analysis of fluid flow and wall shear stress patterns inside...

Analysis of Fluid Flow and Wall Shear Stress Patterns Inside Partially

Filled Agitated Culture Well Plates

M. MEHDI SALEK,1,2 POORIA SATTARI,2 and ROBERT J. MARTINUZZI1,2

1Biofilm Engineering Research Group, Calgary Center for Innovative Technology, University of Calgary, Calgary, AB, Canada;and 2Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive N.W.,

Calgary, AB T2N 1N4, Canada

(Received 8 September 2011; accepted 10 October 2011)

Associate Editor Konstantinos Konstantopoulo oversaw the review of this article.

Abstract—The appearance of highly resistant bacterial bio-films in both community and hospitals environments is amajor challenge in modern clinical medicine. The biofilmstructural morphology, believed to be an important factoraffecting the behavioral properties of these ‘‘super bugs’’, isstrongly influenced by the local hydrodynamics over themicrocolonies. Despite the common use of agitated wellplates in the biology community, they have been used ratherblindly without knowing the flow characteristics and influ-ence of the rotational speed and fluid volume in thesecontainers. The main purpose of this study is to characterizethe flow in these high-throughput devices to link localhydrodynamics to observed behavior in cell cultures. In thiswork, the flow and wall shear stress distribution in six-wellculture plates under planar orbital translation is simulatedusing Computational Fluid Dynamics (CFD). Free surface,flow pattern and wall shear stress for two shaker speeds (100and 200 rpm) and two volumes of fluid (2 and 4 mL) wereinvestigated. Measurements with a non-intrusive opticalshear stress sensor and High Frame-rate Particle ImagingVelocimetry (HFPIV) are used to validate CFD predictions.An analytical model to predict the free surface shape isproposed. Results show a complex three-dimensional flowpattern, varying in both time and space. The distribution ofwall shear stress in these culture plates has been related to thetopology of flow. This understanding helps explain observedendothelial cell orientation and bacterial biofilm distributionsobserved in culture dishes. The results suggest that the meansurface stress field is insufficient to capture the underlyingdynamics mitigating biological processes.

Keywords—Six-well culture plates, CFD, Hydrodynamics,

Wall shear stress, Optical shear stress sensor, PIV, Biofilms,

Endothelial cells.

INTRODUCTION

The appearance of biofilms, communities of bacte-rial cells developed on either an inert or living surfaceand encased in a self-generated exopolysaccharide(EPS) matrix, is one of the major challenges in modernclinical medicine. Because of the natural resistance ofthese complex structures against different types ofantimicrobial agents, they are associated with chronicwound infections, a variety of recalcitrant diseases(e.g., cystic fibrosis) and infections of many medicaldevices (e.g., heart valves, implants, and urinarycatheters) often resulting in device replacement andincreased treatment costs.4,5,8,18–23 Hence, a largenumber of studies have been devoted to finding newsolutions to eradicate these super bugs and preventtheir formation.

Several studies have confirmed the importance ofthe surrounding hydrodynamics on the formationprocess and ensuing biofilm morphology.6,24,25,32 Inturn, the morphology has been associated with differ-ent phenotypic expression.12,25,29,30 Hence, an accurateknowledge of the local hydrodynamic conditions isimportant for correctly interpreting biological obser-vations.

The prediction of the flow patterns and parameterssuch as wall shear stress in different types of ductsand parallel plate flow cells is not a difficult task;however, the use of these flow cells is not practicalwhen a large number of concurrent tests or identicalreplicates are needed. Furthermore, a major clinicalchallenge is to identify both the bacterial strain andan effective antibiotic combination under the correctflow conditions, which may require several hundredindependent tests within a short time (i.e., before theinfection develops beyond control, usually 24 h).Various types of standard high-throughput devices

Address correspondence to Robert J. Martinuzzi, Department of

Mechanical and Manufacturing Engineering, University of Calgary,

2500 University Drive N.W., Calgary, AB T2N 1N4, Canada.

Electronic mail: [email protected]

Annals of Biomedical Engineering (� 2011)

DOI: 10.1007/s10439-011-0444-9

� 2011 Biomedical Engineering Society

are commonly used for this purpose allowing rapidproduction of biofilms and running multiple tests inparallel. These devices usually consist of a plate witha number of identical wells. When the plate is trans-lated by an orbital shaker, it provides the samehydrodynamic condition in all of the wells. Anadditional advantage of these devices is that thehydrodynamic conditions can be altered by changingthe shaker motion.

Six-well plates are one type of the high-throughputdevices widely used in the microbiology communitydue to their practicality.2,15,28,31 These compactdevices, facilitate multi-sample testing, provide oscil-latory fluid motion with adjustable speeds and do notrequire bulky pump-flow cell-tubing setups and prob-lems associated with them in terms of manufacturingand sealing. Also, the relatively large and flat surfacesof each well provide easy access for optical measure-ments, microscopy and other analysis. Therefore, uti-lizing these dishes for hydrodynamic studies, includingparameters such as shear stress and mass transport, ispotentially of great advantage. However, unless thefluid mechanics in these containers is well resolved,interpreting and comparing the effect of parameterswith confidence is difficult.

