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Transport of viscous fluid through porous media is a direct consequence of the pore structure. Here we investigate transport through a specific class of two-dimensional porous geometries, namely those formed by fluid-mechanical erosion. We investigate the tortuosity and dispersion by analyzing the first two statistical moments of tracer trajectories. For most initial configurations, tortuosity decreases in time as a result of erosion increasing the porosity. However, we find that tortuosity can also increase transiently in certain cases. The porosity-tortuosity relationships that result from our simulations are compared with models available in the literature. Asymptotic dispersion rates are also strongly affected by the erosion process, as well as by the number and distribution of the eroding bodies. Finally, we analyze the pore size distribution of an eroding geometry. The simulations are performed by combining a boundary integral equation solver for the fluid equations, a second-order stable time-stepping method to simulate erosion, and high-order numerical methods to stably and accurately resolve nearly touching eroded bodies and particle trajectories near the eroding bodies.

We investigate the work fluctuations in an overdamped non-equilibrium process that is stopped at a stochastic time. The latter is characterised by a first passage event that marks the completion of the non-equilibrium process. In particular, we consider a particle diffusing in one dimension in the presence of a time-dependent potential U ( x , t ) = k | x − v t | n / n , where k  > 0 is the stiffness and n  > 0 is the order of the potential. Moreover, the particle is confined between two absorbing walls, located at L ± ( t ) , that move with a constant velocity v and are initially located at L ± ( 0 ) = ± L . As soon as the particle reaches any of the boundaries, the process is said to be completed and here, we compute the work done W by the particle in the modulated trap upto this random time. Employing the Feynman–Kac path integral approach, we find that the typical values of the work scale with L with a crucial dependence on the order n . While for n  > 1, we show that ⟨ W ⟩ ∼ L 1 − n   exp k L n / n − v L / D for large L , we get an algebraic scaling of the form ⟨ W ⟩ ∼ L n for the n  < 1 case. The marginal case of n  = 1 is exactly solvable and our analysis unravels three distinct scaling behaviours: (i) ⟨ W ⟩ ∼ L for v  >  k , (ii) ⟨ W ⟩ ∼ L 2 for v  =  k and (iii) ⟨ W ⟩ ∼ exp − ( v − k ) L for v  <  k . For all cases, we also obtain the probability distribution associated with the typical values of W . Finally, we observe an interesting set of relations between the relative fluctuations of the work done and the first-passage time for different n —which we argue physically. Our results are well supported by the numerical simulations.

In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids.

The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman–Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap $b(t)$ is increased more rapidly: $b(t)=\\left (1-({7}/{2})\\tau \\mathcal {C} t\\right )^{-{2}/{7}}$, where $\\tau$ is the surface tension and $\\mathcal {C}$ is a function of the interface perturbation mode $k$. Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behaviour of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al., Phys. Fluids, vol. 23, 2011, 123101). When we use the shape invariant gap $b(t)$, our nonlinear results reveal the existence of $k$-fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost unity at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode $k$ of the vanishing interface and the control parameter $\\mathcal {C}$.

An informative look at the theory, computer implementation, and application of the scaled boundary finite element method   This reliable resource, complete with MATLAB, is an easy-to-understand introduction to the fundamental principles of the scaled boundary finite element method.

We adopt a boundary integral method to study the dynamics of a translating droplet confined in a Hele-Shaw cell in the Stokes regime. The droplet is driven by the motion of the ambient fluid with the same viscosity. We characterize the three-dimensional (3D) nature of the droplet interface and of the flow field. The interface develops an arc-shaped ridge near the rear-half rim with a protrusion in the rear and a laterally symmetric pair of higher peaks; this pair of protrusions has been identified by recent experiments (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501) and predicted asymptotically (Burgess & Foster, Phys. Fluids A, vol. 2 (7), 1990, pp. 1105–1117). The mean film thickness is well predicted by the extended Bretherton model (Klaseboer et al., Phys. Fluids, vol. 26 (3), 2014, 032107) with fitting parameters. The flow in the streamwise wall-normal middle plane is featured with recirculating zones, which are partitioned by stagnation points closely resembling those of a two-dimensional droplet in a channel. Recirculation is absent in the wall-parallel, unconfined planes, in sharp contrast to the interior flow inside a moving droplet in free space. The preferred orientation of the recirculation results from the anisotropic confinement of the Hele-Shaw cell. On these planes, we identify a dipolar disturbance flow field induced by the travelling droplet and its $1/r^{2}$ spatial decay is confirmed numerically. We pinpoint counter-rotating streamwise vortex structures near the lateral interface of the droplet, further highlighting the complex 3D flow pattern.

