Simple Shear Flow Research Articles

A hybrid lattice Boltzmann and finite difference method for two-phase flows with soluble surfactants

A hybrid method is developed to simulate two-phase flows with soluble surfactants. In this method, the interface and bulk surfactant concentration equations of diffuse-interface form, which include source terms to consider surfactant adsorption and desorption dynamics, are solved in the entire fluid domain by the finite difference method, while two-phase flows are solved by a lattice Boltzmann color-gradient model, which can accurately simulate binary fluids with unequal densities. The flow and interface surfactant concentration fields are coupled by a modified Langmuir equation of state, which allows for surfactant concentration beyond critical micelle concentration. The capability and accuracy of the hybrid method are first validated by simulating three numerical examples, including the adsorption of bulk surfactants onto the interface of a stationary droplet, the droplet migration in a constant surfactant gradient, and the deformation of a surfactant-laden droplet in a simple shear flow, in which the numerical results are compared with theoretical solutions and available literature data. Then, the hybrid method is applied to simulate the buoyancy-driven bubble rise in a surfactant solution, in which the influence of surfactants is identified for varying wall confinement, density ratio, Eotvos number and Biot number. It is found that surfactants exhibit a retardation effect on the bubble rise due to the Marangoni stress that resists interface motion, and the retardation effect weakens as the Eotvos or Biot number increases. We further show that the weakened retardation effect at higher Biot numbers is attributed to a decreased non-uniform effect of surfactants at the interface. By comparing with the Cahn-Hilliard phase-field method, we also show that the present method conserves the mass for each fluid, improves numerical stability especially at high density ratio and Eotvos number, and does not need the selection of free parameters, thus breaking the limitations of the existing method.

Rotational Taylor dispersion in linear flows

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared with diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection–diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly spherical particles, weak flow and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

Master curves for unidirectional flows of FENE-P fluids in rectilinear and curvilinear geometries

We demonstrate that velocity profiles for steady, unidirectional shear flows of the FENE-P (Finitely-Extensible Nonlinear Elastic, with Peterlin closure) fluid, undergoing canonical rectilinear (pressure-driven flow in a rectangular channel or a circular pipe) or curvilinear (in Taylor–Couette or Dean configurations) flows, obey universal master curves that are a function only of the ratio Wi/L , for a fixed solvent to solution viscosity parameter β. Here, Wi is the Weissenberg number defined as the product of the dumbbell relaxation time and an appropriate shear rate, while L is the ratio of the maximum extension of the polymer to its equilibrium root-mean-square end-to-end distance. The data collapse and the resulting master curves for the velocity profile is a generalization of the recent demonstration of master curves for polymer viscosity and first normal stress coefficient for a FENE-P fluid under steady simple shear flow (Yamani and McKinley, 2023). For pressure-driven channel and pipe flows, we derive simple analytical expressions for the velocity profiles, in the high shear-rate regime of Wi/L≫1, that readily elucidate the role of finite extensibility of the polymer on the velocity profiles. In the Wi/L≫1 regime, for all the flows considered, the limit of zero solvent (β=0) is shown to be singular, owing to the absence of a high-shear plateau in the total solution viscosity, resulting in very different velocity profiles for β=0 and β→0.

Ideal Collision Rate of Symmetric Hexapods in a Simple Shear Flow.

An ideal collision rate (ICR) is defined as the average rate of contact between two particles that translate and rotate with the imposed fluid flow in the absence of interparticle interactions. ICR in a simple shear flow provides an estimate of the collision rate in a flowing dilute particle suspension, and its value is known only for traditional convex shapes such as spheres and cylinders. In this work, we compute the ICR for a family of symmetric hexapods (shown in Figure 1) that are particles composed of three orthogonal cylinders with coinciding centers (forming six arms) with at least two cylinders having the same length. We employed the finite-element method to obtain the rotational dynamics of hexapods and Monte Carlo simulations to calculate their ICR. Our results indicate that the ICR for hexapods is not directly proportional to the volume of the particle, in contrast to what is observed for convex shapes such as spheres and cylinders. For hexapods of the same size, the ICR could be similar for particles with order-of-magnitude differences in their volumes and it could also vary by an order of magnitude for particles with similar volumes. Specifically for hexapods termed branched fibers, which have one longer cylinder, the ICR changes by an order of magnitude even when the shorter cylinders are significantly smaller than the longer cylinder. This change is attributed to the increase in the tumbling frequency of the particle due to hydrodynamic forces acting on the shorter arms, in addition to the increased probability of collision afforded by them. Using asymptotic theory for high-aspect ratio cylinders, we showed that the tumbling period of hexapods was proportional to an algebraic power of the ratio of its arm lengths and has a weaker logarithm relationship with the aspect ratio of its arms. The collision cross-section provided the relative cross-streamline particle separations for the most likely binary collisions in the suspensions, and its value was sensitive to the arm lengths, as well. The collision rate of hexapods also could not be estimated within an order of magnitude from existing geometric models, such as the ICR of a circumscribing convex shape, ICR for one of the individual cylinders of the hexapod, or using the particle volume times the shear rate. Our work indicates that collision rate for non-convex particles in shear flows critically depends on the shape of the particle due to nontrivial changes in the particle's orientational dynamics, and the ICR calculation serves as a more reliable method for estimating their true collision rates in the suspension.

