Simple Shear Flow Research Articles

Collision Modes of Two Eccentric Compound Droplets

A compound droplet with its single inner droplet appears in a broad range of applications and has received much attention in recent years. However, the role of the inner droplet location on the dynamical behaviors of the compound droplet is still not completely understood. Accordingly, the present study numerically deals with the eccentricity of the compound droplet affecting its colliding behaviors with the other droplet in a simple shear flow. The solving method is a front-tracking technique that treats the droplet interface as connected elements moving on a rectangular fixed grid. Initially, two compound droplets assumed circular are placed at a distance symmetrically to the domain center and they come into contact, because of the shear flow, when time progresses. During the collision process, the inner droplet that is initially located at a distance to its outer droplet center circulates around this center. It is found that this rotation also contributes to the formation of the collision modes including the reversing, passing-over and merging ones. Starting from a passing-over mode, a transition to a reversing mode or a merging mode can appear when the inner droplets, in terms of their centroids, are closer than their outer droplets. However, the location of the inner droplet within the outer droplet only has an effect when the value of the Capillary number Ca is varied from 0.01 to 0.08. For Ca < 0.01 corresponding to the merging mode and Ca ≥ 0.16 corresponding to the passing-over mode, the inner droplet position has almost no impact on the collision behaviors of two compound droplets.

Medium amplitude parallel superposition (MAPS) rheology. Part 1: Mathematical framework and theoretical examples

A new mathematical representation for nonlinear viscoelasticity is presented based on the application of the Volterra series expansion to the general nonlinear relationship between shear stress and shear strain history. This theoretical and experimental framework, which we call medium amplitude parallel superposition (MAPS) rheology, reveals a new material property, the third-order complex modulus, which describes completely the weakly nonlinear response of a viscoelastic material in an arbitrary simple shear flow. In this first part, we discuss several theoretical aspects of this mathematical formulation and new material property. For example, we show how MAPS measurements can be performed in strain- or stress-controlled contexts and provide relationships between the weakly nonlinear response functions measured in each case. We show that the MAPS response function is a superset of the response functions that have been previously reported in medium amplitude oscillatory shear and parallel superposition rheology experiments. We also show how to exploit inherent symmetries of the MAPS response function to reduce it to a minimal domain for straightforward measurement and visualization. We compute this material property for a few constitutive models to illustrate the potential richness of the datasets generated by MAPS experiments. Finally, we discuss the MAPS framework in the context of some other nonlinear, time-dependent rheological probes and explain how the MAPS methodology has a distinct advantage over these others because it generates data embedded in a very high-dimensional space without driving fluid mechanical instabilities, and is agnostic to the flow protocol.

Numerical study of collision modes of multi-core compound droplets in simple shear flow

Collisions of multi-core compound droplets have generated substantial interest in recent years because of their applications in the industry and academia. This study uses the front-tracking method to simulate the transition between the two collision modes of multi-core compound droplets in a simple shear flow. Compound droplets initially assumed identical have two sub-droplets of equal size. Given the shear flow, the droplets collide with one another and behave in two main modes, namely, passing-over and reversing. In the passing-over mode, the droplets pass over one another after coming into contact. The reversing mode appears with two compound droplets returning to their initial sides after the collision. During collision, the sub-droplets circulate approximately at the center of their enclosing outer droplets. Some parameters, including capillary number Ca, viscosity ratios μio and μmo, radius ratio Rio of the sub-droplets to the outer droplets, and sub-droplet angle a0, are investigated to determine their impact on these modes of collisions. We find that the transition from a reversing to passing-over mode occurs when we increase the value of Ca from 0.01 to 0.63}, {tiRio from 0.20 to 0.475, and μio and μmo in the range of 0.16-6.3. However, an increase in the value of a0 between -75° and 90° leads to a change from a passing-over to reversing mode. Diagrams of the colliding modes are also presented in this research.

