We present a numerical study of non-colloidal 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 <= Phi <= 0.4) and particle Reynolds numbers (0.1 <= Re<INF>p</INF><INF></INF> <= 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 the latter's mean value with the values estimated from the homogenisation theory of Chateau et al. (J. Rheol., vol. 52, 2008, pp. 489-506), indicating a reasonable agreement in the Stokesian regime. Both the mean value and standard deviation of the local strain rates increase primarily with the solid volume fraction and secondarily with the Re<INF>p</INF>. The wide spectrum of the local shear rate and its dependency on Phi and Re<INF>p</INF> point to the deficiencies of the mean value of the local shear rates in estimating the rheology of these non-colloidal complex suspensions. Finally, we show that in the presence of inertia, the effective viscosity of these non-colloidal 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<INF>p</INF>) similar to 10.

SUMMARY

We study numerically the entire migration dynamics of spherical and oblate particles in straight rectangular and square cross sectional ducts. The reported results can help in design of straight duct channel based microfluidic systems.

KEYWORDS: Inertial microfluidics, Lateral migration, Oblate particles, Straight particles.

INTRODUCTION

We  simulate spherical and oblate rigid particles in straight ducts of different aspect ratios using an Immersed Boundary Method. To the best of our knowledge, this is the first time not only the equilibrium position of particles is described, but also the entire migration dynamics of the particle from the initial to final position, including particle trajectory, velocity, rotation and orientation, are investigated.

EXPERIMENTAL

 The fluid is considered incompressible and its motion is governed by the Navier Stokes and Continuity equations. The numerical approach employed is an Immersed Boundary Method (IBM) with two sets of grid points: an equispaced Eulerian mesh for the fluid flow, and Lagrangian grid points uniformly distributed on the surface of the particle. The flow is set up in square and rectangular cross section ducts with no slip and no penetration boundary conditions (Fig.1).

RESULTS AND DISCUSSION

We examine the lateral motion of spherical and oblate particles using the IBM method mentioned above. While simulating three different spheres in a square duct of duct width to sphere diameter ratio H/Ds= [3.5, 5, 10], we find that the particles focus at closest face-cantered equilibrium position from their point of introduction(Fig.2a). We also show the downstream length needed for a sphere to focus, focusing length, as a function of the distance from the vertical duct symmetry line and as a function of Reynolds number(Fig.2b and c respectively). Spherical particles in rectangular duct tend to move laterally toward the longer length wall and then slowly moves towards the equilibrium position at the face-centre along the long wall(fig.3a). We also observe that the focusing length is longer for spherical particles in a rectangular duct, about three times longer than that in square duct (fig. 3b). In case of an oblate particle flowing through a square duct, the lateral motion towards the face centred equilibrium position is similar to that of a sphere (fig.4a), however there is significant tumbling motion of the particle as it tries to reach equilibrium(fig.4b).In a rectangular duct of aspect ratio 2, the oblate particle reaches a steady configuration on the duct symmetry line at the center of the different faces (fig.5a). The focusing length surprisingly is shorter in a rectangular duct for an oblate particle in contrast to its focusing length in a square duct. This is attributed to the higher lateral velocity of the oblate in the second stage of the migration, that with negligible tumbling(fig.5b). The behavior of three oblate particles in a square duct of duct width to longer diameter ratio H/Ds= [3.5, 5, 10] is different compared to a sphere as the largest oblate tend to focus at the duct cross section diagonals compared to the other two which are at face centred equilibrium as in case of a sphere(fig.6a). We attribute this to the rotation rate of the larger particle which is initially increasing and then decreasing(fig.6b).When it comes to focusing lengths, the smaller particles need longer times to reach their final equilibrium(fig.6c). Another interesting behavior we see is the effect of Reynolds number, where it can be seen that the oblate particles show a tilt of 21 degrees when focusing at equilibrium at certain high Reynolds number (fig.7).

CONCLUSION

The results presented employ a highly accurate interface-resolved numerical algorithm, based on the Immersed Boundary Method to study the entire inertial migration of an oblate particle in both square and rectangular ducts and compare it with that of a single sphere. Currently, we apply a volume penalization method and polymeric drag component to the code to solve for viscoelastic effects in circular microcapillaries.

ACKNOWLEDGEMENTS

This work was supported by the European Research Council Grant no. ERC-2013-CoG-616186, TRITOS and by the Swedish Research Council Grant no. VR 2014-5001, COST Action MP1305: Flowing matter, and computation time from SNIC.

 REFERENCES : Lashgari, Iman, et al. Journal of Fluid Mechanics 819 (2017): 540-561.

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 elastic material obeying a Neo-Hookean constitutive law. The fluid-structure coupling is obtained using an immersed boundary method. 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 itsmiddle 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 modelling the nucleus as an inner stiffer membrane with those obtained using a rigid particle.

We study turbulent channel flow of a binary mixture of finite-sized 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 diameters 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 number 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 the highest collision kernels due to a higher approaching speed.

