Creeping Flow Conditions Research Articles

A novel pseudo-rigid body approach to the non-linear dynamics of soft micro-particles in dilute viscous flow

We propose a novel, demonstrably effective, utmost versatile and computationally highly efficient pseudo-rigid body approach for tracking the barycenter and shape dynamics of soft, i.e. non-linearly deformable micro-particles dilutely suspended in viscous flow. Pseudo-rigid bodies are characterized by affine deformation and thus represent a first-order extension to the kinematics of rigid bodies. Soft particles in viscous flow are ubiquitous in nature and sciences, prominent examples, among others, are cells, vesicles or bacteria. Typically, soft particles deform severely due to the mechanical loads exerted by the fluid flow. Since the shape dynamics of a soft particle - a terminology that shall here also include its orientation dynamics - also affects its barycenter dynamics, the resulting particle trajectory as a consequence is markedly altered as compared to a rigid particle. Here, we consider soft micro-particles of initially spherical shape that affinely deform into an ellipsoidal shape. These kinematic conditions are commensurate with i) the affine deformation assumption inherent to a pseudo-rigid body and ii) the celebrated Jeffery-Roscoe model for the traction exerted on an ellipsoidal particle due to creeping flow conditions around the particle. Without loss of generality, we here focus on non-linear hyperelastic particles for the sake of demonstration. Our novel numerical approach proves to accurately capture the particular deformation pattern of soft particles in viscous flow, such as for example tank-treading, thereby being completely general regarding the flow conditions at the macro-scale and, as an option, the constitutive behavior of the particle. Moreover, our computational method is highly efficient and allows straightforward integration into established Lagrangian tracking algorithms as employed for the point-particle approach to track rigid particles in dilute viscous flow.

A triple emulsion droplet in a uniform flow

The fluid mechanics problem of a triple emulsion droplet in a uniform creeping flow is theoretically studied. The triple emulsion, submerged in an external fluid 1, is composed of an inner spherical drop (fluid 4), covered by a spherical shell (fluid 3), which is subsequently covered by another spherical shell (fluid 2). In the absence of a body force, eight parameters are governed: the Capillary number (Ca), three viscosity ratios (λ21, λ32, λ43), two radii ratios (K, k), and two-surface tension ratios (M, m). If all surfaces remain spherical and concentric, five parameters (λ21, λ32, λ43, K, k) are sufficient. Expressions for the stream functions and the drag forces were obtained to calculate the terminal velocity of the triple emulsion falling or rising in a stagnant fluid under the influence of gravity. With gravity, four additional parameters are needed: the ratio of the buoyancy force to the external viscous force (B), and three density ratios (γ21, γ31, γ41). To remain concentric, all surfaces must have the same velocity (magnitude and direction). This results in two restricted parameters, but even then, not every combination of parameters leads to a physical solution. Our findings suggest that the terminal velocity of a triple emulsion is equal to or lower than that of a double emulsion, which is subsequently equal to or lower than that of a single drop. Similar to single and double emulsion droplets, no deformation takes place in a concentric spherical triple emulsion subjected to a gravitational field under creeping flow conditions.

Spontaneous locomotion of a symmetric squirmer

The squirmer is a popular model to analyse the fluid mechanics of a self-propelled object, such as a micro-organism. We demonstrate that some fore–aft symmetric squirmers can spontaneously self-propel above a critical Reynolds number. Specifically, we numerically study the effects of inertia on spherical squirmers characterised by an axially and fore–aft symmetric ‘quadrupolar’ distribution of surface-slip velocity; under creeping-flow conditions, such squirmers generate a pure stresslet flow, the stresslet sign classifying the squirmer as either a ‘pusher’ or ‘puller’. Assuming axial symmetry, and over the examined range of the Reynolds number $Re$ (defined based upon the magnitude of the quadrupolar squirming), we find that spontaneous symmetry breaking occurs in the puller case above $Re \approx 14.3$ , with steady swimming emerging from that threshold consistently with a supercritical pitchfork bifurcation and with the swimming speed growing monotonically with $Re$ .