Among the few attempts to clarify the hydrody-namic characteristics in six-well plates, Azevedo et al.2

modeled the flow inside these wells to study the effectsof shear stress on the adhesion ofHelicobacter pylori tostainless steel. This study was limited to the estimationof an overall mean wall shear stresses at the bottom ofthe well. As will be discussed later, the periodic flowinside these wells is complex due to the presence of sidewalls. Noticeable differences exist between regions ofthe well both in terms of the local mean and fluctuatingwall shear stress amplitude. It has been shown in anearlier study15 that biofilm deposition and morphologyin these plates is non-uniform and the biofilm charac-teristics correlate strongly with local shear stress meanand fluctuation levels.

The flow in a six-well container undergoing orbitalmotion in a plane normal to the gravitational vector iscomplicated by the presence of the container side wallsand the shallow liquid fill. For an unbounded fluid, thefluid motion induced by the orbiting plate is a simplevariation of Stokes’ Second Problem, which is gov-erned by a linear differential equation. Hence, the fluidmotion induced by plate undergoing orbital translationis obtained simply by a superposition of the solution inthe two orthogonal directions (Sherman27). Briefly,letting x, z represent the co-ordinates in the horizontalplane, ~i, ~k, the corresponding unit vectors and y theco-ordinate in the vertical direction, for a solidplate executing the motion u 0; tð Þ ¼ �U0 cosðXtð Þ~iþsinðXtÞ~kÞ the velocity distribution is given by:

uðy; tÞ ¼ �U0 cosðXt� kyÞ~iþ sinðXt� kyÞ~kh i

such that at any point on the plate, the magnitude ofthe velocity and shear stress are independent of time:

kUðyÞk ¼ U0e�ky sw ¼ �lU0

Xm

� �1=2

where k ¼ffiffiffiffiffiffiffiffiffiffiffiX=2m

p, X is the angular orbital speed; U0 is

the maximum plate speed; l; m are the kinematic anddynamic viscosities. Note that the viscous effects areconcentrated in the Ekman layer, d ffi 4:61

k (defined forUðdÞk k ¼ 0:01U0), above which the fluid behaves as

inviscid.The movement of the vertical side walls of a con-

fined fluid in a container induces a rotation of the fluid.While the problem of a fluid in a container undergoingconstant solid body rotation has been extensivelystudied for laminar10,11 and turbulent13,14 regimes, thepresent problem is more related to the spin-up prob-lem, since the solid boundaries are rotating relative tothe fluid. The impulsively started transient problem inlaminar regime has been analytically resolved10,11,13 fordeep fluids (fluid heights much greater than the Ekmanlayer). It has been shown that the viscous effects aremainly concentrated in the near wall layers, such thatmost of the fluid can be treated as inviscid with acolumnar velocity distribution. In the case of thepresent geometry, however, the fluid layer is shallowand the shape of the free surface plays a preponderantrole in determining the flow patterns such that earliertreatments are not directly amenable.

Six-well plates are also commonly used in endo-thelial cell studies. Dardik et al.7 used these plates tostudy the effects of orbital shear stress on endothelialcells. Experimental data over a range of shaker rota-tion speeds was reported, but due to low data rates, thetime variation in the shear stress could not be resolved.Berson et al.3 performed a numerical study of flowwithin these culture dishes to investigate the influenceof wall shear stress on endothelial cells. This work didshow that mean shear stress levels at the bottom of theplate were not uniform and did not correspond well toa simple analytical ‘‘oscillating plate’’ model and thatthe rotational speed influenced the radial shear stressdistribution. Although this work and that of Kostenkoet al.15 provided some insight into the wall shear stressdistribution on the bottom faces of six-well plates, theyalso highlight an understanding of flow physics leadingto the observed shear stress patterns is lacking. Forexample, morphological differences observed inregions with similar mean shear stress levels werefound to correlate better with shear fluctuation levels,for which the distribution appeared related to theorbital speed and liquid fill volume affecting the flow

SALEK et al.

pattern. A systematic investigation to characterize howthe basic operating parameters (i.e., shaker speed andliquid volume) influence the flow patterns and wallshear stress inside the wells could thus be valuable forcontrolling or predicting the hydrodynamic conditionsin an orbiting well. This capability can allow for betterexperimental design and more reliable interpretation ofbiological test results.

The instantaneous flow pattern and wall shear stressdistribution inside a well translated by an orbital sha-ker was investigated for different shaker speeds andvolumes of fluid typically used in microbiology studies.The numerical simulations of the flow field were vali-dated by experimental measurements. The mechanismsgiving rise to the shear stress distribution and shape ofthe free surface are discussed in detail. These findingsare then related to the orientation of endothelial cellsand distribution of bacterial biofilms cultured in six-well plates.

METHODS

Numerical Model and Governing Equations

The unsteady, free-surface flow inside a well of a six-well culture plate was simulated using the finite volumebased, commercial Computational Fluid Dynamics(CFD) software FLUENT ver. 6.3.9 Two rotationalspeeds of the orbital shaker, i.e., 100 and 200 rpm, andtwo liquid volumes, i.e., 2 and 4 mL (four cases intotal) were simulated. The free surface motion in theliquid–air interface, the flow patterns inside the welland the mean and instantaneous wall shear stresseswere analyzed.