Examination of flow birefringence induced by the shear components along the optical axis using a parallel-plate-type rheometer

In this study, the concept of rheo-optics was applied to explore the flow birefringence induced by the stress components along the camera’s optical axis since it is often overlooked in the traditional theories of photoelastic flow measurement. A novel aspect of this research is that it involved conducting polarization measurements on simple shear flows, specifically from a perspective in which a shear-velocity gradient exists along the camera’s optical axis. A parallel-plate-type rheometer and a polarization camera are employed for these systematic measurements. The experimental findings for dilute aqueous cellulose nanocrystal suspensions demonstrate that the flow birefringence can be expressed as a power law based on the power of the second invariant of the deformation-rate tensor. This suggests that flow birefringence can be universally characterized by the coordinate-independent invariants and a pre-factor determined by the direction of polarization measurement. By adjusting the nonlinear term in the stress-optic law, its applicability could be expanded to include three-dimensional fluid stress fields in which the stress is distributed along the camera’s optical axis.

Evolution of local relaxed states and the modeling of viscoelastic fluids

We introduce a class of continuum mechanical models aimed at describing the behavior of viscoelastic fluids by incorporating concepts originated in the theory of solid plasticity. Within this class, even a simple model with constant material parameters is able to qualitatively reproduce a number of experimental observations in both simple shear and extensional flows, including linear viscoelastic properties, the rate dependence of steady-state material functions, the stress overshoot in incipient shear flows, and the difference in shear and extensional rheological curves. Furthermore, by allowing the relaxation time of the model to depend on the total strain, we can reproduce some experimental observations of the non-attainability of steady flows in uniaxial extension and link this to a concept of polymeric jamming or effective solidification. Remarkably, this modeling framework helps in understanding the interplay between different mechanisms that may compete in determining the rheology of non-Newtonian materials.

DEM analysis of cohesive granular shear flow using dynamic adhesion force model – Model validation for contact-dominated regime

When the DEM simulations are performed, the particle stiffness of the Discrete Element Method (DEM) is often reduced to reduce the computational cost. The Dynamic Adhesion/Cohesion Force Model (DAFM/DCFM) was proposed for adhesive/cohesive particles to make the effect of adhesion/cohesion force on the collisional motion of a particle with a reduced particle stiffness equivalent to that of the original system.This study validates the applicability of DCFM to the contact-dominated regime by means of a series of DEM simulations of two-dimensional simple shear flow of cohesive spherical particles. The results demonstrate that DCFM accurately represents the cohesive nature of the original system with respect to the fluid-to-solid phase transition of granular medium due to a reduction in shear rate, as well as the shear stress of the granular medium. It is demonstrated that the bond-breaking model proposed in the present study describes well the validity of DCFM in the contact-dominated regime.

Dynamics of a driven spheroid in a slow oscillating creeping shear flow

We report the orientation dynamics of a sinusoidally driven spheroid suspended in a slow and weak/strong oscillatory shear flow without Brownian and inertial forces, derive the governing equations, find the classical Jeffery orbits, and then solve them numerically. These equations describe Jeffery's orbits for no external force and no flow oscillations. When the external forces are small, and there are no oscillations, they can be seen as perturbations of the equations that result in Jeffery's orbits. The small perturbations disturb the Jeffery orbits. We also analyze the chaotic and regular dynamics regimes in nearly quiescent, simple shear, and weak/strong and slow oscillating shear flows. We observe quantitative and qualitative differences in the particle dynamics for an oscillating shear flow compared to simple shear flow, as seen from the Poincaré sections, attractors, phase diagrams, time series, and Lyapunov exponents. The analysis indicates that the slow oscillations reduce the complexity of the dynamics of the particle compared to simple shear flow. The steady-state solutions for both prolate and oblate spheroids remain in the flow gradient plane in the case of strong oscillatory shear. At the same time, there is some disturbance from the flow gradient plane for weak oscillations due to the external force instead of inertial forces reported earlier in the literature. In addition, we propose a mechanism to improve particle separation based on shape using a combination of simple and oscillating shear flows, offering significant advantages in separating particles from a colloidal mixture that would otherwise be impossible.

