Inertial Regime Research Articles

Particle migration in channel flow of an elastoviscoplastic fluid

We study the dynamics of a neutrally buoyant rigid sphere carried by an elastoviscoplastic fluid in a pressure-driven channel flow numerically. The yielding to flow is marked by the yield stress which splits the flow into two main regions: the core unyielded region and two sheared yielded regions close to the walls. The particles which are initially in the plug region are observed to translate with the same velocity as the plug without any rotation/migration. Keeping the Reynolds number fixed, we study the effect of elasticity (Weissenberg number) and plasticity (Bingham number) of the fluid on the particle migration inside the sheared regions. In the viscoelastic limit, in the range of studied parameters (low elasticity), inertia is dominant and the particle finds its equilibrium position between the centreline and the wall. The same happens in the viscoplastic limit, yet the yield surface plays the role of centreline. However, the combination of elasticity and plasticity of the suspending fluid (elastoviscoplasticity) trigger particle-focusing: in the elastoviscoplastic flow, for a certain range of Weissenberg numbers (≈0.5), isolated particles migrate all the way to the centreline by entering into the core plug region. This behaviour suggests a particle-focusing process for inertial regimes which was not previously found in a viscoelastic or viscoplastic carrying fluid.

The effect of viscosity ratio on drop pinch-off dynamics in two-fluid flow

Drop pinch-off draws a lot of interest from scientists and engineers because of widely practical applications as well as the fascinating mechanism of finite-time singularities and self-similar behavior. This work experimentally investigates the effect of viscosity ratio on the local pinch-off mechanism of liquid drops in both air and viscous fluids, with a high-speed camera. The results show that for 56 ≤ λ ≤ 6.3×103 and 0.009 ≤ λ ≤ 0.061, drop pinch-off exhibits self-similar profile which is asymmetric and conical away from hmin. Drop pinch-off in air with low viscosity ratios shows either in inertial regime (I) or transition from inertial to viscous regime (I→V). But the kinetics undergoes the inertial regime (I) to inertial-viscous regime (IV) through an intermediate viscous regime (V) when the viscosity ratio becomes larger. Drop pinch-off in viscous fluids displays the transition from inertial to viscous regime (I→V). These results agree well with Eggers’s universal solution until it becomes unstable and the previous literature. The viscosity ratio indeed affects the drop pinch-off dynamics as well as interface deformation. Our experiments enrich the understanding of the interaction between the internal and external fluids on drop pinch-off behavior near the singularity point.

Microstructure-informed probability-driven point-particle model for hydrodynamic forces and torques in particle-laden flows

Abstract

Darcy and inertial fluid flow simulations in porous media using the non-orthogonal central moments lattice Boltzmann method

In this study, two-dimensional (2D) and three-dimensional (3D) lattice Boltzmann method (LBM) has been investigated using the non-orthogonal central moments collision method in order to study the Darcy and inertial (non-Darcy) fluid flow in porous media. A GPU parallel CUDA code has been developed and is employed to study the proposed method. A comparative study has been performed between the non-orthogonal central moments method and the Bhatnagar–Groos–Krook (BGK) method to measure the permeability. The effect of selecting sets of relaxation times and no-slip boundary conditions has been investigated using the 2D hexagonal and 3D body-centered cubic (BCC) structure, and an appropriate model has been presented for flow simulation in porous media. The comparison of the permeability obtained by the proposed numerical method with the analytical solutions shows that the proposed model is highly accurate and eliminates the dependence of permeability on the viscosity, which is one of the problems in the LBM. This model’s capability has been evaluated to simulate Darcy and inertial flows using 2D hexagonal geometry, 2D granular porous media with a random structure, 3D BCC geometry, 3D porous media with a random structure, and real porous media created by using computed tomography (CT) images. In addition, the apparent permeability, the non-Darcy β factor, and the onset of non-Darcy flow have been predicted to demonstrate the capability of the proposed method. The results show the high capability of the proposed model for fluid flow simulation in porous media with complex geometries; so, this model can be used as a powerful and stable tool with the capability of fluid flow simulation in porous media from the Darcy regime to large Reynolds numbers and the inertial regime.

A Deep Look into the Dynamics of Saltwater Imbibition in a Calcite Nanochannel: Temperature Impacts Capillarity Regimes.

