Abstract
A lubrication equation is obtained for a simplified shear-thinning fluid. The simplified rheology consists of a piecewise linear stress tensor, resulting in a two-viscosity model. This can be interpreted as a modified Bingham fluid, which can be recovered in a specific limit. The lubrication equation is obtained in two steps. First two scalings are performed on the incompressible Navier-Stokes equations, namely the long-wave scaling and the slow motion scaling. Second, the resulting equations are averaged along the vertical direction. Numerical illustrations are provided, bringing to light the different possible behaviours.
Highlights
The lubrication equation is quite a classical simplification of the incompressible Navier-Stokes system. It is obtained for thin films of fluid, when viscous effects balance the pressure force
The lubrication equation requires another assumption of balance between the viscous effects and the pressure effects, which amounts to neglect all kinematic effects
Mimicking the Bingham model, which is based on a threshold on the shear stress, we consider a model with a threshold on the strain rate: the viscosity is equal to some large μB for small deformations, that is γ < γc, where γc > 0 is a given constant, and to another value μ for large deformations, γ > γc
Summary
The lubrication equation is quite a classical simplification of the incompressible Navier-Stokes system It is obtained for thin films of fluid, when viscous effects balance the pressure force. A model which is widely used is the so-called Bingham-plastic model This model involves a yield stress, namely a threshold on strain rate: for values of the strain rate above this threshold the fluid behaves like a viscous fluid, for values below, it looks like a solid. We refer to the papers by Liu and Mei [7] and Balmforth et al [2] for the study of such fluids in the lubrication approximation. Liu and Mei introduced in [8] a perturbed Bingham model, which is a two viscosities model, with a high viscosity for small deformations When this viscosity goes to ∞ the Bingham model is recovered, giving a fluid mechanics interpretation of this solid behaviour. We provide a few numerical illustrations based on a finite volume scheme
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