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Multiphase shear flows often show banded structures that affect the global behavior of complex fluids e.g. in microdevices. Here we investigate numerically the banding of emulsions, i.e. the formation of regions of high and low volume fraction, alternated in the vorticity direction and aligned with the flow (shear bands). These bands are associated with a decrease of the effective viscosity of the system. To understand the mechanism of banding experimentally observed we have performed interface resolved simulations of the two-fluid system. The experiments were perfomed starting with a random distribution of droplets which, under the applied shear, evolves in time resulting in a phase separation. To numerically reproduce this process, the banded structures are initialized in a narrow channel confined by two walls moving in opposite direction. We find that the initial banded distribution is stable when droplets are free to merge and unstable when coalescence is prevented. In this case, additionally, the effective viscosity of the system increases, resembling the rheological behavior of suspensions of deformable particles. Droplets coalescence, on the other hand, allows emulsions to reduce the total surface of the system and hence the energy dissipation associated to the deformation, which in turn reduces the effective viscosity.

We simulate the flow of two immiscible and incompressible fluids separated by an interface in a homogeneous turbulent shear flow at a shear Reynolds number equal to 15200. The viscosity and density of the two fluids are equal, and various surface tensions and initial droplet diameters are considered in the present study. We show that the two-phase flow reaches a statistically stationary turbulent state sustained by a non-zero mean turbulent production rate due to the presence of the mean shear. Compared to single-phase flow, we find that the resulting steady state conditions exhibit reduced Taylor microscale Reynolds numbers owing to the presence of the dispersed phase, which acts as a sink of turbulent kinetic energy for the carrier fluid. At steady state, the mean power of surface tension is zero and the turbulent production rate is in balance with the turbulent dissipation rate, with their values being larger than in the reference single-phase case. The interface modifies the energy spectrum by introducing energy at small-scales, with the difference from the single-phase case reducing as the Weber number increases. This is caused by both the number of droplets in the domain and the total surface area increasing monotonically with the Weber number. This reflects also in the droplets size distribution which changes with the Weber number, with the peak of the distribution moving to smaller sizes as the Weber number increases. We show that the Hinze estimate for the maximum droplet size, obtained considering breakup in homogeneous isotropic turbulence, provides an excellent estimate notwithstanding the action of significant coalescence and the presence of a mean shear.

We perform direct numerical simulations of the flow through a model of a deformable porous medium. Our model is a two-dimensional hexagonal lattice, with defects, of soft elastic cylindrical pillars, with elastic shear modulus $G$, immersed in a liquid. We use a two-phase approach: the liquid phase is a viscous fluid and the solid phase is modeled as an incompressible viscoelastic material, whose complete nonlinear structural response is considered. We observe that the Darcy flux ($q$) is a nonlinear function -- steeper than linear -- of the pressure-difference ($\Delta P$) across the medium. Furthermore, the flux is larger for a softer medium (smaller $G$). We construct a theory of this super-linear behavior by modelling the channels between the solid cylinders as elastic channels whose walls are made of a material with a linear constitutive relation but can undergo large deformation. Our theory further predicts that the flow permeability is a universal function of $\Delta P/G$, which is confirmed by the present simulations., Comment: 6 pages, 3 figures, Some minor changes (including the title) from the previous submission

The flapping states of a flexible fiber fully coupled to a three-dimensional turbulent flow are investigated via state-of-the-art numerical methods. Two distinct flapping regimes are predicted by the phenomenological theory recently proposed by Rosti et al. [Phys. Rev. Lett. 121, 044501, 2018]: the under-damped regime, where the elasticity strongly affects the fiber dynamics, and the over-damped regime, where the elastic effects are strongly inhibited. In both cases we can identify a critical value of the bending rigidity of the fiber by a resonance condition, which further provides a distinction between different flapping behaviors, especially in the under-damped case. We validate the theory by means of direct numerical simulations and find that, both for the over-damped regime and for the under-damped one, fibers are effectively slaved to the turbulent fluctuations and can therefore be used as a proxy to measure various two-point statistics of turbulence. Finally, we show that this holds true also in the case of a passive fiber, without any feedback force on the fluid.

We experimentally investigate the influence of finite-size spherical particles in turbulent flows of a Newtonian and a drag reducing viscoelastic fluid at varying particle volume fractions and fixed Reynolds number. Experiments are performed in a square duct at a Reynolds number $Re_{2H}$ of nearly $1.1\times10^4$, Weissenberg number $Wi$ for single-phase flow is between 1-2 and results in a drag-reduction of 43% compared to a Newtonian flow (at the same $Re_{2H}$). Particles are almost neutrally-buoyant hydrogel spheres having a density ratio of 1.0035$\pm$0.0003 and a duct height ${2H}$ to particle diameter $d_p$ ratio of around 10. We measure flow statistics for four different volume fractions $\phi$ namely 5, 10, 15 and 20% by using refractive-index-matched Particle Image Velocimetry (PIV). For both Newtonian Fluid (NF) and Viscoelastic Fluid (VEF), the drag monotonically increases with $\phi$. For NF, the magnitude of drag increase due to particle addition can be reasonably estimated using a concentration dependent effective viscosity for volume fractions below 10%. The drag increase is, however, underestimated at higher $\phi$. For VEF, the absolute value of drag is lower than NF but, its rate of increase with $\phi$ is higher. Similar to particles in a NF, particles in VEF tend to migrate towards the center of the duct and form a layer of high concentration at the wall. Interestingly, relatively higher migration towards the center and lower migration towards the walls is observed for VEF. The primary Reynolds shear stress reduces with increasing $\phi$ throughout the duct height for both types of fluid.

Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) have been employed to investigate the dynamics of finite-size spherical particles, slightly heavier than the carrier fluid, in a horizontal turbulent square duct flow. Interface resolved Direct Numerical Simulations (DNS) have also been performed with the Immersed Boundary Method (IBM) at the same experimental conditions, bulk Reynolds number $Re_{2H}$ = 5600, duct height to particle size ratio $2H/d_p$ = 14.5, particle volume fraction $\Phi$ = 1% and particle to fluid density ratio $\rho_p/\rho_f$ = 1.0035. A good agreement has been observed between experiments and simulations in terms of the overall pressure drop, concentration distribution and turbulent statistics of the two phases. Additional experimental results considering two particle sizes, $2H/d_p$ = 14.5 and 9 and multiple $\Phi$ = 1, 2, 3, 4 and 5% are reported at the same $Re_{2H}$. The pressure drop monotonically increases with the volume fraction, almost linearly and nearly independently of the particle size for the above parameters. However, despite the similar pressure drop, the microscopic picture, the fluid velocity statistics, differs significantly with the particle size. This one-to-one comparison between simulations and experiments extends the validity of interface resolved DNS in complex turbulent multiphase flows and highlights the ability of experiments to investigate such flows in considerable details, even in regions where the local volume fraction is relatively high.

We present a numerical analysis of the rheology of a suspension of red blood cells (RBCs) in a wall-bounded shear flow. The flow is assumed as almost inertialess. The suspension of RBCs, modeled as biconcave capsules whose membrane follows the Skalak constitutive law, is simulated for a wide range of viscosity ratios between the cytoplasm and plasma: $\lambda$ = 0.1-10, for volume fractions up to $\phi$ = 0.41 and for different capillary numbers ($Ca$). Our numerical results show that an RBC at low $Ca$ tends to orient to the shear plane and exhibits the so-called rolling motion, a stable mode with higher intrinsic viscosity than the so-called tumbling motion. As $Ca$ increases, the mode shifts from the rolling to the swinging motion. Hydrodynamic interactions (higher volume fraction) also allows RBCs to exhibit both tumbling or swinging motions resulting in a drop of the intrinsic viscosity for dilute and semi-dilute suspensions. Because of this mode change, conventional ways of modeling the relative viscosity as a polynomial function of $\phi$ cannot be simply applied in suspensions of RBCs at low volume fractions. The relative viscosity for high volume fractions, however, can be well described as a function of an effective volume fraction, defined by the volume of spheres of radius equal to the semi-middle axis of the deformed RBC. We find that the relative viscosity successfully collapses on a single non-linear curve independently of $\lambda$ except for the case with $Ca \geq$ 0.4, where the fit works only in the case of low/moderate volume fraction, and fails in the case of a fully dense suspension.

We have performed direct numerical simulation of the turbulent flow of a polymer solution in a square duct, with the FENE-P model used to simulate the presence of polymers. First, a simulation at a fixed moderate Reynolds number is performed and its results compared with those of a Newtonian fluid to understand the mechanism of drag reduction and how the secondary motion, typical of the turbulent flow in non-axisymmetric ducts, is affected by polymer additives. Our study shows that the Prandtl's secondary flow is modified by the polymers: the circulation of the streamwise main vortices increases and the location of the maximum vorticity move towards the center of the duct. In-plane fluctuations are reduced while the streamwise ones are enhanced in the center of the duct and dumped in the corners due to a substantial modification of the quasi-streamwise vortices and the associated near-wall low- and high-speed streaks; these grow in size and depart from the walls, their streamwise coherency increasing. Finally, we investigated the effect of the parameters defining the viscoelastic behavior of the flow and found that the Weissenberg number strongly influences the flow, with the cross-stream vortical structures growing in size and the in-plane velocity fluctuations reducing for increasing flow elasticity.

We report experimental observations of turbulent flow with spherical particles in a square duct. Three particle sizes namely: $2H/d_{p}$ = 40, 16 and 9 ($2H$ being the duct full height and $d_{p}$ being the particle diameter) are investigated. The particles are nearly neutrally-buoyant with respect to the suspending fluid. Refractive Index Matched - Particle Image Velocimetry is used for fluid velocity measurement even at the highest particle volume fraction (20%) and Particle Tracking Velocimetry for the particle velocity statistics for the flows seeded with particles of the two largest sizes, whereas only pressure measurements are reported for the smallest particles. Settling effects are seen at the lowest bulk Reynolds number $Re_{2H}\approx$ 10000 whereas, at the highest $Re_{2H}\approx$ 27000, particles are in almost full suspension. The friction factor of the suspensions is found to be significantly larger than that of single-phase duct flow at the lower $Re_{2H}$ investigated; however, the difference decreases when increasing the flow rate and the total drag approaches the values of the single phase flow at the higher Reynolds number considered. The pressure drop is found to decrease with the particle diameter for volume fractions lower than $\phi$ = 10% for nearly all $Re_{2H}$. However, at the highest volume fraction $\phi$ = 20%, we report a peculiar non-monotonic behavior: the pressure drop first decreases and then increases with increasing particle size. The decrease of the turbulent drag with particle size at the lowest volume fractions is related to an attenuation of the turbulence. The drag increase for the two largest particle sizes at $\phi$ = 20%, however, occurs despite this large reduction of the turbulent stresses, and it is therefore due to significant particle-induced stresses.