Rheo-optics of giant micelles: SALS patterns of cetyltrimethylammonium tosylate solutions in presence of sodium bromide

In this work, we present a systematic study based on Small-Angle Light Scattering (SALS) patterns of the simple shear flow response of semi-diluted solutions of cetyltrimethylammonium tosylate (CTAT; 5.5 wt.% - 0.12 M) in the presence of sodium bromide (NaBr) at different [NaBr]={0,0.12,0.19,0.25,0.3} M concentrations. We evidence a relationship between rheological and light scattering data that reveals a transition into a fast-breaking regime in the dynamics of wormlike micelles formed by the CTAT/NaBr system (Macías et al., 2011; Fierro et al., 2021). This transition into a micellar fast-breaking regime with salt addition ([NaBr]≥0) appears marked by the following features: (i) a decrease in the relaxation time of the material λ0, accompanied by (ii) a decrease of the viscosity level at low shear rates η0 (Macías et al., 2011; Fierro et al., 2021; Schubert et al., 2003; Alkschbirs et al., 2015; Bandyopadhyay et al., 2003). With these, (iii) the formation of butterfly-like patterns is recorded originating from concentration fluctuations, evolution that is accompanied by: (iv) shear banding in the form of non-monotonic flow curves and (v) slow oscillatory transient responses in start-up flow tests captured theoretically with the Bautista–Manero–Puig (BMP) model. In addition, the Cox–Merz rule is fulfilled at molar salt-to-surfactant ratios of R≥1.5. This results in shorter structure-recovery time-scales than the characteristic-time of the flow (Macías et al., 2011; Fierro et al., 2021; Manero et al., 2002). In the case of the elastic modulus G0, the variation was small, which suggests a transition from an entangled to a multiconnected network, as suggested by Kadoma & van Egmond (1997), Kadoma et al. (1997) and Fierro et al. (2021). From a theoretical perspective, we provide predictions for the shear–stress and the first normal-stress growth coefficients in transient start-up simple shear flow using the BMP model. Here, banding R=1.5 solutions display overshot responses at relatively high shear rates (γ̇=10−30s−1), in-line with experimental findings on the start-up flow of wormlike micellar solutions (Soltero et al., 1999; Lerouge et al., 2004; Hu & Lips 2005; Pipe et al., 2010; Mohammadigoushki et al., 2019). Our results are consistent with those reported in other investigations (Macías et al., 2011; Fierro et al., 2021; Schubert et al., 2003; Alkschbirs et al., 2015; Bandyopadhyay et al., 2003; Kadoma et al., 1997; Soltero et al., 1999; Lerouge et al., 2004; Hu & Lips 2005; López-Barrón et al., 2014) and reveal the influence of the Br−-ion used on the mechanical and optical response of the CTAT-NaBr system.

Experimental Analysis Of Flow Birefringence In Jeffery-Hamel Flow

The photoelastic method, a stress field measurement method in solid mechanics, is being considered for application to fluids. In previous studies, simple shear flow and uniaxial extensional flow experiments have shown a relationship between the measured phase retardation and the velocity field. However, no clear relationship has been shown for extensional and shear combined flow fields. The objective of the present study is to clarify the relationship between the velocity field and the measured phase retardation in an extensional-shear combined flow. For this objective, photoelastic measurements were conducted in a steady flow field using the Jeffery-Hamel flow, which is an extensional-shear combined flow with an analytical solution for the velocity field. Comparison with the analytical velocity field showed that birefringence was proportional to the 0.88 and 0.92 power of the deformation in the shear or extensional-dominated region, respectively. The results show that the birefringence Δ_n followed the power law of extensional rate ε^{0.95} where it is dominant. Whereas shear is dominant, Δ_n proportional to a power-law of γ^{0.88} holds. These results are consistent with previous studies using shear flow and uniaxial extensional flow. Furthermore, it is shown that in the extensional-shear combined flow, the sum-of-squares root of two equations suggests that the theory developed mainly for solids could also be applied to fluids.

