Extended Reynolds lubrication model for incompressible Newtonian fluid

An extended lubrication model is proposed for improvement of the lubrication theory by taking into account a larger surface-to-surface distance than that for the Reynolds lubrication theory. The analysis shows that when considering the non-negligible pressure gradient in the surface-normal direction, the local pressure is separated into (i) a base component satisfying the Reynolds lubrication theory and (ii) an adjusting component varying in the surface-normal direction, which is found to take the form proportional to the longitudinal derivative of the local velocity of the Couette-Poiseuille flow. Comparison of the results obtained by analytical and numerical methods for the lubrication between a moving curved object and stationary object shows that the proposed lubrication model reproduces the pressure distribution in both wall-normal and longitudinal directions. In a problem of a spherical particle approaching to a plane wall, the hydrodynamic force calculated by the proposed model exhibits an inverse-proportional trend to the surface-to-surface distance. The results suggest extended applicability of the lubrication theory to a non-Reynolds regime.

Figure 1

Schematic of a narrow gap between two moving solid surfaces.

Figure 2

Schematic of a fixed cylinder above a moving plate.

Figure 3

Comparison of the cross-channel profiles of the pressures obtained by the present model [Eq. (18)] and the Wannier solution [Eq. (19)] plotted relative to $p_{\mathrm{Re}}$ [Eq. (20)] at three longitudinal positions ($x/a=0.2,0.5$, and 0.8) for three different cylinder locations (a) $d/a=1.01$, (b) 1.18, and (c) 1.5. The pressure value is normalized by $\mu U_0/h_0$.

Figure 4

Comparison of the pressures obtained by Eqs. (18) and (19) for $d/a=1.5$. (a) Extended lubrication model, Eq. (18). (b) Wannier solution [27], Eq. (19).

Figure 5

Schematic of a moving corrugated plate in a parallel channel.

Figure 6

Comparison of the pressure distributions along $y=(H_0-\delta H_0)/2$ obtained by the extended lubrication model (solid lines) and by DNS (broken lines) for different values of $\delta$ and aspect ratio. Ex. Lubr. Model denotes Eq. (18) with $p_{\mathrm{Re}}$ given by Eq. (22).

Figure 7

Schematic of a spherical particle approaching to a flat wall.