We present numerical simulations of laminar and turbulent channel flow of an elastoviscoplastic fluid. The non-Newtonian flow is simulated by solving the full incompressible Navier-Stokes equations coupled with the evolution equation for the elastoviscoplastic stress tensor. The laminar simulations are carried out for a wide range of Reynolds numbers, Bingham numbers and ratios of the fluid and total viscosity, while the turbulent flow simulations are performed at a fixed bulk Reynolds number equal to 2800 and weak elasticity. We show that in the laminar flow regime the friction factor increases monotonically with the Bingham number (yield stress) and decreases with the viscosity ratio, while in the turbulent regime the the friction factor is almost independent of the viscosity ratio and decreases with the Bingham number, until the flow eventually returns to a fully laminar condition for large enough yield stresses. Three main regimes are found in the turbulent case, depending on the Bingham number: for low values, the friction Reynolds number and the turbulent flow statistics only slightly differ from those of a Newtonian fluid; for intermediate values of the Bingham number, the fluctuations increase and the inertial equilibrium range is lost. Finally, for higher values the flow completely laminarises. These different behaviors are associated with a progressive increases of the volume where the fluid is not yielded, growing from the centerline towards the walls as the Bingham number increases. The unyielded region interacts with the near-wall structures, forming preferentially above the high speed streaks. In particular, the near-wall streaks and the associated quasi-streamwise vortices are strongly enhanced in an highly elastoviscoplastic fluid and the flow becomes more correlated in the streamwise direction.

We present a modification of a recently developed volume of fluid method for multiphase problems, so that it can be used in conjunction with a fractional step-method and fast Poisson solver, and validate it with standard benchmark problems. We then consider emulsions of two-fluid systems and study their rheology in a plane Couette flow in the limit of vanishing inertia. We examine the dependency of the effective viscosity on the volume-fraction (from 10% to 30%) and the Capillary number (from 0.1 to 0.4) for the case of density and viscosity ratio 1. We show that the effective viscosity decreases with the deformation and the applied shear (shear-thinning) while exhibits a non-monotonic behavior with respect to the volume fraction. We report the appearance of a maximum in the effective viscosity curve and compare the results with those of suspensions of rigid and deformable particles and capsules. We show that the flow in the solvent is mostly a shear flow, while it is mostly rotational in the suspended phase; moreover this behavior tends to reverse as the volume fraction increases. Finally, we evaluate the contributions to the total shear stress of the viscous stresses in the two fluids and of the interfacial force between them.

Motivated by the emerging applications of liquid-infused surfaces (LIS), we study the drag reduction and robustness of transverse flows over two-dimensional microcavities partially filled with an oily lubricant. Using separate simulations at different scales, characteristic contact line velocities at the fluid-solid intersection are first extracted from nano-scale phase field simulations and then applied to micron-scale two-phase flows, thus introducing a multiscale numerical framework to model the interface displacement and deformation within the cavities. As we explore the various effects of the lubricant-to-outer-fluid viscosity ratio $\tilde{\mu}_2/\tilde{\mu}_1$, the capillary number Ca, the static contact angle $\theta_s$, and the filling fraction of the cavity $\delta$, we find that the effective slip is most sensitive to the parameter $\delta$. The effects of $\tilde{\mu}_2/\tilde{\mu}_1$ and $\theta_s$ are generally intertwined, but weakened if $\delta < 1$. Moreover, for an initial filling fraction $\delta =0.94$, our results show that the effective slip is nearly independent of the capillary number, when it is small. Further increasing Ca to about $0.01 \tilde{\mu}_1/\tilde{\mu}_2$, we identify a possible failure mode, associated with lubricants draining from the LIS, for $\tilde{\mu}_2/\tilde{\mu}_1 \lesssim 0.1$. Very viscous lubricants (\eg $\tilde{\mu}_2/\tilde{\mu}_1 >1$), on the other hand, are immune to such failure due to their generally larger contact line velocity., Comment: Phys. Rev. Fluids (2018)

In this paper, a three-dimensional numerical solver is developed for suspensions of rigid and soft particles and droplets in viscoelastic and elastoviscoplastic (EVP) fluids. The presented algorithm is designed to allow for the first time three-dimensional simulations of inertial and turbulent EVP fluids with a large number particles and droplets. This is achieved by combining fast and highly scalable methods such as an FFT-based pressure solver, with the evolution equation for non-Newtonian (including elastoviscoplastic) stresses. In this flexible computational framework, the fluid can be modelled by either Oldroyd-B, neo-Hookean, FENE-P, and Saramito EVP models, and the additional equations for the non-Newtonian stresses are fully coupled with the flow. The rigid particles are discretized on a moving Lagrangian grid while the flow equations are solved on a fixed Eulerian grid. The solid particles are represented by an Immersed Boundary method (IBM) with a computationally efficient direct forcing method allowing simulations of a large numbers of particles. The immersed boundary force is computed at the particle surface and then included in the momentum equations as a body force. The droplets and soft particles on the other hand are simulated in a fully Eulerian framework, the former with a level-set method to capture the moving interface and the latter with an indicator function. The solver is first validated for various benchmark single-phase and two-phase elastoviscoplastic flow problems through comparison with data from the literature. Finally, we present new results on the dynamics of a buoyancy-driven drop in an elastoviscoplastic fluid.

