Low Reynolds Number Limit Research Articles

Viscosimetric squeeze flow of suspensions.

The rheology of particle suspensions has been extensively explored in the case of a simple shear flow, but less in other flow configurations which are also important in practice. Here we investigate the behavior of a suspension in a squeeze flow, which we revisit using local pressure measurements to deduce the effective viscosity. The flow is generated by approaching a moving disk to a fixed wall at constant velocity in the low Reynolds number limit. We measure the evolution of the pressure field at the wall and deduce the effective viscosity from the radial pressure drop. After validation of our device using a Newtonian fluid, we measure the effective viscosity of a suspension for different squeezing speeds and volume fractions of particles. We find results in agreement with the Maron-Pierce law, an empirical expression for the viscosity of suspensions that was established for simple shear flows. We prove that this method to determine viscosity remains valid in the limit of large gap width. This makes it possible to study the rheology of suspensions within this limit and therefore suspensions composed of large particles, in contrast to Couette flow cells which require small gaps.

Eccentricity-induced dielectrophoretic migration of a compound drop in a uniform external electric field

Eccentric compound drops, which are ubiquitous in many naturally inspired and engineering systems, can migrate under the sole presence of a uniform electric field, unlike the case of isolated single drops. Here, we report the migration of eccentric compound drops under a uniform electric field, imposed parallel to the line of centres of the constituting drops, by developing an approximate analytical model that applies to low Reynolds number limits under negligible droplet deformation, following axisymmetric considerations. In contrast to the sole influence of the electrohydrodynamic forces that has thus far been established to be emphatic for the eccentric configuration, here we report the additional effects induced by the dielectrophoretic forces to result in decisive manipulation in the drop migration. We show that the relative velocity between the inner and outer drops, which is a function of the eccentricity itself, dictates the dynamical evolution of the eccentricity variation under the competing electrohydrodynamic and dielectrophoretic interactions. This brings out four distinct regimes of the migration characteristics of the two drops based on their relative electro-physical properties. Our results reveal that an increase in eccentricity and the size ratio of the inner and outer droplets may induce monotonic or non-monotonic variation in the drop velocities, depending on the operating regime. We show how the interplay of various properties holds the control of selectively increasing or suppressing the eccentricity with time. These findings open up various avenues of electrically manipulative motion of encapsulated fluidic phases in various applications encompassing engineering and biology.

Natural frequencies of a bubble near a solid sphere

This paper presents an analytical and computational method to describe natural frequencies of a spherical bubble residing near a solid sphere of an arbitrary size in an otherwise unbounded fluid. Under low capillary and Reynolds number limits, the relevant hydrodynamic fields are converted into time-invariant but frequency-dependent quantities by temporal Fourier transform. Then, the spatial variations in the velocity and the pressure can be expressed in terms of two sets of harmonic basis functions involving spherical coordinates centered around the particle and the bubble. A subsequent derivation of transformation coefficients between the aforementioned two sets allows a matrix equation relating the unknown amplitudes to the boundary conditions at all interfaces. Finally, natural frequencies corresponding to different modes of pulsation are obtained from the eigenvalues of the constructed matrix. The results show fast convergence of the computed frequencies with the increasing number of basis functions. These values change significantly with the distance of the bubble from the particle and even decay to zero for some modes when their surface-to-surface separation vanishes. Furthermore, bubble oscillation near a solid plate is also discussed when the radius of the solid sphere is increased to an infinitely large dimension. Thus, this article renders a comprehensive study of naturally pulsating submerged bubbles in the presence of a nearby solid surface of various kinds.

Propulsion of Magnetic Beads Asymmetrically Covered with DNA Origami Appendages

AbstractEukaryotic cells that swim by the beating of nanoscale elastic filaments (flagella) present a promising locomotion paradigm for man‐made analogues essential for next‐generation in‐vivo treatments and for the study of collective phenomena at the low Reynolds number limit. However, artificial analogues have been limited to many microns in size due to the engineering challenges of fabricating actable flexible filaments at the nanoscale—thereby narrowing the application scope. Here, made‐to‐order nanoscale filaments designed on the molecular level are fabricated using the DNA‐origami technique. It is found that magnetic beads anisotropically covered with such bundles move in a ballistic fashion when wagged back and forth under an external magnetic field. Furthermore, by comparing bead dynamics at a range of bundle coverages and driving frequencies, compelling evidence is amassed to suggest that this ballistic motion is imparted by the beating of the DNA origami filaments as synthetic flagella. This proof‐of‐concept work opens up avenues for further made‐for‐purpose appendages designed using DNA self‐assembly and with it ever more complex locomotion on the nano and microscale.

