The motion of glaciers over their bedrock or drops of fluid along a solid surface can become unstable when these substrates are lubricated. Previous studies modelled such systems as coupled gravity currents (GCs), consisting of one fluid that lubricates the flow of another fluid, and having two propagating fronts. When both fluids are Newtonian and discharged at constant flux, global similarity solutions were found. However, when the top fluid is strain-rate softening, experiments have shown that each fluid front evolved with a different exponent. Here, we explore theoretically and numerically such lubricated GCs consisting of axisymmetric spreading of a power-law fluid on top of a Newtonian fluid, where each fluid volume grows in time like $t^{\alpha }$. We find that the structure imposed by the non-Newtonian flow precludes general self-similarity, unlike purely Newtonian GCs. Consequently, we identify outstripping solutions in which the inner fluid front outstrips the outer fluid front. Despite the absence of a general global similarity solution, we find similarity solutions in several asymptotic limits. These include the purely Newtonian limit for any $\alpha$, the case of $\alpha =5$ for a general power-law fluid, asymptotic limits in the viscosity ratio, and in the vicinity of the fluid fronts. Many of our theoretical predictions are found to be consistent with recent laboratory experiments. Discrepancies suggest the presence of hydrofracturing or wall slip near the fronts, and potentially, a progressive significance of extensional stresses as front outstripping is approached.

The interactions between upstream-propagating guided jet waves and shear-layer instability waves near the nozzle of subsonic and nearly ideally expanded supersonic, isothermal free jets are investigated for jets at Mach numbers between 0.50 and 2 with fully laminar exit boundary layers of different thicknesses. The velocity spectra in the shear layers downstream of the nozzle exhibit strong narrow peaks for the first azimuthal modes, associated with growing Kelvin–Helmholtz instability waves. The frequencies of the predominant peaks are close, but not necessarily equal, to those of the most amplified instability waves predicted from the mean flow fields using linear stability analysis. They also fall in most cases within or very near the allowable frequency bands of the free-stream upstream-propagating guided jet waves obtained using a vortex-sheet model and jump from one band to another as the Mach number increases. At these frequencies, moreover, high levels organized into elongated stripes are found in the jet potential core and standing-wave patterns are visible at the edges of the shear layers in the power spectral densities of pressure and velocity fluctuations. Therefore, the free-stream upstream-propagating guided jet waves appear to interact with and excite the instability waves near the nozzle of the present jets, as in screeching and impinging jets. This explains the disparities of the frequencies and azimuthal modes of the instability waves dominating early on in the shear layers of initially laminar jets and their discontinuous changes and staging behaviours as the jet velocity varies.

We investigate the scaling properties of the primary flow modes and their sensitivity to aspect ratio in a liquid gallium (Prandtl number $Pr = 0.02$) convection system through combined laboratory experiments and numerical simulations. We survey cylindrical aspect ratios $1.4 \le \varGamma \le 3$ and Rayleigh numbers $10^{4} \lesssim Ra \lesssim 10^{6}$. In this range the flow is dominated by a large-scale circulation (LSC) subject to low-frequency oscillations. In line with previous studies, we show robust scaling of the Reynolds number $Re$ with $Ra$ and we confirm that the LSC flow is dominated by a jump-rope vortex (JRV) mode whose signature frequency is present in velocity and temperature measurements. We further show that both $Re$ and JRV frequency scaling trends are relatively insensitive to container geometry. The temperature and velocity spectra consistently show peaks at the JRV frequency, its harmonic and a secondary mode. The relative strength of these peaks changes and the presence of the secondary peak depend highly on aspect ratio, indicating that, despite having a minimal effect on typical velocities and frequencies, the aspect ratio has a significant effect on the underlying dynamics. Applying a bandpass filter at the secondary frequency to velocity measurements reveals that a clockwise twist in the upper half of the fluid layer coincides with a counterclockwise twist in the bottom half, indicating a torsional mode. For aspect ratio $\varGamma = 3$, the unified LSC structure breaks down into multiple rolls in both simulation and experiment.