Modeling fluid flow in fractured porous media: a comparative analysis between Darcy–Darcy model and Stokes–Brinkman model

AbstractThere are limited comparative studies on modeling fluid transport in fractured porous media. Hence, this paper systematically compares the steady-state creeping flow Stokes–Brinkman and Darcy–Darcy models for computational efficiency and accuracy. Sensitivity analyses were also conducted on the effect of fracture orientations, fracture sizes, mesh resolution, and fractures with Local Grid Refinement (LGR) under the FEniCS computational framework. Both models were validated numerically, and the accuracy of their solution is compared using the R-squared metric and L2 norm estimates. Key results showed that both models have similar pressure and velocity field solutions for a given fracture orientation. The computational time required for solving the Stokes–Brinkman models for a single fracture case was unusually lower than that of the Darcy–Darcy model when the pressure and velocity terms in the Darcy–Darcy model were solved simultaneously using two equations, contrary to where only one equation solves for the pressure and the velocity is obtained by projecting the gradient of pressure onto a vector space. The Stokes–Brinkman model is more sensitive to mesh resolution, and as a result, the Darcy–Darcy model tends to be more accurate than the Stokes–Brinkman model at low resolutions. Local Grid Refinement (LGR) can improve the Stokes–Brinkman model's accuracy at low mesh resolution. Furthermore, both models showed similar results when compared for complex fracture systems such as multiple fracture cases: interconnecting and isolated fractured porous media systems under low-velocity and steady-state creeping flow conditions. The FEniCS code in this paper is shared for future researchers to reproduce results or extend the research work.

Wavy Marangoni effect and strategic slip-pattern-driven species transport in micro-confined sandwiched liquid films

In microfluidic systems, analyte/reagent mixing is essential to achieve rapid chemical/biochemical reactions. The low Reynolds number (Re) flow, however, makes passive mixing difficult and necessitates the use of some external stimulus to cause disturbance in the system. Here, we report how a periodic thermocapillary effect may interact with strategically patterned wall wettability and boost the mixing dynamics in a micro-confined ternary-liquid film system. Approximate, yet without compromise on the physics involved, analytical solutions to the energy and Navier–Stokes equations under creeping flow conditions are obtained to comprehend the thermal and hydrodynamic characteristics of the thermo-capillarity. In the binary fluid limit, our model is validated with the findings of Pendse & Esmaeeli (Intl J. Therm. Sci., vol. 49, 2010, pp. 1147–1155), and good agreement is found. The flow characteristics in the ternary-liquid system due to discrete wall temperatures is also demonstrated via finite-element-based numerical simulations, and agreement with the semi-analytical solution is noted. The qualitative study of the flow pattern reveals that enhanced mixing is obtained as a result of vortical motion created by the interplay of the periodic thermo-capillarity-driven interfacial flow and wall slip. The observation is further consolidated from the numerical simulation of the species transport equation. The species distribution so obtained is compared with fully developed laminar flow for the same inlet concentration. Our study further investigates the effects of the fluids’ relative thermal conductivity, wall slip length, relative film thickness and thermal (or slip) phase differences on the mixing efficiency of the proposed arrangement. While wall slip, relative film thickness and relative thermal conductivity regulate mixing in the top and bottom layers, phase difference determines how well the middle fluid layer mixes. Slip at the walls creates vortex distortions to get a strong churning effect. The lower thermal conductivity of the top and bottom fluids weakens the thermocapillary flow; however, the dominance of the patterned slip improves mixing in such cases.

A further study of history force effect on particle clustering in turbulence

We examine the effects of the history force on the preferential concentration of small particles in homogeneous isotropic turbulence under a wide range of the particle-to-fluid mass density ratio γ (from 0 to 10000) by direct simulations of the Navier–Stokes equations. As to the history integral kernel, we introduce two expressions: i.e., the classical Basset kernel valid only in the limit of creeping flow conditions and for short times, and the Mei–Adrian kernel (Mei and Adrian, 1992) applicable for finite Reynolds numbers and large times. It is revealed that the presence of the history force weakens the level of preferential concentration to some extent, specifically under the conditions of the mass density ratio of around 1.5–10 for heavy particles (e.g., solid particles in liquid) and smaller than 0.7 for light particles (e.g., bubbles in liquid) and that the overprediction of particle accumulation by neglecting the history force is much more remarkable for light particles. A comparison between the Basset kernel and the Mei–Adrian kernel has shown that the Basset kernel overestimates the history effect but only slightly.

