boundary layer separation does not occur in a short period of time. The main technical tool is the geometric theory of incompressible flows developed by T. Ma and S. Wang in [15].
Citation: Quan Wang, Hong Luo, Tian Ma. Boundary layer separation of 2-D incompressible Dirichlet flows. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 675-682. doi: 10.3934/dcdsb.2015.20.675
##### References:
 [1] P. Blanchonette and M. A. Page, Boundary-Layer separation in the two-layer flow past a cylinder in a rotating frame, Theoret. Comput. Fluid Dynamics, 11 (1998), 95-108. [2] Chorin, J. Alexandre and J. E. Marsden, A mathematical Introduction to Fluid Mechanics, Third edition. Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0883-9. [3] W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation, Commun. Pure Appl. Math, 50 (1997), 1287-1293. doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4. [4] 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. [5] M. Ghil, J. Liu, C. Wang and S. Wang, Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow, Physica D, 197 (2004), 149-173. doi: 10.1016/j.physd.2004.06.012. [6] M. Ghil and T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: Applications to boundary-layer separation, SIAM J. Applied Math, 65 (2005), 1576-1596. doi: 10.1137/S0036139903438818. [7] H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Analysis: Real World Applications, 22 (2015), 336-341. doi: 10.1016/j.nonrwa.2014.09.007. [8] O. B. Larin and V. A. Levin, Boundary layer separation in a laminar supersonic flow with energy supply source, Technical Physics Letters, 34 (2008), 181-183. doi: 10.1134/S1063785008030012. [9] O. B. Larin and V. A. Levin, Effect of energy supply to a gas on laminar boundary layer separation, Journal of Applied Mechanics and Technical Physics, 51 (2010), 11-15. doi: 10.1007/s10808-010-0003-4. [10] J. Liu and Z. Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rat. Mech. Anal, 135 (1996), 61-105. doi: 10.1007/BF02198435. [11] T. Ma and S. Wang, Rigorous characterization of boundary layer separations, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, (2003), 1008-1010. doi: 10.1016/B978-008044046-0/50246-3. [12] T. Ma and S. Wang, Interior structural bifurcation and separation of $2-D$ incompressible flows, J. Math. Phy, 45 (2004), 1762-1776. doi: 10.1063/1.1689005. [13] T. Ma and S. Wang, Asymptotic structure for solutions of the Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 11 (2004), 189-204. doi: 10.3934/dcds.2004.11.189. [14] T. Ma and S. Wang, Boundary Layer Separation and Structural Bifurcation for 2-D Incompressible Fluid Flows, Discrete and Continuous Dynamical Systems, 10 (2004), 459-472. doi: 10.3934/dcds.2004.10.459. [15] 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. x+234 pp. doi: 10.1090/surv/119. [16] T. Ma and S. Wang, Boundary layer and interior separations in the Taylor-Couette-Poiseuille flow, Journal of Mathematical Physics, 50 (2009), 1-29. doi: 10.1063/1.3093268. [17] O. A. Oleinik, On the mathematical theory of boundary layer for unsteady flow of incompressible fluid, J. Appl. Math. Mech, 30 (1966), 951-974. doi: 10.1016/0021-8928(66)90001-3. [18] O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, Boca Raton, FL, 1999. [19] L. Prandtl, On the motion of fluids with very little friction, in Verhandlungen des dritten internationalen Mathematiker-Konggresses, Heidelberg, 1904, Leipeizig, (1905), 484-491. [20] H. Schlichting, Boundary Layer Theory, eighth ed., Springer, Berlin-Heidelberg, 2000. doi: 10.1007/978-3-642-85829-1. [21] F.T. Smith and S.N. Brown, Boundary-Layer Separation, Proceedings of the IUTAM Symposium London, 1986, Springer -Verlag, 1987. doi: 10.1007/978-3-642-83000-6. [22] Quan. Wang, Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders, Discrete and Continuous Dynamical Systems-B, 19 (2014), 543-563. doi: 10.3934/dcdsb.2014.19.543.

