Advective trapping occurs when solute enters low velocity zones in heterogeneous porous media. Classical local modelling approaches combine the impact of slow advection and diffusion into a hydrodynamic dispersion coefficient and many temporally non-local approaches lump these mechanisms into a single memory function. This joint treatment makes parameterization difficult and thus prediction of large-scale transport a challenge. Here, we investigate the mechanisms of advective trapping and their impact on transport in media composed of a high conductivity background and isolated low permeability inclusions. Breakthrough curves show that effective transport changes from a streamtube-like behaviour to genuine random trapping as the degree of disorder of the inclusion arrangement increases. We upscale this behaviour using a Lagrangian view point, in which idealized solute particles transition over a fixed distance at random advection times combined with Poissonian advective trapping events. We discuss the mathematical formulation of the upscaled model in the continuous time random walk and mobile–immobile mass transfer frameworks, and derive a model for large-scale solute non-Fickian dispersion. These findings give new insight into transport in highly heterogeneous media.

The focus of this work is to dissect the physical mechanisms that drive and sustain flow-induced, transverse vibrations of cylinders. The influence of different mechanisms is quantified by using a method to partition the fluid dynamic force on the cylinder into distinct, physically relevant components. In conjunction with this force partitioning, calculations of the energy extracted by the oscillating body from the flow are used to make a direct connection between the phenomena responsible for force generation and their effect on driving flow-induced oscillations. These tools are demonstrated in a study of the effect of cylinder shape on flow-induced vibrations. Relatively small increases in cylinder aspect ratio are found to have a significant influence on the amplitude of oscillation, resulting in a large drop in oscillation amplitude at reduced velocities that correspond to the upper range of the synchronization regime. By mapping out the energy transfer between the fluid and structure as a function of aspect ratio, we identify the existence of a low-amplitude stationary state as the cause of the drop in amplitude. Partitioning the fluid dynamic forces on cylinders of varying aspect ratio then allows us to uncover the physical mechanisms behind the appearance of the underlying bifurcation. The analysis also suggests that while vortex shedding in the wake is necessary to initiate oscillations, it is the vorticity associated with the boundary layer over the cylinder that is responsible for the sustenance of flow-induced vibrations.

We present a quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) diffuse interface model for two-phase fluid flows with variable physical properties that maintains thermodynamic consistency. Then, we couple the diffuse domain method with this two-phase fluid model – yielding a new q-NSCH-DD model – to simulate the two-phase flows with moving contact lines in complex geometries. The original complex domain is extended to a larger regular domain, usually a cuboid, and the complex domain boundary is replaced by an interfacial region with finite thickness. A phase-field function is introduced to approximate the characteristic function of the original domain of interest. The original fluid model, q-NSCH, is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the solid surface. We show that the q-NSCH-DD system converges to the q-NSCH system asymptotically as the thickness of the diffuse domain interface introduced by the phase-field function shrinks to zero ($\epsilon \rightarrow 0$) with $\mathcal {O}(\epsilon )$. Our analytic results are confirmed numerically by measuring the errors in both $L^{2}$ and $L^{\infty }$ norms. In addition, we show that the q-NSCH-DD system not only allows the contact line to move on curved boundaries, but also makes the fluid–fluid interface intersect the solid object at an angle that is consistent with the prescribed contact angle.

The ‘coffee stain ring’ is a particle deposit, that forms naturally, when the liquid of a suspension drop evaporates, leaving the particles at the edge of the deposit. Although observed in coffee cups in everyday life, such deposits appear in a wide range of liquid, particle and surface combinations and have attracted vivid research attention. Previous studies focused on the fluidics of evaporating suspension droplets on rigid materials, where the ring formation was shown to occur for pinned contact lines, and possible suppression of the coffee stain effect with surfactants, or other externally driven means, was investigated. Here, we show that, on soft materials, we can control the topography of the deposit on demand – promoting or suppressing the coffee ring effect – by simply changing the environmental humidity, regulating the evaporative flux. We perform particle tracking of droplets drying on soft substrates at varied environmental conditions and show with experimental observations and theoretical analysis that, at an expedited contact line velocity, particles are advected towards the receding contact line. We relate this advection to the viscous dissipation within the soft solid, retarding the contact line motion. The coffee ring formation in the presence of a receding contact line and its control by the environmental humidity, bring a new perspective to the conditions of the manifestation of this frequent deposit topography. We demonstrate the importance of our findings during the printing of a colloidal line, showing the ability to trigger line bifurcation on soft substrates by regulating the evaporative flux, introducing another degree of controllability for contact printing.

In a back pressured duct with incident shocks, the shock train exhibits violent oscillations or even a rapid movement when it passes through a shock-wave–boundary-layer interaction (SWBLI) region. In this study, the dynamics of a shock train system was investigated. Linear stability analysis was used to identify the underlying cause of the unstable behaviour. Results from the eigenvalue analysis indicated that as the shock train enters the SWBLI region, the divergent vibration, which is the outcome of a Hopf bifurcation, emerges. An analysis based on the feedback mechanism identified a criterion for this instability, i.e. the sign of the gradient of the maximal pressure that the boundary layer can sustain. Different unstable motions were also investigated according to the condition of the non-existence of a limit cycle. These motions were associated with the speed of the shock train and the configurations of the flow parameter gradients. It was shown in the controllability matrix that the rapid movement is uncontrollable, which indicates that there is a low correlation between the shock train motion and the flap actuator in the SWBLI region. However, for the remaining part of the unstable motion, a fast-response actuator is required. According to the observability analysis, the shock train movement contributes more to the variation in the pressure behind the first separation shock than the backpressure further downstream, which confirms that monitoring the pressure change along the tunnel is a better method for shock train detection rather than a polynomial model using the backpressure.