The six-well culture plate consists of six identicalwells shown schematically in Fig. 1a. Each well is18 mm deep and has a radius of R = 17.5 mm. Thewells are mounted on an orbital shaker and undergo atwo-dimensional linear translation in the horizontalplane (i.e., XZ plane). The radius of gyration is

FIGURE 1. (a) Isometric diagram of six-well culture plate and schematic of a single well geometry; (b) view of computational gridin a single well of a six-well plate. X, Y, Z denote the stationary reference frame and x, y, z denote the moving reference frame.

Analysis of Fluid Flow and Wall Shear Stress Patterns

Rg = 9.5 mm. The properties of water are used tomodel the liquid phase and air to model the gas phase.All properties are at 20 �C and 1 atm: q = 998.2 kg/m3, l = 0.001003 Pa s. Each well is filled with 2 or4 mL of liquid medium, corresponding to approxi-mately 2 and 4 mm liquid height when at rest. TheReynolds number is defined as Re = qXRg

2/l followingDardik et al.7 The maximum Re in this study is 1886such that the flow is expected to remain in the laminarregime.

To simulate this two-phase flow, the solutiondomain is divided into three regions: two regions cor-responding to the homogeneous single-phase fluids (airand water) and a two-phase water–air interface. Thethree-dimensional unsteady mass and momentumconservation equations in the homogeneous domainare written, respectively, as:

@q@tþr � q~mð Þ ¼ 0

@

@tq~mð Þ þ r � q~m~mð Þ ¼ �rpþr lr~mð Þ þ q~F

where ~m, q, l, p, and ~F are velocity, density, dynamicviscosity, pressure, and external force (per unit mass)for the corresponding single-phase, respectively.

To capture the free surface flow, the volume of fluid(VOF) method9 was used, such that the three domainscould be combined into a continuous domain for thesolution procedure. This approach insures that thedynamic condition (shear stress at the free-surface isequal for both phases) is satisfied. It is assumed thateach fluid phase domain is simply connected. In eachcomputational finite volume at the interface, the vol-ume fraction for each phase, ai, is introduced satisfying:

Xni¼1

ai ¼ 1

where i represents air or water (i.e., n = 2). The con-tinuity and momentum equations are then solved usinga modified definition for the fluid properties:

w ¼Xi¼air;water

aiwi

where w can be the density or the dynamic viscosity.The system of equations is then closed9 by satisfying:

@ai@tþ~m � rai ¼ 0

The geometric Reconstruct Scheme, which is themostaccurate interface tracking technique,9 was employed forthe calculation of the transient VOF model. It uses aPiecewise-linear approach assuming that the interfacebetween the two phases has a linear slope in each cell.35

In the aforementioned treatment, it is assumed thatthe influence of surface tension (r = 0.072 N/m forwater–air interface at 20 �C) is negligible. Thisassumption is valid if the gravitational forces andinertial forces on the liquid phase, expressed through

the Bond Bo ¼ ðqwater�qairÞgð2RÞ2r

� �and Webber

We ¼ qwaterU2ð2RÞ

r

� �numbers, respectively, are signifi-

cantly larger than the capillary forces (i.e., Bo,We � 1). In the present study, Bo and We are typi-cally of the order of 100.

MOTION OF THE SIX-WELL PLATE AND

BOUNDARY CONDITIONS

The orbital shaker imparts the same two-dimen-sional, in-plane movement to all points on the six-wellplate. The velocity of the plate walls are thus given by:

~U ¼Ux

Uy

Uz

24

35 ¼

�RgX sinXt0

�RgX cosXt

24

35

where Rg = 9.5 mm is the orbital radius of agitationand X is its rotational speed.

The solution domain encloses both fluid phases andextends to the upper lip of the well, although only theresults in the liquid layer are presented. A no-slipboundary condition was imposed at the solid walls ofthe six-well container (i.e., no relative velocity betweenthe fluid and the solid surfaces).

The equations of motion were solved in both astationary and in a moving frame of reference forcomparison and validation. For the stationary refer-ence frame, the Dynamic Mesh technique was used,where the entire mesh moves with the velocity imposedby the orbital shaker, ~U. The motion of the movingwell was defined using an external C++ user-definedfunction (UDF) linked to FLUENT. In this frame ofreference, the force vector is:

~F ¼ ~g

The relative frame of reference is non-inertial andtranslating with the speed imposed by the orbitalshaker such that the walls have zero-velocity relative tothe frame. The influence of the plate motion is intro-duced through additional (pseudo-) force terms33 suchthat the force vector can be stated as:

~F ¼ ~g� d~U

dt� d~x

dt� ~Rp � 2~x�~mrel � ~x� ð~x� ~RpÞ

where ~Rp is the position vector from the origin in themoving frame and x is the angular velocity of therotation about the vertical axis of the reference frame.d~U=dt, d~x=dt, 2~x�~mrel, and ~x� ð~x� ~RpÞ are the

SALEK et al.

acceleration of the moving reference frame, the angularacceleration effect, Coriolis and centripetal accelera-tion, respectively. In orbital motion ~x ¼ 0 and the lastthree terms in ~F vanish.

Both methods were implemented to verify thevalidity of the original assumptions. While the simu-lation results were in agreement within the numericalaccuracy, it was found that the convergence rate wasmuch faster and a coarser grid was sufficient whenusing the Dynamic Mesh technique. The time steps andgrids needed in the moving frame of reference wereapproximately 6–8 and 2 times finer, respectively. Allresults are reported in the relative coordinate framewith origin at the center of the well.

