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A concise review is given of astrophysically motivated experimental and theoretical research on Taylor–Couette flow. The flows of interest rotate differentially with the inner cylinder faster than the outer, but are linearly stable against Rayleigh’s inviscid centrifugal instability. At shear Reynolds numbers as large as 10 6 , hydrodynamic flows of this type (quasi-Keplerian) appear to be nonlinearly stable: no turbulence is seen that cannot be attributed to interaction with the axial boundaries, rather than the radial shear itself. Direct numerical simulations agree, although they cannot yet reach such high Reynolds numbers. This result indicates that accretion-disc turbulence is not purely hydrodynamic in origin, at least insofar as it is driven by radial shear. Theory, however, predicts linear magnetohydrodynamic (MHD) instabilities in astrophysical discs: in particular, the standard magnetorotational instability (SMRI). MHD Taylor–Couette experiments aimed at SMRI are challenged by the low magnetic Prandtl numbers of liquid metals. High fluid Reynolds numbers and careful control of the axial boundaries are required. The quest for laboratory SMRI has been rewarded with the discovery of some interesting inductionless cousins of SMRI, and with the recently reported success in demonstrating SMRI itself using conducting axial boundaries. Some outstanding questions and near-future prospects are discussed, especially in connection with astrophysics. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 2)’. 

Abstract Gamma-ray binaries are luminous in gamma rays, composed of a compact object orbiting a massive companion star. The interaction between these two objects can drive relativistic outflows, either jets or winds, in which particles can be accelerated to energies reaching hundreds of teraelectronvolts (TeV). However, it is still debated where and under which physical conditions particles are accelerated in these objects and ultimately whether protons can be accelerated up to PeV energies. Among the well-known gamma-ray binaries, LS 5039 is a high-mass X-ray binary with an orbital period of 3.9 days that has been observed up to TeV energies by the High Energy Stereoscopic System. We present new observations of LS 5039 obtained with the High Altitude Water Cherenkov (HAWC) observatory. Our data reveal that the gamma-ray spectrum of LS 5039 extends up to 200 TeV with no apparent spectral cutoff. Furthermore, we confirm, with a confidence level of 4.7σ, that the emission between 2 and 118 TeV is modulated by the orbital motion of the system, and find a 2.2σhint of variability above 100 TeV. This indicates that these photons are likely produced within or near the binary orbit, where they can undergo absorption by the stellar photons. In a leptonic scenario, the highest energy photons detected by HAWC can be emitted by ∼200 TeV electrons inverse Compton scattering stellar photons, which would require an extremely efficient acceleration mechanism operating within LS 5039. Alternatively, a hadronic scenario could explain the data through proton–proton or proton–gamma collisions of protons accelerated to petaelectronvolt energies.