Complex materials with internal microstructure such as suspensions and emulsions exhibit time-dependent rheology characterised by viscoelasticity and thixotropy. In many large-scale applications such as turbulent pipe flow, the elastic response occurs on a much shorter time scale than the thixotropy, hence these flows are purely thixotropic. The fundamental dynamics of thixotropic turbulence is poorly understood, particularly the interplay between microstructural state, rheology and turbulence structure. To address this gap, we conduct direct numerical simulations (DNS) of fully developed turbulent pipe flow of a model thixotropic (Moore) fluid as a function of the thixoviscous number $\Lambda$, which characterises the thixotropic kinetic rate relative to turbulence eddy turnover time, ranging from slow ($\Lambda \ll 1$) to fast ($\Lambda \gg 1$) kinetics. Analysis of DNS results in the Lagrangian frame shows that, as expected, in the limits of slow and fast kinetics, these time-dependent flows behave as time-independent purely viscous (generalised Newtonian) analogues. For intermediate kinetics ($\Lambda \sim 1$), the rheology is governed by a path integral of the thixotropic fading memory kernel over the distribution of Lagrangian shear history, the latter of which is modelled via a simple stochastic model for the radially non-stationary pipe flow. The DNS computations based on this effective viscosity closure exhibit excellent agreement with the fully thixotropic model for $\Lambda =1$, indicating that the purely viscous (generalised Newtonian) analogue persists for arbitrary values of $\Lambda \in (0,\infty ^+)$ and across nonlinear rheology models. These results significantly simplify our understanding of turbulent thixotropic flow, and provide insights into the structure of these complex time-dependent flows.

Deformation occurs in a thin liquid film when it is subjected to a non-uniform electric field, which is referred to as the electrohydrodynamic patterning. Due to the development of a non-uniform electrical force along the surface, the film would evolve into microstructures/nanostructures. In this work, a linear and a nonlinear model are proposed to thoroughly investigate the steady state (i.e. equilibrium state) of the electrohydrodynamic deformation of thin liquid film. It is found that the deformation is closely dependent on the electric Bond number BoE. Interestingly, when BoE is larger than a critical value, the film would be deformed remarkably and get in contact with the top template. To model the ‘contact’ between the liquid film and the solid template, the disjoining pressure is incorporated into the numerical model. From the nonlinear numerical model, a hysteresis deformation is revealed, i.e. the film may have different equilibrium states depending on whether the voltage is increased or decreased. To analyse the stability of these multiple equilibrium states, the Lyapunov functional is employed to characterise the system’s free energy. According to the Lyapunov functional analysis, at most three equilibrium states can be formed. Among them, one is stable, another is metastable and the third one is unstable. Finally, the model is extended to study the three-dimensional deformation of the electrohydrodynamic patterning.

This study presents a novel approach for constructing turbulence models using the kinetic Fokker–Planck equation. By leveraging the inherent similarities between Brownian motion and turbulent dynamics, we formulate a Fokker–Planck equation tailored for turbulence at the hydrodynamic level. In this model, turbulent energy plays a role analogous to temperature in molecular thermodynamics, and the large-scale structures are characterised by a turbulent relaxation time. This model aligns with the framework of Pope’s generalised Langevin model, with the first moment recovering the Reynolds-averaged Navier–Stokes (RANS) equations, and the second moment yielding a partially modelled Reynolds stress transport equation. Utilising the Chapman–Enskog expansion, we derive asymptotic solutions for this turbulent Fokker–Planck equation. With an appropriate choice of relaxation time, we obtain a linear eddy viscosity model at first order, and a quadratic Reynolds stress constitutive relationship at second order. Comparative analysis of the coefficients of the quadratic expression with typical nonlinear viscosity models reveals qualitative consistency. To further validate this kinetic-based nonlinear viscosity model, we integrate it as a RANS model within computational fluid dynamics codes, and calculate three typical cases. The results demonstrate that this quadratic eddy viscosity model outperforms the linear model and shows comparability to a cubic model for two-dimensional flows, without the introduction of ad hoc parameters in the Reynolds stress constitutive relationship.

The interaction of a swimmer with unsteady vortices in complex flows remains a topic of interest and open discussion. The present study, employing the immersed boundary method with a flexible fin model, explores swimming behaviours behind a circular cylinder with vortex-induced vibration (VIV). Five distinct swimming modes are identified on the $U_r$–$G_0$ plane, where $U_r$ denotes the reduced velocity, and $G_0$ represents the fin’s initial position. These modes include drifting upstream I/II (DU-I/II), Karman gait I/II (KG-I/II), and large oscillation (LO), with the DU-II, KG-II and LO modes being newly reported. The fin can either move around or cross through the vortex cores in the KG-I and KG-II modes, respectively, for energy saving and maintaining a stable position. When the upstream cylinder vibrates with its maximum amplitude, a double-row vortex shedding forms in the wake, allowing the DU-II mode to occur with the fin to achieve high-speed locomotion. This is attributed to a significant reduction in the streamwise velocity caused by vortex-induced velocity. Furthermore, a symmetry breaking is observed in the fin’s wake in the DU-II mode, potentially also contributing to high-speed locomotion. Overall, compared to the case without an upstream cylinder, we demonstrate that a self-propelled fin gains hydrodynamic advantages with various swimming modes in different VIV wakes. Interestingly, increased power transferred from flows by the oscillating cylinder leads to a more favourable environment for the downstream fin’s propulsion, indicating that a fin in VIV wakes obtains more advantages compared to the vortex street generated by a stationary cylinder.