This research concerns fundamentals of spontaneous transport of saltwater (1 mol·dm-3 NaCl solution) in nanopores of calcium carbonates. A fully atomistic model was adopted to scrutinize the temperature dependence of flow regimes during solution transport under CaCO3 nanoconfinement. The early time of capillary filling is inertia-dominated, and the solution penetrates with a near-planar meniscus at constant velocity. Following a transition period, the meniscus angle falls to a stabilized value, characterizing the capillary-viscous advancement in the calcite channel. At this stage, brine displacement follows a parabolic relationship consistent with the classical Lucas-Washburn (LW) theory. Approaching the slit outlet, the meniscus contact lines spread widely on the solid substrate and brine leaves the channel at a constant rate, in oppose to the LW law. The brine imbibition rate considerably increases at higher temperatures as a result of lower viscosity and greater tendency to form wetting layers on slit walls. We also pointed out a longer primary inertial regime and delayed onset of the viscous-capillary regime at higher temperatures. Throughout the whole span of capillary displacement, transport of sodium and chloride ions is tied to dynamics and diffusion of the water phase, even at the mineral interface. The results presented in this study are of broad implications in diverse science and technological applications.

Experimental investigation on macroscopic behavior and microfluidic field of nonlinear flow in rough-walled artificial fracture models

The understanding of nonlinear flow in rough-walled rock fractures is critical for underground engineering and geological sciences. The macroscopic behavior and microfluidic field of flow in rough-walled artificial fracture models was investigated through microscopic visualization of hydraulic experiment. The results indicate that there are three states of flow regime in fracture, namely linear, weak inertial and strong inertial. In linear flow regime, the relationship between pressure drops and flow rate is linear, and all the streamlines in the microfluidic field are parallel. In weak inertial flow regime, pressure drop and flow become nonlinear, and the inertial pressure loss is less than the viscous pressure loss. Moreover, backflow occurs near the tortuous wall in the microfluidic field. In strong inertial flow regime, the inertial pressure loss is greater than viscous pressure loss, and the backflow begins to detach from the wall and eventually develop into a significantly large eddy far away from the wall. However, the flow in the mainstream remains laminar rather than turbulent in this experiment. This result indicates that the main cause of the nonlinearity of flow in fracture is eddies due to the increase of the inertia term and the variation of aperture. The shrinking of effective channel and losing of kinetic energy caused by eddies make fluid flow nonlinear. Moreover, the area and number of eddies depend on the fracture roughness and the Reynolds number.

A comprehensive stress model for gas-particle flows in dense and dilute regimes

A comprehensive solid stress model including the whole flow regimes for gas-particle flows is proposed. In the inertial regime characterized by low solid concentrations, solid stress is closed by kinetic theory based on particle collision mechanisms. In the quasi-static regime characterized by high particle concentrations, solid stress is closed by the frictional stress model based on soil mechanics. In the intermediate regime, solid stress is closed by an inertial number model without the need to specify a threshold of particle concentration for the transition. This new model is validated with experimental data from bubbling fluidized beds with a central jet, and compared with simulation results predicted by kinetic theories. Results show that this new model can give a more accurate prediction of bubble size, bubble shape and detachment time than through kinetic theory or other previous models. The problem of specifying a threshold value when employing previous models is solved.

Inertial capillary uptake of drops.

Uptake of liquid drops into capillary tubes has been experimentally studied and quantitatively analyzed. In experiments, drops of water and aqueous glycerol (≤50 wt%) were drawn into cylindrical borosilicate glass and quartz tubes with an inner diameter of 0.50-0.75mm. The meniscus height rise was measured using high-speed images captured at 4000 frames per second, and results within a conservatively defined inertial regime indicate constant uptake velocity. An increase in the inertial velocity with drop curvature was observed due to increasing Laplace pressure in the drop, as drop sizes were comparable to the width of the capillary tubes. Measured velocities were slower than predicted by a purely inertial-capillary model and best described by introducing a contact line friction, consistent with the observed variability and viscosity dependence of the results. Mean friction coefficients in borosilicate capillaries ranged from 169±1 for 50 wt% glycerol drops to 218±1 for water drops. Peaks in the instantaneous Laplace pressure caused by surface oscillations were also measured. Correlations with uptake velocity were qualitatively apparent, with a delay between peaks of similar magnitude to the inertial-capillary oscillation time.

Early stage of liquid drop spreading on tunable nanostructured surfaces

In this study, the early stage of liquid droplet spreading on tunable nanostructures was investigated through high resolved and fast imaging visualization. It is well-known that the early stage of water droplet spreading on a solid surface features an inertia dominance on resistive droplet motion in a short period (~0.01 sec). The spreading dynamic law of this stage differs greatly from those of the classical spreading stage, after the inertia regime, being determined by the balance between a spreading force and viscous friction. The effect of triple line physics associated with the surface properties is generally underestimated in the early inertial regime by a relatively short period and strong inertia momentum, and such physics result in the nature of the spreading (Dt1/2). Here, we designed tunable nanostructures that can modify the nature law of the early droplet spreading. The results suggest that the increase of initial spreading momentum attribute to the capillary wicking effect controlled by the nanostructure morphology.

