The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Bénard convection for non-Newtonian inelastic fluids. The rheological model considered here is the Darcy–Carreau model, which is an extension to porous media of Carreau rheological model usually used in clear fluid media. The linear stability approach showed that the critical Rayleigh number and wave number corresponding to the onset of convection are the same as for Newtonian fluids. By employing weakly nonlinear theory, we derived a cubic Landau equation that describes the temporal evolution of the amplitude of convection rolls in the unstable regime. It is found that the bifurcation from the conduction state to convection rolls is always supercritical for dilatant fluids. For pseudoplastic fluids, however, the interplay between the macroscale properties of the porous media and the rheological characteristics of the fluid determines the supercritical or subcritical nature of the bifurcation. In the parameter range where the bifurcation is supercritical, we determined and discussed the combined effects of the fluid properties and the porous medium characteristics on the amplitude of convection rolls and the corresponding average heat transfer for both pseudoplastic and dilatant fluids. Remarkably, we found that the curves describing these effects collapse onto the universal curve for Newtonian fluids, provided the average apparent viscosity is used to define Rayleigh number.

中文翻译：

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多孔介质中拟塑性流体和膨胀流体的 Darcy-Carreau 模型和非线性自然对流

进行线性和弱非线性稳定性分析以研究非牛顿非弹性流体的达西-贝纳德对流的不稳定性。这里考虑的流变模型是 Darcy-Carreau 模型，它是 Carreau 流变模型的多孔介质的扩展，通常用于透明流体介质。线性稳定性方法表明，对应于对流开始的临界瑞利数和波数与牛顿流体相同。通过采用弱非线性理论，我们推导出了描述不稳定状态下对流横滚幅度的时间演变的三次朗道方程。发现从传导状态到对流辊的分岔对于膨胀流体总是超临界的。然而，对于假塑性流体，多孔介质的宏观特性与流体的流变特性之间的相互作用决定了分叉的超临界或亚临界性质。在分岔超临界的参数范围内，我们确定并讨论了流体性质和多孔介质特性对假塑性流体和膨胀流体的对流辊振幅和相应平均传热的综合影响。值得注意的是，如果使用平均表观粘度来定义瑞利数，我们发现描述这些效应的曲线会折叠到牛顿流体的通用曲线上。在分岔超临界的参数范围内，我们确定并讨论了流体性质和多孔介质特性对假塑性流体和膨胀流体的对流辊振幅和相应平均传热的综合影响。值得注意的是，如果使用平均表观粘度来定义瑞利数，我们发现描述这些效应的曲线会折叠到牛顿流体的通用曲线上。在分岔超临界的参数范围内，我们确定并讨论了流体性质和多孔介质特性对假塑性流体和膨胀流体的对流辊振幅和相应平均传热的综合影响。值得注意的是，如果使用平均表观粘度来定义瑞利数，我们发现描述这些效应的曲线会折叠到牛顿流体的通用曲线上。