The effect of the variations of the permeability tensor on the close-to-the-wall behaviour of a turbulent channel flow bounded by porous walls is explored using a set of direct numerical simulations. It is found that the total drag can be either reduced or increased by more than $20\%$ by adjusting the permeability directional properties. Drag reduction is achieved for the case of materials with permeability in the vertical direction lower than the one in the wall-parallel planes. This configuration limits the wall normal velocity at the interface while promoting an increase of the tangential slip velocity leading to an almost "one-component" turbulence where the low- and high-speed streaks coherence is strongly enhanced. On the other hand, strong drag increase is found when a high wall-normal and low wall-parallel permeabilities are prescribed. In this condition, the enhancement of the wall-normal fluctuations due to the reduced wall-blocking effect triggers the onset of structures which are strongly correlated in the spanwise direction, a phenomenon observed by other authors in flows over isotropic porous layers or over ribletted walls with large protrusion heights. The use of anisotropic porous walls for drag reduction is particularly attractive since equal gains can be achieved at different Reynolds numbers by rescaling the magnitude of the permeability only.

We consider suspensions of deformable particles in a Newtonian fluid by means of fully Eulerian numerical simulations with a one-continuum formulation. We study the rheology of the visco-elastic suspension in plane Couette flow in the limit of vanishing inertia and examine the dependency of the effective viscosity $\mu$ on the solid volume-fraction $\Phi$, the capillary number $\mbox{Ca}$, and the solid to fluid viscosity ratio $\mbox{K}$. The suspension viscosity decreases with deformation and applied shear (shear-thinning) while still increasing with volume fraction. We show that $\mu$ collapses to an universal function, $\mu \left( \Phi^{\rm e} \right)$, with an effective volume fraction $\Phi^{\rm e}$, lower than the nominal one owing to the particle deformation. This universal function is well described by the Eilers fit, which well approximate the rheology of suspension of rigid spheres at all $\Phi$. We provide a closure for the effective volume fraction $\Phi^{\rm e}$ as function of volume fraction $\Phi$ and capillary number $\mbox{Ca}$ and demonstrate it also applies to data in literature for suspensions of capsules and red-blood cells. In addition, we show that the normal stress differences exhibit a non-linear behavior, with a similar trend as in polymer and filament suspensions. The total stress budgets reveals that the particle-induced stress contribution increases with the volume fraction $\Phi$ and decreases with deformability.

We study the dynamics of a flexible fiber freely moving in a three-dimensional fully-developed turbulent field and present a phenomenological theory to describe the interaction between the fiber elasticity and the turbulent flow. This theory leads to the identification of two distinct regimes of flapping, which we validate against Direct Numerical Simulations (DNS) fully resolving the fiber dynamics. The main result of our analysis is the identification of a flapping regime where the fiber, despite its elasticity, is slaved to the turbulent fluctuations. In this regime the fiber can be used to measure two-point statistical observables of turbulence, including scaling exponents of velocity structure functions, the sign of the energy cascade and the energy flux of turbulence, as well as the characteristic times of the eddies within the inertial range of scales. Our results are expected to have a deep impact on the experimental turbulence research as a new way, accurate and efficient, to measure two-point, and more generally multi-point, statistics of turbulence.

We construct a boundary integral representation for the low-Reynolds-number flow in a channel in the presence of freely-suspended particles (or droplets) of arbitrary size and shape. We demonstrate that lubrication theory holds away from the particles at horizontal distances exceeding the channel height and derive a multipole expansion of the flow which is dipolar to the leading approximation. We show that the dipole moment of an arbitrary particle is a weighted integral of the stress and the flow at the particle surface, which can be determined numerically. We introduce the equation of motion that describes hydrodynamic interactions between arbitrary, possibly different, distant particles, with interactions determined by the product of the mobility matrix and the dipole moment. Further, the problem of three identical interacting spheres initially aligned in the streamwise direction is considered and the experimentally observed "pair exchange" phenomenon is derived analytically and confirmed numerically. For non-aligned particles, we demonstrate the formation of a configuration with one particle separating from a stable pair. Our results suggest that in a dilute initially homogeneous particulate suspension flowing in a channel the particles will eventually separate into singlets and pairs., Comment: 15 pages, 4 figures, accepted in Physical Review E

We study the settling of rigid oblates in quiescent fluid using interface-resolved Direct Numerical Simulations. In particular, an immersed boundary method is used to account for the dispersed solid phase together with lubrication correction and collision models to account for short-range particle-particle interactions. We consider semi-dilute suspensions of oblate particles with aspect ratio AR=1/3 and solid volume fractions $\phi=0.5\%-10\%$. The solid-to-fluid density ratio $R=1.5$ and the Galileo number (i.e. the ratio between buoyancy and viscous forces) based on the diameter of a sphere with equivalent volume $Ga=60$. With this choice of parameters, an isolated oblate falls vertically with a steady wake with its broad side perpendicular to the gravity direction. At this $Ga$, the mean settling speed of spheres is a decreasing function of the volume $\phi$ and is always smaller than the terminal velocity of the isolated particle, $V_t$. On the contrary, we show here that the mean settling speed of oblate particles increases with $\phi$ in dilute conditions and is $33\%$ larger than $V_t$. At higher concentrations, the mean settling speed decreases becoming smaller than the terminal velocity $V_t$ between $\phi=5\%$ and $10\%$. The increase of the mean settling speed is due to the formation of particle clusters that for $\phi=0.5\%-1\%$ appear as columnar-like structures. From the pair-distribution function we observe that it is most probable to find particle-pairs almost vertically aligned. However, the pair-distribution function is non-negligible all around the reference particle indicating that there is a substantial amount of clustering at radial distances between 2 and $6c$ (with $c$ the polar radius of the oblate)., Comment: Submitted to Journal of Fluid Mechanics