Hydrodynamic clustering and emergent phase separation of spherical spinners

In this article, the authors demonstrate a spontaneous assembly of spinner vortices from an initially uniform suspension. The clustering is observed at weakly inertial regime due to hydrodynamic interactions between the spinning particles. The results highlight the importance of inertial interactions and the third dimension in a hydrodynamic assembly of spinner materials.

Direct numerical simulations of forced homogeneous isotropic turbulence in a dense gas

ABSTRACTDirect Numerical Simulations (DNS) of forced homogeneous isotropic turbulence in a dense gas (FC-70), accurately described by a complex EoS, are computed for a turbulent Mach number of 0.8. In a numerical experiment, results are compared to the ones obtained when considering the fluid as a perfect gas. It is found that the dense gas displays a deeply modified shocklets' structure. The amplitude of compression shocklets jumps in pressure, density and entropy is divided by an order of magnitude with respect to the perfect gas. Moreover, expansion shocklets are found in the dense gas flow, also associated with small jumps in pressure, density and entropy. Comparing TKE spectra, the same inertial range is found regardless of the EoS. By comparing the terms of the filtered TKE equation for the dense and perfect gas EoS, it is found that for FC-70 and the present turbulent Mach and Taylor Reynolds numbers, the SGS deformation work is the only significant term in the inertial regime and does not significantly change with the EoS. A preliminary analysis of the flux terms responsible for the total energy conservation shows that the pressure term PDF is however significantly modified by the thermodynamic properties of the fluid.

Unifying viscous and inertial regimes of discontinuous shear thickening suspensions

The rheology of dense suspensions in the viscous regime can be characterized by the viscous number J and granular flows by the inertia number I, both relating the shear rate γ ˙ to the confining pressure P. Furthermore, several works have shown that the rheology of suspensions and granular flows can be unified, and this unification is used to characterize suspensions with non-negligible particle inertia from the viscous to the inertial regime (i.e., granular flows). There have also been recent works that apply this unification to suspensions where the number of frictional contacts increases upon the shear rate with constant packing fraction, a model that has successfully described discontinuous shear thickening in suspension flows. We and other researchers have previously shown that the fraction of frictional contacts χ f is a key control parameter for friction-driven shear thickening in the viscous regime. It is, however, difficult to control and study the effect of the number of frictional contacts on the rheology of dense suspensions at constant packing fraction as this quantity increases sharply around a threshold shear rate. In this work, we extend our previous work and use numerical simulations to study particle flows in both the viscous and inertial regimes as well as for suspensions in the crossover between these regimes with varying χ f. By having pressure-controlled simulations, we are able to keep χ f at values between 0 and 1, hence critically testing the validity of a unification at intermediate values. With the help of a binary model composed of nonfrictional and frictional particles, where χ f is well-controlled, we manage to obtain expressions for constitutive laws in both limits. For the critical load model, we find a simple relationship between χ f and the pressure-rescaled threshold force f ^ applicable for both the viscous and inertial regimes. Combining these expressions, we then show that suspensions in the crossover between these regimes can be characterized by four dimensionless numbers, f ^, K, K μ, and K z, where the last three are simple combinations of J and I and verified against numerical simulations. These expressions can be further simplified by approximating K μ = K, which we show is a good approximation close to shear jamming or at low Stokes numbers. In the end, we construct a dimensionless parameter encoding various shear protocols and show predictions of behaviors of suspensions under different shear conditions. Having derived constitutive relations from constant pressure simulations, we finally test our relations under constant volume assumption which indeed well captures the various discontinuous shear thickening behaviors seen at different Stokes numbers. Finally, we discuss the role of having a varying microscopic friction coefficient μ p as a function of the normal force.

Hydrodynamic forces on monodisperse assemblies of axisymmetric elongated particles: Orientation and voidage effects

We investigate the average drag, lift, and torque on static assemblies of capsule‐like particles of aspect ratio 4. The performed simulations are from Stokes flow to high Reynolds numbers (0.1 ≤ Re ≤ 1,000) at different solids volume fraction (0.1 ≤ ɛ s ≤ 0.5). Individual particle forces as a function of the incident angle ϕ with respect to the average flow are scattered. However, the average particle force as a function of ϕ is found to be independent of mutual particle orientations for all but the highest volume fractions. On average, a sine‐squared scaling of drag and sine‐cosine scaling of lift holds for static multiparticle systems of elongated particles. For a packed bed, our findings can be utilized to compute the pressure drop with knowledge of the particle‐orientation distribution, and the average particle drag at ϕ = 0° and 90°. We propose closures for average forces to be used in Euler–Lagrange simulations of particles of aspect ratio 4.

