The flow in a rapidly rotating cylinder forced by the harmonic oscillations of a small sphere along the rotation axis is explored numerically. For oscillation frequencies less than twice the cylinder rotation frequency, the forced response flows feature conical shear layers emitted from the critical latitudes of the sphere. These latitudes are where the characteristics of the hyperbolic system, arrived at by ignoring nonlinear, viscous and forcing terms in the governing equations, are tangential to the sphere. These conical shear layers vary continuously with the forcing frequency so long as it remains inertial. At certain values of the forcing frequency, linear inviscid inertial modes of the cylinder are resonated. Of all possible inertial modes, only those whose symmetries are compatible with the symmetry of the forced system are resonated. This all occurs even in the linear limit of vanishingly small forcing amplitude. As the forcing amplitude is increased, nonlinearity leads to non-harmonic oscillations and a non-zero mean flow which features a Taylor columnar structure extending from the sphere to the two endwalls in an axially invariant fashion.

Pulsatile fluid flows through straight pipes undergo a sudden transition to turbulence that is extremely difficult to predict. The difficulty stems here from the linear Floquet stability of the laminar flow up to large Reynolds numbers, well above experimental observations of turbulent flow. This makes the instability problem fully nonlinear and thus dependent on the shape and amplitude of the flow perturbation, in addition to the Reynolds and Womersley numbers and the pulsation amplitude. This problem can be tackled by optimising over the space of all admissible perturbations to the laminar flow. In this paper, we present an adjoint optimisation code, based on a GPU implementation of the pseudo-spectral Navier–Stokes solver nspipe, which incorporates an automatic, optimal checkpointing strategy. We leverage this code to show that the flow is susceptible to two distinct instability routes: one in the deceleration phase, where the flow is prone to oblique instabilities, and another during the acceleration phase with similar mechanisms as in steady pipe flow. Instability is energetically more likely in the deceleration phase. Specifically, localised oblique perturbations can optimally exploit nonlinear effects to gain over nine orders of magnitude in energy at a peak Reynolds number of ${\textit{Re}}_{\textit{max}}\approx 4000$. These oblique perturbations saturate into regular flow patterns that decay in the acceleration phase or break down to turbulence depending on the flow parameters. In the acceleration phase, optimal perturbations are substantially less amplified, but generally trigger turbulence if their amplitude is sufficiently large.

In this paper, we consider the flow of a nematic liquid crystal in the domain exterior to a small spherical particle. We work within the framework of the $\unicode{x1D64C}$-tensor model, taking into account the orientational elasticity of the medium. Under a suitable regime of physical parameters, the governing equations can be reduced to a system of linear partial differential equations. Our focus is on precise far-field asymptotics of the flow velocity with an emphasis on its anisotropic behaviour. We are able to analytically characterize the flow pattern and compare it with that of the classical isotropic Stokes flow. The expression for velocity away from the particle can be computed numerically or symbolically.

Uniform momentum zones (UMZs) are widely used to describe and model the coherent structure of wall-bounded turbulent flows, but their detection has traditionally relied on relatively narrow fields of view which preclude fully resolving features at the scale of large-scale motions (LSMs). We refine and extend recent proposals to detect UMZs with moving-window fields of view by including physically motivated coherency criteria. Using synthetic data, we show how this updated moving-window approach can eliminate noise contamination that is likely responsible for the previously reported, high fractal dimension of UMZ interfaces. By applying the approach to channel flow direct numerical simulation (DNS), we identify a significant number of previously undetected, large-scale UMZ interfaces, including a small fraction of highly linear interfaces with well-defined streamwise inclination angles. We show that the inclination angles vary inversely with the size of the UMZ interfaces and that this relationship can be modelled by the opposing effects of shear-induced inclination and vortex-induced lift-up on hairpin packets. These geometric properties of large-scale UMZ interfaces play an important role in the development of improved stochastic models of wall-bounded turbulence.

We derive effective Boussinesq and Korteweg–de Vries equations governing nonlinear wave propagation over a structured bathymetry using a three-scale homogenization approach. The model captures the anisotropic effects induced by the bathymetry, leading to significant modifications in soliton dynamics. Homogenized parameters, determined from elementary cell problems, reveal strong directional dependencies in wave speed and dispersion. Our results provide new insights into nonlinear wave propagation in structured shallow-water environments, and consequently motivate further fundamental and applied studies in wave hydrodynamics and coastal engineering.

We investigate flow instability produced by viscosity and density discontinuities at the interface separating two Newtonian fluids in generalised Couette–Poiseuille (GCP) flow. The base flow, driven by counter-moving plates and an inclined pressure gradient at angle $0^\circ \leqslant \phi \leqslant 90^\circ$, exhibits a twisted, two-component velocity profile across the layers, characterised by the Couette–Poiseuille magnitude parameter $0^\circ \leqslant \theta \leqslant 90^\circ$. Plane Couette–Poiseuille (PCP) flow at $ \phi = 0^\circ$ is considered as a special case. Flow/geometry parameters are $(\phi ,\theta )$, a Reynolds number $Re$ and the viscosity, depth and density ratios $(m,n,r)$, respectively. A mapping from the GCP to PCP extended Orr–Sommerfeld equations is found that simplifies the numerical study of interfacial-mode instabilities, including determination of shear-mode critical parameters. For interfacial modes, unstable regions in $(m,n,r)$ space are delineated by three distinct surfaces found via long-wave analysis, with the exception of strict Couette flow where the $(m,n)$ surface asymptotically vanishes with $\theta \rightarrow 0^\circ$. In interfacial stable regions but with unstable shear modes, one-layer PCP stability can be identified with a cut-off $\theta$ that conforms to canonical PCP stability. Competition between the interfacial-mode reversal phenomenon and the shear-mode cut-off behaviour is discussed. Extending to the full GCP configuration with the mapping algorithms applied, we systematically chart how pressure-gradient inclination and perturbation wavefront angle shift the balance between interfacial and shear instabilities in a specific case.

We present theoretical models for flow and diffusion in microfluidic polygonal mixers of arbitrary shapes. Combining work based on Boussinesq coordinates with modern methods for the calculation of the Schwarz–Christoffel transform, we present an integrated method that yields analytical solutions for both flow and concentration profiles everywhere in microfluidic mixers with arbitrary numbers of inlets. We illustrate how the problem can be reduced to a sequence of conformal maps to a known domain, where the advection–diffusion problem can be readily solved, and map back the solution to the geometry of interest. We use the method to model a number of previously published microfluidic mixer geometries, used in lipid nanoparticle synthesis, among others. The method is also applicable to other problems described by planar transport equations in polygonal domains, for instance, in groundwater flows or electrokinetics.