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number ${{{Pm}}} < 1$. We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.

A two-domain approach is used to investigate the thermal convection of Poiseuille flow in an anisotropic and inhomogeneous porous domain underlying a fluid domain. The flow of the Newtonian fluid is regulated by Darcy's law in the porous domain with the implementation of the Beavers–Joseph condition at the interface. The impact of medium anisotropy and inhomogeneity is inspected by virtue of linear stability analysis along with other governing parameters such as depth ratio (ratio of depth of fluid domain to porous domain), Darcy number, Reynolds number and Prandtl number concerning the stability of the fluid–porous system. The neutral curves are found to be bimodal and unimodal in nature with the anisotropy and inhomogeneity leaving its imprint on parametric variation. An increase in anisotropy or decrease in the inhomogeneity parameter follows the modal change from unimodal (porous) to bimodal (both porous and fluid). Also, it has been identified that, irrespective of the considered variations in anisotropy and inhomogeneity, the least stable mode for the depth ratio ${<}0.05$ is porous and for the depth ratio ${>}0.16$ is fluid. Furthermore, energy budget analysis is carried out to classify the type of instability and validate the type of mode. The instability is found to be thermal–buoyant in nature with omission of low Reynolds numbers along with very low values of the ratio of permeability in the horizontal to vertical direction, where thermal–shear instability is witnessed. Likewise, secondary flow patterns in the context of the streamfunction and temperature contour are analysed to validate the least stable mode and the type of prevailing instability in the fluid–porous system. The presented numerical results find good experimental support from the results of Chen & Chen (J. Fluid Mech., vol. 207, 1989, pp. 311–321) in the limit of natural convection with an isotropic and homogeneous porous domain.

To investigate the attenuation of turbulence in a periodic cube due to the addition of spherical solid particles, we conduct direct numerical simulations using an immersed boundary method with resolving flow around each particle. Numerical results with systematically changing particle diameters and Stokes numbers for a fixed volume fraction $\varLambda$ show that the additional energy dissipation rate in the wake of particles determines the degree of the attenuation of turbulent kinetic energy. On the basis of this observation, we propose formulae describing the condition and degree of the attenuation of turbulence intensity. We conclude that particles with size proportional to $\lambda /\sqrt {\gamma }$, where $\lambda$ and $\gamma$ are the Taylor length and the mass density ratio between particles and fluid, most significantly reduce the intensity of developed turbulence under the condition that $\gamma$ and $\varLambda$ are fixed.

The adhesion of two surfaces relies on the propagation of an adhesion front. What is the dynamics of the front when both surfaces are coated with a thin layer of viscous liquid? Standard criteria from fingering instabilities would predict a stable front since viscous fluid pushes away air of low viscosity. Surprisingly, the front propagation may be unstable and generally leads to growing fingers. We demonstrate with model experiments where the two adhering surfaces are slightly tilted by an angle $\alpha$ that the origin of this interfacial instability relies on feeding the front from the surrounding thin film. We show experimentally that the typical wavelength of the instability is mainly dictated by the thickness of the oil layers $h$. In this wedge geometry, the propagation dynamics is found to follow a $t^{1/2}$ dependence and to saturate for an extension length of the order of $h/\alpha$.