Fluid Dynamics of Ballistic Strategies in Nematocyst Firing

Nematocysts are stinging organelles used by members of the phylum Cnidaria (e.g., jellyfish, anemones, hydrozoans) for a variety of important functions including capturing prey and defense. Nematocysts are the fastest-known accelerating structures in the animal world. The small scale (microns) coupled with rapid acceleration (in excess of 5 million g) present significant challenges in imaging that prevent detailed descriptions of their kinematics. The immersed boundary method was used to numerically simulate the dynamics of a barb-like structure accelerating a short distance across Reynolds numbers ranging from 0.9–900 towards a passive elastic target in two dimensions. Results indicate that acceleration followed by coasting at lower Reynolds numbers is not sufficient for a nematocyst to reach its target. The nematocyst’s barb-like projectile requires high accelerations in order to transition to the inertial regime and overcome the viscous damping effects normally encountered at small cellular scales. The longer the barb is in the inertial regime, the higher the final velocity of the projectile when it touches its target. We find the size of the target prey does not dramatically affect the barb’s approach for large enough values of the Reynolds number, however longer barbs are able to accelerate a larger amount of surrounding fluid, which in turn allows the barb to remain in the inertial regime for a longer period of time. Since the final velocity is proportional to the force available for piercing the membrane of the prey, high accelerations that allow the system to persist in the inertial regime have implications for the nematocyst’s ability to puncture surfaces such as cellular membranes or even crustacean cuticle.

From droplets to waves: periodic instability patterns in highly viscous microfluidic flows

We experimentally study the transition from droplet to wave regimes in microfluidic liquid–liquid multiphase flows having large differences in viscosity. A unified approach based on periodic pattern analysis is employed to study relationships between dispersed and separated flow regimes, including dripping, jetting, capillary waves, inertial waves and core–annular flows over a wide range of flow rates and viscosity contrasts. We examine the morphology and dynamics of each flow regime based on wavelength, frequency and velocity of repeating unit cells to elucidate their connections and to develop predictive capabilities based on dimensionless control parameters. We demonstrate in particular that pattern selection is contingent upon the propagation velocity of droplets and waves at the transition. We also investigate microfluidic wave breaking phenomena with the formation of ligaments and droplets from wave crests in both capillary and inertial wave regimes. This work expands conventional multiphase flow regimes observed in microchannels and shows new routes to disperse highly viscous materials using interfacial waves dynamics in confined microsystems.

Flow, turbulence and potential droplet break up mechanisms in an in-line Silverson 150/250 high shear mixer

Angle resolved 2-D PIV measurements were performed to characterise the flow and turbulence as well as indicate potential droplet break up mechanisms in an in-line Silverson 150/250 high shear mixer, using water as the working fluid in the turbulent regime (120,000 < Re < 420,000). Distributions of Reynolds stresses, turbulent kinetic energy (TKE), and energy dissipation rates (ε) were examined. The regions of interest (ROI) were: A – jet emanating from a stator hole and B – the rotor swept volume. The complex flow pattern can cause droplet break up under either laminar or turbulent conditions depending on the characteristic length and velocity in the ROI; break up due to turbulence in the inertial regime was identified as the dominant mechanism in this study. Evaluated energy dissipation rates obtained assuming either a fully resolved velocity field (DE) or using the Smagorinsky closure model (SGS) were found to depend on rotor speed e.g. ε∝Nb with b exponents of 1.59–1.90 (DE) and 2.42–2.84 (SGS), which are comparable to existing literature values. The influence on ε of the rotor speed, external pump flow rate and induced backpressure on the mixer outlet, were also investigated. Analysis revealed that the intensity and propensity of ε is dictated by the dominant flow in the mixing head e.g. radial flow at high pump flow rates, prominent in ROI A or tangential flow at high rotor speeds and when backpressure is induced, prominent in ROI B.

