二维不可压缩 Navier-Stokes-Allen-Cahn 系统的边界层分离

数学物理学报 ›› 2023, Vol. 43 ›› Issue (4): 1123-1132.

二维不可压缩 Navier-Stokes-Allen-Cahn 系统的边界层分离

陈敏(),胡碧艳(),罗宏^*()

四川师范大学数学科学学院成都 610068

收稿日期:2022-07-17 修回日期:2023-02-11 出版日期:2023-08-26 发布日期:2023-07-03
通讯作者: 罗宏 E-mail:1653637845@qq.com;2838954298@qq.com;lhscnu@163.com
作者简介:陈敏，E-mail: 1653637845@qq.com;|胡碧艳，E-mail: 2838954298@qq.com
基金资助:
国家自然科学基金(12171343);四川省科研厅科研基金(22CXTD0029)

Boundary Layer Separation of 2-D Incompressible Navier-Stokes-Allen-Cahn System

Chen Min(),Hu Biyan(),Luo Hong^*()

School of Mathematics Science, Sichuan Normal University, Chengdu 610068

Received:2022-07-17 Revised:2023-02-11 Online:2023-08-26 Published:2023-07-03
Contact: Hong Luo E-mail:1653637845@qq.com;2838954298@qq.com;lhscnu@163.com
Supported by:
NSFC(12171343);Scientific Research Fund of the Science and Technology Department of Sichuan Province(22CXTD0029)

摘要/Abstract

摘要：

该文研究了二维不可压缩 Navier-Stokes-Allen-Cahn 系统的边界层分离. 首先利用不可压缩流的几何理论以及泰勒展开式, 得到平直边界下边界层分离的条件; 其次求出边界奇点, 得到弯曲边界下边界层分离的条件. 这些条件由初值和外力决定, 可以预测 Navier-Stokes-Allen-Cahn 系统发生边界层分离的时间和地点.

关键词: 边界层分离, Navier-Stokes-Allen-Cahn 系统, 分歧, 二维不可压缩流

Abstract:

In this paper, boundary layer separation of 2-D incompressible Navier-Stokes-Allen-Cahn system is considered. Firstly, the condition of boundary layer separation under flat boundary is obtained with the help of the geometric theory of incompressible flow and Taylor expansion. Secondly, the expression for boundary singularity is presented and the condition of boundary layer separation under curved boundary is discovered. The conditions, determined by initial values and external forces, can predict when and where boundary layer separation for Navier-Stokes-Allen-Cahn system will occur.

Key words: Boundary layer separation, Navier-Stokes-Allen-Cahn system, Bifurcation, 2-D incompressible flow

中图分类号:

O175.29

陈敏,胡碧艳,罗宏. 二维不可压缩 Navier-Stokes-Allen-Cahn 系统的边界层分离[J]. 数学物理学报, 2023, 43(4): 1123-1132.

Chen Min,Hu Biyan,Luo Hong. Boundary Layer Separation of 2-D Incompressible Navier-Stokes-Allen-Cahn System[J]. Acta mathematica scientia,Series A, 2023, 43(4): 1123-1132.

