![]() As will be shown in § 3.1, for the rolling flows considered here, the film splitting is complicated still further by the generation of substantial negative pressures that likely induce cavitation or de-gassing. In particular, the associated two-dimensional flow is known to be prone to the so-called printer's instability, which generates a complicated three-dimensional filamentary structure (Pearson Reference Pearson1960 Pitts & Greiller Reference Pitts and Greiller1961). Reference Becerra, Romero, Azevedo and Carvalho2007). The splitting of the film at the downstream meniscus also features in a great many other coating problems (Greener & Middleman Reference Greener and Middleman1975 Benkreira, Edwards & Wilkinson Reference Benkreira, Edwards and Wilkinson1981 Coyle, Macosko & Scriven Reference Coyle, Macosko and Scriven1986 Decré, Gailly & Buchlin Reference Decré, Gailly and Buchlin1995 Weinstein & Ruschak Reference Weinstein and Ruschak2004 Ascanio & Ruiz Reference Ascanio and Ruiz2006 Becerra et al. ( Reference Dalwadi, Cimpeanu, Ockendon, Ockendon and Mullin2021), wherein a solid object is held aloft by a vertical moving belt coated with a thin layer of viscous fluid. The lubricated rolling process illustrated in figure 1 is similar to the levitation problems considered by Eggers, Kerswell & Mullin ( Reference Eggers, Kerswell and Mullin2013), Mullin, Ockendon & Ockendon ( Reference Mullin, Ockendon and Ockendon2020) and Dalwadi et al. The axial direction is defined as the direction of the cylinder/substrate motion and the lateral direction is perpendicular to the page, along the cylinder width ($z$-axis). Sketches of the geometry for lubricated rolling over a viscous fluid layer, showing (a) the wheel rolling into the initial pool, and (b) the details of the lubrication film, with various physical parameters indicated. The split film adhering to the wheel is conveyed further along where it may be deposited back onto the track in a subsequent carry-down process.įigure 1. The fluid passes through this gap and splits at a downstream meniscus, with part of the fluid adhering to the wheel and part remaining on the track ahead of the gap, the fluid accumulates in a bow wave ( figure 1). When the approaching wheel contacts this liquid pool, the lubrication pressure developed can be sufficient to raise the wheel slightly off the track. Reference Stock, Stanlake, Hardwick, Yu, Eadie and Lewis2016 Rahmani & Green Reference Rahmani and Green2017). Some of this liquid is picked up by the passing wheel and can reduce wheel and rail wear, noise and fuel consumption (Harmon & Lewis Reference Harmon and Lewis2016 Stock et al. In rail transport, which primarily motivated this work, a pool of ‘liquid friction modifier’, a viscous liquid containing small amounts of microscopic solid lubricants like graphite, is deposited on the track ahead of the approaching train. The interaction of a rolling cylinder or wheel with a pool of liquid resting on a substrate is relevant to many practical problems such as roll coating, lubrication of bearings and rail transport. Conversely, if there is insufficient adhered fluid, no contiguous lubrication film is formed instead, the pattern from the printer's instability ‘prints’ from the cylinder to the substrate. If the pool length is less than the cylinder circumference, the fluid adhering to the cylinder is rotated back into contact with the substrate, and when there is sufficient adhered fluid a lubrication film forms that can again be modelled by the theory. The printer's instability arises during the splitting process, patterning the residual fluid films on the substrate and cylinder. For lubricated rolling, the film splits evenly between the cylinder and substrate downstream of the nip. Once this parameter is calibrated against experiment, the theory predicts peak lubrication pressures, gap sizes and film thicknesses to within approximately ten per cent. The theory approximates the flow by the one-dimensional Reynolds equation with the addition of one term, with an adjustable parameter, to account for the flux of fluid to the cylinder sides. For the former situation, a lubrication theory is presented that describes the pressure underneath the cylinder and the thickness of the film. Depending on the conditions, the cylinder will either ride on a lubrication film or remain in solid contact with the underlying substrate. The speed, width and loading of the cylinder are varied along with the initial depth and length of the viscous pool. Experiments are conducted to explore the rolling of a cylinder over a pool of viscous fluid.
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