Perspective on the description of viscoelastic flows via continuum elastic dumbbell models

Non-Newtonian fluid mechanics and computational rheology widely exploit elastic dumbbell models such as Oldroyd-B and FENE-P for a continuum description of viscoelastic fluid flows. However, these constitutive equations fail to accurately capture some characteristics of realistic polymers, such as the steady extension in simple shear and extensional flows, thus questioning the ability of continuum-level modeling to predict the hydrodynamic behavior of viscoelastic fluids in more complex flows. Here, we present seven elastic dumbbell models, which include different microstructurally inspired terms, i.e., (i) the finite polymer extensibility, (ii) the conformation-dependent friction coefficient, and (iii) the conformation-dependent non-affine deformation. We provide the expressions for the steady dumbbell extension in shear and extensional flows and the corresponding viscosities for various elastic dumbbell models incorporating different microscopic features. We show the necessity of including these microscopic features in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows, highlighting their potential significance in accurately modeling viscoelastic channel flow with mixed kinematics.

The morphology of cell spheroids in simple shear flow

Cell spheroids are a widely used model to investigate cell-cell and cell-matrix interactions in a 3D microenvironment in vitro. Most research on cell spheroids has been focused on their response to various stimuli under static conditions. Recently, the effect of flow on cell spheroids has been investigated in the context of tumor invasion in interstitial space. In particular, microfluidic perfusion of cell spheroids embedded in a collagen matrix has been shown to modulate cell-cell adhesion and to represent a possible mechanism promoting tumor invasion by interstitial flow. However, studies on the effects of well-defined flow fields on cell spheroids are lacking in the literature. Here, we apply simple shear flow to cell spheroids in a parallel plate apparatus while observing their morphology by optical microscopy. By using image analysis techniques, we show that cell spheroids rotate under flow as rigid prolate ellipsoids. As time goes on, cells from the outer layer detach from the sheared cell spheroids and are carried away by the flow. Hence, the size of cell spheroids declines with time at a rate increasing with the external shear stress, which can be used to estimate cell-cell adhesion. The technique proposed in this work allows one to correlate flow-induced effects with microscopy imaging of cell spheroids in a well-established shear flow field, thus providing a method to obtain quantitative results which are relevant in the general field of mechanobiology.

Modelling the effect of soluble surfactants on droplet deformation and breakup in simple shear flow

A hybrid lattice Boltzmann and finite difference method is applied to study the influence of soluble surfactants on droplet deformation and breakup in simple shear flow. First, the influence of bulk surfactant parameters on droplet deformation in two-dimensional shear flow is investigated, and the surfactant solubility is found to influence droplet deformation by changing average interface surfactant concentration and non-uniform effects induced by non-uniform interfacial tension and Marangoni forces. In addition, the droplet deformation first increases and then decreases with Biot number, increases significantly with adsorption number $k$ and decreases with Péclet number or adsorption depth; and among the parameters, $k$ is the most influential one. Then, we consider three-dimensional shear flow and investigate the roles of surfactants on droplet deformation and breakup for different capillary numbers and viscosity ratios. Results show that in the soluble case with $k=0.429$ , the droplet exhibits nearly the same deformation as in the insoluble case due to the balance between surfactant adsorption and desorption; upon increasing $k$ from 0.429 to 1, the average interface surfactant concentration is greatly enhanced, leading to significant increase in droplet deformation. The critical capillary number of droplet breakup $C{{a}_{cr}}$ is identified for varying viscosity ratios in clean, insoluble and soluble ( $k=0.429$ and 1) systems. As the viscosity ratio increases, $C{{a}_{cr}}$ first decreases and then increases rapidly in all systems. The addition of surfactants always favours droplet breakup, and increasing solubility or $k$ could further reduce $C{{a}_{cr}}$ by increasing average interface surfactant concentration and local surfactant concentration near the neck during the necking stage.

Rheology of a suspension of deformable spheres in a weakly viscoelastic fluid

In this work, we consider a suspension of weakly deformable solid particles within a weakly viscoelastic fluid. The fluid phase is modelled as a second-order fluid, and particles within the suspended phase are assumed linearly elastic and relatively dilute. We apply a cell model as a proxy for mean field flow, and solve analytically within a cellular fluid layer and its enclosed particle. We use an ensemble averaging process to derive analytical results for the bulk stress in suspension, and evaluate the macroscopic properties in both shear and extensional flow. Our viscometric functions align with existing literature over a surprisingly broad range of fluid and solid elasticities.The suspension behaves macroscopically as a second-order fluid, and we give simple formulae by which the reader can calculate the parameters of this effective fluid, for use in more complex simulations. We additionally calculate the particle shape and orientation, and in simple shear flow show that the leading-order modifications to the angle of inclination ζ act to align the particle towards the flow direction, giving ζ=π/4−3Cae/4+α0Wi/2α1 where Cae is the elastic capillary number, Wi is the Weissenberg number, and αi are material properties of the suspending second-order fluid, for which the ratio α0/α1 is negative.

