Here we extend the results in Gordillo et al. ( J. Fluid Mech. , vol. 866, 2019, pp. 298–315), where the spreading of drops impacting perpendicularly a solid wall was analysed, to predict the time-varying flow field and the thickness of the liquid film created when a spherical drop of a low viscosity fluid, like water or ethanol, spreads over a smooth dry surface at arbitrary values of the angle formed between the drop impact direction and the substrate. Our theoretical results accurately predict the time evolving asymmetric shape of the border of the thin liquid film extending over the substrate during the initial instants of the drop spreading process. In addition, the particularization of the ordinary differential equations governing the unsteady flow when the rim velocity vanishes provides an algebraic equation for the asymmetric final shapes of the liquid stains remaining after the impact, valid for low values of the inclination angle. For larger values of the inclination angle, the final shape of the drop can be approximated by an ellipse whose major and minor semiaxes can also be calculated by making use of the present theory. The predicted final shapes agree with the observed remaining stains, excluding the fact that a liquid rivulet develops from the bottom part of the drop. The limitations of the present theory to describe the emergence of the rivulet are also discussed.

中文翻译：

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下降的倾斜影响

在这里，我们扩展了 Gordillo 等人的结果。( J. Fluid Mech. , vol. 866, 2019, pp. 298–315)，其中分析了垂直撞击固体壁的液滴的扩散，以预测时变流场和液膜厚度低粘度流体（如水或乙醇）的球形液滴以液滴撞击方向和基材之间形成的任意角度值在光滑干燥的表面上扩散。我们的理论结果准确地预测了在液滴扩散过程的初始时刻，在基板上延伸的薄液膜边界的时间演变不对称形状。此外，当边缘速度消失时，控制非定常流动的常微分方程的特殊化为撞击后残留的液体污渍的不对称最终形状提供了代数方程，适用于低倾角值。对于较大的倾角值，液滴的最终形状可以近似为椭圆，其长、短半轴也可以利用现有理论计算。预测的最终形状与观察到的剩余污渍一致，不包括从液滴底部形成液体小溪的事实。还讨论了目前描述小溪出现的理论的局限性。对于较大的倾角值，液滴的最终形状可以近似为椭圆，其长半轴和短半轴也可以利用现有理论来计算。预测的最终形状与观察到的剩余污渍一致，排除了液体小溪从液滴底部形成的事实。还讨论了目前描述小溪出现的理论的局限性。对于较大的倾角值，液滴的最终形状可以近似为椭圆，其长、短半轴也可以利用现有理论计算。预测的最终形状与观察到的剩余污渍一致，排除了液体小溪从液滴底部形成的事实。还讨论了目前描述小溪出现的理论的局限性。