Based on the kernel-free boundary integral method proposed by Ying and Henriquez (J Comput Phys 227(2):1046–1074, 2007 ), which is a second-order accurate method for general elliptic partial differential equations, this work develops it to be a fourth-order accurate version for the modified Helmholtz equation. The updated method is in line with the original one. Unlike the traditional boundary integral method, it does not need to know any analytical expression of the fundamental solution or Green’s function in evaluation of boundary or volume integrals. Boundary value problems under consideration are reformulated into Fredholm boundary integral equations of the second kind, whose corresponding discrete forms are solved with the simplest Krylov subspace iterative method, the Richardson iteration. During each iteration, a Cartesian grid based nine-point compact difference scheme is used to discretize the simple interface problem whose solution is the boundary or volume integral in the BIEs. The resulting linear system is solved by a fast Fourier transform based solver, whose computational work is roughly proportional to the number of grid nodes in the Cartesian grid used. As the discrete boundary integral equations are well-conditioned, the iteration converges within an essentially fixed number of steps, independent of the mesh parameter. Numerical results are presented to verify the solution accuracy and demonstrate the algorithm efficiency.

We study the pairwise interactions of drops in an applied uniform DC electric field within the framework of the leaky dielectric model. We develop three-dimensional numerical simulations using the boundary integral method and an analytical theory assuming small drop deformations. We apply the simulations and the theory to explore the electrohydrodynamic interactions between two identical drops with arbitrary orientation of their line of centres relative to the applied field direction. Our results show a complex dynamics depending on the conductivities and permittivities of the drops and suspending fluids, and the initial drop pair alignment with the applied electric field.

The inertialess motion of lipid-bilayer vesicles flowing through a circular tube is investigated via direct numerical simulation and lubrication theory. A fully three-dimensional boundary integral equation method, previously used to study unbounded and wall-bounded Stokes flows around freely suspended vesicles, is extended to study the hindered mobility of vesicles through conduits of arbitrary cross-section. This study focuses on the motion of a periodic train of vesicles positioned concentrically inside a circular tube, with particular attention given to the effects of tube confinement, vesicle deformation and membrane bending elasticity. When the tube diameter is comparable to the transverse dimension of the vesicle, axisymmetric lubrication theory provides an approximate solution to the full Stokes-flow problem. By combining the present numerical results with a previously reported asymptotic theory (Barakat & Shaqfeh, J. Fluid Mech., vol. 835, 2018, pp. 721–761), useful correlations are developed for the vesicle velocity $U$ and extra pressure drop $\\unicode[STIX]{x0394}p^{+}$ . When bending elasticity is relatively weak, these correlations are solely functions of the geometry of the system (independent of the imposed flow rate). The prediction of Barakat & Shaqfeh (2018) supplies the correct limiting behaviour of $U$ and $\\unicode[STIX]{x0394}p^{+}$ near maximal confinement, whereas the present study extends this result to all regimes of confinement. Vesicle–vesicle interactions, shape transitions induced by symmetry breaking, and unsteadiness introduce quantitative changes to $U$ and $\\unicode[STIX]{x0394}p^{+}$ . By contrast, membrane bending elasticity can qualitatively affect the hydrodynamics at sufficiently low flow rates. The dependence of $U$ and $\\unicode[STIX]{x0394}p^{+}$ on the membrane bending stiffness (relative to a characteristic viscous stress scale) is found to be rather complex. In particular, the competition between viscous forces and bending forces can hinder or enhance the vesicle’s mobility, depending on the geometry and flow conditions.

A variety of numerical methods exist for the study of deformable particles in dense suspensions. None of the standard tools, however, currently include volume-changing objects such as oscillating microbubbles in three-dimensional periodic domains. In the first part of this work, we develop a novel method to include such entities based on the boundary integral method. We show that the well-known boundary integral equation must be amended with two additional terms containing the volume flux through the bubble surface. We rigorously prove the existence and uniqueness of the solution. Our proof contains as a subset the simpler boundary integral equation without volume-changing objects (such as red blood cell or capsule suspensions) which is widely used but for which a formal proof in periodic domains has not been published to date. In the second part, we apply our method to study microbubbles for targeted drug delivery. The ideal drug delivery agent should stay away from the biochemically active vessel walls during circulation. However, upon reaching its target it should attain a near-wall position for efficient drug uptake. Though seemingly contradictory, we show that lipid-coated microbubbles in conjunction with a localized ultrasound pulse possess precisely these two properties. This ultrasound-triggered margination is due to hydrodynamic interactions between the red blood cells and the oscillating lipid-coated microbubbles which alternate between a soft and a stiff state. We find that the effect is very robust, existing even if the duration in the stiff state is more than three times lower than the opposing time in the soft state.