Steady circular hydraulic jump on a rotating disk

Abstract

The paper deals with the steady axially symmetric flow of a viscous liquid layer over a rotating disk. The liquid is fed near the axis of rotation and spreads due to inertia and the centrifugal force. The viscous shallow-water approach gives a system of ordinary differential equations governing the flow. We consider inertia, gravity, centrifugal and Coriolis forces and estimate the effect of surface tension. We found four qualitatively different flow regimes. Transition through these regimes shows the continuous evolution of the flow structure from a hydraulic jump on a static disk to a monotonic thickness decrease on a fast rotating one. We show that, in the absence of surface tension, the intensity of the jump gradually vanishes at a finite distance from the axis of rotation while the angular velocity increases. The surface tension decreases the jump radius and destroys the steady solution for a certain range of parameters.

A coherent composite approach for the continuous circular hydraulic jump and vortex structure

“…An integral approach of the KP type (Schlichting & Gersten 2000) is adopted in the developing boundary-layer and fully developed viscous regions. The KP method has been widely adopted in the literature for steady and transient jumps, not only when the thin-film equations are parabolic (Watson 1964 (Tani 1949;Bohr et al 1993Bohr et al , 1997Watanabe et al 2003;Kasimov 2008;Fernandez-Feria et al 2019;Wang & Khayat 2019;Dhar et al 2020;Ipatova, Smirnov & Mogilevskiy 2021). The problem becomes weakly elliptic when the relatively weak effect of gravity upstream of the jump is not neglected in the analysis.…”

“…2019; Wang & Khayat 2019; Dhar et al. 2020; Ipatova, Smirnov & Mogilevskiy 2021). The problem becomes weakly elliptic when the relatively weak effect of gravity upstream of the jump is not neglected in the analysis.…”

The transient spread of a circular liquid jet and hydraulic jump formation

“…The classification of the equations as parabolic is the result of neglecting the effect of gravity upstream of the jump (Scheichl, Bowles & Pasias 2018). We also note that the KP method has been widely adopted in the literature for steady-state jumps, not only when the thin-film equations are parabolic (Watson 1964;Bush & Aristoff 2003;Dressaire et al 2010;Prince et al 2012;Wang & Khayat 2018) but also when the equations are weakly elliptic as well (Tani 1949;Bohr et al 1993Bohr et al , 1997Watanabe et al 2003;Kasimov 2008;Fernandez-Feria et al 2019;Wang & Khayat 2019;Dhar, Das & Das 2020;Ipatova et al 2021). The problem becomes weakly elliptic when the relatively weak effect of gravity upstream of the jump is not neglected in the analysis.…”

“…2019; Wang & Khayat 2019; Dhar, Das & Das 2020; Ipatova et al. 2021). The problem becomes weakly elliptic when the relatively weak effect of gravity upstream of the jump is not neglected in the analysis.…”

“…The surface tension correction was shown to be small for jumps of laboratory scale but appreciable for a small jump. Several extensions of the classical circular hydraulic jump problem were later implemented theoretically and numerically, including the jet impingement on a rotating disk (Wang & Khayat 2018;Scheichl & Kluwick 2019;Ipatova, Smirnov & Mogilevskiy 2021) and on a circular heated disk (Wang & Khayat 2020), and the influence of slip (Prince, Maynes & Crockett 2012), gravity (Wang & Khayat 2019) and surface tension (Bhagat et al 2018). The latter study was, later, shown to be formally flawed and misinterpreted results (Duchesne, Andersen & Bohr 2019;Bohr & Scheichl 2021;Duchesne & Limat 2022).…”