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The fully developed, steady, incompressible, laminar flow in a channel delimited by rough and/or permeable walls is considered. The influence of the micro-structured bounding surfaces on the channel flow behavior is mimicked by imposing high-order effective boundary conditions which stem from homogenization theory and do not contain empirical parameters. A closed-form solution of the Navier–Stokes equations is found for the flow in the channel, with conditions at each virtual boundary linking the slip velocities to shear stress and streamwise pressure gradient. The boundary condition for the longitudinal velocity coincides with a little-noticed extension of the so-called Beavers-Joseph condition, first derived by Saffman (1971); it applies to both permeable and rough surfaces, including the case of separated flow (Couette-Poiseuille motion with adverse pressure gradient). The analytical solution obtained for the velocity distribution in the channel is validated against full feature-resolving simulations of the flow for either highly ordered or random textures, highlighting the accuracy and the applicability range of the model. The Stokes-based model used to identify slip and interface permeability coefficients in the effective boundary conditions is found to be reliable and accurate up to ϵ R e τ ≈ 10 , with ϵ ratio of microscopic to macroscopic length scales and R e τ the shear-velocity Reynolds number. Above this threshold, the coefficients must account for advective effects: a new upscaling procedure, based on an Oseen's approximation , is thus proposed and validated, extending considerably beyond the Stokes regime. • Laminar Poiseuille and Couette-Poiseuille flows in channels with non-smooth boundaries are studied. • Effective boundary conditions, based on homogenization theory, mimic flow interaction with textured walls. • Analytical solutions for channel flows are derived and extensively validated. • A new approach is proposed where macroscopic coefficients are advection-sensitive. • Improved model is highly accurate, beyond the Stokes regime. [ABSTRACT FROM AUTHOR]

We study a generalization of the Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to nonisothermal flows in an infinite plane channel under the influence of magnetic field. For the linearized problem (when the basic solution is an analogue of the classical Poiseuille flow for a viscous fluid described by the Navier–Stokes equations) we find a formal asymptotic representation for the eigenvalues under the growth of their modulus. We obtain a necessary condition for the asymptotic stability of the Poiseuille-type shear flow. For analysis we use new result, that generalizes Birkhoff theorem on the case, when the coefficient matrix of the eigenvalue itself has zero with multiplicity greater than one as an eigenvalue. We also get the necessary condition for Lyapunov stability of the shear Poiseuille-type flow as a result of acquired representation. [ABSTRACT FROM AUTHOR]

The nonlinear dynamics of two immiscible superposed viscous fluid layers in a channel is examined using asymptotic modelling and direct numerical simulations (DNS). The flow is driven by an imposed axial pressure gradient. Working on the assumption that one of the layers is thin, a weakly-nonlinear evolution equation for the interfacial shape is derived that couples the dynamics in the two layers via a nonlocal integral term whose kernel is determined by solving the linearised Navier–Stokes equations in the thicker fluid. The model equation incorporates salient physical effects including inertia, gravity, and surface tension, and allows for comparison with DNS at finite Reynolds numbers. Direct comparison of travelling-wave solutions obtained from the model equation and from DNS show good agreement for both stably and unstably stratified flows. Both the model and the DNS indicate regions in parameter space where unimodal, bimodal and trimodal waves co-exist. Nevertheless, the asymptotic model cannot capture the dynamics for a sufficiently strong unstable density stratification when interfacial break-up and eventual dripping occurs. In this case, complicated interfacial dynamics arise from the dominance of the gravitational force over the shear force due to the underlying flow, and this is investigated in detail using DNS. [ABSTRACT FROM AUTHOR]

Abstract We consider the Newtonian Poiseuille flow in a duct the cross section of which is either a circular or an annular sector assuming that Navier slip occurs either along both the cylindrical walls or only along the outer cylindrical wall. A general analytical solution is derived and the results for the latter case are discussed and the effects of the angle of the sector, the radii ratio and the slip number are analysed. Highlights • The Newtonian Poiseuille flow in ducts of annular-sector or circular-sector cross-sections is considered. • Navier slip is assumed to occur along the circular walls. • A general analytical solution is derived for the above flow. • The effects of the sector angle, the radii ratio and the slip number are analysed. [ABSTRACT FROM AUTHOR]