Since the motion of well of the culture plate isidentical, simulations were conducted for a single well.The numerical grids were generated in GAMBIT. Anexample is shown in Fig. 1b. The grids generated inthis geometry make it possible to refine the grids nearthe solid boundaries optimally: 90% of the total gridshad a skewness of less than 0.2 and none of the cellshad high skewness.9 The resolution of grids was

increased at the walls with a geometric expansioncoefficient of 1.04. Grid sensitivity was assessed bynominally doubling the grid density in successive sim-ulations. The wall shear stress was found to be themost sensitive parameter to the change of grid density.The changes in shear stress were negligible betweengrids of 64,000 and 121,500 cells using dynamic meshtechniques. All results are reported for the geometrywith 64,000 grids and the relative coordinate system(x, y, z).

WALL SHEAR STRESS MEASUREMENT USING

MICRO SENSOR

The magnitudes of the instantaneous wall shearstress components at the bottom of three modifiedPlexiglas wells were measured with a non-intrusive,optical MicroS sensor (Measurement Science Enter-prise). The sensor is flush-mounted with the bottomsurface of the well through a sealed hole as shownin Fig. 2. Radial and tangential components were

FIGURE 2. Shear rate sensor mounted flush with the well surface and schematic of sensor.


measured independently at radial locations of 1, 6, and12 mm from the well center. The sensor operates basedon a modified Laser Doppler Anemometry technique.As shown schematically in Fig. 2, a linearly diverginginterference fringe pattern is produced when a laserbeam is incident on two closely spaced parallel slits.Approximating the velocity profile to be linear close tothe wall (approximately within 150 lm), the frequencyof the scattered light from seed particles in the directvicinity of the wall is then independent of particle’sposition and directly related to the wall shear stress viaswj j ¼ lfdC, where l is the fluid viscosity, fd is theDoppler frequency, and C is a sensor characteristic(0.0526 for water). Shear stress magnitudes as low as0.013 Pa can be measured with this device. The lowerlimit of the measurable wall shear stress is dictated bythe high pass filter settings required for signal pro-cessing. To verify the accuracy of the sensor, it wastested in a laminar duct flow with known theoreticalwall shear rate. The accuracy was within ±10% of theinstantaneous local wall shear rate. Silicon carbide(SiC) particles with mean diameter of 8 lm were usedto seed the flow. Data rate and signal-to-noise rationwere higher than 15 Hz and 5 dB, respectively. Rawand filtered signals were monitored on an oscilloscope.The band-pass filters were set such to optimize thesignal-to-noise ratio and data rate. Details of sensoroperation have been described in more detail by Sat-tari26 and Wilson et al.34

PLANAR FLOW FIELD MEASUREMENT USING

PARTICLE IMAGE VELOCIMETRY

The planar velocity field in a horizontal plane 2 mmfrom the bottom surface of the well was measured witha LaVision high frame-rate particle imaging Velocim-eter (HFPIV). For this purpose a single well wasmanufactured from Plexiglas with the dimensions ofthe actual wells in culture plates. The bottom surfaceof the well was painted black to reduce reflections. A527 nm, 20 mJ/pulse Photonics Industries Nd-YLFpulsed laser provided a 1-mm-thick laser sheet with a12� light fan. A HighSpeedStar camera (1024 9 1024pixels) was used with a 50 mm Nikkor macrolensobjective to capture images in the single-frame (cine-matographic) mode at 1000 Hz sampling rate. Inter-rogation windows of 16 9 16 pixel with 50% overlapresulted in a spatial resolution of 3.0 mm and a vectorseparation of 1.5 mm. The flow was seeded using Sil-icon Carbide particles with 8 lm mean diameter. Aschematic of the HFPIV experimental setup is pre-sented in Fig. 3.

To analyze the velocity vector field inside the well,the circular inner boundary of the well was detectedfrom the recorded snapshots. Since the position of thecamera was fixed, the translational velocity of theshaker was subtracted from the raw measured velocityfield to obtain velocities in the frame of reference of themoving well. Averages of the velocity field at a given

FIGURE 3. Schematic of experimental setup for High Frame-rate Particle Imaging Velocimetry (HFPIV).

SALEK et al.

phase were calculated over approximately 10 cycles.The same pattern repeats with a pure rotation at othertimes within a cycle.

FREE-SURFACE VISUALIZATION

The motion of the free-surface was captured using aSONY model DSC-H5 digital camera mounted on theshaker and translated together with the well. Therecording rate of the camera was 24 frames-per-secondand hence approximately 14 images within a cyclecould be captured at the shaker speed of 100 rpm.Food coloring was added to the fluid (water) for betterpicture contrast.

RESULTS AND DISCUSSIONS

Model Validation

The flow patterns inside a well undergoing two-dimensional orbital motion in a plane (XZ) perpen-dicular to gravity are investigated using CFD tounderstand how the wall shear stress patterns arise

along the bottom face of the well. Results are com-pared for two rotational speeds (X = 100 and200 rpm) and two liquid fill volumes (2 and 4 mL). Thevalidity of the numerical simulations is verified bycomparison to experimental observations.