We consider suspensions of neutrally-buoyant finite-size rigid spherical particles in channel flow and investigate the relation between the particle dynamics and the mean bulk behavior of the mixture for Reynolds numbers 500 ≤ Re ≤ 5000 and particle volume fraction 0 ≤ Φ ≤ 0.3, via fully resolved numerical simulations. Analysis of the momentum balance reveals the existence of three different regimes: laminar, turbulent and inertial shear-thickening depending on which of the stress terms, viscous, Reynolds or particle stress, is the major responsible for the momentum transfer across the channel. We show that both Reynolds and particle stress dominated flows fall into the Bagnoldian inertial regime and that the Bagnold number can predict the bulk behavior although this is due to two distinct physical mechanisms. A turbulent flow is characterized by larger particle dispersion and a more uniform particle distribution, whereas the particulate-dominated flows is associated with a significant particle migration towards the channel center where the flow is smooth laminar-like and dispersion low. Interestingly, the collision kernel shows similar values in the different regimes, although the relative particle velocity and clustering clearly vary with inertia and particle concentration.

Direct numerical simulations of the fully developed turbulent flow through a porous channel and duct are performed based on the spectral element code Nek5000. The volume-averaged Navier-Stokes (VANS) equations are implemented in order to describe the flow in the composite medium. The numerical simulations of the VANS equations are carried out at a constant value of the bulk Reynolds number when the porosity, or equivalently, the permeability of the medium varies successively. The mean and turbulent energy budgets are computed and the effect of porosity on the secondary flow in a duct is examined.

Direct numerical simulations of the fully developed turbulent flow through a porous square duct are performed to study the effect of the permeable wall on the secondary cross-stream flow. The volume-averaged Navier-Stokes equations are used to describe the flow in the porous phase, a packed bed with porosity epsilon(c) = 0.95. The porous square duct is computed at Re-b similar or equal to 5000 and compared with the numerical simulations of a turbulent duct with four solid walls. The two boundary layers on the top wall and porous interface merge close to the centre of the duct, as opposed to the channel, because the sidewall boundary layers inhibit the growth of the shear layer over the porous interface. The most relevant feature in the porous duct is the enhanced magnitude of the secondary flow, which exceeds that of a regular duct by a factor of four. This is related to the increased vertical velocity, and the different interaction between the ejections from the sidewalls and the porous medium. We also report a significant decrease in the streamwise turbulence intensity over the porous wall of the duct (which is also observed in a porous channel), and the appearance of short spanwise rollers in the buffer layer, replacing the streaky structures of wall-bounded turbulence. These spanwise rollers most probably result from a Kelvin-Helmholtz type of instability, and their width is limited by the presence of the sidewalls.

Inertial regimes in a channel flow of suspension of finite-size neutrally buoyant particles are studied for a wide range of Reynolds numbers: 500 ≤ Re ≤ 5000, and particle volume fractions: 0 ≤ Φ ≤ 0.3. The flow is classified in three different regimes according to the phase-averaged stress budget across the channel [2]. The laminar viscous regime at low Re and Φ where the viscous stress is the dominating term in the budget, the turbulent regime at high Re and relatively low Φ where the momentum is mainly transferred by the action of the Reynolds stress and the inertial shear-thickening regime where the particle stress contributes the most to the significant enhancement of the wall shear stress. Particle distribution and dispersion properties provide additional evidence for the existence of the three different regimes. 

We study the transition to turbulence of channel flow of finite-size particle suspensions at low volume fraction, i.e., Φ ≈ 0.001. The critical Reynolds number above which turbulence is sustained reduces to Re ≈ 1675, in the presence of few particles, independently of the initial condition, a value lower than that of the corresponding single-phase flow, i.e., Re ≈ 1775. In the dilute suspension, the initial arrangement of the particles is important to trigger the transition at a fixed Reynolds number and particle volume fraction. As in single phase flows, streamwise elongated disturbances are initially induced in the flow. If particles can induce oblique disturbances with high enough energy within a certain time, the streaks breakdown, flow experiences the transition to turbulence and the particle trajectories become chaotic. Otherwise, the streaks decay in time and the particles immigrate towards the channel core in a laminar flow. 

We study the transition from laminar to turbulent flow in a channel seeded with finite-size neutrally buoyant particles. A fixed ratio of 10 between the channel height and the particle diameter is considered. The flow is examined in the range of Reynolds numbers 500 ≤ Re ≤; 5000 and the particle volume fractions 0.001 ≤ Φ ≤; 0.3. We report a non-monotonic behavior of the threshold value of the Reynolds number above which the flow becomes turbulent, in agreement with previous experimental studies. The mean square velocity fluctuations and Reynolds shear stress of the fluid phase are reduced by increasing the particle volume fraction at a fixed Re=1500, while the mean square velocities of the solid phase are enhanced monotonically suggesting a transition from fluid to particle dominated dynamics at high volume fraction.