doi:10.3785/j.issn.1008-9497.2022.02.003

Journal of Zhejiang University (Science Edition)

2022, Vol. 49

Issue (2): 151-158 DOI: 10.3785/j.issn.1008-9497.2022.02.003

Mathematics and Computer Science

Jincheng SHI¹(

),Shengzhong XIAO²(

)

^1.School of Data Science，Guangzhou Huashang College，Guangzhou 511300，China
^2.Scientific Research Department，Guangdong AIB Polytechnic College，Guangzhou 510507，China） Convergence of solutions for Brinkman equations and Darcy equations interacting in porous media

Download:

HTML( 13 )

PDF(531KB)
Export: BibTeX | EndNote (RIS)

Abstract

The convergence of solutions for the Brinkman fluid interfacing with a Darcy fluid in a bounded region in $? R 3$ is studied. We assume that the velocity of fluid is slow and it is governed by the Brinkman equations in $Ω 1$ , while in $Ω 2$ ,the saturated flow satisfies the Darcy equations. With the aid of the maximum of the temperature $T$ and some other a priori bounds, we formulate an energy expression, and the expression satisfies a differential inequality. By integrating, we are able to demonstrate the convergence result for the boundary coefficient.

Key words： convergence Brinkman equations Darcy equations boundary coefficient

Received: 06 January 2021 Published: 22 March 2022

CLC:

O 175.29

Corresponding Authors: Shengzhong XIAO E-mail: 0818@163.com;172013444@qq.com

	Service
	E-mail this article
	Add to my bookshelf
	Add to citation manager
	E-mail Alert
	RSS
	Articles by authors
	Jincheng SHI
	Shengzhong XIAO

Cite this article:

Jincheng SHI,Shengzhong XIAO. . Journal of Zhejiang University (Science Edition), 2022, 49(2): 151-158.

URL:

https://www.zjujournals.com/sci/EN/Y2022/V49/I2/151

多孔介质中相互作用的Brinkman方程组与Darcy方程组解的收敛性

研究了在 $R 3$ 有界区域内多孔介质中相互作用的Brinkman流体方程组与Darcy流体方程组解的收敛性。假设在 $Ω 1$ 中，流体速度较慢满足Brinkman方程组，而在 $Ω 2$ 中，饱和流体满足Darcy方程组，借助温度 $T$ 的最大值以及其他界，构造了能量表达式，得到了满足该能量表达式的微分不等式和Brinkman-Darcy流体方程组的解对边界系数的收敛性结果。

关键词： 收敛性, Brinkman方程组, Darcy方程组, 边界系数


[1]	AMES K A， STRAUGHAN B. Non-Standard and Improperly Posed Problems［M］. San Diego： Academic Press， 1997. doi:10.1016/s0076-5392(97)80007-0 doi: 10.1016/s0076-5392(97)80007-0

[2]	STRAUGHAN B. Continuous dependence on the heat source in resonant porous penetrative convection［J］. Studies in Applied Mathematics， 2011， 127（3）： 302-314. DOI：10.1111/j.1467-9590.2011.00521.x doi: 10.1111/j.1467-9590.2011.00521.x

[3]	SCOTT N L. Continuous dependence on boundary reaction terms in a porous medium of Darcy type［J］. Journal of Mathematical Analysis and Applications， 2013， 399（2）： 667-675. DOI：10.1016/j. jmaa. 2012.10.054 doi: 10.1016/j. jmaa. 2012.10.054

[4]	SCOTT N L， STRAUGHAN B. Continuous dependence on the reaction terms in porous convection with surface reactions［J］. Quarterly of Applied Mathematics， 2013， 71（3）： 501-508. DOI：10.1090/S0033-569X-2013-01289-X doi: 10.1090/S0033-569X-2013-01289-X

[5]	HARFASH J A. Structural stability for two convection models in a reacting fluid with magnetic field effect［J］. Annales Henri Poincaré， 2014， 15（12）： 2441-2465. DOI：10.1007/s00023-013-0307-z doi: 10.1007/s00023-013-0307-z

