A spherical capsule (radius $R$) is suspended in a viscous liquid (viscosity $\mu$) and exposed to a uniaxial extensional flow of strain rate $E$. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus $G$ and an area dilatation modulus $K$. Dimensional arguments imply that the slenderness $\epsilon$ of the deformed capsule depends only upon $K/G$ and the elastic capillary number ${Ca}=\mu R E/G$. We address the coupled flow–deformation problem in the limit of strong flow, ${Ca}\gg 1$, where large deformation allows for the use of approximation methods in the limit $\epsilon \ll 1$. The key conceptual challenge, encountered at the very formulation of the problem, is in describing the Lagrangian mapping from the spherical reference state in a manner compatible with hydrodynamic slender-body formulation. Scaling analysis reveals that $\epsilon$ is proportional to ${Ca}^{-2/3}$, with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon $\ln \epsilon$. Going beyond scaling arguments, we employ asymptotic methods to obtain a reduced formulation, consisting of a differential equation governing a mapping field and an integral equation governing the axial tension distribution. The leading-order deformation is independent of the ratio $K/G$; in particular, we find the approximation $\epsilon ^{2/3} {Ca}\approx 0.2753\ln (2/\epsilon ^2)$ for the relation between $\epsilon$ and $Ca$. A scaling analysis for the neo-Hookean constitutive law reveals the impossibility of a steady slender shape, in agreement with existing numerical simulations. More generally, the present asymptotic paradigm allows us to rigorously discriminate between strain-softening and strain-hardening models.

We study the effects of Prandtl number $\mathit {Pr}$ and Rayleigh number $\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range $10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$, the viscous dissipation scales as $\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$, which is based on the observation that enstrophy $\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$, and the Nusselt number tends to follow the ‘ultimate’ scaling $\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of $\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum $\hat {E}_u(k) \propto k^{-2.3}$, which is close to $k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum $\hat {E}_\theta (k) \propto k^{-1.2}$, which deviates from the expected $k^{-7/5}$. Finally, at $\mathit {Pr} \to 0$ and $\infty$, we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.

We study the stability of steady convection rolls in two-dimensional Rayleigh–Bénard convection with free-slip boundaries and horizontal periodicity over 12 orders of magnitude in the Prandtl number $(10^{-6} \leq Pr \leq 10^6)$ and 6 orders of magnitude in the Rayleigh number $(8{\rm \pi} ^4 < Ra \leq 10^8)$. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio $\varGamma = 2$, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the $(Pr,Ra)$ plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit $Pr \to 0$ and find that the steady convection rolls become unstable almost instantaneously, eventually leading to nonlinear relaxation osculations and bursts, which we can explain with a weakly nonlinear analysis. In the complementary large-$Pr$ limit, we observe that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.

We derive an accurate estimate for the diffusive evaporation rates of multiple droplets of different sizes and arbitrary contact angles placed on a horizontal substrate. The derivation, which is based on a combination of Green's second identity and the method of reflections, simply makes use of the solution for the evaporation of a single droplet. The theoretical results can serve as a guide for future computational and experimental studies on the collective evaporation of arrays of droplets, as well as similar multi-body, diffusion-dominated transport problems.

A rare subclass of Type Ia supernovae (SNe Ia) shows evidence of strong interaction with a hydrogen-rich circumstellar medium (CSM); these objects are referred to as SNe Ia-CSM. PTF11kx began life as a SN Ia, but after a month it began to show indications of significant interaction with its CSM. This well-studied object solidified the connection between SNe Ia-CSM and more typical SNe Ia, despite their spectral similarity to Type IIn SNe (which likely come from massive star progenitors, as opposed to the white dwarf progenitors for the SNe Ia-CSM). The spectra of all ~20 known SNe Ia-CSM are dominated by Hα emission (with widths of ~2000 km s−1) and exhibit large Hα/Hβ intensity ratios; moreover, they have an almost complete lack of He I emission (see left panel of Figure 1). They also show possible evidence of dust formation through a decrease in the red wing of Hα 75–100 days past maximum brightness. The absolute magnitudes of SNe Ia-CSM are found to be -21.3 mag ≤ M R ≤ −19 mag (see right panel of Figure 1), and they also show ultraviolet emission at early times and strong infrared emission at late times (but no detected radio or X-ray emission). Finally, the host galaxies of SNe Ia-CSM imply that these objects come from a relatively young stellar population.

We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radius R(t) tends to infinity in finite time. As time t approaches the critical time t*, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thickness h in the central region is described by a similarity solution of the second kind, with h ∝ (t* − t)α where the exponent α satisfies a nonlinear eigenvalue problem. Finally, for non-zero surface tension, we find that the exponent increases rapidly to infinity at a critical value of the rotational Bond number B = 1/4. For B > 1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than to stretch, with the limiting behaviour described by a similarity solution of the first kind.