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 引用本文: 康建宏, 谭文长. 多孔介质内黏弹性流体的热对流稳定性研究[J]. 力学学报, 2018, 50(6): 1436-1457.
Kang Jianhong, Tan Wenchang. THERMAL INSTABILITY OF VISCOELASTIC FLUIDS IN POROUS MEDIA[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(6): 1436-1457.
 Citation: Kang Jianhong, Tan Wenchang. THERMAL INSTABILITY OF VISCOELASTIC FLUIDS IN POROUS MEDIA[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(6): 1436-1457.

## THERMAL INSTABILITY OF VISCOELASTIC FLUIDS IN POROUS MEDIA

• 摘要: 基于修正的Darcy模型, 介绍了多孔介质内黏弹性流体热对流稳定性研究的现状和主要进展. 通过线性稳定性理论, 分析计算多孔介质几何形状(水平多孔介质层、多孔圆柱以及多孔方腔)、热边界条件(底部等温加热、底部等热流加热、底部对流换热以及顶部自由开口边界)、黏弹性流体的流动模型(Darcy-Jeffrey, Darcy-Brinkman-Oldroyd以及Darcy-Brinkman -Maxwell模型)、局部热非平衡效应以及旋转效应对黏弹性流体热对流失稳的临界Rayleigh数的影响. 利用弱非线性分析方法, 揭示失稳临界点附近热对流流动的分叉情况, 以及失稳临界点附近黏弹性流体换热Nusselt数的解析表达式. 采用数值模拟方法, 研究高Rayleigh数下黏弹性流体换热Nusselt数和流场的演化规律,分析各参数对黏弹性流体热对流失稳和对流换热速率的影响.主要结果：(1)流体的黏弹性能够促进振荡对流的发生;(2)旋转效应、流体与多孔介质间的传热能够抑制黏弹性流体的热对流失稳;(3)在临界Rayleigh数附近,静态对流分叉解是超临界稳定的, 而振荡对流分叉可能是超临界或者亚临界的,主要取决于流体的黏弹性参数、Prandtl数以及Darcy数;(4)随着Rayleigh数的增加,热对流的流场从单个涡胞逐渐演化为多个不规则单元涡胞, 最后发展为混沌状态.

Abstract: Based on the modified Darcy model, the status and progress in research of thermal instability of viscoelastic fluids in porous media are reviewed. By using the method of linear stability analysis, the effects of the geometry of porous media (i.e. horizontal porous layer, porous cylinder and porous cavity), thermal boundary conditions (i.e. bottom heated with constant temperature, bottom heated with constant heat flux, bottom with Newtonian heating and open top), flow model of viscoelastic fluids (i.e. modified Darcy-Jeffrey, Darcy-Brinkman-Oldroyd and Darcy-Brinkman- Maxwell models), local thermal non-equilibrium and rotation on the critical Rayleigh number of thermal instability of viscoelastic fluids can be calculated. By using the method of weakly non-linear analysis, the bifurcation from the basic state and the analytical solution of Nusselt number at the neighborhood of critical point can be obtained. By the numerical simulation method, the evolution of flow pattern as well as the variations of Nusselt number at high Rayleigh number can be revealed. It has been found that (1) the elasticity of viscoelastic fluids can destabilize the oscillatory convection; (2) the rotation effect and local thermal non-equilibrium effect can suppress the thermal instability of viscoelastic fluids; (3) at the neighborhood of critical point, the bifurcation from the basic state for stationary convection is supercritical, while the bifurcation for the oscillatory can be supercritical or subcritical, mainly depending on the values of viscoelastic parameters, Prandtl number and Darcy number; (4) with the increasing Rayleigh number, the flow pattern of thermal convection evolve from one-cell pattern into multi-cell roll pattern, and finally a chaotic pattern.

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