This work investigates the thin film instability of a non-Newtonian second-grade fluid flowing down a heated inclined plane subject to linear variation of physical properties such as density, viscosity, thermal diffusivity, and surface tension concerning temperature. A nonlinear evolution equation for the description of the free surface is derived using long-wave approximation. The critical conditions for the onset of instability, such as critical Reynolds number, critical wave number, and the linear phase speed of wave propagation, are obtained by linear stability analysis through the normal mode approach. Further, the investigation on the weakly nonlinear stability characteristic of the fluid using a multi-scale approach reveals that both supercritical stability and subcritical instability exist. The influence of the non-Newtonian parameter and the variable fluid properties on subcritical, supercritical, unconditional, and explosive zones are discussed. It also discusses the amplitude of disturbances in subcritical and supercritical regions. The results show the destabilizing nature of the second-grade parameter.

这项工作研究了非牛顿第二级流体向下流经加热的倾斜平面时的薄膜不稳定性，该流体在物理性质（例如密度，粘度，热扩散系数和与温度有关的表面张力）线性变化的情况下会发生变化。使用长波近似推导了用于描述自由表面的非线性演化方程。通过正常模式的线性稳定性分析获得了不稳定开始的临界条件，例如临界雷诺数，临界波数和波传播的线性相速度。此外，使用多尺度方法研究流体的弱非线性稳定性特征表明，超临界稳定性和亚临界不稳定性都存在。讨论了非牛顿参数和可变流体特性对亚临界，超临界，无条件和爆炸区域的影响。它还讨论了亚临界和超临界区域的扰动幅度。结果显示了二级参数的不稳定性质。