We revisit the model problem of Squires & Brady (Phys. Fluids, vol. 17, 2005, 073101), where a Brownian probe is dragged through a dilute dispersion of Brownian bath particles. In this problem, the microrheology due to excluded-volume interactions is represented by an effective viscosity, with the nonlinearity in the driving force entering via the dependence of the viscosity increment (relative to the viscosity of a pure solvent) upon the deformation of the dispersion microstructure. Our interest is in the limit of large Péclet numbers, $ P{\kern-1pt}e\gg 1$, where the microstructural deformation adopts the form of a boundary layer about the upstream hemisphere of the probe. We show that the boundary-layer solution breaks down at the equator of the probe and identify a transition region about the equator, connecting the layer to a downstream wake. The microstructural deformation in this region is governed by a universal boundary-value problem in a semi-bounded two-dimensional domain. The equatorial region continues downstream as a transition layer, which separates the wake of the probe from the undisturbed ambient; in that layer, the microstructure is governed by a one-dimensional heat-like equation. Accounting for the combined contributions from the respective asymptotic provinces we find the approximation $ ({1}/{2})[1+ (\ln P{\kern-1pt}e + 1.046)/ P{\kern-1pt}e]$ for the ratio of the large-$ P{\kern-1pt}e$ viscosity increment to the corresponding linear-response increment. Our asymptotic approximation is in excellent agreement with the increment predicted by a finite-difference numerical calculation of the microstructure deformation, tailored to the large-$ P{\kern-1pt}e$ topology.

A typical dielectric-barrier-discharge plasma actuator operating in burst mode generates periodic vortices resembling the starting vortex. This paper presents the three-dimensional (3-D) characteristics and instability mechanism of these vortices. The experimental investigation is carried out using smoke visualisation and time-resolved particle image velocimetry techniques in three orthogonal measurement planes. The size of the vortices decreases with an increase in burst signal frequency, $ f_{b}$, at a constant duty cycle, $ \alpha$. At higher burst frequencies, dipole vortices are formed due to the roll-up of the wall boundary layer. The angle of travel also decreases with an increase in $ f_{b}$. The evolution of $ \lambda _{2}$-criterion clearly demonstrates the vortex merging of co-rotating vortices. The vortex merging occurs at a critical ratio $ a_{c}/l_{c}$ of core size, $ a_{c}$, and separation distance, $ l_{c}$, equal to $ 0.22\pm 0.01$ which is close to $ a_{c}/l_{c} = 0.24\pm 0.01$ reported by Meunier et al. (Phys. Fluids,vol.14, 2002, pp. 2757–2766) for merging of a pair of equal two-dimensional co-rotating vortices. The periodic vortices are self-similar in nature and the vorticity distribution inside their core region follows the Lamb–Oseen vortex model. Cell structures form in the spanwise direction, which develops wave-like behaviour with an increase in burst frequency. Subsequently, these cell-like structures separate from each other, whose size and spacing correlate well with that of vorticity patches. The alternating sign of vorticity indicates that the circular cells have rotational motion in opposite sense with respect to each other. These cells grow downstream and appear in pairs of counter-rotating vortices (vortex dipole) akin to mushroom-like structures. At low values of $ \alpha$ and $ f_{b}$, the periodic vortex is subjected to a very weak strain and centrifugal instability dominates. The vortices are subjected to a higher strain at elevated burst frequencies, leading to the elliptic instability phenomenon similar to that observed in counter-rotating (Leweke & Williamson, J. Fluid Mech. 1998, vol. 360, pp. 85–119) and co-rotating (Meunier & Leweke, J. Fluid Mech.2005, vol. 533, pp. 125–159) vortex pair generated in water. The present experimental results based on the cross-cut visualisation, Galilean streamlines and vorticity decomposition confirm the role of the instability mechanism on the 3-D vortical structures generated by the dielectric-barrier-discharge plasma actuator.

This study explores an interesting fluid–structure interaction scenario: the flow past a flexible filament fixed at two ends. The dynamic performance of the filament under various inclination angles ($\theta$) was numerically investigated using the immersed boundary method. The motion of the filament in the $\theta$–$Lr$ space was categorised into three flapping modes and two stationary modes, where $Lr$ is the ratio of filament length to the distance between its two ends. The flow fields for each mode and their transitions were introduced. A more in-depth analysis was carried out for flapping at a large angle (FLA mode), which is widely present in the $\theta$–$Lr$ space. The maximum width $W$ of the time-averaged shape of the filament has been shown to strongly correlate with the flapping frequency. After non-dimensionalising based on $W$, the flapping frequency shows little variation across different $Lr$ and $\theta$. Moreover, two types of lift variation process were also identified. Finally, the total lift, drag and lift-to-drag ratio of the system were studied. Short filaments, such as those with $Lr\leqslant 1.5$, were shown to significantly increase lift and the lift-to-drag ratio over a wide range of $\theta$ compared with a rigid plate. Flow field analysis concluded that the increases in pressure difference on both sides of the filament, along with the upper part of the flexible filament having a normal direction closer to the $y$ direction, were the primary reasons for the increase in lift and lift-to-drag ratio. This study can provide some guidance for the potential applications of flexible structures.

The stability characteristics of a Mach $5.35$ boundary-layer flow over a flat plate with parametrised two-dimensional sinusoidal surface roughness are investigated. The investigation involves varying the roughness height from $10\,\%$ to $44\,\%$ of the boundary-layer thickness and exploring wavelengths ranging between $0.44$ and $3.56$ times the dominant second-mode wavelength in the region. The introduction of surface roughness leads to notable variations in the mean flow, resulting in separation behind the roughness elements and the propagation of local compression and expansion waves into the free stream. Stability investigations involved the utilisation of wave packet tracking in a linear disturbance simulation (LDS) framework and linear stability theory. The findings revealed significant effects on Mack modes including a reduction in frequency corresponding to maximum amplification with increased roughness height. Proper scaling of the dominant wavelength facilitates a collapse of the growth rate data. In contrast to the commonly reported stabilisation effects for roughness wavelengths significantly larger than the instability mode’s wavelength, the findings primarily revealed destabilisation compared with the smooth-wall case, except for cases with very small roughness wavelengths and large amplitudes approaching the threshold of being classified as porous media. The LDS findings depicted lobed wall pressure amplitude plots, indicating potential undiscovered instability mechanisms or differences compared with the smooth wall. A detailed stability analysis elucidates these LDS findings, establishing a connection between the lobed amplitude structures and substantial changes in local stability characteristics, along with the emergence of Mack’s first, second and third modes.