Flow properties of surfactant solutions of rod-like micelles passing through a small slit

Pressure drops in flows through a small slit were experimentally measured at constant flow rates to investigate flows with an abrupt contraction and expansion. The flow properties of water, 50 wt% glycerol solution, and two types of surfactant solutions (Lipoquad and Lipothoquad) with a counterion (NaSal) were examined. The slit heights used were 480 µm, 222 µm, and 129 µm, with estimated hydraulic diameters of 938 µm, 439 µm, and 256 µm, respectively. Viscosity was measured using a capillary-type viscosity meter. Water and the 50 wt% glycerol solution exhibited Newtonian viscosity, with the viscosity of the glycerol solution being 10 times that of water. By contrast, the surfactant solutions, which contained rod-like micelles, exhibited non-Newtonian viscosity. A power law model was therefore applied for the relationship between shear viscosity and shear rate on the wall. The observed flow properties of water (and the 50 wt% glycerol solution) and the numerical predictions of the Navier–Stokes equations were found to agree to within the experimental errors. In plots of the dimensionless pressure drop against the generalized Reynolds number, however, anomalous flow properties of the surfactant solutions were observed. The pressure drops of Lipoquad/NaSal agreed with those of water (and the 50 wt% glycerol solution) at low apparent strain rate (Reynolds number) but were larger at high apparent strain rate (Reynolds number). Furthermore, the pressure drops of Lipothoquad/NaSal were greater than those of water in the investigated range of the generalized Reynolds number. To examine these experimental results further, flows in a planar channel with an abrupt contraction and expansion were observed using a polarization measurement system based on a high-speed polarization camera. In contrast to polymer solutions (in which polymer chains were oriented along the flow direction), the phase difference for Lipoquad/NaSal before and in the slit region was high in regions with a strong elongational component. In addition, it was high along the centerline after the slit region. These results also differ from those of simple shear flows. Furthermore, the phase difference of Lipothoquad/NaSal was very unstable. In addition, the relationship between the resultant pressure drop and the measured phase differences was investigated, and a dependence was found on the apparent strain rate (Weissenberg number).

An immersed boundary method for mass transfer through porous biomembranes under large deformations

In this paper, a novel immersed boundary method for modeling convection and diffusion of mass transfer through porous membranes under large deformations is proposed. An algebraic function consisted of transmembrane velocity and diffusion coefficient of membrane is derived for modeling local concentration jumps across porous membranes. An additional equation is introduced to describe the temporal evolution of diffusive flux. Numerical validations show that the present method is robust to predict the concentration field under various fluid-porous membrane coupling conditions such as membrane oscillations in an elastic force driven flow, tank-treading and swinging motion of red blood cells in simple shear flow, and parachuting motion when cells squeeze in narrow channels down to 4 micrometers. The conservations of mass and volume in single cell are examined, and the relative errors of mass conservation are negligible. Numerical results suggest that the proposed method is capable of predicting mass transfer through porous membranes while providing detailed hydrodynamics around cells in micro-circulatory systems.

Minuet motion of a pair of capsules interacting in simple shear flow

We study the three-dimensional hydrodynamic interaction of a pair of identical, initially spherical capsules freely suspended in a simple shear flow under Stokes flow conditions. The capsules are filled with a Newtonian liquid (same density and viscosity as the suspending fluid). Their membranes satisfy the neo-Hookean constitutive law. We consider the rarely studied case where the capsule centres are initially located in (or near) the plane defined by the flow direction and the vorticity vector, i.e. in two different shear planes. The motion and deformation of the capsules are modelled by means of a boundary integral technique to compute the flows, coupled to a finite element method to calculate the force exerted by the membranes on the fluids. We follow the motion and deformation of the capsules as they are convected towards each other after a sudden start of the flow. Our main finding is that, depending on their initial position and deformability, the two capsules may oscillate slowly about the flow gradient axis, get nearer to each other at each oscillation to finally interact strongly and separate. This minuet motion had not been identified previously. We identify the regions of space where either simple crossing or minuet occurs. This phenomenon has a marked influence on the irreversible trajectory drift of two capsules after crossing: the minuet process leads to a significant trajectory displacement along the flow gradient when none was expected, based on the previous studies where the two capsules had a significant relative velocity.