On a variational principle for the Upper Convected Maxwell model

A variational principle for the Upper Convected Maxwell model is presented. The stationary value of the appropriate functional is the drag on an immersed object. From the principle, a formula is derived for the derivative of the drag with respect to the Deborah number for an arbitrarily shaped particle in a circular duct under creeping flow conditions. The formalism is compared with the conventional reciprocal theorem. Whereas the reciprocal theorem gives the drag as a volume integral involving the Stokesian stress tensor, the variational principle involves the stress from the adjoint equation. For low Deborah numbers both approaches provide the correction to the Stokes drag as a volume integral involving only the Stokesian rate-of-strain tensor, in line with second-order fluid theory. • A variational principle for the Upper Convected Maxwell model is presented. • At stationarity the functional equals the drag on a partcle in a duct. • For low Deborah numbers the drag correction involves only the Stokesian strain rate. • This is in correspondence with the reciprocal theorem and second-order fluid theory.

Shear-induced motion of a bead on regular substrates at small particle Reynolds numbers

We study experimentally the impact of substrate topology on shear-flow-induced motion of a single bead at low particle Reynolds numbers. The substrates are regular quadratic and triangular arrangements of fixed spherical particles. Their topology is varied by using different spacings between the spheres. Here, we show that it has a strong impact not only on the critical Shields number for incipient bead motion but also on its motion above threshold. We focus on Shields numbers where the bead velocity is smaller than the settling velocity. For the different substrates, the data on the average bead velocity collapse on a master curve, showing the impact of the critical Shields number on the bead motion. To describe the bead motion, we develop a model for creeping flows based on expressions by Goldman, Cox and Brenner for the flow-induced forces and torques on a moving bead near a plane. Our model considers rolling and sliding motion. The bead detaches from the substrate on the downhill side at larger substrate spacing or higher Shields numbers, and flies through the interstices of the substrate until hitting the neighbouring substrate spheres. While sliding has only a minor effect on the average bead velocity, detachment has a strong impact. At large substrate spacings, it leads to a bistability, usually associated with inertial flows, even for adhesionless particles under creeping-flow conditions. The model shows good agreement with the experimental results.

Numerical simulation of viscoelastic effects in peristaltic transport of drops

• Under creeping-flow conditions, elasticity of a drop slightly hinders its peristaltic transport. • Elasticity of the liquid surrounding the drop has a significant but non-monotonic effect on the drop's peristaltic transport. • Viscosity of the bulk liquid can have an accelerating or decelerating effect on the drop's velocity. • Peristalsis can be regarded as a viable option for the transport of elastic drops in microfluidic systems. Peristalsis is known to be a viable option for the transport of Newtonian drops suspended in another Newtonian liquid. In certain applications, the drop or the bulk liquid is known to be, more or less, viscoelastic. In the present work the effect of a liquid's elasticity is investigated on the peristaltic transport of a single, circular drop. Having assumed that the liquid obeys the Oldroyd-B model, numerical results have been obtained at Deborah numbers up to 15 for two different cases: (1) the drop is viscoelastic, and (2) the liquid surrounding the drop is viscoelastic. Based on the results obtained in this work, it is predicted that under creeping-flow conditions, a drop's elasticity has a negligible effect on its transport velocity provided it is placed on the centerline. But for off-center viscoelastic drops the effect of elasticity is significant and delay their migration to the centerline. For the case in which only the liquid surrounding the drop is viscoelastic, elastic stresses are predicted to have a significant but non-monotonic effect on the drop velocity. In both cases, the response of a drop is predicted to strongly depend on its size and also on the cross-viscosity ratio.