Enhanced settling and dispersion of inertial particles in surface waves

Particulate matter in the environment, such as sediment, marine debris and plankton, is transported by surface waves. The transport of these inertial particles is different from that of fluid parcels described by Stokes drift. In this study, we consider the transport of negatively buoyant particles that settle in flow induced by surface waves as described by linear wave theory in arbitrary depth. We consider particles that fall under both a linear drag regime in the low Reynolds number limit and in a nonlinear drag regime in the transitional Reynolds number range. Based on an analysis of typical applications, we find that the nonlinear regime is the most widely applicable. From an expansion in the particle Stokes number, we find kinematic expressions for inertial particle motion in waves, and from a multiscale expansion in the dimensionless wave amplitude, we find expressions for the wave-averaged drift velocities. These drift velocities are analogous to Stokes drift and can be used in large-scale models that do not resolve surface waves. We find that the horizontal drift velocity is reduced relative to the Stokes drift of fluid parcels and that the vertical drift velocity is enhanced relative to the particle terminal settling velocity. We also demonstrate that a cloud of settling particles released simultaneously will disperse in the horizontal direction. Finally, we discuss the accuracy of our expressions by comparing against numerical simulations, which show excellent agreement, and against experimental data, which show the same trends.

The discrete Green's function paradigm for two-way coupled Euler–Lagrange simulation

We outline a methodology for the simulation of two-way coupled particle-laden flows. The drag force that couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. In addition to hydrodynamics, the method can be extended to other physics present in particle-laden flows such as heat transfer and electromagnetism. The discrete Green's functions for the Navier–Stokes equations are obtained at low particle Reynolds number in a two-plane channel geometry. We perform verification at different Reynolds numbers for a particle settling under gravity parallel to a plane wall, for different wall-normal separations. Compared with other point-particle schemes, the Stokesian discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations. To account for degradation in accuracy away from the wall at finite Reynolds number, we extend the present methodology to an Oseen-like discrete Green's function. The extended discrete Green's function method is found to be accurate within $6\,\%$ at all wall-normal separations for particle Reynolds numbers up to 24. The discrete Green's function approach is well suited to dilute systems with significant mass loading and this is highlighted by comparison against other Euler–Lagrange as well as particle-resolved simulations of gas–solid turbulent channel flow. Strong particle–turbulence coupling is observed in the form of turbulence modification and turbophoresis suppression, and these observations are placed in context of the different methods.

Particle migration in channel flow of an elastoviscoplastic fluid

We study the dynamics of a neutrally buoyant rigid sphere carried by an elastoviscoplastic fluid during 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 on the brink of the walls. The particles which are initially within the plug region are observed to translate with an equivalent velocity because the plug with none 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. within the viscoelastic limit, within the range of studied parameters (low elasticity), inertia is dominant and therefore the refore the particle finds its equilibrium position between the centreline and the wall. An equivalent happens within the viscoplastic limit, yet the yield surface plays the role of centreline. However, the mixture of elasticity and plasticity of the suspending fluid (elastoviscoplasticity) trigger particle-focusing: within the elastoviscoplastic flow, for a particular range of Weissenberg numbers isolated particles migrate all the thanks to the centreline by getting into the core plug region. This behaviour suggests a particle-focusing process for inertial regimes which wasn't previously found during a viscoelastic or viscoplastic carrying fluid.Transporting suspension of particles during a yield-stress fluid may be a crucial problem to be understood thanks to intrinsic complexities from the carrying fluid rheology to the particle dynamics. a brief list of applications may include, but isn't limited to, construction and oil & gas industries. Efforts of Segre & Silberber and other scholars uncovered the behaviour of particles suspended during a Newtonian fluid Poiseuille flow from theoretical frameworks that extensively revealed features of this problem at the low Reynolds number limit, to experimental validations/extensions, Further studies uncovered the features of the matter in non-cylindrical conduits and migration of deformable and non-spherical particles.