The magnitude of the mean radial and tangentialwall shear stress components measured at three radiallocations relative to the center of the well (r = 1, 6, and12 mm) agreed within experimental uncertainty withthose predicted by CFD. Sample comparisons ofinstantaneous magnitudes at two locations are pro-vided in Fig. 4. In this figure, spectra obtained fromthe shear rate sensor are also shown. The peak poweroccurs at the frequency corresponding to the orbitalshaker forcing frequency, while the lower-power sec-ondary peaks correspond to harmonics. Since thesensor measures relative to the moving plate, theseresults indicate that the shear stress field is movingrelative to the plate. As will be shown below, this isconsistent with the observation that the entire flowfield rotates relative to the well at the forcing fre-quency.

HFPIV measurements were made in a plane 2 mmabove and parallel to the bottom face of the well.

FIGURE 4. Comparison of the predicted and measured tangential wall shear stress magnitude as a function of time and theirpower spectrum for (a) and (b) 100 rpm, 4 mL at (r 5 12 mm); (c) and (d) 200 rpm, 4 mL (r 5 6 mm). Dashed line indicates the high-pass filter limit for the optical shear sensor. Note that time t 5 t0 is arbitrary.


Planar flow field from PIV measurements are com-pared with CFD results in Fig. 5 for one of fourstudied cases, i.e., 100 rpm and 4 mL fluid volume.The dashed circle indicates the well boundaries. Closeto the side wall (less than 5 mm) the velocity vectorscould not be resolved due to reflections. Sample pro-files of the z-component velocity profiles along x = 0and z = 0 are also shown. Instantaneous PIV resultsshow a good agreement between experiments and CFDresults both qualitatively (flow pattern) and quantita-tively (velocity magnitudes) as shown in Fig. 5.

The numerically predicted and experimentallyobserved free surface motion also agrees well. Samplecomparisons for X = 100 rpm and 4 mL are at dif-ferent times of the shaker cycle are shown in Fig. 6.The free surface is characterized by a traveling waveundergoing solid body rotation at the rate of X aboutthe center axis of the well. The shape and elevation ofthe predicted and observed free-surface agree well,

validating the underlying assumptions (with respect tosurface tension) and suggesting an approach formodeling of the free-surface dynamics. Generally, theshape of the free-surface resembles an inclined horse-shoe over the elevated fluid region. As will be discussedlater, regions of low liquid level, or troughs, are asso-ciated with a thin film of rapidly moving fluid. Thehigh fluid level regions lagging the leading wave crestgive rise to a complex system of counter-rotating vor-tices. In addition to the wall shear stress on the bottomwall, the shape of the free surface is also an importantconsideration in interpreting biological results. Forexample, at X = 100 rpm, the fluid covers the entirebottom face of the well for both fill volumes. AtX = 200 rpm, a small portion of the well bottom faceis exposed near the distal corners in the trough of thehorseshoe. This exposure to air during portion of thecycle is expected to affect the culture area (e.g., biofilmgrowth) in these regions.

0.01

0.02

0.03

0.04

0.05

z (mm)

Vel

ocity

(m

/s)

CFD

PIV

0.01

0.02

0.03

0.04

0.05

-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15

x (mm)

Vel

ocity

(m

/s)

CFD

PIV

(a) (b)

(c) (d)

FIGURE 5. Comparison of CFD and HFPIV velocity vector snapshot at y 5 2 mm (a) CFD; (b) PIV; (c) comparison of z-componentof velocity at x 5 0; (d) comparison of z-component of velocity at z 5 0. The reference vector magnitude is 0.1 m/s.

SALEK et al.

Prediction of Free Surface Level

The shape of the free-surface and motion of thehorseshoe liquid torus, generally located in high liquidelevation regions, can be approximated using a sim-plified heuristic argument. Based on the observationthat the torus undergoes a rigid rotation about thecenter axis of the well, it is advantageous to considerthe problem in a translating, non-inertial reference

frame with origin fixed at the well center. Assumingthat viscous effects are restricted to the immediatevicinity of the solid boundaries, the net effect of thewall shear stress is to induce the solid rotation aboutthe well center such that the relative tangential com-ponent of velocity (averaged over a vertical line in theliquid layer) is approximated simply as:

�uh ¼ Xr

0

2

4

6

8

10

12

14

16

-20 -15 -10 -5 0 5 10 15 20x (mm)

free

sur

face

hei

ght (

mm

)

Experiment

CFD

(a)

(b)

(c)

FIGURE 6. Visualization of free surface at 100 rpm and 4 mL at (a) t 5 t0; (b) t 5 t0 + T/2; (c) comparison of CFD and experiment.T is the period of oscillation.


where r is the radius from the well center. Assumingthat the vertical component of velocity, which isbounded by the well and free surfaces, is small com-pared to �uh, it is found on grounds of continuity thatthe radial component of the fluid velocity must also besmall. Hence, in this reference frame, the transientterms approximately vanish and the shape of the freesurface is governed by the pressure distribution, P,which can be described by a simplified form of themomentum equations in the vertical, y, and radialdirections, respectively:

0 ¼ � @P@y� qg

�q�u2hr¼ � @P

@r� qarel

where q is the liquid density and arel the acceleration ofthe translating reference frame:

arel ¼ �RgX2 cos h� /ð Þ þ sin h� /ð Þ½ �

where h = Xt corresponds to the angular orientationof the acceleration vector at an arbitrary time t. Themotion of the fluid over an oscillating well lags themotion of the well bottom wall (cf. Stokes’ secondproblem27) by a phase /, which is determined empiri-cally.

