Conditions of boundary layer separation for Boussinesq equations
Tóm tắt
In this paper, we analyze structural bifurcation of solutions to 2-D incompressible Boussinesq equations, where no-slip boundary condition for velocity and nonhomogenous Dirichlet boundary condition for temperature are considered. We get two conditions for boundary layer separation by Taylor expansion of the functions in Boussinesq equations and structural bifurcation theory for flows with Dirichlet boundary conditions. Furthermore, the conditions, determined by initial values, the external force and the temperature on the boundary, can predict when and where boundary layer separation of the Boussinesq equations will occur. The basic theory on boundary layer separation in this manuscript comes from the book Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics written by Ma and Wang.
Tài liệu tham khảo
Abidi, H., Hmidi, T.: On the global well-posedness for Boussinesq system. J. Differ. Equ. 233, 199–220 (2007)
Adhikari, D., Cao, C., Shang, H., Wu, J., Xu, X., Ye, Z.: Global regularity results for the 2D Boussinesq equations with partial dissipation. J. Differ. Equ. 260, 1893–1917 (2016)
Chae, D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203, 497–513 (2006)
Chorin, A., Marsden, J.: A Mathematical Introduction to Fluid Mechanics. Springer (1997)
Constantin, P., Doering, C.: Infinite Prandtl number convection. J. Stat. Phys. 94, 159–172 (1999)
Danchin, R., Paicu, M.: Les theoremes de Leray et de Fujita-Kato pour le systeme de Boussinesq partiellement visqueux. Bull. Soc. Math. France 136, 261–309 (2008)
Elgindi, T., Widmayer, K.: Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems. SIAM J. Math. Anal. 47, 4672–4684 (2015)
Gargano, F., Sammartino, M., Sciacca, V.: High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex. Comput. Fluids 52, 73–91 (2011)
Ghil, M., Liu, J., Wang, C., Wang, S.: Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow. Phys. D 197, 149–173 (2004)
Ghil, M., Ma, T., Wang, S.: Structural bifurcation of 2-D incompressible flows with Dirichlet boundary conditions: applications to boundary-layer separation. SIAM J Appl. Math. 65, 1576–1596 (2005)
Gill, A.: Atmosphere-Ocean Dynamics. Academic Press, London (1982)
Hmidi, T., Keraani, S.: On the global well-posedness of the Boussinesq system with zero viscosity. Indiana Univ. Math. J. 58, 1591–1618 (2009)
Hmidi, T., Rousset, F.: Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data. Ann. Inst. H. Poincare Anal. Non Lineaire. 27, 1227–1246 (2010)
Hmidi, T., Rousset, F.: Global well-posedness for the Euler–Boussinesq system with axisymmetric data. J. Funct. Anal. 260, 745–796 (2011)
Hou, T., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12, 1–12 (2005)
Hu, W., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equations in a bounded domain. J. Math. Phys. 54, 081507 (2013)
Hu, W., Wang, Y., Wu, J., Xiao, B., Yuan, J.: Partially dissipative 2D Boussinesq equations with Navier type boundary conditions. Phys. D 376–377, 39–48 (2018)
Kluwick, A., Braun, S.: Unified description of bifurcation processes associated with laminar boundary-layer separation. Proc. Appl. Math. Mech. 15, 479–480 (2015)
Lai, M., Pan, R., Zhao, K.: Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch. Ration. Mech. Anal. 199, 739–760 (2011)
Lamb, K.: On boundary-layer separation and internal wave generation at the Knight Inlet sill. Proc. R. Soc. A Math. Phys. 460, 2305–2337 (2004)
Larin, O., Levin, V.: Boundary layer separation in a laminar supersonic flow with energy supply source. Tech. Phys. Lett. 34, 181–183 (2008)
Liu, Y., Qin, X., Shi, J., Zhi, W.: Structural stability of the Boussinesq fluid interfacing with a Darcy fluid in a bounded region in \(R^{2}\). Appl. Math. Comput. 411, 126488 (2021)
Luo, H., Wang, Q., Ma, T.: A predicable condition for boundary layer separation of 2-D incompressible fluid flows. Nonlinear Anal. Real. 22, 336–341 (2015)
Ma, T., Wang, S.: Rigorous characterization of boundary layer separations. In: Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, Cambridge (2003)
Ma, T., Wang, S.: Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Discrete Contin. Dyn. Syst. 10, 459–472 (2004)
Ma, T., Wang, S.: Geometric theory of incompressible flows with applications to fluid dynamics. AMS Mathematical Surveys and Monographs Series (2005)
Majda, A.: Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes in Mathematics 9, AMS/CIMS (2003)
Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1987)
Shen, W., Wang, Y., Zhang, Z.: Boundary layer separation and local behavior for the steady Prandtl equation. Adv. Math. 389, 107896 (2021)
Simpson, R.: Turbulent boundary-layer separation. Annu. Rev. Fluid Mech. 21, 205–234 (1989)
Smith, F.: Steady and unsteady boundary-layer separation. Annu. Rev. Fluid Mech. 18, 197–220 (1986)
Sun, Y., Zhang, Z.: Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity. J. Differ. Equ. 255, 1069–1085 (2013)
Wang, C., Zhang, Z.: Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. Adv. Math. 228, 43–62 (2011)
Wang, Q., Luo, H., Ma, T.: Boundary layer separation of 2-D incompressible Dirichlet flows. Discrete Contin. Dyn. Syst. B 20, 675–682 (2015)
Wu, J., Zhang, Q.: Stability and optimal decay for a system of 3D anisotropic Boussinesq equations. Nonlinearity 34, 5456–5484 (2021)