[6]	LI Y F， LIN C H. Continuous dependence for the nonhomogeneous Brinkman-Forchheimer equations in a semi-infinite pipe［J］. Applied Mathematics and Computation， 2014， 244（1）： 201-208. doi:10.1016/j.amc.2014.06.082 doi: 10.1016/j.amc.2014.06.082

[7]	CIARLETTA M， STRAUGHAN B， TIBULLO V. Structural stability for a thermal convection model with temperature-dependent solubility［J］. Nonlinear Analysis： Real World Applications， 2015， 22： 34-43. DOI：10.1016/j.nonrwa.2014.07.012 doi: 10.1016/j.nonrwa.2014.07.012

[8]	CHEN W H， LIU Y. Structural stability for a Brinkman-Forchheimer type model with temperature-dependent solubility［J］. Boundary Value Problems， 2016， 2016（55）： 1-14. DOI：10.1186/S13661-016-0558-Y doi: 10.1186/S13661-016-0558-Y

[9]	LIU Y. Continuous dependence for a thermal convection model with temperature-dependent solubility［J］. Applied Mathematics and Computation， 2017， 308： 18-30. DOI：10.1016/j.amc.2017.03.004 doi: 10.1016/j.amc.2017.03.004

[10]	李远飞. 大尺度海洋大气动力学三维黏性原始方程对边界参数的连续依赖性［J］. 吉林大学学报（理学版）， 2019， 57（5）： 1053-1059. DOI：10.13413/j.cnki.jdxblxb.2019038 LI Y F. Continuous dependence on boundary parameters for three-dimensional viscous primitive equation of large-scale ocean atmospheric dynamics［J］. Journal of Jilin University（Science Edition）， 2019， 57（5）： 1053-1059. DOI：10.13413/j.cnki.jdxblxb.2019038 doi: 10.13413/j.cnki.jdxblxb.2019038

[11]	李远飞.原始方程组对黏性系数的连续依赖性［J］. 山东大学学报（理学版）， 2019， 54（12）： 12-23. DOI：10.6040/j.issn.1671-9352.0.2019.539 LI Y F. Continuous dependence on the viscosity coefficient for the primitive equations［J］. Journal of Shandong University（Science Edition）， 2019， 54（12）： 12-23. DOI：10.6040/j.issn.1671-9352.0.2019.539 doi: 10.6040/j.issn.1671-9352.0.2019.539

[12]	李远飞，郭连红.具有边界反应Brinkman-Forchheimer型多孔介质的结构稳定性［J］. 高校应用数学学报，2019， 34（3）： 315-324. DOI：10.13299/j.cnki.amjcu.002084 LI Y F， GUO L H. Structural stability on boundary reaction terms in a porous medium of Brinkman-Forchheimer type［J］. Applied Mathematics-A Journal of Chinese Universities， 2019， 34（3）： 315-324. DOI：10.13299/j.cnki.amjcu.002084 doi: 10.13299/j.cnki.amjcu.002084

[13]	CICHO? M， STRAUGHAN B， YANTIR A. On continuous dependence of solutions of dynamic equations［J］. Applied Mathematics and Computation， 2015， 252： 473-483. DOI：10.1016/j.amc.2014.12.047 doi: 10.1016/j.amc.2014.12.047

[14]	MA H， LIU B. Exact controllability and continuous dependence of fractional neutral integro-differential equations with state dependent delay［J］. Acta Mathematica Scientia， 2017， 37（1）： 235-258. DOI：10.1016/S0252-9602（16）30128-X doi: 10.1016/S0252-9602（16）30128-X

[15]	WU H L， REN Y， HU F. Continuous dependence property of BSDE with constraints［J］. Applied Mathematics Letters， 2015， 45： 41-46. DOI：10.1016/j.aml.2015.01.002 doi: 10.1016/j.aml.2015.01.002

[16]	PAYNE L E， STRAUGHAN B. Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions［J］. Journal de Mathématiques Pures et Appliqués， 1998， 77（4）： 317-354. DOI：10.1016/S0021-7824（98）80102-5 doi: 10.1016/S0021-7824（98）80102-5

