In this paper we study the self-induced low-Reynolds-number flow generated by a cylinder immersed in a stratified fluid. In the low Péclet limit, where the Péclet number is the ratio of the radius of the cylinder and the Phillips length scale, the flow is captured by a set of linear equations obtained by linearising the governing equations with respect to the prescribed far field conditions. We specifically focus on the low Péclet regime and develop a Green's function approach to solve the linearised equations governing the flow over the cylinder. We cross check our analytical solution against numerical solution of the nonlinear equations to obtain the range of the Péclet numbers for which the linear solution is valid. We then take advantage of the analytical solution to find explicit far-field decay rates of the flow. Our detailed analysis points out that the streamfunction and the velocity field decays algebraically in the far field. Intriguingly, this algebraic decay of the flow is much slower when compared with the exponential decay of the flow generated by a slow moving cylinder in the homogeneous Stokes regime, in the absence of stratification. Consequently, the flow generated by a cylinder in the stratified Stokes regime will have a larger domain of influence when compared with the flow generated by a cylinder in the homogeneous Stokes regime.

The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier–Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian. A recent model handles these unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson & Meneveau, Phys. Rev. Fluids, vol. 2 (7), 2017, 072601). This model is in remarkable agreement with direct numerical simulations (DNS) data and works for arbitrary Taylor Reynolds numbers $\textit {Re}_\lambda$. Inspired by this, we develop a Lagrangian model for passive scalar gradients in isotropic turbulence. The equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term, which we model using the RDGF approach. However, comparisons of the statistics obtained from this model with DNS data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to $\textit {Re}_\lambda \approx 500$, beyond this, the model breaks down.

We study the influence of side walls on the stability of falling liquid films. We combine a temporal biglobal stability analysis based on the linearised Navier–Stokes equations with experiments measuring the spatial growth rate of sinusoidal waves flowing downstream an inclined channel. Very good agreement was found when comparing the theoretical and experimental results. Strong lateral confinement of the channel stabilises the flow. In the wavenumber-Reynolds number space, the instability region experiences a fragmentation due to selective damping of moderate wavenumbers. For this range of parameters, the three-dimensional confined problem shows several prominent stability modes which are classified into two categories, the well-known Kapitza hydrodynamic instability mode (H-mode) and a new unstable mode, we refer to it as wall-mode (W-mode). The two mode types are stabilised differently, where the H-modes are stabilised at small wavenumbers, while the W-modes experience stabilisation at high wavenumbers, and at sufficiently small channel widths, only the W-mode is observed. The reason behind the unique H-modes stabilisation is that they become analogous to waveguide modes, which can not propagate below a certain cut-off wavenumber. The spatial structure of the eigenmodes experiences significant restructuring at wavenumbers smaller than the most damped wavenumber. The mode switching preserves the spatial symmetry of the unstable mode.

We investigate numerically the spreading dynamics of a bubble coming into contact with a smooth solid substrate in a viscous liquid. The substrate is partially wettable, and the singularity of the moving contact line is relieved by adopting the Navier-slip model. The Stokes equations are solved by employing a boundary element method coupled with adaptive mesh refinement. This allows us to realize sufficiently small slip lengths down to $O(10^{-5})$ in dimensionless form, which is crucial to resolve the local interface structures at the early stage of spreading. The results show that the early-stage spreading of the bubble is always characterized by the growth and propulsion of a dewetting liquid rim close to the contact line, while the macroscopic interface remains unchanged. The evolution of the contact line and the morphology of the rim depends on the wettability and the slip length, and a parametric investigation is performed. Based on mass conservation, a relation between the rim size and the spreading radius is established. We also propose an analytical prediction of the temporal variation in the contact line radius at the early stage of spreading, which is found to follow a logarithmically corrected linear relation rather than a pure power law. Moreover, the early stages of bubble spreading are qualitatively similar for two-dimensional and axisymmetric configurations.

Ice-infiltrated sediment, or frozen fringe, is responsible for phenomena such as frost heave, ice lenses and metres of debris-rich ice under glaciers. Understanding frozen fringes is important as frost heave is responsible for damaging infrastructure at high latitudes and sediment freeze-on at the base of glaciers can modulate subglacial friction, influencing the rate of global sea level rise. Here we describe the thermomechanics of liquid water flow and freezing in ice-saturated sediments, focusing on the conditions relevant for subglacial environments. The force balance that governs the frozen fringe thickness depends on the weight of the overlying material, the thermomolecular force between ice and sediments across liquid premelted films and the water pressure required by Darcy flow. We combine this mechanical model with an enthalpy method that conserves energy across phase change interfaces on a fixed computational grid. The force balance and enthalpy model together determine the evolution of the frozen fringe thickness and our simulations predict frost heave rates and ice lens spacing. Our model accounts for premelting at ice–sediment contacts, partial ice saturation of the pore space, water flow through the fringe, the thermodynamics of the ice–water–sediment interface and vertical force balance. We explicitly account for the formation of ice lenses, regions of pure ice that cleave the fringe at the depth where the interparticle force vanishes. Our model results allow us to predict the thickness of a frozen fringe and the spacing of ice lenses in subaerial and subglacial sediments.