An expression for the pressure field is obtained uponintegration of the momentum equations:

P¼qX2r2

2�qgyþqRgX

2 rcos h�/ð Þþrsin h�/ð Þ½ �þC0

where C0 is a constant of integration. Denoting the freesurface location, where the gauge pressure is zero bydefinition, by y = h:

h ¼ X2

gRg r cos h� /ð Þ þ r sin h� /ð Þ þ Rð Þ þ r2

2

� �þ C

where C is a constant of integration.Further insight can be gained by recasting this

relationship in terms of a Cartesian coordinate system(x¢, z¢) with origin at the center of the well and alignedwith the direction of arel, which is equivalent torotating the coordinate system through an angle ofh 2 / such that the expression for the free-surfacebecomes:

h ¼ X2

gRg x0 þ Rð Þ þ

x02 þ z02

2

� �þ C

This relationship can be further modified by com-pleting the squares to obtain:

h ¼ X2

2gðx0 þ RgÞ2 þ z02 � R2

g þ 2RRg

h iþ C

Hence, the free-surface describes a parabola withfocus at x¢ = 2Rg, z¢ = 0 and the focus rotates aboutthe center of the well at a rate X.

The constant of integration, C, is determined byrequiring that the volume enclosed under the free-surface equal the total volume of liquid, V, present inthe well. If the free-surface does not intersect the bot-tom surface of the well, this condition yields:

C ¼ V

pR2� X2

gR Rg þ

R

4

� �

If the free-surface intersects the well bottom, byvirtue of parabolic distribution described above, thecontact point will describe a circle of radius Ra aboutx¢ = 2Rg, z¢ = 0. Thus, the volume of fluid under thefree-surface (and above the plate bottom surface) isgiven by:

V ¼ pX2

4g2R2 R2

2þ R2

g

� �� 2R2

a þ R4a

� �

and, the integration constant is given by:

C ¼ X2

4gR2R4

a � 2R2a

� �

where

R2a ¼ R2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR4 � Bp

B ¼ 2R2 R2

2þ R2

g

� �� 4g

pX2V Ra<Rþ Rg

The comparison of the numerically predicted freesurface with theoretical equations has been summa-rized in Fig. 13. The results are in excellent agreement,accurately capturing: the locations where the bottomof the well is exposed (important for biological tests);the bulk motion of the fluid around the well; the cur-vature of the free surface (and thus the main surfacedynamics) as well as the general sloshing behavior.This analytical form conveniently expresses the freesurface behavior in closed form (all inputs are deter-mined a priori by the user) (Fig. 7).

Topology of Flow Field and Wall Shear StressDistribution

For an arbitrary, fixed point on the well surface, theshear stress fluctuates about the mean with a frequencyof X, as determined from the power spectra of thefluctuating component. Examples of this behavior areshown for two cases in Fig. 4. This periodic behavior is

SALEK et al.

consistent with the observed solid body rotation of thefree surface about the well center.

An overview of the mean wall shear stress field on thebottom of the well is given in terms of the radial distri-bution in Fig. 8. The mean and fluctuating fields arepredicted to be axisymmetric about the center of thewell, which is consistent with the observed surfacemotion. Together, these observations suggest a solid bodyrotation of the fluid field about the center of the well.

The mean wall shear stress field is shown in Fig. 8a.Generally, the magnitude of the wall shear stress, sw,increases as the rotational rate X increases. The trendsobserved in the radial distribution change as a functionof X and the liquid fill volume.

A two-dimensional extension of Stokes’ secondproblem27) for an infinite, immersed plate undergoingcircular (orbital) motion has been used exten-sively7,16,17 to estimate the magnitude of the wall shearstress, sw, at the bottom of the plate. For a constantrotational rate X, Stokes’ approximation yields thatthe magnitude of the wall shear stress is constant overthe entire plate surface and has the value:

sw ¼ RgX3=2 ffiffiffiffiffiffi

qlp

This approach has been used ad hoc,3,7,16,17 but itsapplicability warrants further consideration since itignores the influence of the well walls and finite volumefill.

At X = 100 rpm, the time averaged wall shear stressmagnitude sw is nearly constant over the entire wellbottom, but the absolute level changes with volume fill.Whereas the Stokes’ approximation predicts a value of0.32 Pa for the present test conditions, the observedvalues in Fig. 8a are 0.28 and 0.15 Pa, for 2 and 4 mLvolume fill, respectively. At X = 200 rpm, sw isapproximately constant only near the center(r< 2 mm), where the value is close to 0.83 Pa ascompared to 0.91 Pa as predicted from Stokes’approximation. The average shear stress magnitudeover the bottom of the well at X = 200 rpm is 0.87 and1.02 Pa for 2 and 4 mL volume fill, respectively. Forthis higher rotation rate, the influence of the volume fillis increasingly observed as the container distal wall isapproached, such that the wall shear stress on the wellbottom may not be suitably represented by a single,average value.

These results suggest that Stokes’ approximationcan lead to erroneous conclusions when comparing

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results at different X or when the influence of thevolume fill is neglected. Note that in this work, thevolume fills are typical of most biological experiments.Moreover, sw is not constant (in time) on the wellbottom as can be seen from the standard deviation ofthe fluctuations shown in Fig. 8c. In fact, Fig. 8apresents the time averaged of mean wall shear stressmagnitude (Pa) as a function of the radial location.The fluctuation distribution changes with both rota-tional rate and volume fill, indicating an importantinfluence of the dynamics.15 Thus, in consideringresults from biological experiments and especially forcomparison purposes, further attention should begiven to the changes in the flow pattern as a function of

the operating parameters. A description of the flowfield and its topology follows, from which severalfeatures observed in the distributions of the Fig. 8 areelucidated.