A drop in a nonlinear shear flow

The fluid mechanics problem of an initially spherical drop subjected to a two-directional nonlinear cubic shear flow, under creeping flow conditions, is theoretically studied. This physical situation is described by four dimensionless numbers: the capillary number (Ca), the viscosity ratio (λ), the linear velocity ratio (G) and the nonlinear intensity of the flow (E). Analytical expressions for the external and internal flows of a spherical drop as well as for the shape of the deformed drop were obtained by applying Lamb's general solution to the Stokes equations. The internal and external streamlines of a spherical drop (Ca = 0) suggest interesting and unusual shapes which are sometimes also accompanied by many stagnation points. Whenλ = ∞, the drop becomes a solid sphere with an angular velocity$\omega ={-} (1 - G)/2$independent ofE. For the familiar simple shear flow in thex-direction (G = 0,E = 0), Taylor's theory, valid at$Ca \ll 1$andλ = O(1), predicts an ellipsoidal drop oriented at an azimuthal angleϕ = 45°; asCaincreases, the deformation increases and the drop attains a peanut shape. For other linear or nonlinear cases, the drop is aligned atϕ = ±45°, and a spherical drop with no deformation is possible atCa ≠ 0. In a highly viscous drop$(\lambda \gg 1)$, the drop remains almost spherical even atCa = O(1), oriented atϕ = 0° or 90° (G ≠ 1), and its deformation is independent ofCa.

Peristaltic transport of elliptic particles: A numerical study

Peristaltic transport of elliptic particles suspended in Newtonian fluids is numerically investigated in a planar channel formed between two flexible membranes. Numerical results were obtained under creeping-flow conditions for centered and off-center particles using the lattice Boltzmann method. The results demonstrate the importance of aspect ratio and initial inclination angle on peristaltic transport of solid particles. For a domain comprising just one wave, it was shown that, in free-pumping mode, circular particles move faster than elliptic particles and experience less shear stress. They also resist a larger adverse pressure gradient before they are finally brought to rest. Above a critical Reynolds number, however, elliptic particles are predicted to move faster than circular particles. The effect was attributed to the vulnerability of circular particles to hydrodynamic instability, which is exhibited by the particle detaching itself from the centerline, thereby adopting a longer trajectory. This is the first time that peristaltic transport of elliptic particles is being numerically studied, and the results can be used for designing peristalsis-based micro-swimmers or microfluidic systems deemed for single-cell studies.

Rheo-PIV analysis of the steady torsional parallel-plate flow of a viscoplastic microgel with wall slip

The steady torsional parallel-plate flow with apparent slip of a viscoplastic microgel with 0.12 wt. % of poly(acrylic acid), Carbopol® 940, is analyzed under isothermal and creeping flow conditions by simultaneous rheometrical and particle image velocimetry measurements (rheo-PIV). This analysis brings out the complexity of this non-homogeneous flow to evaluate the rheological properties of yield-stress fluids with apparent slip. Overall, velocity distributions of the microgel are linear across the gap and evidence that plug and steady shear flow cannot coexist in the parallel-plate geometry. Also, velocity distributions compare favorably with the predictions of the analytical solutions of the motion equation for the steady parallel-plate flow of a Herschel–Bulkley (H–B) fluid subject to apparent slip. However, edge fracture results in non-linear velocity distributions, i.e., non-rheometrical flow. A slip yield stress is calculated for the microgel-parallel-plate combination, and the slip velocity beyond this stress is described by two distinct relationships, namely, one almost linear as a function of the excess stress (pure plug flow) and the one power-law type as a function of the viscous stress (shear flow with slip). On the other hand, we show how the yield stress of the microgel can be determined from various new methodologies using slip velocities and rheometrical data. Finally, we demonstrate that the ordinate to the origin in the Mooney method becomes negative when the fluid undergoes plug flow. Despite this fact, we confirm the validity of the Mooney and Yoshimura and Prud'homme (Y–P) analyses and highlight their benefits in the characterization of yield-stress fluids.