Steady flow of a viscoelastic film over an inclined plane featuring periodic slits

We consider the steady flow of a viscoelastic film over an inclined plane featuring a periodic arrangement of slits, which are oriented normal to the main direction of flow. The film creates a second gas-liquid interface connecting the two hydrophilic sidewalls of a slit. This interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. We develop a computational model and carry out detailed numerical simulations, based on the finite element method to investigate this flow. To this end, we solve the two-dimensional momentum and mass conservation equations, while employing the Phan-Thien-Tanner (PTT) constitutive model to account for the rheology of the viscoelastic material. An elliptic grid generation scheme is used to follow the large deformation of the film shape. We perform a thorough parametric analysis to investigate the combined effects of elastic, inertia, capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that increasing fluid elasticity decreases the two wetting lengths along each side wall of a slit. On the contrary, the wetting of the slit is enhanced by the shear-thinning of the fluid. Multiple steady solutions connected by turning points forming a hysteresis loop and transcritical bifurcations as well as isolated solution branches are revealed by pseudo-arc-length continuation. In particular, it is predicted that under certain conditions, the transition from the capillary to the inertia regime is not smooth; instead a hysteresis loop arises. This is the signature of an abrupt decrease of the film penetration with increasing flow rate since higher deformations cannot be sustained. Additionally, we have performed calculations for a wide range of the geometrical characteristics of the substrate. We find that the elastic effects of the fluid become more pronounced when the slits are closer together in their periodic arrangement and of width decreasing to become almost comparable to the capillary length of the liquid. These can lead to almost no wetting of the slit, i.e. the Cassie-Baxter state, even in a hydrophilic surface.

Interfacial viscoelasticity and aging effect on droplet formation and breakup

Interfacial rheology of interfaces stabilized by food ingredients has been studied extensively within the limits of small and slow deformations, within the linear viscoelastic regime (LVR). However, in practice, products such as foams or emulsions undergo fast and large deformations during food processes such as chewing. emulsifying, or foaming. In this study, we used capillary break-up and droplet formation to observe and quantify large and fast deformations of such interfaces, in the presence of surface viscoelasticity. We studied the inertial thinning of the liquid neck during drop formation using high-speed imaging. We tested solutions of several surface active molecules, including two low molecular weight surfactants (SDS and Tween 20), two milk proteins (β-Lactoglobulin, Bovine Serum Albumin) and the saponin Escin. Escin showed the most significant effect on the shape of the neck and neck thinning dynamics. Interfacial shear rheology measurements indicated that the unusual breakup observation of Escin solutions was related to the high viscoelastic surface shear moduli. These high moduli, and the resulting more symmetric and shorter neck shape observed for the proteins and Escin, are associated with the formation of solid-like interfacial structures. However, the presence of a viscoelastic interface did not affect the inertial thinning regime, which indicates that the solid structures were broken down locally at the area of minimum radius (Rmin), and affected only the shape outside this region. The results of this work therefore, show that interfacial measurements within the LVR are not entirely representative for fast and large deformations well beyond linearity.

Periodic folding of a falling viscoelastic sheet.

A viscoelastic solid sheet fed from a certain height towards a rigid horizontal plane folds on itself provided that there is no slip. This phenomenon commonly occurs in the manufacturing process of textile and paper products. In this paper we apply a particle dynamics model to investigate this phenomenon. At a low feeding velocity and low viscosity, the inertial effect and the viscous dissipation within the sheet are negligible, and our model successfully reproduces the existing quasistatic results in the gravitational regime. As the feeding velocity and the viscosity of the sheet increase, the folding process changes significantly. The length of the folds decrease and the "rolling back" motion of the sheet vanishes. In the inertial regime, a scaling law between the fold length and the feeding velocity is derived by balancing the kinetic energy and the elastic bending energy involved in folding, which is verified by the simulation. It is found that above a critical feeding velocity, the folding morphology transforms from line contact into point contact with the sheet exhibiting a lemniscate-like pattern. Finally, a phase diagram for the folding morphology is constructed. The results presented in this work may offer some insights into the high-speed manufacturing of paper and fabric sheets.

Modeling Turbulent Flows in Porous Media

Turbulent flows in porous media occur in a wide variety of applications, from catalysis in packed beds to heat exchange in nuclear reactor vessels. In this review, we summarize the current state of the literature on methods to model such flows. We focus on a range of Reynolds numbers, covering the inertial regime through the asymptotic turbulent regime. The review emphasizes both numerical modeling and the development of averaged (spatially filtered) balances over representative volumes of media. For modeling the pore scale, we examine the recent literature on Reynolds-averaged Navier–Stokes (RANS) models, large-eddy simulation (LES) models, and direct numerical simulations (DNS). We focus on the role of DNS and discuss how spatially averaged models might be closed using data computed from DNS simulations. A Darcy–Forchheimer-type law is derived, and a prior computation of the permeability and Forchheimer coefficient is presented and compared with existing data.