Like the motion of the free surface, the entire flowfield undergoes a solid body rotation about the wellcenter axis. This behavior is confirmed by inspection ofthe shear stress field at different time points in theshaker cycle and thus a view of the flow field at a singletime point is sufficient. In Fig. 9, the wall shear stressmagnitude distribution on the bottom face of the wellis shown for all four cases investigated at an arbitrarytime (t = t0) during the orbital cycle. The corre-sponding elevation contours of the liquid levels and 3Dstructure of the free surfaces are shown in Figs. 10 and11, respectively. In all cases, the shear stress fieldrotates counter-clockwise about the well center axis inthe direction and at the rotational rate of the orbitalshaker. The surface-stress streamlines (tangential tothe surface shear stress vector at all points) aresuperimposed to assist in the interpretation of the flowpatterns in subsequent sections. The shear stress vectoris oriented in the direction of the flow in the immediateproximity of the bottom surface. It will be noted thatfor any radial position, the same cyclical (generally notsinusoidal, e.g., Figs. 13b, 13c) fluctuation of the wallshear stress level is observed with a period corre-sponding to the shaker rotation frequency, albeit witha phase difference which is purely a linear function ofthe tangential location. Consequently, the mean flowand wall stress fields are axisymmetric, but theinstantaneous flow field contains significant tangentialgradients and thus differs fundamentally from a purerotation.

The shear stress distribution can be related to thetopology of the fluid flow structure. Consider first thesimpler structure case (X = 100 rpm, 4 mL). Inspec-tion of the velocity vectors in the perpendicular planesshown in Fig. 12a, together with the surface stress andfree surface height (Figs. 10a, 11a), shows that thebottom surface of the well is never exposed to air. Alarge (counter-clockwise rotating) semi-torus vortexcharacterizes the bulk of the liquid motion in the ele-vated fluid region. Generally, the flow is directed fromhigh to low fluid elevation along the well bottom andthe wall shear stress magnitude increases radially out-ward along this structure. The flow impinges in thedistal corner accounting for the low wall shear stresslevels observed in the region (r fi R). The fluidmotion is helical about the tangential axis of the torus,forcing fluid along the bottom surface from high liquidlevel locations to the trough region, giving rise to aconvergent node (focus) in the corner region(x ~ 215 mm; z ~ 215 mm) along the bottom surface.Along the bifurcation line originating at the node, the

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SALEK et al.

tangential component of the shear stress changes signand the node is located between the local magnitudemaxima (a positive and a negative, see Fig. 9a).

These flow structures will cross any fixed radius onthe plate during one cycle of rotation, which results ina cyclical fluctuating shear stress when observed at afixed point as shown in Fig. 13. Near the plate center(Fig. 13a), the liquid motion is almost sinusoidal,resembling that over an oscillating plate. The waveform is gradually distorted by the rotation of the flowfield as r is increased (for example in Figs. 13b and 13cat the location r = 6 mm and r = 12 mm). Togetherwith Fig. 9a, a local magnitude maximum (positivesign) of the tangential shear stress occurs behind (lag-ging) the bifurcation line in the elevated liquid region.

This shear component decreases under the semi-torus(horseshoe) vortex and through the trough region,reaching a magnitude maximum (negative sign)directly ahead of the wave crest. The shear stress valuesjump abruptly across the bifurcation line, whichcoincides with a zero tangential stress line and a sud-den flow reversal.

An increase in the rotational speed of the shakercauses approximately a 3-fold increase of the meanshear levels over the entire plate (Fig. 8a), but mayresult in up to a 10- to 12-fold increase in the maximumshear levels (Figs. 8b and 9b, mainly in the distalcorners) and significant changes in the liquid levelacross the well (Fig. 10b). The horseshoe shape of theelevated liquid regions becomes more distinct. In the

FIGURE 9. Contours of the magnitude of wall shear stress (Pa) on the bottom face with surface ‘‘streamlines’’ superimposed att 5 t0 for (a) 100 rpm, 4 mL; (b) 200 rpm, 4 mL; (c) 100 rpm, 2 mL; and (d) 200 rpm, 2 mL. Curved arrow indicates sense of rotation.The units are in Pa and the reference vector magnitude is 1 Pa.


well region surrounded by the horseshoe, a deeptrough forms, resulting in a thin film of liquid movingrapidly away from the wall (contiguous with the openend of the horseshoe). A region exposed to air alsoappears near the vertical (distal) wall (Figs. 10b, 11b).

At the center of the well, the time series of the shearstress components again show a sinusoidal fluctuation.However, outside of the immediate vicinity of thecenter, the shear stress evolution resembles more thatshown in Fig. 13c for r = 12 mm. Generally, overmuch of the cycle, the bottom face is covered by thehorseshoe structure and the tangential stress compo-nent is positive and changes little. The trailing edge ofthe wake is characterized by a rapid decrease of theshear, becoming negative in the thin film layer andreaching a magnitude maximum (negative sign)immediately ahead of the bifurcation line. The bifur-cation line, similarly to the previous case, is found tolead the crest of the wave. The flow and tangentialshear stress abruptly change direction across thebifurcation line.