We study the turbulent square duct flow of dense suspensions of neutrally-buoyant spherical particles. Direct numerical simulations (DNS) are performed in the range of volume fractions $\phi=0-0.2$, using the immersed boundary method (IBM) to account for the dispersed phase. Based on the hydraulic diameter a Reynolds number of $5600$ is considered. We report flow features and particle statistics specific to this geometry, and compare the results to the case of two-dimensional channel flows. In particular, we observe that for $\phi=0.05$ and $0.1$, particles preferentially accumulate on the corner bisectors, close to the duct corners as also observed for laminar square duct flows of same duct-to-particle size ratios. At the highest volume fraction, particles preferentially accumulate in the core region. For channel flows, in the absence of lateral confinement particles are found instead to be uniformily distributed across the channel. We also observe that the intensity of the cross-stream secondary flows increases (with respect to the unladen case) with the volume fraction up to $\phi=0.1$, as a consequence of the high concentration of particles along the corner bisector. For $\phi=0.2$ the turbulence activity is strongly reduced and the intensity of the secondary flows reduces below that of the unladen case. The friction Reynolds number increases with $\phi$ in dilute conditions, as observed for channel flows. However, for $\phi=0.2$ the mean friction Reynolds number decreases below the value for $\phi=0.1$., Comment: Submitted to Journal of Fluid Mechanics

We present a numerical study of noncolloidal spherical and rigid particles suspended in Newtonian, shear thinning and shear thickening fluids employing an Immersed Boundary Method. We consider a linear Couette configuration to explore a wide range of solid volume fractions ($0.1\le \Phi \le 0.4$) and particle Reynolds Numbers ($0.1\le Re_p \le 10$). We report the distribution of solid and fluid phase velocity and solid volume fraction and show that close to the boundaries inertial effects result in a significant slip velocity between the solid and fluid phase. The local solid volume fraction profiles indicate particle layering close to the walls, which increases with the nominal $\Phi$. This feature is associated with the confinement effects. We calculate the probability density function of local strain rates and compare their mean value with the values estimated from the homogenization theory of \cite{Chateau08}, indicating a reasonable agreement in the Stokesian regimes. Both the mean value and standard deviation of the local strain rates increase primarily with the solid volume fraction and secondarily with the $Re_p$. The wide spectrum of the local shear rate and its dependency on $\Phi$ and $Re_p$ points to the deficiencies of the mean value of the local shear rates in estimating the rheology of these noncolloidal complex suspensions. Finally, we show that in the presence of inertia, the effective viscosity of these noncolloidal suspensions deviates from that of Stokesian suspensions. We discuss how inertia affects the microstructure and provide a scaling argument to give a closure for the suspension shear stress for both Newtonian and power-law suspending fluids. The stress closure is valid for moderate particle Reynolds numbers, $ O(Re_p)\sim 10$.

We study suspensions of deformable (viscoelastic) spheres in a Newtonian solvent in plane Couette geometry, by means of direct numerical simulations. We find that in the limit of vanishing inertia the effective viscosity $\mu$ of the suspension increases as the volume-fraction occupied by the spheres $\Phi$ increases and decreases as the elastic modulus of the spheres $G$ decreases; the function $\mu(\Phi,G)$ collapses to an universal function, $\mu(\Phi_e)$, with a reduced effective volume fraction $\Phi_e(\Phi,G)$. Remarkably, the function $\mu(\Phi_e)$ is the well-known Eilers fit that describes the rheology of suspension of rigid spheres at all $\Phi$. Our results suggest new ways to interpret macro-rheology of blood.

To study the dynamics of particles in turbulence when their sizes are comparable to the smallest eddies in the flow, the Kolmogorov length scale, efficient and accurate numerical models for the particle-fluid interaction are still missing. Therefore, we here extend the treatment of the particle feedback on the fluid based on the volume-averaged fluid equations (VA simulation) in the previous study of the present authors, by estimating the fluid force correlated with the disturbed flow. We validate the model against interface-resolved simulations using the immersed-boundary method. Simulations of single particles show that the history effect is well captured by the present estimation method based on the disturbed flow. Similarly, the simulation of the flow around a rotating particle demonstrates that the lift force is also well captured by the proposed method. We also consider the interaction between non-negligible size particles and an array of Taylor-Green vortices. For density ratios $\rho_d/\rho_c\geq$ 10, the results show that the particle motion captured by the VA approach is closer to that of the fully-resolved simulations than that obtained with a traditional two-way coupling simulation. The flow disturbance is also well represented by the VA simulation. In particular, it is found that history effects enhance the curvature of the trajectory in vortices and this enhancement increases with the particle size. Furthermore, the flow field generated by a neighboring particle at distances of around ten particle diameters significantly influences particle trajectories. The computational cost of the VA simulation proposed here is considerably lower than that of the interface-resolved simulation., Comment: 54 pages, 20 figures

The shear-induced deformation of a capsule with a stiff nucleus, a model of eukaryotic cells, is studied numerically. The membrane of the cell and of its nucleus are modelled as a thin and impermeable elastic material obeying a Neo-Hookean constitutive law. The membranes are discretised by a Lagrangian mesh and their governing equations are solved in spectral space using spherical harmonics, while the fluid equations are solved on a staggered grid using a second-order finite differences scheme. The fluid-structure coupling is obtained using an immersed boundary method. The numerical approach is presented and validated for the case of a single capsule in a shear flow. The variations induced by the presence of the nucleus on the cell deformation are investigated when varying the viscosity ratio between the inner and outer fluids, the membrane elasticity and its bending stiffness. The deformation of the eukaryotic cell is smaller than that of the prokaryotic one. The reduction in deformation increases for larger values of the capillary number. The eukaryotic cell remains thicker in its middle part compared to the prokaryotic one, thus making it less flexible to pass through narrow capillaries. For a viscosity ratio of 5, the deformation of the cell is smaller than in the case of uniform viscosity. In addition, for non-zero bending stiffness of the membrane, the deformation decreases and the shape is closer to an ellipsoid. Finally, we compare the results obtained modeling the nucleus as an inner stiffer membrane with those obtained using a rigid particle.

