A solid long slender body with curved centreline is held at rest in a fluid undergoing a uniform flow. Assuming that the Reynolds number Re based on body length is fixed, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio κ of the cross-sectional radius to body length. In the limit of large Re, this result is no longer valid and an asymptotic expansion in κRe is necessary. A uniformly valid solution is obtained from these two expansions. The total force and torque acting on a body with a straight centreline are explicitly determined. The limiting cases of small and large Re are studied in detail.

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The influence of inertia and elasticity on the onset and stability of Rayleigh–Bénard thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The Galerkin projection method is used to obtain the departure from the conduction state. The solution is capable of displaying complex dynamical behaviour for viscoelastic fluids in the elastic and inertio-elastic ranges, which correspond to ${\it Ra} \,{<}\, {\it Ra}_c^s$ and ${\it Ra} \,{>}\, {\it Ra}_c^s $, respectively, ${\it Ra}_c^s $ being the critical Rayleigh number at which stationary thermal convection emerges. This behaviour is reminiscent of that observed experimentally for viscoelastic Taylor–Couette flow. For a given ${\it Ra}$ in the pre-critical range, finite-amplitude periodic oscillatory convection emerges when the elasticity number, $E$, exceeds a threshold. Periodicity is lost as $E$ increases, leading to a $T^{2}$ quasi-periodic behaviour, and the breakup of the torus as $E$ increases further. Although no experimental data are available for direct comparison, this scenario is reminiscent of the flow sequence observed by Muller et al. (1993) in the Taylor–Couette flow of a Boger fluid. Stationary thermal convection emerges, via a supercritical bifurcation, when ${\it Ra}$ exceeds ${\it Ra}_c^s $. The amplitude of motion is found to be little influenced by fluid elasticity or retardation time, especially as the Rayleigh number increases. However, the range of stability of the stationary thermal convection narrows considerably for viscoelastic fluids. In this case, oscillatory thermal convection is favoured. The onset and the stability of other steady convective patterns, namely hexagons and squares, are studied in the inertio-elastic range by using an amplitude equation approach. The range of stability of each pattern is examined, simultaneously allowing the validation of the two-dimensional picture.

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The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$, where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.