Flow-induced resonant shear-wave instability between a viscoelastic fluid and an elastic solid

Linear stability analysis of plane Couette flow of a viscoelastic, upper-convected Maxwell (UCM) fluid past a deformable elastic solid is carried out in the low Reynolds number limit using both numerical and asymptotic techniques. The UCM fluid is characterized by its viscosity η, density ρ, and relaxation time τR, whereas the deformable solid is considered to be a linear elastic solid of shear modulus G. The asymptotic analysis is performed in the Re ≪ 1 limit, where Re = ρVR/μ is the Reynolds number, V is the top plate velocity, and R is the thickness of the fluid. Both asymptotic and numerical approaches are used to understand the effect of solid elasticity, represented by the dimensionless parameter Γ, and fluid elasticity, characterized by the Weissenberg number W, on the growth rate of a class of modes with high frequencies (compared to the imposed shear rate, termed high-frequency Gorodtsov-Leonov, or “HFGL” modes) in the Re ≪ 1 limit. Here, the dimensionless groups are defined as W = τRV/R and Γ = ηV/GR. The results obtained from the numerical analysis show that there is an interaction between the shear waves in the fluid and the elastic solid, which are coupled via the continuity conditions at the interface. The interaction is particularly pronounced when W = Γ, strongly reminiscent of resonance. The resonance-induced interaction leads to shear waves in the coupled system with a decay rate of ci = −1/[2k(W + Γ)]. In this case, it is not possible to differentiate the fluid and solid shear waves individually and the coupled fluid-solid system behaves as a single composite material. The leading order asymptotic analysis suggests that the growth rate of the HFGL modes is proportional to W2 for W ≪ 1. The asymptotic analysis, up to first correction, shows an oscillating behavior of ci with an increase in Γ, in agreement with the results from our numerical approach. In addition, we also carry out an asymptotic analysis in the no-flow, but nonzero inertia limits to illustrate the role played by the imposed flow in the instability of the shear waves. It is found that, at the leading order, the wave speed for the coupled fluid-solid problem is neutrally stable in the absence of flow. Thus, the unstable resonant modes in the coupled fluid-solid system are shown to be driven by the imposed flow.

Numerical Study of the Magnetohydrodynamic Heat Transfer Peristaltic Flow in Tube Against High Reynolds Number

In the present investigation, we have studied the magnetohydrodynamic (MHD) heat transfer of peristaltic flow in a tube. The analysis is made without imposing any assumption to obtain the streamline and isothermal line directly. Galerkin’s finite element method has been used on the governing Navier-Stoke’s equation in the form of ψ-ω. The graphs of the computed longitudinal velocity, temperature and pressure are plotted against different value of the emerging parameter by using the stream function and vorticity. The results are valid beyond the long wavelength and the low Reynolds number limits. We conclude that higher values of the parameters are not independent of the time mean flow rate.

Effect of high Reynolds number on hydromagnetic peristaltic flow in an inclined channel using finite element method

An analysis of the peristaltic flow in an inclined channel for different wave forms is carried out in this paper. The developed mathematical model is represented by a set of partial differential equations. The finite element method is implemented to solve the governing equations for stream function and vorticity. The obtained results are valid beyond the long wavelength and low Reynolds number limits. Important features of peristaltic transport are discussed for the variation of magnetic field, Reynolds and wave numbers. The obtained results, when compared with the results available in literature are in good agreement.

STRD MARTINI: Simulating Quasi-2D Hydrodynamics with Chemically Detailed Lipid Models

Hydrodynamic interactions are an important component of lipid membrane dynamics. A proper molecular dynamics simulation of lateral diffusion in a membrane requires coupling to hydrodynamic flows in the surrounding solvent. In the low Reynolds number limit, the interactions mediated by these flows are long-ranged. Camley, et al recently generalized the Saffman-Delbruck result to periodic boundary conditions, and showed that hydrodynamic interactions introduce significant error into lateral diffusion of lipids and proteins in typical molecular dynamics simulations. The system sizes required to overcome the finite size effect are computationally prohibitive when using explicit solvent models in both all-atom and coarse-grained simulation. Implicit solvent models offer a potential solution, but entirely neglect hydrodynamic momentum transport. To remedy this, we have supplemented the Dry MARTINI implicit solvent model with an efficient mesoscopic particle-based hydrodynamic model called Stochastic Rotation Dynamics (SRD). Our implementation allows for fine control over fluid properties of the solvent, such as viscosity and Reynolds number, and includes a thermostat for the solvent that produces canonical energy fluctuations without interfering with the mesoscopic hydrodynamic flows. The approach, dubbed “STRD Martini”, has been implemented in an in-house version of Gromacs v 5.1.