The Navier–Stokes–Cahn–Hilliard model with a high-order polynomial free energy

In this paper, we present a high-order polynomial free energy for the phase-field model of two-phase incompressible fluids. The model consists of the Navier–Stokes (NS) equation and the Cahn–Hilliard (CH) equation with a high-order polynomial free energy potential. In practice, a quartic polynomial has been used for the bulk free energy in the CH equation. It is well known that the CH equation does not satisfy the maximum principle and the phase-field variable takes shifted values in the bulk phases instead of taking the minimum values of the double-well potential. This phenomenon substantially changes the original volume enclosed by the isosurface of the phase-field function. Furthermore, it requires fine resolution to keep small shapes. To overcome these drawbacks, we propose high-order (higher than fourth order) polynomial free energy potentials. The proposed model is tested for an equilibrium droplet shape in a spherical symmetric configuration and a droplet deformation under a simple shear flow in a fully three-dimensional fluid flow. The computational results demonstrate the superiority of the proposed model with a high-order polynomial potential to the quartic polynomial function in volume conservation property.

The rotation of two-dimensional elliptical porous particles in a simple shear flow with fluid inertia

The motion of porous particles in fluid flow is of fundamental importance in both natural and industrial processes. Recent work shows that fluid inertia can essentially alter the rotation of spherical porous particles in a simple shear flow. In this contribution, we examined the influence of fluid inertia on the rotation of elliptical porous particles in shear flow by solving the volume-averaged macroscopic equations with a two-dimensional lattice Boltzmann model. It is confirmed that the Darcy number Da has only a minor effect on the rotation of elliptical porous particles if fluid inertia is neglected. At finite fluid inertia, the elliptical porous particles, however, manifested time-periodic rotation with a non-uniform angular rate. For particles with small to intermediate Da, the period of rotation increases with Reynolds number Re up to a critical Rec above which the particle would stop rotating. It is shown that the maximum and minimum angular rates, as well as the inclination angle at which the particle has a minimum angular rate, are significantly affected by Da. A scaling law for the period of rotation initially proposed for solid impermeable particles can be extended to elliptical porous particles at finite fluid inertia. For a highly permeable ellipse, however, Rec has not been observed, and thus, the scaling law breaks down. We calculated the relative viscosity and intrinsic viscosity for simple shear flow containing elliptical porous particles. A formula developed for suspensions with vanishing Re can also be extended to correlate the intrinsic viscosity to Da at finite Re.

Helicoidal particles and swimmers in a flow at low Reynolds number

In this paper, we consider the dynamics of a helicoidal object, which can be either a passive particle or an active swimmer, with an arbitrary shape in a linear background flow at low Reynolds number, and derive a generalized version of the Jeffery equations for the angular dynamics of the object, including a new constant from the chirality of the object as well as the Bretherton constant. The new constant appears from the inhomogeneous chirality distribution along the axis of the helicoidal symmetry, whereas the overall chirality of the object contributes to the drift velocity. Further investigations are made for an object in a simple shear flow, and it is found that the chirality effects generate non-closed trajectories of the director vector which will be stably directed parallel or anti-parallel to the background vorticity vector depending on the sign of the chirality. A bacterial swimmer is considered as an example of a helicoidal object, and we calculate the values of the constants in the generalized Jeffery equations for a typical morphology of Escherichia coli. Our results provide useful expressions for the studies of microparticles and biological fluids, and emphasize the significance of the symmetry of an object on its motion at low Reynolds number.

On modelling shear layers in dense granular flows

Shear bands are common in dense quasi-static granular flows. They can appear in the interior of the flowing material or at confining boundaries and are typically of the order of ten particle diameters in thickness. Deformation tends to be localized in shear bands separating non-deforming or weakly deforming regions. Dilatancy and sharp velocity variation are typical in these shear layers. Much work has been reported in the literature concerning the development of non-local quasi-static rheological models to predict the flow behaviour in shear layers. In a recent article, Dsouza & Nott (J. Fluid Mech., vol. 888, 2020, R3) derive a non-local extension to a classical plasticity model by postulating that some local quantities appearing in the yield function, which stipulates the relationship between different components of the stress for the material to undergo sustained yielding, and the flow rule which provides information on the rate of deformation tensor to within an arbitrary multiplicative constant, should be replaced by their local averages. They then obtain an explicit non-local model which does not involve new microstructural variables and they show that the model captures velocity and volume fraction fields in simple shear flows, although some model parameters must be fitted to achieve quantitative agreement. This article discusses the work of Dsouza & Nott (2020) and comments on work ahead for further testing and developing the model.

