We analyze stationary Stokes flow of a Navier–Stokes fluid in a thin tube with a variable cross-section. The objective is to derive a simplified model by examining the asymptotic behavior as the tube's thickness approaches zero. The flow is driven by a pressure gradient between the inlet and outlet, which is modeled by prescribing the normal component of the stress tensor at the tube's ends. Using multiple-scale asymptotic expansions, we first obtain an approximate solution. This inner approximation is accurate for thin tubes, except in thin boundary layers near the inlet and outlet. To address this, we refine the approximation by introducing boundary layer correctors. Finally, we rigorously prove an error estimate for the difference between the exact solution and the improved approximate solution.

We consider Stokes flow past cylindrical obstacles in a generalized Hele-Shaw cell, i.e. a thin three-dimensional domain confined between two surfaces. The flow is assumed to be driven by an external pressure gradient, which is modeled as a normal stress condition on the lateral boundary of the cell. On the remaining part of the boundary we assume that the velocity is zero. We derive a divergence-free (volume preserving) approximation of the flow by studying its asymptotic behavior as the thickness of the domain tends to zero. The approximation is verified by error estimates for both the velocity and pressure in H¹- and L²-norms, respectively.

We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, i.e. a thin three-dimensional domain which is confined between two curved surfaces and contains a cylindrical obstacle. The fluid is assumed to be of power-law type defined by the exponent 1< p<∞. By letting the thickness of the domain tend to zero we obtain a generalized form of the Poiseuille law, i.e. the limit velocity is a nonlinear function of the limit pressure gradient. The flow is assumed to be driven by an external pressure which is applied as a normal stress along the lateral part of the boundary. On the remaining part of the boundary we impose a no-slip condition. The two-dimensional limit problem for the pressure is a generalized form of the p′-Laplace equation, 1/p+1/p'=1, with a coefficient called ‘flow factor’, which depends on the geometry as well as the power-law exponent. The boundary conditions are preserved in the limit as a Dirichlet condition for the pressure on the lateral boundary and as a Neumann condition for the pressure on the solid obstacle.

It has been over a century since the interest in inventing the optimal topology for bearings arose. A significant achievement was published by Lord Rayleigh, who found the step-bearing geometry which maximise the load-carrying capacity when the classical Reynolds equation is used to model thin film flow of an iso-viscous and incompressible fluid. Since then, new optimisation methods considering some variants of governing equations for finding the best possible bearings have surfaced, one of which will be presented in this paper. Here, two different formulations for compressible flow, i.e. ideal gas and constant bulk modulus compressibility, as well as the classical Reynolds formulation will be used in combination with the method of moving asymptotes for topological optimisation. All three of these problem formulations provide us with unique geometries, which either maximise the load-carrying capacity or minimise friction, for fluids with a wide variety of compressibility.

We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuille law, i.e. the limit velocity field is a linear function of the pressure gradient. By prescribing the external pressure as a normal stress condition, we recover a Dirichlet condition for the limit pressure. In contrast, a Dirichlet condition for the velocity yields a Neumann condition for the limit pressure.

We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in . Particular attention is given to the mixed boundary condition: a Dirichlet condition is imposed for the velocity on one part of the boundary while a Neumann condition for the stress tensor is imposed on the remaining part. Some minor modifications to the standard theory are therefore required. The most noteworthy result is that both pressure and velocity are unique.

This work relates to previous studies concerning the asymptotic behavior of Stokes flow in a narrow gap between two surfaces in relative motion. It is assumed that one of the surfaces is rough, with small roughness wavelength l, so that the film thickness h becomes rapidly oscillating. Depending on the limit of the ratio h/l, denoted as k, three different lubrication regimes exist: Reynolds roughness (k-0), Stokes roughness (0<γ<1), and high-frequency roughness (γ = ∞). In each regime, the pressure field is governed by a generalized Reynolds equation, whose coefficients (so-called flow factors) depend on k. To investigate the accuracy and applicability of the limit regimes, we compute the Stokes flow factors for various roughness patterns by varying the parameter k. The results show that there are realistic surface textures for which the Reynolds roughness is not accurate and the Stokes roughness must be used instead.

This paper is devoted to homogenization of different models of flow in thin domains with a microstructure. The focus is on applications connected to the effect of surface roughness in full film lubrication, but a parallel to flow in thin porous media is also discussed. Mathematical models of such flows naturally include two small parameters. One is connected to the fluid film thickness and the other to the microstructure. The corresponding asymptotic analysis is a delicate problem, since the result depends on how fast the two small parameters tend to zero relative to each other. We give a review of the current status in this area and point out some future challenges.

We homogenize stationary incompressible Stokes flow in a periodic porous medium. The fluid is assumed to satisfy a no-slip condition on the boundary of solid inclusions and a normal stress (traction) condition on the global boundary. Under these assumptions, the homogenized equation becomes the classical Darcy law with a Dirichlet condition for the pressure.