Hydrodynamic instability and formation of waves for thin viscous film flowing down a slippery inclined substrate with broken time-reversal-symmetry have been discussed in this present study. The effect of slip on the substrate is modeled using the Navier slip boundary condition. We have derived a nonlinear evolution model in the framework of the long-wave expansion technique. The one equation model can track the free surface evolution and involve the viscosity, gravity, surface tension, and the effect of the slip of the wall. The odd part of the Cauchy stress tensor with an odd-viscosity coefficient significantly modifies the characteristic of the film flow. We performed the linear stability analysis with the Orr Sommerfeld technique and classical temporal analysis from the model. Comparing the results shows a satisfactory agreement between Orr Sommerfeld and the model when the velocity scale is chosen as twice the free surface velocity. Analyzing the traveling wave solutions and studying different bifurcation analyzes and phase diagrams, we focused on and discussed two significant wave families, namely ${\gamma }_1$ (low-wave-speed) and ${\gamma }_2$ (high-wave-speed). The spatio-temporal study of the model has been performed numerically for different odd-viscosity and slip parameters. Results show that the odd-viscosity plays a stabilizing role while the slip on the substrate gives a destabilizing effect within the parameter range of our consideration.

本研究讨论了流体动力学不稳定性和薄粘性薄膜在时间反转对称性被破坏的光滑倾斜基底上流动的波浪形成。使用 Navier 滑动边界条件模拟滑动对基底的影响。我们在长波扩展技术的框架内推导出非线性演化模型。一方程模型可以跟踪自由表面演化，并涉及粘度、重力、表面张力和壁面滑移的影响。具有奇粘度系数的柯西应力张量的奇数部分显着改变了薄膜流动的特性。我们使用 Orr Sommerfeld 技术和模型的经典时间分析进行了线性稳定性分析。比较结果表明，当速度尺度选择为自由表面速度的两倍时，Orr Sommerfeld 与模型之间具有令人满意的一致性。分析行波解并研究不同的分岔分析和相图，我们重点讨论了两个重要的波族，即${\gamma }_1$ （低波速）和 ${\gamma }_2$（高波速度）。该模型的时空研究已经针对不同的奇数粘度和滑移参数进行了数值计算。结果表明，在我们考虑的参数范围内，奇数粘度起稳定作用，而基材上的滑移会产生不稳定效应。