In this paper, we numerically investigate the orbit dynamics of three-dimensional symmetric Janus drops in shear flow using an improved ternary-fluids phase field method, focusing on how drop deformation and initial orientation affect the orbit drift of two configurations of Janus drops: dumbbell-shaped and near-spherical. We find that the motion of dumbbell-shaped drops eventually evolves into tumbling, while near-spherical drops attain stable spinning. We attribute this bifurcation in orbit drift to contrasting deformation dynamics and shape-dependent hydrodynamics of the two configurations. Specifically, the drift bifurcation is closely related to the aspect ratio of Janus drops at equilibrium, giving rise to two distinct mechanisms: (1) coupling between outer interface deformation and the surrounding flow field; and (2) interplay between inner interface deformation and vortices enclosed within the drop. In addition, we observe that for the dumbbell-shaped Janus drops with different aspect ratios, their tumbling dynamics resembles ellipsoids in shear flow. Moreover, the trajectories of the dumbbell-shaped Janus drops during orbit drift collapse onto a universal curve, independent of their initial orientations, and significant deformation and inertia accelerate the orbit transition. To quantitatively evaluate the effect of drop deformation on the orbit drift of the dumbbell-shaped Janus drops, we propose an effective aspect ratio model based on the drop shapes at equilibrium and at the maximum elongation. By incorporating the effective aspect ratio into Jeffery’s theory for solid particles, we accurately predict the rotation period and angular velocity of Janus drops in the tumbling regime and during the orbit drift, especially for drops with linear deformation. Moreover, the orbit parameter $C$ is found to vary exponentially with time for drops with linear deformation, while the time variation of $C$ transits from one exponential function to another for drops with nonlinear deformation.

We investigate a novel Marangoni-induced instability that arises exclusively in diffuse fluid interfaces, that is absent in classical sharp-interface models. Using a validated phase-field Navier–Stokes–Allen–Cahn framework, we linearise the governing equations to analyse the onset and development of interfacial instability driven by solute-induced surface tension gradients. A critical interfacial thickness scaling inversely with the Marangoni number, $\delta _{\textit{cr}} \sim \textit{Ma}^{-1}$, emerges from the balance between advective and diffusive transport. Unlike sharp-interface scenarios where matched viscosity and diffusivity stabilise the interface, finite thickness induces asymmetric solute distributions and tangential velocity shifts that destabilise the system. We identify universal power-law scalings of velocity and concentration offsets with a modified Marangoni number $\textit{Ma}_\delta$, independent of capillary number and interfacial mobility. A critical crossover at $ \textit{Ma}_\delta \approx 590$ distinguishes diffusion-dominated stabilisation from advection-driven destabilisation. These findings highlight the importance of diffuse-interface effects in multiphase flows, with implications for miscible fluids, soft matter, and microfluidics where interfacial thickness and coupled transport phenomena are non-negligible.

In this paper, we will investigate the following inequality

\begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*}

which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.

In this paper, the upper bounds of non-real eigenvalues of indefinite Sturm–Liouville (S-L) problems with boundary conditions depend on the eigenparameter are studied. The upper bounds of real parts, imaginary parts and absolute values of non-real eigenvalues are given under the condition that the coefficients are integrable.

Flapping-wing robots, inspired by natural flyers, have gained significant attention for surveillance and environmental monitoring applications. This study presents the design and analysis of a bat-inspired flapping-wing robot with foldable wings, aiming to enhance flight efficiency and maneuverability. The robot features silicone-based, stretchable membrane wings, with a wingspan of 1.4 m and a total mass of 620 g. A one-degree-of-freedom (DOF) revolute-spherical-spherical-revolute mechanism is used to reproduce the flapping motion, while a one-DOF Watt six-bar linkage mechanism enables dynamic wing folding, allowing adaptive wing shape modulation during flight. Explicit solutions for joint angle of the wing were expressed through analytical method. Flight tests were conducted to validate the effectiveness of the flapping-folding mechanism. Results show that the robot successfully replicates bat wing kinematics, with folding during the upstroke and unfolding during the downstroke. This research offers insights into bio-inspired wing designs for next-generation flapping-wing robots.