High-fidelity scaling relationships for determining dissipative particle dynamics parameters from atomistic molecular dynamics simulations of polymeric liquids

An optimized Dissipative Particle Dynamics (DPD) model with simple scaling rules was developed for simulating entangled linear polyethylene melts. The scaling method, which can be used for mapping dimensionless (reduced units) DPD simulation data to physical units, was based on scaling factors for three fundamental physical units; namely, length, time, and viscosity. The scaling factors were obtained as ratios of equilibrium Molecular Dynamics (MD) simulation data in physical units and equivalent DPD simulation data for relevant quantities. Specifically, the time scaling factor was determined as the ratio of longest relaxation times, the length scaling factor was obtained as the ratio of the equilibrium end-to-end distances, and the viscosity scaling factor was calculated as the ratio of zero-shear viscosities, each as obtained from the MD (in physical units) and DPD (reduced units) simulations. The scaling method was verified for three MD/DPD model liquid pairs under several different nonequilibrium conditions, including transient and steady-state simple shear and planar elongational flows. Comparison of the MD simulation results with those of the scaled DPD simulations revealed that the optimized DPD model, expressed in terms of the proposed scaling method, successfully reproduced the computationally expensive MD results using relatively cheaper DPD simulations.

Minimization principle for shear alignment of liquid crystals.

If a static perturbation is applied to a liquid crystal, then the director configuration changes to minimize the free energy. If a shear flow is applied to a liquid crystal, then one might ask: Does the director configuration change to minimize any effective potential? To address that question, we derive the Leslie-Ericksen equations for dissipative dynamics and determine whether they can be expressed as relaxation toward a minimum. The answer may be yes or no, depending on the number of degrees of freedom. Using theory and simulations, we consider two specific examples, reverse tilt domains under simple shear flow and dowser configurations under plane Poiseuille flow, and we demonstrate that each example shows relaxation toward the minimum of an effective potential.

Multiplicity of stable orbits for deformable prolate capsules in shear flow.

This work investigates the orbital dynamics of a fluid-filled deformable prolate capsule in unbounded simple shear flow at zero Reynolds number using direct simulations. The motion of the capsule is simulated using a model that incorporates shear elasticity, area dilatation, and bending resistance. Here the deformability of the capsule is characterized by the nondimensional capillary number Ca, which represents the ratio of viscous stresses to elastic restoring stresses on the capsule. For a capsule with small bending stiffness, at a given Ca, the orientation converges over time towards a unique stable orbit independent of the initial orientation. With increasing Ca, four dynamical modes are found for the stable orbit, namely, rolling, wobbling, oscillating-swinging, and swinging. On the other hand, for a capsule with large bending stiffness, multiplicity in the orbit dynamics is observed. When the viscosity ratio λ ≲ 1, the long-axis of the capsule always tends towards a stable orbit in the flow-gradient plane, either tumbling or swinging, depending on Ca. When λ ≳ 1, the stable orbit of the capsule is a tumbling motion at low Ca, irrespective of the initial orientation. Upon increasing Ca, there is a symmetry-breaking bifurcation away from the tumbling orbit, and the capsule is observed to adopt multiple stable orbital modes including nonsymmetric precessing and rolling, depending on the initial orientation. As Ca further increases, the nonsymmetric stable orbit loses existence at a saddle-node bifurcation, and rolling becomes the only attractor at high Ca, whereas the rolling state coexists with the nonsymmetric state at intermediate values of Ca. A symmetry-breaking bifurcation away from the rolling orbit is also found upon decreasing Ca. The regime with multiple attractors becomes broader as the aspect ratio of the capsule increases, while narrowing as viscosity ratio increases. We also report the particle contribution to the stress, which also displays multiplicity.

