A simplified model is derived for pressure-driven flow between adjacent surfaces of materials modeled as seemingly viscoplastic or truly viscoplastic. The material response to external forces is traditionally described by constitutive relations in which the extra stress tensor $S$ is expressed as a function of the symmetric part of the velocity gradient $D$. However, for viscoplastic materials, $S$ cannot, in general, be written as a function of $D$, whereas $D$ can be expressed in terms of $S$. Motivated by this observation, a model based on constitutive relations of the form $D = f(S)$ is proposed, leading to a system of first-order partial differential equations. A local Poiseuille law is also formulated, and a reduced-dimensional equation for the pressure is derived. Explicit velocity profiles are obtained for selected cases.

In modeling flows of compressible fluids, it is common practice to set the bulk viscosity equal to zero. This assumption, known as Stokes’ hypothesis, has been the subject of debate since its introduction by Stokes in 1845. In this work, we present continuum-mechanical, experimental, and atomistic arguments showing that Stokes’ hypothesis is not valid for any real fluid. Nevertheless, it is well known that modeling approaches based on this assumption often yield results of acceptable accuracy in practical applications. We show that this apparent success is not primarily due to the bulk viscosity being zero or negligibly small. Rather, whether bulk-viscous effects may be neglected depends on a careful analysis of the flow under consideration. We present a novel kinematic criterion that replaces Stokes’ constitutive hypothesis and determines in which flows bulk-viscous effects may be neglected.

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 pressure-driven flow between adjacent surfaces, where the fluid is assumed to have constant density. The main novelty lies in using implicit algebraic constitutive relations to describe the fluid’s response to external stimuli, enabling the modeling of fluids whose responses cannot be accurately captured by conventional methods. When the implicit algebraic constitutive relations cannot be solved for the Cauchy stress in terms of the symmetric part of the velocity gradient, the traditional approach of inserting the expression for the Cauchy stress into the equation for the balance of linear momentum to derive the governing equation for the velocity becomes inapplicable. Instead, a non-standard system of first-order equations governs the flow. This system is highly complex, making it important to develop simplified models. Our primary contribution is the development of a framework for achieving this. Additionally, we apply our findings to a fluid that exhibits an S-shaped curve in the shear stress versus shear rate plot, as observed in some colloidal solutions.

Flows of incompressible Navier–Stokes (Newtonian) fluids between adjacent surfaces are encountered in numerous practical applications, such as seal leakage and bearing lubrication. In seals, the flow is primarily pressure-driven, whereas, in bearings, the dominating driving force is due to shear. The governing Navier–Stokes system of equations can be significantly simplified due to the small distance between the surfaces compared to their size. From the simplified system, it is possible to derive a single lower-dimensional equation, known as the Reynolds equation, which describes the pressure field. Once the pressure field is computed, it can be used to determine the velocity field. This computational algorithm is much simpler to implement than a direct numerical solution of the Navier–Stokes equations and is therefore widely employed by engineers. The primary objective of this article is to investigate the possibility of deriving a type of Reynolds equation also for non-Newtonian fluids, using the balance of linear momentum. By considering power-law fluids we demonstrate that it is not possible for shear-driven flows, whereas it is feasible for pressure-driven flows. Additionally, we demonstrate that in the full 3D model, a normal stress boundary condition at the inlet/outlet implies a Dirichlet condition for the pressure in the Reynolds equation associated with pressure-driven flow. Furthermore, we establish that a Dirichlet condition for the velocity at the inlet/outlet in the 3D model results in a Neumann condition for the pressure in the Reynolds equation.

This paper constitutes the third part of a series of works on lower-dimensional models in lubrication. In Part A, it was shown that implicit constitutive theory must be used in the modelling of incompressible fluids with pressure-dependent viscosity and that it is not possible to obtain a lower-dimensional model for the pressure just by letting the film thickness go to zero, as in the proof of the classical Reynolds equation. In Part B, a new method for deriving lower-dimensional models of thin-film flow of fluids with pressure-dependent viscosity was presented. Here, in Part C, we also incorporate the energy equation so as to include fluids with both temperature and pressure dependent viscosity. By asymptotic analysis of this system, as the film thickness goes to zero, we derive a simplified model of the flow. We also carry out an asymptotic analysis of the boundary condition, in the case where the normal stress is specified on one part of the boundary and the velocity on the remaining part.

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.