Thin, viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitchlike patterns including meanders, W patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasistatic geometrical model which captures the patterns, consisting of three coupled ordinary differential equations for the radial deflection, the orientation, and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the belt speed.

We identify 'minimal seeds' for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density E c that trigger turbulence from the laminar state, in stratified plane Couette flow, the flow between two horizontal plates of separation 2H, moving with relative velocity 2 U, across which a constant density difference 2 ρ from a reference density ρ r is maintained. To find minimal seeds, we use the 'direct-adjoint-looping' (DAL) method for finding nonlinear optimal perturbations that optimise the time-averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system's 'edge state'. Therefore, the trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (Phys. Fluids, vol. 9, 1997, pp. 883-900), or equivalently finite Reynolds number solutions of the vortex-wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666). The stratified coherent states we identify at moderate Reynolds number display an altered form from their unstratified counterparts for bulk Richardson numbers Ri B = g ρH/(ρ r U 2 ) = O(Re −1 ), and exhibit chaotic motion for larger Ri B . We demonstrate that at hith Reynolds number the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when Ri B = O(Re −2 ).

A combined theoretical and experimental study is presented for the flow-induced compaction of a onedimensional fibrous porous medium near its gel point for deformation at low and high rates. The theory is based on a two-phase model in which the permeability is a function of local solid fraction, and the deformation of the solid is resisted by both a compressive yield stress and a rate-dependent bulk viscosity. All three material properties are parameterized and calibrated for cellulose fibres using sedimentation, permeation and filtration experiments. It is shown that the incorporation of rate-dependence in the solid stress significantly improves the agreement between theory and experiment when the drainage flow is relatively rapid. The model is extended to rates outside the range where it was calibrated to understand the dynamics of a standard test for pulp suspensions: the Canadian Standard Freeness test. The model adequately captures all of the experimental findings, including the score of the freeness test, which is found to be sensitively controlled by the bulk solid viscosity and to a lesser degree by the permeability law, but depends only weakly on the compressive yield stress.

We present the linear stability properties and nonlinear evolution of two-dimensional plane Couette flow for a statically stable Boussinesq three-layer fluid of total depth$2h$between two horizontal plates driven at constant velocity$\pm \unicode[STIX]{x0394}U$. Initially the three layers have equal depth$2h/3$and densities$\unicode[STIX]{x1D70C}_{0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$,$\unicode[STIX]{x1D70C}_{0}$and$\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, such that$\unicode[STIX]{x1D70C}_{0}\gg \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$. At finite Reynolds and Prandtl number, we demonstrate that this flow is susceptible to two distinct primary linear instabilities for sufficiently large bulk Richardson number$Ri_{B}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}h/(\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x0394}U^{2})$. For a given bulk Richardson number$Ri_{B}$, the zero phase speed ‘Taylor’ instability is always predicted to have the largest growth rate and to be an inherently two-dimensional instability. An inherently viscous instability, reminiscent of the ‘Holmboe’ instability, is also predicted to have a non-zero growth rate. For flows with Prandtl number$Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, where$\unicode[STIX]{x1D708}$is the kinematic viscosity, and$\unicode[STIX]{x1D705}$is the diffusivity of the density distribution, we find that the most unstable Taylor instability, maximized across wavenumber and$Ri_{B}$, has a (linear) growth rate which is a non-monotonic function of Reynolds number$Re=\unicode[STIX]{x0394}Uh/\unicode[STIX]{x1D708}$, with a global maximum at$Re=700$over 50 % larger than the growth rate as$Re\rightarrow \infty$. In a fully nonlinear evolution of the flows with$Re=700$and$Pr=1$, the two interfaces between the three density layers diffuse more rapidly than the underlying instabilities can grow from small amplitude. Therefore, we investigate numerically the nonlinear evolution of the flow at$Re=600$and$Pr=300$, and at$Re=5000$and$Pr=70$in two-dimensional domains with streamwise extent equal to two wavelengths of the Taylor instability with the largest growth rate. At both sets of parameter values, the primary Taylor instability undergoes a period of identifiable exponential ‘linear’ growth. However, we demonstrate that, unlike the so-called ‘Kelvin–Helmholtz’ instability that it superficially resembles, the Taylor instability’s finite-amplitude state of an elliptical vortex in the middle layer appears not to saturate into a quasiequilibrium state, but is rapidly destroyed by the background shear. The decay process reveals$Re$-dependent secondary processes. For the$Re=600$simulation, this decay allows the development to finite amplitude of the co-existing primary ‘viscous Holmboe wave instability’, which has a substantially smaller linear growth rate. For the$Re=5000$simulation, the Taylor instability decay induces a non-trivial modification of the mean velocity and density distributions, which nonlinearly develops into more classical finite-amplitude Holmboe waves. In both cases, the saturated nonlinear Holmboe waves are robust and long-lived in two-dimensional flow.

We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number Pr, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, Re = 400. In the two asymptotic limits Pr → 0 and Pr → ∞, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the Pr → 0 limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number Ri b O(Re −2 Pr −1 ). Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as Pr → ∞. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing Ri b . An asymptotic analysis is presented in the passive scalar regime Ri b O(Re −2 ), which reveals O(Pr −1/3 )-thick stratified boundary layers with O(Pr −2/9 )wide eruptions, giving rise to density fingers of O(Pr −1/9 ) length and O(Pr −4/9 ) width that invade an otherwise homogeneous interior. Finally, increasing Re to 10 5 in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.