The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.

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Most of the problems in lubrication are studied within the context of Reynolds’ equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero.

Boundedness of the maximal operator and the Calderón–Zygmund singular integral operators in central Morrey–Orlicz spaces were proved in papers (Maligranda et al. in Colloq Math 138:165–181, 2015; Maligranda et al. in Tohoku Math J 72:235–259, 2020) by the second and third authors. The weak-type estimates have also been proven. Here we show boundedness of the Riesz potential in central Morrey–Orlicz spaces and the corresponding weak-type version.

We prove the boundedness of the Riesz fractional integration operator from a generalized Morrey space