Rheology of granular particles immersed in a molecular gas under uniform shear flow.

Non-Newtonian transport properties of a dilute gas of inelastic hard spheres immersed in a molecular gas are determined. We assume that the granular gas is sufficiently rarefied, and hence the state of the molecular gas is not disturbed by the presence of the solid particles. In this situation, one can treat the molecular gas as a bath (or thermostat) of elastic hard spheres at a given temperature. Moreover, in spite of the fact that the number density of grains is quite small, we take into account their inelastic collisions among themselves in its kinetic equation. The system (granular gas plus a bath of elastic hard spheres) is subjected to a simple (or uniform) shear flow. In the low-density regime, the rheological properties of the granular gas are determined by solving the Boltzmann kinetic equationby means of Grad's moment method. These properties turn out to be highly nonlinear functions of the shear rate and the remaining parameters of the system. Our results show that the kinetic granular temperature and the non-Newtonian viscosity present a discontinuous shear thickening effect for sufficiently high values of the mass ratio m/m_{g} (m and m_{g} being the mass of grains and gas particles, respectively). This effect becomes more pronounced as the mass ratio m/m_{g} increases. In particular, in the Brownian limit (m/m_{g}→∞) the expressions of the non-Newtonian transport properties derived here are consistent with those previously obtained by considering a coarse-grained approach where the effect of gas phase on grains is through an effective force. Theoretical results are compared against computer simulations, showing an excellent agreement.

Simulating dynamics of ellipsoidal particles using lattice Boltzmann method.

Anisotropic particles are often encountered in different fields of soft matter and complex fluids. In this work, we present an implementation of the coupled hydrodynamics of solid ellipsoidal particles and the surrounding fluid using the lattice Boltzmann method. A standard link-based mechanism is used to implement the solid-fluid boundary conditions. We develop an implicit method to update the position and orientation of the ellipsoid. This exploits the relations between the quaternion which describes the ellipsoid's orientation and the ellipsoid's angular velocity to obtain a stable and robust dynamic update. The proposed algorithm is validated by looking at four scenarios: (i) the steady translational velocity of a spheroid subject to an external force in different orientations, (ii) the drift of an inclined spheroid subject to an imposed force, (iii) three-dimensional rotational motions in a simple shear flow (Jeffrey's orbits), and (iv) developed fluid flows and self-propulsion exhibited by a spheroidal microswimmer. In all cases the comparison of numerical results shows good agreement with known analytical solutions, irrespective of the choice of the fluid properties, geometrical parameters, and lattice Boltzmann model, thus demonstrating the robustness of the proposed algorithm.

Numerical investigation on the deformation and breakup of an elastoviscoplastic droplet in simple shear flow

In this paper, direct numerical simulations (DNSs) are performed to investigate the deformation and breakup of an elastoviscoplastic (EVP) droplet in a Newtonian matrix under simple shear flow. The two-phase interface is captured by the volume-of-fluid (VOF) method with adaptive mesh refinement technique. The Saramito model (Bingham model coupled exponential Phan-Thien–Tanner viscoelastic model) is used to characterize the rheological behavior of the droplet. The droplet deformation and conformational state are studied with different Capillary numbers Ca, Weissenberg numbers Wi, and Bingham numbers Bi, which represent the surface tension, elasticity, and yield stress of the droplet, respectively. Our results show that droplet deformation occurs at low Ca, while breakup occurs at high Ca. The droplet non-monotonically deforms with increasing Wi and Bi, while is elongated for higher Ca. In addition, three breakup modes (mid-point pinching, transitional breakup, and homogeneous breakup) are reported for EVP droplets, in which transitional breakup disappears due to the influence of high elasticity. The conformational state of the droplet intuitively demonstrates the change of breakup from horizontal shear to vertical breakup. In spite of the fact that the surface tension always inhibits the deformation of droplets, the present work indicates that Bi has little effect on the deformation with high Wi and high Ca, while the influence is obvious at low Wi and Ca. The observed elastic and plastic effects on droplet deformation and breakup are believed to have significant impacts, as yield stress fluids are widely encountered in industrial applications.

Robust microstructure of self-aligning particles in a simple shear flow

Robust microstructure of self-aligning particles in a simple shear flow

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model.

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad's moment method and with computer simulations performed in the Brownian limiting case (m/mg→∞, where mg and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio m/mg increases.

Simulation of drop deformation and breakup in simple shear flow

Simulation of drop deformation and breakup in simple shear flow