We perform numerical simulations of a turbulent channel flow over an hyper-elastic wall. In the fluid region the flow is governed by the incompressible Navier-Stokes (NS) equations, while the solid is a neo-Hookean material satisfying the incompressible Mooney-Rivlin law. The multiphase flow is solved with a one-continuum formulation, using a monolithic velocity field for both the fluid and solid phase, which allows the use of a fully Eulerian formulation. The simulations are carried out at Reynolds bulk $Re=2800$ and examine the effect of different elasticity and viscosity of the deformable wall. We show that the skin friction increases monotonically with the material elastic modulus. The turbulent flow in the channel is affected by the moving wall even at low values of elasticity since non-zero fluctuations of vertical velocity at the interface influence the flow dynamics. The near-wall streaks and the associated quasi-streamwise vortices are strongly reduced near a highly elastic wall while the flow becomes more correlated in the spanwise direction, similarly to what happens for flows over rough and porous walls. As a consequence, the mean velocity profile in wall units is shifted downwards when shown in logarithmic scale, and the slope of the inertial range increases in comparison to that for the flow over a rigid wall. We propose a correlation between the downward shift of the inertial range, its slope and the wall-normal velocity fluctuations at the wall, extending results for the flow over rough walls. We finally show that the interface deformation is determined by the fluid fluctuations when the viscosity of the elastic layer is low, while when this is high the deformation is limited by the solid properties.

We study turbulent channel flows of monodisperse and polydisperse suspensions of finite-size spheres by means of Direct Numerical Simulations using an immersed boundary method to account for the dispersed phase. Suspensions with 3 different Gaussian distributions of particle radii are considered (i.e. 3 different standard deviations). The distributions are centered on the reference particle radius of the monodisperse suspension. In the most extreme case, the radius of the largest particles is 4 times that of the smaller particles. We consider two different solid volume fractions, 2% and 10%. We find that for all polydisperse cases, both fluid and particles statistics are not substantially altered with respect to those of the monodisperse case. Mean streamwise fluid and particle velocity profiles are almost perfectly overlapping. Slightly larger differences are found for particle velocity fluctuations. These increase close to the wall and decrease towards the centerline as the standard deviation of the distribution is increased. Hence, the behavior of the suspension is mostly governed by excluded volume effects regardless of particle size distribution (at least for the radii here studied). Due to turbulent mixing, particles are uniformly distributed across the channel. However, smaller particles can penetrate more into the viscous and buffer layer and velocity fluctuations are therein altered. Non trivial results are presented for particle-pair statistics., Comment: Under review in the European Journal of Mechanics/B - Fluids

We present the results of a Direct Numerical Simulation of a particle-laden spatially developing turbulent boundary layer up to Re{\theta} = 2500. Two main features differentiate the behavior of inertial particles in a zero- pressure-gradient turbulent boundary layer from the more commonly studied case of a parallel channel flow. The first is the variation along the streamwise direction of the local dimensionless parameters defining the fluid-particle in- teractions. The second is the coexistence of an irrotational free-stream and a near-wall rotational turbulent flow. As concerns the first issue, an inner and an outer Stokes number can be defined using inner and outer flow units. The inner Stokes number governs the near-wall behavior similarly to the case of channel flow. To understand the effect of a laminar-turbulent interface, we ex- amine the behavior of particles initially released in the free stream and show that they present a distinct behavior with respect to those directly injected inside the boundary layer. A region of minimum concentration occurs inside the turbulent boundary layer at about one displacement thickness from the wall. Its formation is due to the competition between two transport mecha- nisms: a relatively slow turbulent diffusion towards the buffer layer and a fast turbophoretic drift towards the wall.

We use interface-resolved simulations to study finite-size effects in turbulent channel flow of neutrally-buoyant spheres. Two cases with particle sizes differing by a factor of 2, at the same solid volume fraction of 20% and bulk Reynolds number are considered. These are complemented with two reference single-phase flows: the unladen case, and the flow of a Newtonian fluid with the effective suspension viscosity of the same mixture in the laminar regime. As recently highlighted in Costa et al. (PRL 117, 134501), a particle-wall layer is responsible for deviations of the statistics from what is observed in the continuum limit where the suspension is modeled as a Newtonian fluid with an effective viscosity. Here we investigate the fluid and particle dynamics in this layer and in the bulk. In the particle-wall layer, the near wall inhomogeneity has an influence on the suspension micro-structure over a distance proportional to the particle size. In this layer, particles have a significant (apparent) slip velocity that is reflected in the distribution of wall shear stresses. This is characterized by extreme events (both much higher and much lower than the mean). Based on these observations we provide a scaling for the particle-to-fluid apparent slip velocity as a function of the flow parameters. We also extend the flow scaling laws in to second-order Eulerian statistics in the homogeneous suspension region away from the wall. Finite-size effects in the bulk of the channel become important for larger particles, while negligible for lower-order statistics and smaller particles. Finally, we study the particle dynamics along the wall-normal direction. Our results suggest that 1-point dispersion is dominated by particle-turbulence (and not particle-particle) interactions, while differences in 2-point dispersion and collisional dynamics are consistent with a picture of shear-driven interactions.

