The weakly nonlinear stability analysis of plane Poiseuille flow (PPF) and plane Couette flow (PCF) when viscous dissipation is taken into account is considered. The impermeable lower boundary is considered adiabatic, while the impermeable upper boundary is isothermal. The linear stability of this problem has been performed by Barletta and Nield (J. Fluid Mech., vol. 662, 2010, pp. 475–492) for PCF and by Barletta et al. (J. Fluid Mech., vol. 681, 2011, pp. 499–514) for PPF. These authors found that longitudinal rolls are the preferred mode of convection and the onset of instability is described through the governing parameters $\unicode[STIX]{x1D6EC}=Ge\;Pe^{2}$ and $Pr$ , where $Ge$ , $Pe$ and $Pr$ are respectively the Gebhart number, the Peclet number and the Prandtl number. The current study focuses on the near-threshold behaviour of longitudinal rolls by using a weakly nonlinear analysis. We determine numerically up to third order the coefficients of the Landau amplitude equation and investigate in detail the influences on bifurcation characteristics of the different nonlinearities present in the system. The results indicate that for both PPF and PCF configurations (i) the inertial terms have no influence on the nonlinear evolution of the disturbance amplitude (ii) the nonlinear thermal advection terms act in favour of pitchfork supercritical bifurcations and (iii) the nonlinearities associated with viscous dissipation promote subcritical bifurcations. The global impact of the different nonlinear contributions indicate that, independently of the Gebhart number, the bifurcation is subcritical if $Pr<0.25$ ( $Pr<0.77$ ) for PPF (PCF). Otherwise, for higher Prandtl number, there exists a particular value of Gebhart number, $Ge^{\ast }$ such that the bifurcation is supercritical (subcritical) if $GeGe^{\ast }$ ). Finally, for both PPF and PCF, the amplitude analysis indicates that, in the supercritical bifurcation regime, the equilibrium amplitude decreases on increasing $Pr$ and a substantial enhancement (reduction) in heat transfer rate is found for small $Pr$ (moderate or large $Pr$ ).

中文翻译：

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平面 Poiseuille 和平面 Couette 流中粘性耗散热不稳定性的弱非线性分析

考虑了粘性耗散时的平面泊肃叶流(PPF)和平面库埃特流(PCF)的弱非线性稳定性分析。不可渗透的下边界被认为是绝热的，而不可渗透的上边界是等温的。Barletta 和 Nield (J. Fluid Mech., vol. 662, 2010, pp. 475–492) 为 PCF 和 Barletta 等人完成了这个问题的线性稳定性。（J. Fluid Mech.，第 681 卷，2011 年，第 499-514 页）用于 PPF。这些作者发现纵向滚动是首选的对流模式，不稳定的开始通过控制参数 $\unicode[STIX]{x1D6EC}=Ge\;Pe^{2}$ 和 $Pr$ 来描述，其中 $Ge $、$Pe$ 和 $Pr$ 分别是 Gebhart 数、Peclet 数和 Prandtl 数。目前的研究重点是通过使用弱非线性分析来研究纵向滚动的近阈值行为。我们在数值上确定了朗道振幅方程的三阶系数，并详细研究了系统中存在的不同非线性对分岔特性的影响。结果表明，对于 PPF 和 PCF 配置 (i) 惯性项对扰动幅度的非线性演化没有影响 (ii) 非线性热平流项有利于干草叉超临界分叉和 (iii) 与粘性耗散促进亚临界分岔。不同非线性贡献的全局影响表明，独立于 Gebhart 数，如果 $Pr<0，分岔是次临界的。PPF (PCF) 25$ ( $Pr<0.77$ )。否则，对于更高的 Prandtl 数，存在一个特定的 Gebhart 数值 $Ge^{\ast }$ 使得分岔是超临界（亚临界）如果 $GeGe^{\ast }$ ）。最后，对于 PPF 和 PCF，振幅分析表明，在超临界分叉状态下，平衡振幅随着 $Pr$ 的增加而减小，并且发现小的 $Pr$（中等或中等或大 $Pr$）。