The scaling of pressure and vorticity in aquatic swimming can provide insights into the mechanisms of propulsion. This is investigated through self-propelled, wall-resolved, large-eddy simulations of a lamprey (an anguilliform swimmer) and a mackerel (a carangiform swimmer) using the curvilinear immersed boundary method. It is observed that the pressure around the swimmers scales with theoretical fluid acceleration, which includes both local body and the convective acceleration, for anguilliform swimmers, whereas it scales with both acceleration and the angle of attack (AoA) for carangiform swimmers. This indicates that the main mechanism for propulsion in anguilliform swimmers is added mass (unsteady), whereas both lift-based (steady) and added mass (unsteady) are at play for carangiform swimmers. Furthermore, it is observed that the vorticity in the boundary layer of the swimmer initially follows the body rotation at low speeds but not at high speeds during the quasisteady swimming. This is explained by identifying the scaling of vorticity components: one due to body rotation and the other due to shear, which scale with Strouhal number ($St$) and Reynolds number ($\sqrt {Re}$), respectively. Here $St$ (body rotation) dominates at low speeds, but $\sqrt {Re}$ (shear) dominates at high speeds. Finally, it is observed that the pressure decreases as the swimming speed increases. This counterintuitive observation is explained by showing that both fluid acceleration and AoA decrease as swimming speed increases. This suggests that for efficient swimming, the pressure difference across the body should be minimised, but high enough to overcome the viscous drag.

Turbulent mixing driven by the reshocked Richtmyer–Meshkov (RM) instability plays a critical role in numerous natural phenomena and engineering applications. As the most fundamental physical quantity characterizing the mixing process, the mixing width transitions from linear to power-law growth following the initial shock. However, there is a notable absence of quantitative models for predicting the pronounced compression of initial interface perturbations or mixing regions at the moment of shock impact. This gap has restricted the development of integrated algebraic models to only the pre- and post-shock evolution stages. To address this limitation, the present study develops a predictive model for the compression of the mixing width induced by shocks. Based on the general principle of growth rate decomposition proposed by Li et al. (Phy. Rev. E, vol. 103, issue 5, 2021, 053109), two distinct types of shock-induced compression processes are identified, differentiated by the dominant mechanism governing their evolution: light–heavy and heavy–light shock-induced compression. For light–heavy interactions, both stretching (compression) and penetration mechanisms are influential, whereas heavy–light interactions are governed predominantly by the stretching (compression) mechanism. To characterize these mechanisms, the average velocity difference between the extremities of the mixing zone is quantified, and a physical model of RM mixing is utilized. A quantitative theoretical model is subsequently formulated through the independent algebraic modelling of these two mechanisms. The proposed model demonstrates excellent agreement with numerical simulations of reshocked RM mixing, offering valuable insights for the development of integrated algebraic models for mixing width evolution.

Predicting the temperature distribution in laminar two-phase flows is essential in a wide range of engineering applications, like heat dissipation of electronic equipment and thermal design of biological reactors. Motivated by this, we extend the classical Graetz problem, studying the heat transfer between two flowing phases in a core-annular flow configuration. Using a rigorous two-scale asymptotic analysis, we derived two coupled one-dimensional advection–diffusion heat-transfer equations (one for each phase) embedding the effects of advection, diffusion (both axial and transverse) and viscous dissipation. Specifically, the heat-transfer mechanisms are described through effective velocity and effective diffusion coefficients, while the interaction between the phases is accounted for via ad hoc coupling and source terms, respectively. The dynamics of the problem is controlled by seven dimensionless groups: the Péclet and Brinkman numbers, the heat flux, the viscosity, thermal diffusivity and thermal conductivity ratios, and the volume fraction. Our analysis reveals the existence of two main regimes, depending on the disparity in thermal conductivity between the phases. When the conductivity ratio is of order one, the problem is strongly coupled; otherwise, the phases are thermally decoupled. Interestingly, we investigate the evolution of the heat-transfer coefficient in the thin-film limit, shedding light on the most common assumptions underlying extensively used models in the context of film flows. Finally, we derived closed-form scaling laws for the Nusselt number clarifying the impact of the phases topology on heat-transfer dynamics. Since our model has been derived by first principles, we hope that it will improve the understanding of two-phase forced convection.

Plugging of a hydraulic fracture because of particle bridging in the fracture channel is ubiquitous in drilling operations and reservoir stimulations. The particles transported in the fluid and fracture can aggregate under certain conditions and finally form a plug. The plug reduces the permeability of the flow channel and blocks the fluid pressure from reaching the fracture front, leading to fracture arrest. In this paper, a numerical model is developed to describe the plugging process of a hydraulic fracture driven by a slurry of solid particles in a viscous fluid while accounting for the rock deformation, slurry flow in the fracture channel, fracture propagation, particle transport and bridging. Three dimensionless numbers are derived from the governing equations, which reveal two length scales that control the fracture propagation and particle transport behaviour, respectively. The difference in magnitude between the two length scales implies three limiting regimes for fracture propagation, i.e. static regime, fluid-driven regime and slurry-driven regime, which correspond to fracture arrest, fracture driven by clean fluid, and fracture driven by slurry, respectively. Numerical results show that the fracture will sequentially transition through the static regime, fluid-driven regime and slurry-driven regime as the fracture length increases. The transition between regimes is controlled by the ratio between the two length scales. Simulation results also reveal two plugging modes, with the plug located near the fracture tip region and at the fracture inlet. The transition between the two plugging modes is controlled by the ratio of the length scales and the injected particle concentration.