We study numerically the inertial migration of a single rigid sphere and an oblate spheroid in straight square and rectangular ducts. A highly accurate interface-resolved numerical algorithm is employed to analyse the entire migration dynamics of the oblate particle and compare it with that of the sphere. Similarly to the inertial focusing of spheres, the oblate particle reaches one of the four face-centred equilibrium positions, however they are vertically aligned with the axis of symmetry in the spanwise direction. In addition, the lateral trajectories of spheres and oblates collapse into an equilibrium manifold before ending at the equilibrium positions, with the equilibrium manifold tangential to lines of constant background shear for both sphere and oblate particles. The differences between the migration of the oblate and sphere are also presented, in particular the oblate may focus on the diagonal symmetry line of the duct cross-section, close to one of the corners, if its diameter is larger than a certain threshold. Moreover, we show that the final orientation and rotation of the oblate exhibit a chaotic behaviour for Reynolds numbers beyond a critical value. Finally, we document that the lateral motion of the oblate particle is less uniform than that of the spherical particle due to its evident tumbling motion throughout the migration. In a square duct, the strong tumbling motion of the oblate in the first stage of the migration results in a lower lateral velocity and consequently longer focusing length with respect to that of the spherical particle. The opposite is true in a rectangular duct where the higher lateral velocity of the oblate in the second stage of the migration, with negligible tumbling, gives rise to shorter focusing lengths. These results can help the design of microfluidic systems for bio-applications., Comment: 21 pages, 9 figures

Aiming for the simulation of colloidal droplets in microfluidic devices, we present here a numerical method for two-fluid systems subject to surface tension and depletion forces among the suspended droplets. The algorithm is based on an efficient solver for the incompressible two-phase Navier-Stokes equations, and uses a mass-conserving level set method to capture the fluid interface. The four novel ingredients proposed here are, firstly, an interface-correction level set (ICLS) method; global mass conservation is achieved by performing an additional advection near the interface, with a correction velocity obtained by locally solving an algebraic equation, which is easy to implement in both 2D and 3D. Secondly, we report a second-order accurate geometric estimation of the curvature at the interface and, thirdly, the combination of the ghost fluid method with the fast pressure-correction approach enabling an accurate and fast computation even for large density contrasts. Finally, we derive a hydrodynamic model for the interaction forces induced by depletion of surfactant micelles and combine it with a multiple level set approach to study short-range interactions among droplets in the presence of attracting forces.

Previous studies have shown that intermediate surface tension has a counterintuitive destabilizing effect on 2-phase planar jets. Here, the transition process in confined 2D jets of two fluids with varying viscosity ratio is investigated using DNS. Neutral curves for persistent oscillations are found by recording the norm of the velocity residuals in DNS for over 1000 nondimensional time units, or until the signal has reached a constant level in a logarithmic scale - either a converged steady state, or a "statistically steady" oscillatory state. Oscillatory final states are found for all viscosity ratios (0.1-10). For uniform viscosity (m=1), the first bifurcation is through a surface tension-driven global instability. For low viscosity of the outer fluid, there is a mode competition between a steady asymmetric Coanda-type attachment mode and the surface tension-induced mode. At moderate surface tension, the Coanda-type attachment dominates and eventually triggers time-dependent convective bursts. At high surface tension, the surface tension-dominated mode dominates. For high viscosity of the outer fluid, persistent oscillations appear due to a strong convective instability. Finally, the m=1 jet remains unstable far from the inlet when the shear profile is nearly constant. Comparing this to a parallel Couette flow (without inflection points), we show that in both flows, a hidden interfacial mode brought out by surface tension becomes temporally and absolutely unstable in an intermediate Weber and Reynolds regime. An energy analysis of the Couette setup shows that surface tension, although dissipative, induces a velocity field near the interface which extracts energy from the flow through a viscous mechanism. This study highlights the rich dynamics of immiscible planar uniform-density jets, where several self-sustained and convective mechanisms compete depending on the exact parameters.

We study turbulent channel flow of a binary mixture of finite-size neutrally-buoyant rigid particles by means of interface-resolved direct numerical simulations. We fix the bulk Reynolds number and total solid volume fraction, $Re_b = 5600$ and $\Phi=20\%$, and vary the relative fraction of small and large particles. The binary mixture consists of particles of two different sizes, $2h/d_l=20$ and $2h/d_s=30$ where $h$ is the half channel height and $d_l$ and $d_s$ the diameter of the large and small particles. While the particulate flow statistics exhibit a significant alteration of the mean velocity profile and turbulent fluctuations with respect to the unladen flow, the differences between the mono-disperse and bi-disperse cases are small. However, we observe a clear segregation of small particles at the wall in binary mixtures, which affects the dynamics of the near wall region and thus the overall drag. This results in a higher drag in suspensions with a larger amount of large particles. As regards bi-disperse effects on the particle dynamics, a non-monotonic variation of the particle dispersion in the spanwise (homogeneous) direction is observed when increasing the percentage of small/large particles. Finally, we note that particles of the same size tend to cluster more at contact whereas the dynamics of the large particles gives highest collision kernels due to a higher approaching speed., Comment: 22 pages, 12 figures