The concept of elasto-visco-plasticity and its application to a bubble rising in yield stress fluids

• New simulations are presented that include the elasticity of yield stress materials, as oldroyd first suggested. • The inverted teardrop bubble shape and the negative wake are predicted. • Numerical results indicate the importance of elasticity in yield stress materials. • A satisfactory agreement is found between experiments and simulations for the bubble shape and rising velocity. • The critical yield stress value for bubble entrapment increases by elastic forces. Based on the original ideas by Oldroyd in 1947, Saramito in 2007 introduced a generally acceptable model for yield stress fluids that also accounts for their elasticity. We use this model to examine the buoyancy-driven rise of a bubble through Carbopol and predict the inverted teardrop shape that bubbles often attain when they translate through an elastoviscoplastic material. Such shapes are common in viscoelastic materials, but it is impossible to predict them using viscoplastic models without elasticity. Our results also show that generally, the fore-aft symmetry is lost and the negative wake is formed in the trailing edge of the bubble, even when creeping flow conditions are approached. We show that elastic and plastic forces are dominant in the smaller bubbles, whereas significant shear and extensional thinning occurs in the larger ones, making inertia forces dominant. The agreement of our predicted bubble shapes and rise velocities with their experimental counterparts is very satisfactory either for the smaller or the larger bubbles from those we examined. The former have an inverted teardrop shape and the latter have either an oblate or spherical cap shape. A deviation is found in the bubble radius for which the transition from the inverted teardrop shape to the oblate shape occurs. This deviation is attributed primarily to insufficient rheological characterization of the materials, particularly in terms of their nonlinear elasticity and their extensional properties.

Stokesian motion of a spherical particle near a right corner made by tangentially moving walls

The slow motion of a small buoyant sphere near a right dihedral corner made by tangentially sliding walls is investigated. Under creeping-flow conditions the force and torque on the sphere can be decomposed into eleven elementary types of motion involving simple particle translations, particle rotations and wall movements. Force and torque balances are employed to find the velocity and rotation of the particle as functions of its location. Depending on the ratio of the wall velocities and the gravitational settling velocity of the sphere, different dynamical regimes are identified. In particular, a non-trivial line attractor/repeller for the particle motion exists at a location detached from both the walls. The existence, location and stability of the corresponding two-dimensional fixed point are studied depending on the wall velocities and the buoyancy force. The impact of the line attractors/repellers on the motion of small particles in cavities and its relevance for corner cleaning applications are discussed.

Hydrodynamic Entrance Length for Laminar Flow in Microchannels with Rectangular Cross Section

This work presents a detailed numerical investigation on the required development length (L=L/B) in laminar Newtonian fluid flow in microchannels with rectangular cross section and different aspect ratios (AR). The advent of new microfluidic technologies shifted the practical Reynolds numbers (Re) to the range of unitary (and even lower) orders of magnitude, i.e., creeping flow conditions. Therefore, accurate estimations of L at Re≤O(1) are important for microsystem design. At such low Reynolds numbers, in which inertial forces are less dominant than viscous forces, flow characteristics become necessarily different from those at the macroscale where Re is typically much larger. A judicious choice of mesh refinement and adequate numerical methods allowed obtaining accurate results and a general correlation for estimating L, valid in the ranges 0≤Re≤2000 and 0.1≤AR≤1, thus covering applications in both macro and microfluidics.

Flow structure and loads over inclined cylindrical rodlike particles and fibers

The flow past a fixed finite-length circular cylinder, the axis of which makes a nonzero angle with the incoming stream, is studied through fully-resolved simulations, from creeping-flow conditions to strongly inertial regimes. The investigation focuses on the way the body aspect ratio $\chi$ (defined as as the length-to-diameter ratio), the inclination angle $\theta$ with respect to the incoming flow and the Reynolds number $\text{Re}$ (based on the cylinder diameter) affect the flow structure past the body and therefore the hydrodynamic loads acting on it. The configuration $\theta=0^\circ$ (where the cylinder is aligned with the flow) is considered first, from creeping-flow conditions up to $\text{Re}=400$, with aspect ratios up to $20$ ($10$) for $\text{Re}\leq10$ ($\text{Re}\geq10$). In the low-to-moderate Reynolds number regime ($\text{Re}\lesssim5$), influence or the aspect ratio, inclination (from $0^{\circ}$ to $30^{\circ}$), and inertial effects is examined by comparing numerical results for the axial and transverse force components and the spanwise torque with theoretical predictions based on the slender-body approximation, possibly incorporating finite-Reynolds-number corrections. Semiempirical models based on these predictions and incorporating finite-length and inertial corrections extracted from the numerical data are derived. For large enough Reynolds numbers ($\text{Re}\gtrsim10^2$), separation takes place along the upstream part of the lateral surface of the cylinder, deeply influencing the surface stress distribution. Numerical results are used to build empirical models for the force components and the torque, valid for moderately inclined cylinders ($|\theta|\lesssim30^\circ$) of arbitrary aspect ratio up to $\text{Re}\approx300$ and matching those obtained at low-to-moderate Reynolds number.