A non-local constitutive model for slow granular flow that incorporates dilatancy

Over the past two decades several attempts have been made to formulate constitutive models for slow granular flow to remedy the deficiencies of classical plasticity. All the proposed models assume the medium to be incompressible, though it is well known that density change accompanies deformation in granular materials. A particularly important aspect of density change that is distinctive of granular materials is dilatancy, or volume deformation caused by shear deformation. No constitutive model for sustained flow has thus far captured dilatancy. Here we present a non-local constitutive model wherein the deformation rate and density at a point depend on the state of stress in a mesoscopic region around it. Apart from incorporating dilatancy, our model has a physical origin that is distinct from that of the previously proposed non-local models. We test our model on simple shear flow in the absence and presence of gravity, and find its predictions to be in good agreement with particle dynamics simulations.

Shear thickening and history-dependent rheology of monodisperse suspensions with finite inertia via an immersed boundary lattice Boltzmann method

Three-dimensional direct numerical simulations of dense suspensions of monodisperse spherical particles in simple shear flow have been performed at particle Reynolds numbers between 0.1 and 0.6. The particles translate and rotate under the influence of the applied shear. The lattice Boltzmann method was used to solve the flow of the interstitial Newtonian liquid, and an immersed boundary method was used to enforce the no-slip boundary condition at the surface of each particle. Short range spring forces were applied between colliding particles over sub-grid scale distances to prevent overlap. We computed the relative apparent viscosity for solids volume fractions up to 38% for several shear rates and particle concentrations and discuss the effects of these variables on particle rotation and cluster formations. The apparent viscosities increase with increasing particle Reynolds number (shear thickening) and solids fraction. As long as the particle Reynolds number is low (0.1), the computed viscosities are in good agreement with experimental measurements, as well as theoretical and empirical equations. For higher Reynolds numbers, we find much higher viscosities, which we relate to slower particle rotation and clustering. Simulations with a sudden change in shear rate also reveal a history (or hysteresis) effect due to the formation of clusters. We quantify the changes in particle rotation and clustering as a function of the Reynolds number and volume fraction.

Pattern method for higher harmonics from macromolecular orientation in oscillatory shear flow

For a suspension of rigid dumbbells, in any simple shear flow, we must first solve the diffusion equation for the orientation distribution function by a power series expansion in the shear rate. Our recent work has uncovered the pattern in the coefficients of this power series [L. M. Jbara and A. J. Giacomin, “Orientation distribution function pattern for rigid dumbbell suspensions in any simple shear flow,” Macromol. Theory Simul. 28, 1800046-1–1800046-16 (2019)]. Specifically, we have here used this pattern on large-amplitude oscillatory shear (LAOS) flow, for which we have extended the orientation distribution function to the 6th power of the shear rate. In this letter, we embed this extension into the Giesekus expression for the extra stress tensor to arrive at the alternant shear stress response, up to and including the seventh harmonic. We thus demonstrate that the pattern method for macromolecular orientation now allows our harmonic analysis to penetrate the shear stress response to oscillatory shear flow far more deeply than ever.

Effects of chain length and polydispersity on shear banding in simple shear flow of polymeric melts.

The characteristics of shear banding were investigated in entangled, polydisperse, linear polymer melts under steady-state and startup conditions of simple shear flow. This virtual experimentation was conducted using course-grained nonequilibrium dissipative particle dynamics simulations expressed in terms of a force-field representation that faithfully models the atomistic system dynamics. We examined melts with two mean molecular bead numbers of Nn = 2 50 and 400 and polydispersity indexes of 1.0, 1.025, and 1.05. The wide range of relaxation timescales in the polydisperse melts decreased the nonmonotonic character of the steady-state shear stress vs. shear rate profile compared to a monodisperse linear melt. The polydispersity level required to observe a stress plateau in the shear stress profile at intermediate shear rates was correlated with the nominal entanglement density. Startup of shear flow simulations revealed the development of spatial inhomogeneities and dynamic instabilities in polydisperse fluids containing both monotonic and nonmonotonic shear stress flow curves. Although the shape and duration of instabilities were found to be correlated with the monotonicity of the shear stress profile, the onset and underlying mechanism leading to the formation of shear bands were generally universal. The simulations revealed that perturbations arose soon after the occurrence of a large stress overshoot under startup conditions, and that banded structures stemmed from local reorientation and subsequent deconstruction of the entanglement network. Furthermore, data indicated that the inception of strain localization occurred at shear rates near the reciprocal of the Rouse characteristic timescale, [small gamma, Greek, dot above] > τR-1. Transient shear banding was observed in shorter chain melts undergoing startup of shear flow in which instabilities arose after the appearance of a stress overshoot. These instabilities eventually decayed, but only long after the stresses had attained their steady-state values. The longer chain melt exhibited a shear band structure that remained indefinitely after the stresses had attained steady state.