The dynamics of dilute micron-sized spherical inertial particles in turbulent duct flows is studied by means of direct numerical simulations of the carrier phase turbulence with one-way coupled Lagrangian particles. The geometries are a square and a rectangular duct with width-to-height aspect ratio $AR$ of 3 operating at $Re_{\tau,c}=360$ . The present study is designed to determine the effect of turbulence-driven secondary motion on the particle dynamics. Our results show that a weak cross-flow secondary motion significantly changes the cross-sectional map of the particle concentration, mean velocity and fluctuations. As the geometry of the duct is widened from $AR=1$ to 3, the secondary vortex on the horizontal wall significantly expands in the spanwise direction, and although the kinetic energy of the secondary flow increases close to the corner, it decays towards the duct centreplane in the $AR=3$ case so as the turbulent carrier phase approaches the behavior in spanwise-periodic channel flows, a fact that significantly affects the particle statistics. In the square duct the particle concentration in the viscous sublayer is maximum at the duct centreplane, whereas the maximum is found closer to the corner, at a distance of $|z/h|\approx1.25$ from the centreplane, in the $AR=3$ case. Interestingly the centreplane concentration in the rectangular duct is around 3 times lower than that in the square duct. Moreover, a second peak in the accumulation distribution is found right at the corners for both ducts. At this location the concentration increases with particle inertia. The secondary motion changes also the cross-stream map of the particle velocities significantly in comparison to the fluid flow statistics. These directly affect the particle velocity fluctuations such that multiple peaks appear near the duct walls for the particle streamwise and wall-normal velocity fluctuations.

We study heat transfer in plane Couette flow laden with rigid spherical particles by means of direct numerical simulations using a direct-forcing immersed boundary method to account for the dispersed phase. A volume of fluid approach is employed to solve the temperature field inside and outside of the particles. We focus on the variation of the heat transfer with the particle Reynolds number, total volume fraction (number of particles) and the ratio between the particle and fluid thermal diffusivity, quantified in terms of an effective suspension diffusivity. We show that, when inertia at the particle scale is negligible, the heat transfer increases with respect to the unladen case following an empirical correlation recently proposed. In addition, an average composite diffusivity can be used to predict the effective diffusivity of the suspension the inertialess regime when varying the molecular diffusion in the two phases. At finite particle inertia, however, the heat transfer increase is significantly larger, saturating at higher volume fractions. By phase-ensemble averaging we identify the different mechanisms contributing to the total heat transfer and show that the increase of the effective conductivity observed at finite inertia is due to the increase of the transport associated to fluid and particle velocity. We also show that the heat conduction in the solid phase reduces when increasing the particle Reynolds number and so that particles of low thermal diffusivity weakly alter the total heat flux in the suspension at finite particle Reynolds numbers. On the other hand, a higher particle thermal diffusivity significantly increase the total heat transfer.

We study the settling of finite-size rigid spheres in sustained homogeneous isotropic turbulence (HIT) by direct numerical simulations using an immersed boundary method to account for the dispersed solid phase. We study semi-dilute suspensions at different Galileo numbers, Ga. The Galileo number is the ratio between buoyancy and viscous forces, and is here varied via the solid-to-fluid density ratio. The focus is on particles that are slightly heavier than the fluid. We find that in HIT, the mean settling speed is less than that in quiescent fluid; in particular, it reduces by 6%-60% with respect to the terminal velocity of an isolated sphere in quiescent fluid as the ratio between the latter and the turbulent velocity fluctuations is decreased. Analysing the fluid-particle relative motion, we find that the mean settling speed is progressively reduced while reducing the density ratio due to the increase of the vertical drag induced by the particle cross-flow velocity. Unsteady effects contribute to the mean overall drag by about 6%-10%. The probability density functions of particle velocities and accelerations reveal that these are closely related to the features of the turbulent flow. The particle mean-square displacement in the settling direction is found to be similar for all Ga if time is scaled by (2a)/u' (where 2a is the particle diameter and u' is the turbulence velocity root mean square)., Comment: Accepted for publication in Journal of Fluid Mechanics

The gravity-driven motion of rigid particles in a viscous fluid is relevant in many natural and industrial processes, yet this has mainly been investigated for spherical particles. We therefore consider the sedimentation of non-spherical (spheroidal) isolated and particle pairs in a viscous fluid via numerical simulations using the Immersed Boundary Method. The simulations performed here show that the critical Galileo number for the onset of secondary motions decreases as the spheroid aspect ratio departs from 1. Above this critical threshold, oblate particles perform a zigzagging motion whereas prolate particles rotate around the vertical axis while having their broad side facing the falling direction. Instabilities of the vortices in the wake follow when farther increasing the Galileo number. We also study the drafting-kissing-tumbling associated with the settling of particle pairs. We find that the interaction time increases significantly for non-spherical particles and, more interestingly, spheroidal particles are attracted from larger lateral displacements. This has important implications for the estimation of collision kernels and can result in increasing clustering in suspensions of sedimenting spheroids.

The macroscopic behavior of dense suspensions of neutrally-buoyant spheres in turbulent plane channel flow is examined. We show that particles larger than the smallest turbulence scales cause the suspension to deviate from the continuum limit in which its dynamics is well described by an effective suspension viscosity. This deviation is caused by the formation of a particle layer close to the wall with significant slip velocity. By assuming two distinct transport mechanisms in the near-wall layer and the turbulence in the bulk, we define an effective wall location such that the flow in the bulk can still be accurately described by an effective suspension viscosity. We thus propose scaling laws for the mean velocity profile of the suspension flow, together with a master equation able to predict the increase in drag as function of the particle size and volume fraction., Comment: Accepted for publication in PRL. Supplemental material included