In the present paper, we investigate the path instabilities and the transition scenario for oblate homogeneous spheroids falling (or ascending) freely in a quiescent and unconfined Newtonian fluid under the action of gravity, buoyancy and hydrodynamic forces. The problem depends on three independent external parameters: the aspect ratio $\unicode[STIX]{x1D712}=d/a$ , where $d$ is the diameter and $a$ the length of the axisymmetry axis of the spheroid; the non-dimensionalized mass $m^{\ast }=m/(\unicode[STIX]{x1D70C}d^{3})$ , $m$ being the mass of the spheroid and $\unicode[STIX]{x1D70C}$ the fluid density; and the Galileo number, defined as $G=\sqrt{(m^{\ast }-V^{\ast })gd^{3}}/\unicode[STIX]{x1D708}$ . In the definition of the Galileo number, $V^{\ast }=V/d^{3}$ is the non-dimensionalized volume, $g$ the gravitational acceleration and $\unicode[STIX]{x1D708}$ is the kinematic viscosity. Asymptotic solutions (regimes) are investigated in seven $\unicode[STIX]{x1D712}=\text{const}.$ parameter planes going from $\unicode[STIX]{x1D712}=10$ (very flat spheroid) to $\unicode[STIX]{x1D712}=1.1$ (an almost spherical shape), for $m^{\ast }$ going from 0 to 5 and Galileo numbers up to 300 (i.e. Reynolds numbers roughly up to 500). The obtained results provide a link between the known scenario of a homogeneous disk and that, well known, of a sphere. The scenario of the flat spheroid of aspect ratio 10 has many common features with that of an infinitely thin disk, but the finite thickness brings about significant quantitative differences. At the opposite side of the investigated aspect ratio interval, the dynamics of the spheroid of aspect ratio 1.1 is found very close to that of a perfect sphere except for small density ratios (smaller than approximately 0.5). Very light spheroids of aspect ratio 1.1 move along vertical zig-zagging trajectories. At intermediate aspect ratios, the strong subcritical effects and the characteristic zig-zagging and fluttering motion, typical for flat bodies, progressively disappear. The tumbling regime remains remarkably stable and is shown to be present down to $\unicode[STIX]{x1D712}=2$ . An interesting result consists in the evidence of the first two bifurcations typical for the sphere scenario (leading to steady oblique and oblique oscillating trajectories) present both for very flat and thick spheroids but absent at intermediate aspect ratios. The results pertaining to spheroids might be more useful in practical applications than those obtained for too idealized thin disks and perfect spheres.

Local analysis of the onset of instability in flows that are not exactly parallel is considered. Corrections to the Orr–Sommerfeld equation arising as a consequence of the nonparallelism of the unperturbed flow are studied. The quasiparallel hypothesis is quantified on a model of Gaussian plane wave packets. It appears that the characteristic length scales of the downstream dependence of flow field characteristics must be substantially larger than the inverse of the wave number characterizing the instability. A new eigenvalue problem describing the propagation of these Gaussian wave packets is written. The relation between the marginal and absolute instability analysis for marginal Reynolds numbers is discussed. For flows varying slowly in the downstream direction, closed-form corrections of the Orr–Sommerfeld equation terms taking account of the x variation of the flow field and of the extension of the propagating wave packets are derived. A first-order perturbation theory correction of the Orr–Sommerfeld dispersion relation is proposed, allowing the reduction of the calculation of nonparallel corrections of the local instability quantities to quadratures. The proposed theory is applied to two important cases: the Blasius boundary layer and the cylinder wake. For the Blasius boundary layer the basic condition of applicability of the quasiparallel theory is found to be satisfied. However, the nonparallel correction of the critical Reynolds number is found to be non-negligible and provides a good agreement with experimental results. In the cylinder wake case direct bidimensional simulation results are used to assess the downstream variation of the flow field characteristics. The characteristic length scale of this variation in the near wake is found to be of the order of unity, which is also the magnitude of the wave numbers characterizing the local absolute instabilities in this region. Hence, the Orr–Sommerfeld analysis and any corrections based on the propagation of plane waves in the wake can hardly be expected to provide more than qualitative results.

The stability of the Taylor vortex flow in Newtonian and shear-thinning fluids is investigated in the case of a wide gap Taylor–Couette system. The considered radius ratio is $\eta = R_1/R_2=0.4$ . The aspect ratio (length over the gap width) of experimental configuration is 32. Flow visualization and measurements of two-dimensional flow fields with particle image velocimetry are performed in a glycerol aqueous solution (Newtonian fluid) and in xanthan gum aqueous solutions (shear-thinning fluids). The experiments are accompanied by axisymmetric numerical simulations of Taylor–Couette flow in the same gap of a Newtonian and a purely viscous shear-thinning fluid described by the Carreau model. The experimentally observed critical Reynolds and wavenumbers at the onset of Taylor vortices are in very good agreement with that obtained from a linear theory assuming a purely viscous shear-thinning fluid and infinitely long cylinders. They are not affected by the viscoelasticity of the used fluids. For the Newtonian fluid, the Taylor vortex flow (TVF) regime is found to bifurcate into a wavy vortex flow with a high frequency and low amplitude of axial oscillations of the vortices at ${Re} = 5.28 \, {Re}_c$ . At ${Re} = 6.9 \, {Re}_c$ , the frequency of oscillations decreases and the amplitude increases abruptly. For the shear-thinning fluids the secondary instability conserves axisymmetry. The latter is characterized by an instability of the array of vortices leading to a continuous sequence of creation and merging of vortex pairs. Axisymmetric numerical simulations reproduce qualitatively very well the experimentally observed flow behaviour.

This paper presents numerical simulations of the free fall of homogenous cylinders of length-to-diameter ratios $2$ , $3$ and 5 and solid-to-fluid-density ratios $\rho _s/\rho$ going from 0 to 10 in transitional regimes. The path instabilities are shown to be due to two types of transitional states. The well-known fluttering state is a solid mode, characterised by significant oscillations of the cylinder axis due to a strong interaction between the vortex shedding in the wake and the solid degrees of freedom. Weakly oscillating, mostly irregular trajectories, are fluid modes, associated with purely fluid instabilities in the wake. The interplay of solid and fluid modes leads to a varying scenario in which the length-to-diameter and density ratios play an important role. The description is accompanied by the presentation of the identified transitional states in terms of path characteristics and vorticity structure of the wakes and by bifurcation diagrams showing the evolution of asymptotic states with increasing Galileo numbers. There appears to be a strong difference between the behaviour of cylinders of aspect ratio $L/d=3$ and 5. A similar contrast is stated between light cylinders of density ratios $\rho _s/\rho \le 2$ and dense cylinders of density ratios 5 and 10. Finally, the question of the scatter of values of the drag coefficient and of the frequency of oscillations raised in the literature is addressed. It is shown, that in addition to external parameters (Galileo number, density and aspect ratio) the amplitude of oscillations characterising the instability development is to be taken into account to explain this scatter. Fits of the simulation results to simple correlations are proposed. Namely that of the drag coefficient proves to be accurate (better than 1 % of accuracy) but also that of the Strouhal number (a few per cent of accuracy) may be of practical use.