Study on Coagulation Kinetics of Disk-like Particles under Simple Shear Flow.

In this study, a theoretical study is carried out on the collision of two disk-like particles to understand the coagulation of disk-like particles suspended in liquid under a shear flow. The diameter of the particle is fixed at 2 μm while the length is varied so that the aspect ratio (length/diameter) varies from 0.1 to 0.4. The liquid viscosity is changed from 0.01 to 1 Pa s. The minimum Peclet number is 10, and the Brownian motion is considered to be negligible. Both hydrodynamic and van der Waals interactions are included in tracking the position and the orientation of each particle. The Hamaker constant is fixed at 1.06 × 10-20 J. The boundary integral formulation is used to calculate the hydrodynamic interaction. To obtain the kinetic constant of coagulation, the time-independent orientation distribution function for a particle is obtained under the noninteracting condition. The kinetic constant of coagulation is obtained by considering the presence of collision between two particles initially separated by a long distance, the orientations of two particles, and the flux of the liquid flow. The result shows that the kinetic constant of coagulation is reduced to approximately 1/3 of the value for the noninteracting particles when the viscosity is 1 Pa·s. As collision modes, side-side, face-edge, side-edge, and edge-edge are considered. The edge-edge mode is frequently observed in the given range of the aspect ratio. The collision mode does not change much from the noninteracting case except for the side-side mode.

Influence of the Polarity of the Electric Field on Electrorheometry

Uniaxial extensional flow is a canonical flow typically used in rheological characterization to provide complementary information to that obtained by imposing simple shear flow. In spite of the importance of having a full rheological characterization of complex fluids, publications on the rheological characterization of mobile liquids under extensional flow have increased significantly only in the last 20 years. In the case of the rheological characterization of electrorheological fluids, the situation is even more dramatic, as the ERFs have been exclusively determined under simple shear flow, where an electrorheological cell is attached to the rotational rheometer generating an electric field perpendicular to the flow direction and that does not allow for inverting the polarity. The very recent work published by Sadek et al., who developed a new electrorheological cell to be used with the commercial Capillary Breakup Extensional Rheometer (CaBER), allows for the very first time performing electrorheometry under extensional flow. By means of the same experimental setup, this study investigates the influence of the polarity of the imposed electric field on the filament thinning process of a Newtonian and an electrorheological fluid. Results show that a polarity against the gravity results in filament thinning processes that live longer or reach a stable configuration at lower intensities of the applied electric field.

Magnetic field induced ferrofluid droplet breakup in a simple shear flow at a low Reynolds number

The breakup phenomenon of a ferrofluid droplet in a simple shear flow under a uniform magnetic field is numerically investigated in this paper. The numerical simulation, based on the finite element method, uses a level set method to capture the dynamic evolution of the droplet interface between the two phases. Focusing on small Reynolds numbers (i.e., Re ≤ 0.03), systematic numerical simulations are carried out to analyze the effects of magnetic field strength, direction, and viscosity ratio on the breakup phenomenon of the ferrofluid droplet. The results suggest that applying a magnetic field along α = 45° and 90° relative to the flow direction initiates breakup in a ferrofluid droplet at a low capillary number in the Stokes flow regime, where the droplet usually does not break up in a shear flow alone. At α = 0° and 135°, the magnetic field suppresses breakup. Also, there exists a critical magnetic bond number, Bocr, below which the droplet does not rupture, which is also dependent on the direction of the magnetic field. Additionally, the effect of the viscosity ratio on droplet breakup is examined at variable magnetic bond numbers. The results indicate a decrease in the critical magnetic bond number Bocr values for more viscous droplets. Furthermore, more satellite droplets are observed at α = 45° compared to α = 90°, not only at higher magnetic field strengths but also at larger viscosity ratios.