Hartmann duct flow at moderate magnetic Reynolds numbers

AbstractThe application of a uniform external magnetic field on the turbulent duct flow of an electrically conducting fluid leads to several interesting changes in the structure and the mean charateristics of the flow. This is fairly well understood from the existing studies of duct flows in the low magnetic Reynolds number (Rm) limit. In this paper, we present the results for magnetohydrodynamic duct flow at moderate Rm obtained from direct numerical simulations (DNS). Several differences are observed to occur in this case as compared to low Rm flows, such as increased Hartmann layer thickness and enhanced large scale turbulence in the core region of the duct cross‐section due to partial expulsion of magnetic flux. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Flows in slip-patterned micro-channels using boundary element methods

In this study we investigate steady, pressure-driven, two-dimensional flow of Newtonian fluid through slip-patterned, rectangular channels in the low Reynolds number limit. The slip flow regime is modeled using the Navier's slip boundary condition. In this work, we present only in-phase patterned slip. Subsequently, based on the characteristic length of the patterning, we have considered two subcases, namely large and fine patterned slip. Boundary element method (BEM) is used to numerically solve Stokes equation and obtain the streamline profiles. Streamlines, velocity profiles, pressure gradients, and shear stresses are analyzed to gain a proper understanding of the flow mechanics.

Experimental studies on the flow through soft tubes and channels

Experiments conducted in channels/tubes with height/diameter less than 1 mm with soft walls made of polymer gels show that the transition Reynolds number could be significantly lower than the corresponding value of 1200 for a rigid channel or 2100 for a rigid tube. Experiments conducted with very viscous fluids show that there could be an instability even at zero Reynolds number provided the surface is sufficiently soft. Linear stability studies show that the transition Reynolds number is linearly proportional to the wall shear modulus in the low Reynolds number limit, and it increases as the 1/2 and 3/4 power of the shear modulus for the ‘inviscid’ and ‘wall mode’ instabilities at high Reynolds number. While the inviscid instability is similar to that in the flow in a rigid channel, the mechanisms of the viscous and wall mode instabilities are qualitatively different. These involve the transfer of energy from the mean flow to the fluctuations due to the shear work done at the interface. The experimental results for the viscous instability mechanism are in quantitative agreement with theoretical predictions. At high Reynolds number, the instability mechanism has characteristics similar to the wall mode instability. The experimental transition Reynolds number is smaller, by a factor of about 10, than the theoretical prediction for the parabolic flow through rigid tubes and channels. However, if the modification in the tube shape due to the pressure gradient, and the consequent modification in the velocity profile and pressure gradient, are incorporated, there is quantitative agreement between theoretical predictions and experimental results. The transition has important practical consequences, since there is a significant enhancement of mixing after transition.

Controlled splitting and focusing of a stream of nanoparticles in a converging-diverging microchannel.

We demonstrate the potential of a converging-diverging microchannel to split a stream of nanoparticles towards the interfacial region of the dispersed and the carrier phases, introduced through the middle inlet and through the remaining two inlets respectively, while maintaining a low Reynolds number limit (<10) for the flow of both phases. In addition to the splitting of passive tracer particles, such as polystyrene beads as used herein, the present setup has the potential to be utilized for a controlled reaction and thereby the separation of products towards an intended location, as observed from the experimentation with silver-nanoparticles and hydrogen-peroxide solution. Moreover, the microscale dimension of the channel allows controlled deposition of the reaction product over the bottom surface of the channel, allowing the possibility of bottom-up fabrication of microscale features. We unveil the underlying hydrodynamics that lead to such behaviours through numerical simulations, which are consistent with the experimental observations. The phenomenological features are found to be guided by the splitting of the intrinsic streamlines conforming to the flow geometry under consideration.

Three-dimensional deformation of a spherical droplet in a square duct flow at moderate reynolds numbers

