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We examine the translation of a sphere in a stratified ambient in the limit of small Reynolds numbers (
$Re \ll 1$) and viscous Richardson numbers (
$Ri_v \ll 1$); here, 
$Re = {\rho Ua}/{\mu }$ and 
$Ri_v = {\gamma a^3 g}/{\mu U}$, with 
$a$ being the sphere radius, 
$U$ the translation speed, 
$\rho$ and 
$\mu$ the density and viscosity of the stratified ambient, 
$g$ the acceleration due to gravity, and 
$\gamma$ the density gradient characterizing the ambient stratification. In contrast to most earlier efforts, our study considers the convection-dominant limit corresponding to 
$Pe = {Ua}/{D} \gg 1$, 
$D$ being the diffusivity of the stratifying agent. We characterize in detail the velocity and density fields around the particle in what we term the Stokes stratification regime, defined by 
$Re \ll Ri_v^{{1}/{3}} \ll 1$, and corresponding to the dominance of buoyancy over inertial forces. Buoyancy forces associated with the perturbed stratification fundamentally alter the viscously dominated fluid motion at large distances of order the stratification screening length that scales as 
$a\,Ri_v^{-{1}/{3}}$. The motion at these distances transforms from the familiar fore–aft symmetric Stokesian form to a fore–aft asymmetric pattern of recirculating cells with primarily horizontal motion within, except in the vicinity of the rear stagnation streamline. At larger distances, the motion is vanishingly small except within (a) an axisymmetric horizontal wake whose vertical extent grows as 
$O(r_t^{{2}/{5}})$, 
$r_t$ being the distance in the plane perpendicular to translation, and (b) a buoyant reverse jet behind the particle that narrows as the inverse square root of distance downstream. As a result, for 
$Pe = \infty$, the motion close to the rear stagnation streamline starts off pointing in the direction of translation, in the inner region, and decaying as the inverse of the downstream distance; the motion reverses beyond distance 
$1.15a\,Ri_v^{-{1}/{3}}$, with the eventual reverse flow in the far-field buoyant jet again decaying as the inverse of the distance downstream. For large but finite 
$Pe$, the narrowing jet is smeared out beyond a distance of 
$O(a\,Ri_v^{-{1}/{2}}\, Pe^{{1}/{2}})$, leading to an exponential decay of the aforementioned reverse flow.
This paper examines the shape dynamics of deformable elastic and viscoelastic particles in an ambient Newtonian fluid subjected to simple shear. The particles are allowed to undergo large deformation, with the elastic stress determined using the neo-Hookean constitutive relation. We first present a method to determine the shape dynamics of initially ellipsoidal particles that is an extension of the method of Roscoe (J. Fluid Mech., vol. 28, issue 2, 1967, pp. 273–293), originally used to determine the shape at steady state of an initially spherical particle. We show that our method recovers earlier results for the in-plane trembling and tumbling dynamics of initially prolate spheroids in simple shear flow, obtained by a different approach. We then examine the in-plane dynamics of oblate spheroids and triaxial ellipsoids in simple shear flow, and show that they too, like prolate spheroids, exhibit time-periodic tumbling or trembling dynamics, depending on the initial aspect ratios of the particle and the elastic capillary number 
$G \equiv \mu \dot {\gamma }/\eta$, where 
$\mu$ is the viscosity of the fluid, 
$\eta$ is the elastic shear modulus of the particle and 
$\dot {\gamma }$ is the shear rate. In addition, we find a novel state wherein the particle extends indefinitely in time and asymptotically aligns with the flow axis. We demarcate all the dynamical regimes in the parameter space comprising 
$G$ and the initial particle aspect ratios. When the particles are viscoelastic, damped oscillatory dynamics is observed for initially spherical particles, and the tumbling–trembling boundary is altered for initially prolate spheroids so as to favour tumbling.
We analyse the instability of a vortex column in a dilute polymer solution at large 
${{Re}}$ and 
${{De}}$ with 
${{El}} = {{De}}/{{Re}}$, the elasticity number, being finite. Here, 
${{Re}} = \varOmega _0 a^2/\nu$ and 
${{De}} = \varOmega _0 \tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity (
$\varOmega _0$), the radius of the column (
$a$), the total (solvent plus polymer) kinematic viscosity (
$\nu = (\mu _s +\mu _p)/\rho$ with 
$\mu _s$ and 
$\mu _p$ being the solvent and polymer contributions to the viscosity) and the polymeric relaxation time (
$\tau$). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by 
${E} = {{El}}(1-\beta )$, 
$\beta$ being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. Unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite 
${E}$ due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold for this instability. The growth rate is 
$O(\varOmega _0)$ for order unity 
$E$, although it becomes transcendentally small for 
${E} \ll 1$, being 
$O(\varOmega _0 {E}^2{\rm e}^{-1/{E}^{{1}/{2}}})$. An accompanying numerical investigation shows that the instability persists for smooth monotonically decreasing vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain 
${E}$-dependent threshold.
We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this article. Although the literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by 
$We < n^2 + k^2 -1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here, 
$We = \rho \varOmega ^2 a^3 / \gamma$ is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with 
$\varOmega$ being the angular velocity of the rigidly rotating column, 
$a$ the column radius, 
$\rho$ the density of the fluid and 
$\gamma$ the surface tension coefficient; 
$k$ and 
$n$ denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the 
$We$–
$k$ plane. For all 
$n > 1$, the viscously unstable region, corresponding to 
$We > n^2 + k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the 
$We$ axis, persists for 
$n=1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy.
