Dynamics of levitation during rolling over a thin viscous film

TL;DR

This work tackles the problem of hydrodynamic levitation of a wheel rolling over a thin viscous film. The authors derive a coupled model based on the Reynolds lubrication equation for the film and a wheel-force balance, then analyze two asymptotic limits: $W\gg1$ (infinitely wide wheel) and $W\ll1$ (narrow wheel), before bridging them with a finite-width, rectangular lubrication-zone approximation solved by separation of variables. They validate the framework by comparing with rollpool experiments, demonstrating qualitative agreement on lift-off and touch-down and identifying systematic quantitative gaps in predicted load and timing, likely due to bow-wave shape and edge effects. The results provide a principled, analytically tractable basis for hydrodynamic levitation relevant to rail lubrication and related applications, and point to necessary extensions to fully 2D Reynolds solutions and refined boundary conditions.

Abstract

A mathematical model is derived for the dynamics of a cylinder, or wheel, rolling over a thin viscous film. The model combines the Reynolds lubrication equation for the fluid with an equation of motion for the wheel. Two asymptotic limits are studied in detail to interrogate the dynamics of levitation: an infinitely wide wheel and a relatively narrow one. In both cases the front and back of the fluid-filled gap are either straight or nearly so. To bridge the gap between these two asymptotic limits, wheels of finite width are considered, introducing a further simplying approximation: although the front and back are no longer expected to remain straight for a finite width, the footprint of the fluid-filled gap is still taken to be rectangular, with boundary conditions imposed at the front and back in a wheel-averaged sense. The Reynolds equation can then be solved by separation of variables. For wider wheels, with a large amount of incoming flux or a relatively heavy loading of the wheel, the system is prone to flooding by back flow with fluid unable to pass underneath. Otherwise steady planing states are achieved. Both lift-off and touch-down are explored for a wheel rolling over a film of finite length. Theoretical predictions are compared with a set of experimental data.

Figures (18)

• Figure 1: (a) A sketch of the model geometry. (b,c,d) The three phases of evolution arising as the wheel rolls over a film of finite length: lift-off, steady planing and touch-down. Displayed are times series of the minimum gap, $h_0(t)$, scaled by its value during steady planing, $h_*$ (so that $\eta_0(t)=h_0/h_*$), with time made dimensionless using the timescale $U/\sqrt{Rh_*}$, where $U$ is the rolling speed and $R$ the wheel radius. The vertical dashed lines and stars indicate the time at which the wheel reaches the end of the film. Three examples are shown. For each, the red dots show experimental measurements using the laboratory set-up from rollpool; the red solid lines show a running average over 6 data points. The blue solid lines show corresponding predictions of the theoretical model. The parameter values are: (b) $(U,h_*,{\mathbb{W}})=(3$m/s$,0.012$mm$,2$cm$)$, (c) $(U,h_*,{\mathbb{W}})=(0.5$m/s$,0.059$mm$,1$cm$)$, (d) $(U,h_*,{\mathbb{W}})=(0.5$m/s$,0.176$mm$,0.5$cm$)$; in each case, the incoming film depth and length are $(h_{in},L_p)=(0.5,60)$mm and $R=9.55$cm. The dimensionless loads imposed experimentally or predicted by the model (as defined in §\ref{['inbetween']}) are: (b) ${\cal L}_0=0.022$vs.$0.042$, (c) ${\cal L}_0=0.076$vs.$0.197$, (d) ${\cal L}_0=0.129$vs.$0.352$. Note that the kinematic viscosity $\nu$ of the fluid for tests (b,c) was about $10^{-2}$m$^2$/s; that for (d) was about $7\times10^{-4}$m$^2$/s. The Reynolds number based on the gap, $Uh_*/\nu$, is therefore of order $0.1$ or smaller.
• Figure 2: The implications of the constraint \ref{['constraint']}, plotting (a) $X_R$ and (b) $\eta_R/\eta_0=1+\frac{1}{2}X_R^2$ against $X_L$. The dashed lines show the limiting behaviours for $X_L\to-\infty$ ($X_R\to 0.6719$ and $\eta_R\to1.2257\eta_0$) and $X_L\to0$ ($X_R\to -\frac{1}{2}X_L$ and $\eta_R\to1+\frac{1}{8}X_L^2$).
• Figure 3: Model solutions showing lift off, starting with $\eta_0(0)=\varepsilon=10^{-j}$, $j=3,4,5$ (colour coded, from blue to red), with $M=10^{-3}$ and adopting a flux $\eta_{in}$ and load ${\cal L}_0$ such that the steady minimim gap and bow-wave position are $\eta_0=1$ and $\xi_L=\Xi_L=-5$. In (a,b,c), the initial condition (indicated by dashed lines) are chosen so that the lift force due to the displaced pool to the left of the dry contact position reaches the load (equation (\ref{['ics1']})). In (d,e,f), the initial conditions are given by the quasi-steady solution as in \ref{['ics2']} ($\xi_{Rs}$ and $\eta_0\sim\dot\eta_{0s}$ are shown by dot-dashed lines). The dotted lines in (a,b,c) show a further solution in which $\xi_R(0)$ is arbitrarily reset to the centre of the wheel and $\dot\eta_0(0)$ is adjusted to satisfy the constraint accordingly.
• Figure 4: Model solutions showing lift off, starting with \ref{['ics2']} and $\eta_0(0)=\varepsilon=10^{-4}$, for a flux $\eta_{in}$ such that the steady bow-wave position is $\xi_L=\Xi_L=-5$. (a,c) Time series of the positions of the bow wave and lubrication front, $\xi_R(t)$ and $\xi_L(t)$, and (b,d) minimum gap $\eta_0(t)$. Solutions for different mass parameters are presented: (a,b) $M=10^{-j},\frac{1}{4}$, $j=\{4,3,2,1\}$, and (c,d) $M=\frac{1}{4}$, $\frac{1}{2}$, 1, 2, $3.5$ and 5 (in both cases, colour-coded, from red to blue). The dashed lines indicate the steady final planing state, with the triangle marking $t=3t_\infty = 3|\xi_{L\infty}|^5/(16\eta_{in}^2)$. The dot-dashed lines in (a,b) show $\xi_{Rs}$ and $\dot\eta_{0s}t$. The star indicates the time at which the lubrication zone for the solution with largest $M$ shrinks to a point.
• Figure 5: Solutions to \ref{['odes']} for $M=10^{-3}$, the initial conditions in \ref{['ics2']}, and the same load parameter ${\cal L}_0$ as in figure \ref{['sold']}. Six solutions are shown, corresponding to fluxes of $\eta_{in}=\{0.9,1.06,1.22,1.34,1.42,1.52\}$ (colour-coded from blue to red). The dashed lines show the expected steady planing state (§\ref{['steadyflood']}) with the triangle marking $t=3t_\infty$. The dotted lines show the predictions \ref{['floodit1']} and \ref{['floodit2']}.