In this work, we study the dynamics of a three-dimensional neutrally buoyant droplet as it translates and deforms in a square-duct. An emphasis is placed on understanding the transient deformation history. We study the effect of the Reynolds number, capillary number, and droplet to carrier fluid viscosity ratio on deformation. In the low Reynolds number limit, we expect the droplet deformation to be independent of Reynolds number. We examine moderate Reynolds numbers in the range of 10–100 and show that inertia plays a significant role in droplet deformation both in the magnitude of steady-state as well as transient deformations. For Reynolds numbers greater than 25, we observe non-monotonic deformation histories. At higher Reynolds numbers we also observe cavity regions forming at the trailing edge of the droplets. These cavity regions appear to be unstable due to their relatively high interfacial curvature. Interfacial tension therefore plays a significant role at the trailing edge to return the cavity region to a more stable convex shape. We study droplet deformation for moderate capillary numbers in the range of 0.10–0.25 and observe a strong variation in deformation magnitude with capillary number. However, the capillary number does not appear to influence the qualitative transient deformation behavior. We also explore the effect of droplet to carrier fluid viscosity ratio on droplet deformation and observe a more complicated relationship. For small viscosity ratios, the deformation histories are non-monotonic and the steady-state deformation increases as the viscosity ratio is increased. In this regime, a cavity forms at the droplet trailing during early times. The formation of a trailing edge cavity is considerably decreased and vanishes for large droplet viscosity. For large drop viscosity, the system is considerably damped and the deformation histories are monotonic for the period of time investigated in this study.

Active temperature and velocity correlations produced by a swimmer suspension

The agitation produced in a fluid by a suspension of microswimmers in the low Reynolds number limit is studied. In this limit, swimmers are modeled as force dipoles all with equal strength. The agitation is characterized by the active temperature defined, as in kinetic theory, as the mean square velocity, and by the equal-time spatial correlations. Considering the phase in which the swimmers are homogeneously and isotropically distributed in the fluid, it is shown that the active temperature and velocity correlations depend on a single scalar correlation function of the dipole-dipole correlation function. By making a simple medium-range order model, in which the dipole-dipole correlation function is characterized by a single correlation length k(0)(-1) it is possible to make quantitative predictions. It is found that the active temperature depends on the system size, scaling as L(4-d) at large correlation lengths L<<k(0)(-1), while in the opposite limit it saturates in three dimensions and diverges logarithmically with the system size in two dimensions. In three dimensions the velocity correlations decay as 1/r for small correlation lengths, while at large correlation lengths the transverse correlation function becomes negative at maximum separation r~L/2, an effect that disappears as the system increases in size.

Linear stability of plane creeping Couette flow for Burgers fluid

It is well known that plane creeping Couette flow of UCM and Oldroy-B fluids are linearly stable. However, for Burges fluid, which includes UCM and Oldroyd-B fluids as special cases, unstable modes are detected in the present work. The wave speed, critical parameters and perturbation mode are studied for neutral waves. Energy analysis shows that the sustaining of perturbation energy in Poiseuille flow and Couette flow is completely different. At low Reynolds number limit, analytical solutions are obtained for simplified perturbation equations. The essential difference between Burgers fluid and Oldroyd-B fluid is revealed to be the fact that neutral mode exists only in the former.

Hydrodynamic synchronization between objects with cyclic rigid trajectories

Synchronization induced by long-range hydrodynamic interactions is attracting attention as a candidate mechanism behind coordinated beating of cilia and flagella. Here we consider a minimal model of hydrodynamic synchronization in the low Reynolds number limit. The model consists of rotors, each of which assumed to be a rigid bead making a fixed trajectory under periodically varying driving force. By a linear analysis, we derive the necessary and sufficient conditions for a pair of rotors to synchronize in phase. We also derive a non-linear evolution equation for their phase difference, which is reduced to minimization of an effective potential. The effective potential is calculated for a variety of trajectory shapes and geometries (either bulk or substrated), for which the stable and metastable states of the system are identified. Finite size of the trajectory induces asymmetry of the potential, which also depends sensitively on the tilt of the trajectory. Our results show that flexibility of cilia or flagella is not a requisite for their synchronized motion, in contrast to previous expectations. We discuss the possibility to directly implement the model and verify our results by optically driven colloids.

ASYMPTOTIC TRANSITION FROM COSSERAT ROD TO STRING MODELS FOR CURVED VISCOUS INERTIAL JETS

This work deals with the modeling and simulation of slender viscous jets exposed to gravity and rotation, as they occur in rotational spinning processes. In terms of slender-body theory, we show the asymptotic reduction of a viscous Cosserat rod to a string system for vanishing slenderness parameter. We propose two string models, i.e. inertial and viscous-inertial string models, that differ in the closure conditions and hence yield a boundary value problem and an interface problem, respectively. We investigate the existence regimes of the string models in the four-parametric space of Froude, Rossby, Reynolds numbers and jet length. The convergence regimes where the respective string solution is the asymptotic limit to the rod turn out to be disjoint and to cover nearly the whole parameter space. We explore the transition hyperplane and derive analytically low and high Reynolds number limits. Numerical studies of the stationary jet behavior for different parameter ranges complete the work.