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##### References:
 [1] P. Blanchonette and M. A. Page, Boundary-Layer separation in the two-layer flow past a cylinder in a rotating frame, Theoret. Comput. Fluid Dynamics, 11 (1998), 95-108. [2] Chorin, J. Alexandre and J. E. Marsden, A mathematical Introduction to Fluid Mechanics, Third edition. Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0883-9. [3] W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation, Commun. Pure Appl. Math, 50 (1997), 1287-1293. doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4. [4] 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. [5] M. Ghil, J. Liu, C. Wang and S. Wang, Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow, Physica D, 197 (2004), 149-173. doi: 10.1016/j.physd.2004.06.012. [6] M. Ghil and T. Ma and S. Wang, Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: Applications to boundary-layer separation, SIAM J. Applied Math, 65 (2005), 1576-1596. doi: 10.1137/S0036139903438818. [7] H. Luo, Q. Wang and T. Ma, A predicable condition for boundary layer separation of 2-D incompressible fluid flows, Nonlinear Analysis: Real World Applications, 22 (2015), 336-341. doi: 10.1016/j.nonrwa.2014.09.007. [8] O. B. Larin and V. A. Levin, Boundary layer separation in a laminar supersonic flow with energy supply source, Technical Physics Letters, 34 (2008), 181-183. doi: 10.1134/S1063785008030012. [9] O. B. Larin and V. A. Levin, Effect of energy supply to a gas on laminar boundary layer separation, Journal of Applied Mechanics and Technical Physics, 51 (2010), 11-15. doi: 10.1007/s10808-010-0003-4. [10] J. Liu and Z. Xin, Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rat. Mech. Anal, 135 (1996), 61-105. doi: 10.1007/BF02198435. [11] T. Ma and S. Wang, Rigorous characterization of boundary layer separations, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, (2003), 1008-1010. doi: 10.1016/B978-008044046-0/50246-3. [12] T. Ma and S. Wang, Interior structural bifurcation and separation of $2-D$ incompressible flows, J. Math. Phy, 45 (2004), 1762-1776. doi: 10.1063/1.1689005. [13] T. Ma and S. Wang, Asymptotic structure for solutions of the Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 11 (2004), 189-204. doi: 10.3934/dcds.2004.11.189. [14] T. Ma and S. Wang, Boundary Layer Separation and Structural Bifurcation for 2-D Incompressible Fluid Flows, Discrete and Continuous Dynamical Systems, 10 (2004), 459-472. doi: 10.3934/dcds.2004.10.459. [15] 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. x+234 pp. doi: 10.1090/surv/119. [16] T. Ma and S. Wang, Boundary layer and interior separations in the Taylor-Couette-Poiseuille flow, Journal of Mathematical Physics, 50 (2009), 1-29. doi: 10.1063/1.3093268. [17] O. A. Oleinik, On the mathematical theory of boundary layer for unsteady flow of incompressible fluid, J. Appl. Math. Mech, 30 (1966), 951-974. doi: 10.1016/0021-8928(66)90001-3. [18] O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, Boca Raton, FL, 1999. [19] L. Prandtl, On the motion of fluids with very little friction, in Verhandlungen des dritten internationalen Mathematiker-Konggresses, Heidelberg, 1904, Leipeizig, (1905), 484-491. [20] H. Schlichting, Boundary Layer Theory, eighth ed., Springer, Berlin-Heidelberg, 2000. doi: 10.1007/978-3-642-85829-1. [21] F.T. Smith and S.N. Brown, Boundary-Layer Separation, Proceedings of the IUTAM Symposium London, 1986, Springer -Verlag, 1987. doi: 10.1007/978-3-642-83000-6. [22] Quan. Wang, Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders, Discrete and Continuous Dynamical Systems-B, 19 (2014), 543-563. doi: 10.3934/dcdsb.2014.19.543.
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