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Lubrication theory for free-surface flows with finite slopes and fluxes

有限斜率和流量条件下自由面流动的润滑理论

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Abstract

It is known that free-surface flows with small slopes can be described by classical lubrication theory. By replacing the assumption of quasi-parallel flows with local wedge flows, the classical lubrication theory has been generalized to the situation with finite surface slopes and vanishing fluxes, e.g., steady capillary flows with moving contact lines. In this work, this theory is further extended by imposing the contribution of finite fluxes, which can be modeled by a source/sink flow in a wedge. The resulting lubrication equation is used to investigate the surface morphologies observed in dip coating of an inclined plate, including the Landau-Levich-Derjaguin film, dimple and capillary shock. Dependence of these structures on the inclination angle and relative speed with respect to the plate is discussed in detail. Numerical solutions of the lubrication equation agree well with available asymptotic theory.

摘要

众所周知, 自由面流动在小斜率情况下可以用经典润滑理论进行描述. 采用局部楔形流动代替准平行流动假设, 经典润滑理论已被推广到有限表面斜率和零流量的情况, 例如具有移动接触线的定常毛细流动. 在本工作中, 通过引入楔形中的源/汇流动对有限流量效应进行建模, 进一步扩展了润滑理论. 由此得到的润滑方程被用于研究倾斜板浸涂过程中的表面形态, 包括Landau-Levich-Derjaguin液膜、凹坑和毛细间断. 详细讨论了这些结构对平板倾角和相对平板运动速度的依赖性. 润滑方程的数值解与已有的渐近理论吻合较好.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11972340, 11932019, and 11621202).

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Authors and Affiliations

  1. Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230026, China

    He-Wei Du  (杜和伟), Jian Qin  (秦坚) & Peng Gao  (高鹏)

  2. State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui, 230026, China

    Peng Gao  (高鹏)

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  1. He-Wei Du  (杜和伟)
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Correspondence to Peng Gao  (高鹏).

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Author contributions

Peng Gao designed the research. He-Wei Du, Jian Qin, and Peng Gao wrote the first draft of the manuscript. He-Wei Du and Jian Qin performed the derivation and the calculation. Peng Gao revised and edited the final version.

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Du, HW., Qin, J. & Gao, P. Lubrication theory for free-surface flows with finite slopes and fluxes. Acta Mech. Sin. 38, 322131 (2022). https://doi.org/10.1007/s10409-022-22131-x

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