Precise control of the laser focal position in the relativistic laser–plasma interaction is crucial for electron acceleration, inertial confinement fusion, high-order harmonic generation, etc. However, conventional methods are characterized by limited tunability and rapid divergence of the relativistic laser pulse after passing through a single focal point. In this work, we propose a novel plasma lens with a density gradient to achieve laser focusing in a tunable focal volume. The capacity depends on the modification of the phase velocity of the incident seed laser propagating in plasma. By modifying the plasma density gradient, one can even achieve an off-axis focusing plasma lens, allowing the laser to be focused further at an adjustable focus. Based on this new type of optical device, a beam-splitting array is also proposed to leverage this unique focusing mechanism for the generation of strong axial magnetic fields (>1000 T). Three-dimensional particle-in-cell simulations demonstrate that the seed laser with a focal spot of $9\ \unicode{x3bc} \mathrm{m}$ passing through the density varying plasma lens exhibits a focused laser with a focal spot of approximately $2.3\ \unicode{x3bc} \mathrm{m}$ and an 18 times enhancement of the laser intensity. The approach has considerable potential for applications in several areas, including laser-driven particle acceleration, X/$\gamma$-ray emission, strong magnetic field generation, etc.

Large-scale spanwise motions in shock wave–turbulent boundary-layer interactions over a $ 25^{\circ }$ compression ramp at Mach 2.95 are investigated using large-eddy simulations. Spectral proper orthogonal decomposition (SPOD) identifies coherent structures characterised by low-frequency features and a large-scale spanwise wavelength of $ O(15\delta _{0})$, where $ \delta _{0}$ is the incoming boundary-layer thickness. The dominant frequency is at least one order of magnitude lower than that of the shock motions. These large-scale spanwise structures are excited near the shock foot and are sustained along the separation shock. Global stability analysis (GSA) is then employed to investigate the potential mechanisms driving these structures. The GSA identifies a stationary three-dimensional (3-D) mode at a wavelength of $ 15\delta _{0}$ with a similar perturbation field, particularly near the separation shock. Good agreement is achieved between the leading SPOD mode and the 3-D GSA mode both qualitatively and quantitatively, which indicates that global instability is primarily responsible for the large-scale spanwise structures surrounding the shock. The reconstructed turbulent separation bubble (TSB) using the 3-D global mode manifests as spanwise undulations, which directly induce the spanwise rippling of the separation shock. Furthermore, the coupled TSB motions in the streamwise and spanwise directions are examined. The TSB oscillates in the streamwise direction while simultaneously exhibiting spanwise undulations. The filtered wall-pressure signals indicate the dominant role of the streamwise motions.

The incorporation of trace metals into land snail shells may record the ambient environmental conditions, yet this potential remains largely unexplored. In this study, we analyzed modern snail shells (Cathaica sp.) collected from 16 sites across the Chinese Loess Plateau to investigate their trace metal compositions. Our results show that both the Sr/Ca and Ba/Ca ratios exhibit minimal intra-shell variability and small inter-shell variability at individual sites. A significant positive correlation is observed between the shell Sr/Ca and Ba/Ca ratios across the plateau, with higher values being recorded in the northwestern sites where less monsoonal rainfall is received. We propose that shell Sr/Ca and Ba/Ca ratios, which record the composition of soil solution, may be controlled by the Rayleigh distillation in response to prior calcite precipitation. Higher rainfall amounts may lead to a lower degree of Rayleigh distillation and thus lower shell Sr/Ca and Ba/Ca ratios. This is supported by the distinct negative correlation between summer precipitation and shell Sr/Ca and Ba/Ca ratios, enabling us to reconstruct summer precipitation amounts using the Sr/Ca and Ba/Ca ratios of Cathaica sp. shells. The potential application of these novel proxies may also be promising for other terrestrial mollusks living in the loess deposits globally.