We study the two-dimensional Rayleigh-Bénard-Poiseuille flow in a viscoelastic fluid using linear stability analysis and energy analysis. Different from the case of Newtonian fluids, three instability regimes are identified when gradually increasing the shear intensity (characterized by the Reynolds number ). In regime I with a very small , the viscoelasticity has destabilizing effects on the flow, leading to a smaller critical Rayleigh number than that in the Newtonian case. This is due to the generation of oscillatory instability for high elasticity. Energy analysis indicates that (the rate of work done by the polymeric stresses) makes a negligible contribution to the total energy balance and viscous dissipation () becomes the main driving force to reduce the perturbation kinetic energy. In regime II with moderate , in contrast to regime I, the viscoelasticity has stabilizing effects on the flow, and this results in a higher in comparison to the Newtonian case. This can be evidenced by the increase of the negative work rate of the polymeric stresses. In higher regime III, the viscoelasticity is again enhancing flow instability, and this is owing to the increase of . Among all polymer stress components, the streamwise normal stress plays the most important role. In addition, the effects of Weissenberg number , viscosity ratio and Prandtl number on linear instability threshold have been investigated. By fixing and streamwise wavenumber , energy analysis indicates that suppressing the heat energy transferred from perturbed temperature field is the main way to inhibit flow instability at lower , while the disturbance kinetic energy absorbed by shear flow is the major stabilizing factor at larger . In very high or cases, Tollmien–Schlichting instability induced by strong shear and center mode instability caused by strong elasticity are dominant respectively, which are nearly unaffected by the temperature gradient.

中文翻译：

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剪切强度对粘弹性瑞利-贝纳德-泊肃叶流线性不稳定性的影响

我们使用线性稳定性分析和能量分析研究粘弹性流体中的二维瑞利-贝纳德-泊肃叶流。与牛顿流体的情况不同，当逐渐增加剪切强度（以雷诺数为特征）时，会识别出三种不稳定状态。在 非常小的情况下，粘弹性对流动具有不稳定的影响，导致临界瑞利数比牛顿情况下的小。这是由于高弹性振荡不稳定性的产生。能量分析表明，（聚合物应力做功的速率）对总能量平衡的贡献可以忽略不计，粘性耗散（）成为降低扰动动能的主要驱动力。与状态 I 相比，在中等 的状态 II 中，粘弹性对流动具有稳定作用，这导致与牛顿情况相比更高。这可以通过聚合物应力的负功率的增加来证明。在较高的状态 III 中，粘弹性再次增强流动不稳定性，这是由于 的增加。在所有聚合物应力分量中，流向正应力起着最重要的作用。此外，还研究了Weissenberg数、粘度比和Prandtl数对线性不稳定阈值的影响。通过固定波数和流向波数，能量分析表明，在较低温度场抑制扰动温度场传递的热能是抑制流动失稳的主要途径，而在较高温度场则抑制流动失稳的主要稳定因素是剪切流吸收的扰动动能。在非常高的情况下，强剪切引起的Tollmien-Schlichting不稳定性和强弹性引起的中心模态不稳定性分别占主导地位，几乎不受温度梯度的影响。