Linear and weakly nonlinear stability analyses of Brinkman–Bénard convection in water with a dilute concentration of single-walled carbon nanotubes ($SWCNTs$) is studied analytically in the paper. The thermophysical properties of water-$SWCNTs$ nanoliquid and water-$SWCNTs$ nanoliquid-saturated high-porosity medium are calculated using phenomenological relations and the mixture theory. The five-mode Lorenz model is derived under the assumptions of Boussinesq approximation, small-scale convective motion, weak thermophoresis and porous friction. Using the method of multiscales the five-mode Lorenz model is reduced to a cubic Ginzburg–Landau equation the solution of which helps in quantifying the unsteady heat transport. The effects of $SWCNTs$, porous parameter and viscosity ratio on onset of convection and the heat transport are documented. Results of the unicellular-Brinkman–Bénard convection problem is extracted from those of the multicellular-Brinkman–Bénard convection problem. Three different enclosures are considered in the study and their influence on the onset of convection and the heat transport are compared. The single-phase model which incorporates nanoliquid thermophysical properties is shown to be a limiting case of the model proposed in the present paper.

稀浓度单壁碳纳米管在水中的Brinkman-Bénard对流的线性和弱非线性稳定性分析（$\mathrm{小号}\mathrm{w\; ^}CñŤs$）是本文的分析研究。水的热物理性质$\mathrm{小号}\mathrm{w\; ^}CñŤs$ 纳米液体和水$\mathrm{小号}\mathrm{w\; ^}CñŤs$使用现象学关系和混合理论计算了纳米液体饱和的高孔隙率介质。五种模式的Lorenz模型是在Boussinesq逼近，小尺度对流运动，弱热泳和多孔摩擦的假设下得出的。使用多尺度方法，将五模式Lorenz模型简化为三次Ginzburg-Landau方程，该方程的解有助于量化不稳定的热传递。的影响$\mathrm{小号}\mathrm{w\; ^}CñŤs$记录了对流开始和传热时的多孔参数和粘度比。单细胞-布林克曼-贝纳德对流问题的结果是从多细胞布林克曼-贝纳德对流问题的结果中提取的。在研究中考虑了三种不同的围护结构，并比较了它们对对流和传热的影响。结合了纳米液体热物理性质的单相模型被证明是本文提出的模型的一个限制情况。