[17]	LIU Y， XIAO S Z. Structural stability for the Brinkman fluid interfacing with a Darcy fluid in an unbounded domain［J］. Nonlinear Analysis： Real World Applications， 2018，42： 308-333. DOI：10.1016/j.nonrwa.2018.01.007 doi: 10.1016/j.nonrwa.2018.01.007

[18]	LIU Y， XIAO S Z， LIN Y W. Continuous dependence for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded domain［J］. Mathematics and Computers in Simulation， 2018， 150： 66-82. DOI：10.1016/j.matcom.2018.02.009 doi: 10.1016/j.matcom.2018.02.009

[19]	STRAUGHAN B. Stability and Wave Motion in Porous Media［M］. New York： Springer-Verlag，2008. doi:10.1007/978-0-387-76543-3_4 doi: 10.1007/978-0-387-76543-3_4

[1]	Tao WANG,Yan LI. Approximation properties of q-type Lupas-Kantorovich operators[J]. Journal of Zhejiang University (Science Edition), 2023, 50(5): 533-538.

[2]	Hongyan JIN,Gennian SUN,Xingtai ZHANG. Regional differences and convergence of high-quality development of tourism in China[J]. Journal of Zhejiang University (Science Edition), 2023, 50(4): 495-507.

[3]	Xinyu YANG,Yi LU,Panyan SHI,Lizhen ZHOU. The almost convergence of sequence and measurable function[J]. Journal of Zhejiang University (Science Edition), 2023, 50(2): 131-136.

[4]	Ning KANG,Ke JING. Convergence of second derivative of a family of barycentric Hermite rational interpolants[J]. Journal of Zhejiang University (Science Edition), 2022, 49(3): 324-328.

[5]	Min REN. Limit properties for branching process affected by communicable diseases in random environments[J]. Journal of Zhejiang University (Science Edition), 2022, 49(1): 53-59.

[6]	Juan LI. A second-order linearized finite difference method for a higher order convective Cahn-Hilliard type equation[J]. Journal of Zhejiang University (Science Edition), 2022, 49(1): 60-65.

[7]	Yantao YANG,Jingjing CHEN,Haiyun ZHOU. A weak convergence theorem involving the zero point of quasi-inverse strongly monotone operators with application[J]. Journal of Zhejiang University (Science Edition), 2022, 49(1): 49-52.

[8]	ZHANG Qian, CAI Guanghui. Complete convergence for weighted sums of WOD random variables[J]. Journal of Zhejiang University (Science Edition), 2021, 48(4): 435-439.

[9]	SHI Jincheng, LI Yuanfei. Structural stability of solutions for the Darcy equations in porous medium[J]. Journal of Zhejiang University (Science Edition), 2021, 48(3): 298-303.

[10]	ZHU Ping, CHEN Xiaodiao, MA Weiyin, JIANG Nichang. Explicit formulae for progressively computing a real root of the smooth function[J]. Journal of Zhejiang University (Science Edition), 2021, 48(2): 143-150.

[11]	SHI Jincheng, LI Yuanfei. Structural stability of the Forchheimer-Darcy fluid in a bounded domain[J]. Journal of Zhejiang University (Science Edition), 2021, 48(1): 57-63.

[12]	SONG Mingzhu, SHAO Jing, LIU Caiyun. Limiting properties of moving average processes for END random variable sequence[J]. Journal of Zhejiang University (Science Edition), 2020, 47(5): 559-563.

[13]	GAO Yunfeng, ZOU Guangyu. Complete moment convergence for moving average processes generated by NSD sequences[J]. Journal of Zhejiang University (Science Edition), 2020, 47(2): 172-177.

[14]	ZHAO Zhe, ZHANG Tianye, HUANG Yanhao, ZHENG Wenting, CHEN Wei. Simulation-based visual analysis of power grid operation mode[J]. Journal of Zhejiang University (Science Edition), 2020, 47(1): 36-44.

[15]	ZHANG Qian, CAI Guanghui, ZHENG Yuyan. Complete convergence of WOD random variable sequences.[J]. Journal of Zhejiang University (Science Edition), 2019, 46(4): 412-415.

Viewed

Full text

Abstract

Cited

Shared

Discussed