We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences.

我们讨论了一类广义 Navier-Stokes 方程中的流动模型。工作重点是生成热对流模型，详细分析这些模型，并推导对流流体运动开始的临界瑞利和波数。除了线性不稳定理论之外，我们还对完全非线性稳定性理论进行了仔细分析。除了正常的空间导数项之外，所分析的理论都具有双拉普拉斯项。讨论的理论是 Stokes 耦合应力理论、偶极流体理论、Green-Naghdi 理论、Fried-Gurtin-Musesti 理论以及 Fried 和 Gurtin 的第二理论。我们表明 Stokes 耦合应力理论和 Fried-Gurtin-Musesti 理论涉及相同的偏微分方程，而 Green-Naghdi 和偶极理论的偏微分方程相似。然而，