The formation and evolution process of the outer vortex ring (denoted as OV0) in annular starting jets has been studied numerically at stroke ratio $3\leqslant L_m/D\leqslant 8$, $\beta \leqslant 0.8$ and Reynolds number $400\leqslant Re_{\triangle R}\leqslant 1400$, where $D$ is the outer diameter of orifice, $L_m$ is the length of the fluid column, $\beta$ and $\triangle R$ are the ratio and difference between inner and outer radii of the annular orifice, respectively. The initial formation process of OV0 is found to be similar to that of the leading vortex ring in a circular starting jet, but the circulation strength is relatively lower and decreases gradually with increasing $\beta$. The pinch-off process will also appear after OV0 grows to a certain size but is different from that of the circular starting jet. On one hand, OV0 is stretched and deformed by the strain field generated by the trailing jet's instability, and vorticity would be shed. On the other hand, OV0 accelerates the irrotational fluids around it under viscous action so as to form the induced velocity field, which would consume energy causing it to shed vorticity further under the limit of the Kelvin–Benjamin variational principle. Furthermore, OV0 would eventually become a stable vortex ring whose non-dimensional circulation ($\varGamma /(U_{0max}D$)) is found to be between 0.59–0.62 and it would not change with different $L_m/D$ at $\beta =0.5$, which is much less than the 2–2.5 of circular starting jets.

This paper investigates flow and transport in a slender wavy-walled vertical channel subject to a prescribed oscillatory pressure difference between its ends. When the ratio of the stroke length of the pulsatile flow to the channel wavelength is small, the resulting flow velocity is known to include a slow steady-streaming component resulting from the effect of the convective acceleration. Our study considers the additional effect of gravitational forces in configurations with a non-uniform density distribution. Specific attention is given to the slowly evolving buoyancy-modulated flow emerging after the deposition of a finite amount of solute whose density is different from that of the fluid contained in the channel, a relevant problem in connection with drug dispersion in intrathecal drug delivery (ITDD) processes, involving the injection of the drug into the cerebrospinal fluid that fills the spinal canal. It is shown that when the Richardson number is of order unity, the relevant limit in ITDD applications, the resulting buoyancy-induced velocities are comparable to those of steady streaming. As a consequence, the slow time-averaged Lagrangian motion of the fluid, involving the sum of the Stokes drift and the time-averaged Eulerian velocity, is intimately coupled with the transport of the solute, resulting in a slowly evolving problem that can be treated with two-time-scale methods. The asymptotic development leads to a time-averaged, nonlinear integro-differential transport equation that describes the slow dispersion of the solute, thereby circumventing the need to describe the small concentration fluctuations associated with the fast oscillatory motion. The ideas presented here can find application in developing reduced models for future quantitative analyses of drug dispersion in the spinal canal.

Three-dimensional elastic turbulence in Taylor–Couette flows of dilute polymer solutions has been realized and thoroughly investigated via direct numerical simulations. A novel flow transition pathway from elastically dominated turbulence to solitary vortex pairs (or diwhirls) and eventually to elastic turbulence is observed by decreasing the fluid inertia ($Re$) over seven orders of magnitude, i.e. from $Re=1000$ to $0.0001$. The dominant spatio-temporal flow features in the elastic turbulence regime are those of large-scale unsteady diwhirls and small-scale axial and azimuthal travelling waves in the outer and inner halves of the gap, respectively. Moreover, it is conclusively shown that production of turbulent kinetic energy in purely elastic turbulence solely arises due to the stochastic nature of polymer stretch/relaxation. Overall, based on this comprehensive numerical investigation, the differences in the underlying fluid physics that give rise to turbulent fluctuations in elastically dominated and purely elastic turbulence have been delineated.

This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of non-zero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. Our method also provides a non-perturbative procedure to reconstruct the surface elevation from the wave envelope, based on the Birkhoff normal form transformation to eliminate all non-resonant triads. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation derived recently by Curtis, Carter & Kalisch (J. Fluid Mech., vol. 855, 2018, pp. 322–350) in the context of constant vorticity. Very good agreement is found for a range of values of the vorticity.

The action of temporal heating modulation on Marangoni flows in a droplet on a liquid substrate is investigated. The problem is studied numerically in the framework of long-wave amplitude equations and precursor model. It is shown that temporal modulation can lead to a change of the droplet shape. Specifically, rhombic droplets have been obtained. A modulated cooling from below can lead to periodic or quasiperiodic oscillations or the droplet's decomposition.