图/表 1

参考文献 24

[1]	Xu X, Zhao L Y, Liu C. Axisymmetric solution to coupled Navier-Stokes/Allen-Cahn equations. SIAM Journal on Mathematical Analysis, 2009, 41(6): 2246-2282 doi: 10.1137/090754698
[2]	Gal C G, Grasselli M. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete and Continuous Dynamical Systems, 2010, 28(1): 1-39 doi: 10.3934/dcds.2010.28.1
[3]	Zhao L Y, Guo B L, Huang H Y. Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system. Journal of Mathematical Analysis and Applications, 2011, 384(2): 232-245 doi: 10.1016/j.jmaa.2011.05.042
[4]	Medjo T T. On the existence and uniqueness of solution to a stochastic $2$D Allen-Cahn-Navier-Stokes model. Stochastics and Dynamics, 2019, 19(1): 1950007
[5]	Zhao X P. Global well-posedness and decay estimates for three-dimensional compressible Navier-Stokes-Allen-Cahn system. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2022, 152(5): 1291-1322
[6]	Li Y H, Ding S J, Huang M X. Blow-up criterion for an incompressible Navier-Stokes-Allen-Cahn system with different densities. Discrete and Continuous Dynamical Systems, 2016, 21(5): 1507-1523
[7]	Chen M T, Guo X W. Global large solutions for a coupled compressible Navier-Stokes/Allen-Cahn system with initial vacuum. Nonlinear Analysis: Real World Applications, 2017, 37: 350-373 doi: 10.1016/j.nonrwa.2017.03.001
[8]	Li Y H, Huang M X. Strong solutions for an incompressible Navier-Stokes-Allen-Cahn system with different densities. Journal of Applied Mathematics and Physics, 2018, 69(3): 1-18
[9]	Deteix J, Kouamo G, Yakoubi D. Well-posedness of a semi-discrete Navier-Stokes-Allen-Cahn model. Journal of Mathematical Analysis and Applications, 2021, 496(2): 124816 doi: 10.1016/j.jmaa.2020.124816
[10]	Oleinik O. On the mathematical theory of boundary layer for unsteady flow of incompressible fluid. Journal of Applied Mathematics and Mechanics, 1966, 30(5): 951-974 doi: 10.1016/0021-8928(66)90001-3
[11]	Liu J G, Xin Z P. Boundary layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation. Archive for Rational Mechanics and Analysis, 1996, 135(1): 61-105 doi: 10.1007/BF02198435
[12]	Xie X Q. Boundary layers associated with a coupled Navier-Stokes/Allen-Cahn system: the non-characteristic boundary case. Journal of Partial Differential Equations, 2012, 25(1): 66-78 doi: 10.4208/jpde
[13]	FAN L L, HOU M C. Asymptotic stability of a boundary layer and rarefaction wave for the outflow problem of the heat-conductive ideal gas without viscosity. Acta Mathematica Scientia, 2020, 40B(6): 1627-1652
[14]	张浩, 汪娜. 一类弱非线性临界奇摄动积分边界问题. 数学物理学报, 2022, 42A(4): 1060-1073
	Zhang H, Wang N. A class of weakly nonlinear critical singularly perturbed integral boundary problems. Acta Mathematica Scientia, 2022, 42A(4): 1060-1073
[15]	Chorin A J, Marsden J E. A Mathematical Introduction to Fluid Mechanics. New York: Springer-Verlag, 1997
[16]	Ghil M, Ma T, Wang S H. Structural bifurcation of $2$-D incompressible flows with dirichlet boundary conditions: applications to boundary-Layer separation. SIAM Journal on Applied Mathematics, 2005, 65(5): 1576-1596 doi: 10.1137/S0036139903438818
[17]	Ma T, Wang S H. Rigorous Characterization of Boundary Layer Separations. Cambridge: Proceedings Second MIT Conference on Computational Fluid and Solid Mechanics, 2003
[18]	Ghil M, Liu J G, Wang C, Wang S H. Boundary-layer separation and adverse pressure gradient for $2$-D viscous incompressible flow. Physica D: Nonlinear Phenomena, 2004, 197(1): 149-173 doi: 10.1016/j.physd.2004.06.012
[19]	Luo H, Wang Q, Ma T. A predicable condition for boundary layer separation of $2$-D incompressible fluid flows. Nonlinear Analysis: Real World Applications, 2015, 22: 336-341 doi: 10.1016/j.nonrwa.2014.09.007
[20]	Wang Q, Luo H, Ma T. 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
[21]	姜倩. 阻尼 Navier-Stokes 系统的渐近吸引子与边界层分离. 四川师范大学学报, 2020, 182(3): 96-102
	Jiang Q. The asymptotic attractor of and boundary layer separtion of the damped Navier-Stokes system. Journal of Sichuan Normal University, 2020, 182(3): 96-102
[22]	Shen W M, Wang Y, Zhang Z F. Boundary layer separation and local behavior for the steady Prandtl equation. Advances in Mathematics, 2021, 389: 107896 doi: 10.1016/j.aim.2021.107896
[23]	Ma T, Wang S. Boundary layer separation and structural bifurcation for $2$-D incompressible fluid flows. Discrete and Continuous Dynamical Systems, 2004, 10(1/2): 459-472 doi: 10.3934/dcdsa
[24]	Ma T, Wang S H. Geometric theory of incompressible flows with applications to fluid Dynamics. Providence: American Mathematics Society, 2005