Modeling flow in embryonic lymphatic vasculature: what is its role in valve development?

A majority of lymphatic valves tend to form in proximity to vessel junctions, and it is often proposed that disturbed flow at junctions creates oscillating shear stress that leads to accumulation of transcription factors which bring about valvogenesis at these sites. In images of networks of dorsal skin lymphatics from embryonic mice (day E16), we compared simulated fluid flow patterns and observed distributions of the transcription factor Prox1, which is implicated in valve formation. Because of creeping-flow conditions, flow across vessel junctions was not 'disturbed', and within a given vessel, shear stress varied inversely with local conduit width. Prox1 concentration was indeed localised to vessel end-regions, but over three networks was not consistently correlated with the vessel normalised-distance distribution of either fluid shear stress or shear-stress axial gradient. These findings do not support the presently accepted mechanism for the role of flow in valve localisation.

Hydrodynamic mobility of a porous spherical particle with variable permeability in a spherical cavity

The present article deals with the analytical study of translational and rotational motion of a porous spherical particle with quadratically increasing permeability inside a concentric spherical cavity filled with incompressible Newtonian fluid, under the creeping flow conditions. The flow fields in clear fluid and porous regions are governed by Stokes equation and generalized Darcy’s law (Brinkman equation) together with mass conservation, respectively. Closed form solutions for translational and rotational mobilities of the particle are obtained with the help of drag and torque acting on the particle surface. The particle mobility inside a cavity attains a maximum value of 1. However, the presence of cavity wall slows down the particle motion as a result the particle mobility becomes smaller than unity. The effect of cavity wall on the mobility is significant when the gap between the particle surface and cavity wall is less. Various limiting cases are obtained which agree with earlier existing results. The results are explained with the aid of graphs for better clarity.

Micron-sized double emulsions and nematic shells generated via tip streaming

Here we show that the so called Confined Selective Withdrawal technique [1] can be used to produce compound drops whose radii and thicknesses can be easily decreased down to length scales of the order of 5 µ m, and even smaller. This method is based on the generation of a highly stretched thin coaxial jet of two immiscible fluids that flow inside a third bulk co-flowing fluid under creeping flow conditions. The subsequent capillary breakup of such coaxial jet produces, in a single step, compound droplets at rates that can be easily modulated to values of the order of 10 KHz. Using algebraic equations deduced using slender body theory, we report that the production frequency, the diameter and the shell thickness follow well defined scaling laws, enabling us to generate, from milimeter-sized injection tubes, and in a highly controlled way, uniformly-sized double emulsions with the smallest diameters reported up to date. We illustrate our results by producing micron-sized nematic shells with radii below 5 microns and show that the radius of curvature of the surface and the shell thickness can be fixed independently.

History hydrodynamic torque transitions in oscillatory spinning of stick-slip Janus particles

We theoretically investigate the oscillatory spinning of an axisymmetric stick-slip Janus particle (SSJP) under the creeping flow condition. Solving the unsteady Stokes equation together with a matched asymptotic boundary layer theory, we find that such a particle can display unusual viscous torque responses in the high frequency regime depending on the Stokes boundary layer thickness δ, the slip length λ of the slip face, and the coverage of the stick face. Our analysis reveals that an SSJP will always experience a reduced Basset torque of 1/δ decay due to the presence of the slip face, with amplitude smaller than the no-slip counterpart irrespective of the value of λ. If the coverage of the stick face is sufficiently small, the reduced Basset torque can turn into a constant torque plateau due to prevailing slip effects at larger values of δ, representing a new history torque transition prior to the slip-stick transition at δ ∼ λ. All these features are markedly different from those for no-slip and uniform slip particles, providing not only distinctive fingerprints for Janus particles but also a new means for manipulating these particles.