Abstract Manifestation of Kuppers–Lortz secondary instability in rotating Rayleigh–Benard convection in a Newtonian liquid (water) and in homogeneous and heterogeneous nanoliquids is reported for rigid-isothermal and free-isothermal boundaries. Water–alumina and water–alumina–copper are used as representative homogeneous and heterogeneous nanoliquids. Experimental data on thermal conductivity and dynamic viscosity are used and empirical models which represent experimental data faithfully are obtained. In order to study Kuppers–Lortz secondary instability an energy-conserving, ninth-order Lorenz model is derived using the minimal mode truncated Fourier–Galerkin expansion. It is shown that the critical Rayleigh number obtained in the case of rotating Rayleigh–Benard convection in water–alumina–copper and water–alumina nanoliquids is smaller than the value obtained in the case of water, whereas the critical Taylor number obtained in the case of water–alumina–copper and water–alumina nanoliquids is larger than the value obtained in the case of water. Thus, the individual effect of suspending dilute concentrations of homogeneous and heterogeneous nanoparticles in water is to promote formation of the steady, primary instability (longitudinal rolls) and impede the appearance of the Kuppers–Lortz instability (intersecting rolls). It is also shown that compared to a homogeneous nanoliquid, a heterogeneous nanoliquid is more pro-primary-instability as well as anti-secondary-instability. Further, it is found that by slightly increasing the volume fraction of alumina nanoparticles in water one can achieve the same effect as that of alumina and copper in water. In order to validate the results on Kuppers–Lortz instability, we considered the results on Kuppers–Lortz instability in the absence of nanoparticles obtained in the two cases of rigid and free boundaries and compared them with those of previous investigations, and reasonably good agreement is found.

中文翻译：

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以刚性/自由等温边界为界的旋转瑞利-贝纳对流中的 Küppers-Lortz 不稳定性

摘要 刚性等温和自由等温边界在牛顿液体（水）和均质和非均质纳米液体中旋转瑞利-贝纳德对流中库珀斯-洛兹二次不稳定性的表现被报道。水-氧化铝和水-氧化铝-铜被用作代表性的均质和异质纳米液体。使用了热导率和动态粘度的实验数据，并获得了忠实地代表实验数据的经验模型。为了研究 Kuppers-Lortz 二次不稳定性，使用最小模式截断傅立叶-伽辽金展开式导出了能量守恒的九阶 Lorenz 模型。结果表明，在水-氧化铝-铜和水-氧化铝纳米液体中旋转瑞利-贝纳德对流情况下获得的临界瑞利数小于在水情况下获得的值，而在这种情况下获得的临界泰勒数水-氧化铝-铜和水-氧化铝纳米液体的值大于在水的情况下获得的值。因此，将稀释浓度的均匀和异质纳米粒子悬浮在水中的个体效应是促进稳定的初级不稳定性（纵向卷）的形成并阻止 Kuppers-Lortz 不稳定性（交叉卷）的出现。还表明，与均质的纳米液体相比，非均质的纳米液体更利于初级不稳定性和抗次级不稳定性。更多，发现通过稍微增加氧化铝纳米粒子在水中的体积分数，可以达到与水中氧化铝和铜的体积分数相同的效果。为了验证 Kuppers-Lortz 不稳定性的结果，我们考虑了在刚性和自由边界两种情况下获得的在没有纳米粒子的情况下 Kuppers-Lortz 不稳定性的结果，并将它们与以前的研究进行比较，并且相当好的一致性是成立。