Velocity vector plots in two perpendicular verticalplanes are shown in Fig. 12b for the case 200 rpm,

4 mL. The curvature of the free surface is much morepronounced than in the previous case. The base of thetrough consists mainly of a thin film of water movingrapidly from the distal region of the trough to theelevated liquid region. Inside the elevated liquidregions, typically two counter-rotating vortices areobserved (see Fig. 12b). In the plane z = 0, the flownear the bottom surface moves rapidly from the outerwall toward the plate center, where it meets theopposing thin film flow, deflecting the fluid radiallyoutward below the free surface. The bifurcation linelags the leading wave edge slightly.

These vortex structures can also be observed in theplane x = 0. On the trailing side of the wave, z> 0, theflow along the well bottom is forced from the distalwall to the center, where it meets the opposing flowfrom the film layer, thus inducing a radially outwardflow beneath the free-surface. In this plane, the sec-ondary (counter-rotating) vortex in the distal cornerbelow the free-surface is stronger and more easilyrecognized. This vortex induces a radially inward flowalong the free-surface to about a radius of r = 10 mm(z = 10 mm in Fig. 12b) to encounter the radially

FIGURE 10. Elevation contours of the liquid level (free surface) at t 5 t0 for (a) 100 rpm, 4 mL; (b) 200 rpm, 4 mL; (c) 100 rpm,2 mL; and (d) 200 rpm, 2 mL. The units are in mm. Curved arrow showed in the previous figure indicates sense of rotation.

SALEK et al.

outward surface current, originating from the trough,and resulting in a sub-ducted stream. At the conver-gence point of these two streams, a change in curvatureof the free-surface is observed.

On the leading side of the wave (z< 0), the flowalong the bottom surface is radially inward throughthe elevated and thin film region. The elevated liquidregion is dominated by a very strong recirculationwhich entrains fluid upward toward the free-surfaceand resulting in the leading wave crest. Figure 9bshows that a converging (negative) bifurcation lineslightly leads the crest region. The shear stress mag-nitude is small in the vicinity of the bifurcation line andreaches maxima below the core of the vortex and in thefilm region leading the bifurcation line. The radialcomponent of the shear is negative in this region, whilethe tangential component of shear stress changes signsacross the bifurcation line.

The flow and wall shear stress fields for the 2 mLcases are topologically similar to the correspondingcases at 4 mL quantitatively, however, some differencesare observed. The same horseshoe pattern and presenceof a bifurcation line can be recognized (Figs. 9c, 9d).However, due to the lower fluid levels, the troughregions are more pronounced and the thin film regionsextend over a larger area. For the 200 rpm, 2 mL case,the distal region of the plate bottom surface exposed to

air during the cycle increases. These quantitativechanges are reflected by sharper transitions betweenelevated liquid and thin film regions.

The forgoing discussion helps clarify the trendsobserved in the mean field as summarized in Fig. 8.The axisymmetry of the mean stress field resultsbecause the flow field is rotating relative to the centeraxis of the well. Note that the wall shear stress mag-nitude, rather than the components, was selected sinceit is expected that bacteria or mammalian cells will bemore sensitive to the local magnitude of shear level andfluctuations than to the direction. The mean shearstress magnitude, sw, can be interpreted synonymouslyas either the time-average at a point, or tangentiallyaveraged in space at constant radius for a fixed timepoint. As shown in Fig. 8a, sw, varies little across theplate for X = 100 rpm, except close to the distal cor-ners due to the secondary flow. In generally, regions oflarger radial gradients are associated with regions ofsecondary flow structures. Doubling of the rotationrate to X = 200 rpm causes significantly larger radialgradients, which agrees with the appearance of signif-icantly larger secondary flow regions. Thus, not sur-prisingly, in secondary flow regions the mean shearlevels are not well predicted using stokes’ plateapproximation. Conversely, for the case 100 rpm,2 mL, the strength of the secondary motion is much

FIGURE 11. 3D structure of free surfaces at t 5 t0 for (a) 100 rpm, 4 mL; (b) 200 rpm, 4 mL; (c) 100 rpm, 2 mL; and (d) 200 rpm,2 mL.


weaker and Stokes’ approximation yields satisfactoryresults.

Themaximum level and the standard deviation of thefluctuations of the shear level magnitude are shown inFigs. 8b and 8c, respectively. Figure 8c shows that ateach radius from the center of the well, how diverse themean wall shear values are over a cycle. As indicated

above, the maximum shear is associated with the pas-sage of the bifurcation lines in the surface stress field.Near the distal walls, the induced secondary motiongives rise to strong variations in the shear levels. Thelarge peaks observed near the distal corners in Fig. 8bfor X = 200 rpm correlate well with the location of thenear-wall vortex cores observed in Fig. 12b. At

FIGURE 12. Vector plots of the in-plane velocity components in the liquid phase (a) 100 rpm, 4 mL; (b) 200 rpm, 4 mL; (c)100 rpm, 2 mL; and (d) 200 rpm, 4 mL. The reference vector magnitude is 0.1 m/s.

SALEK et al.

X = 100 rpm, 2 mL (Fig. 12c), the vortex is weaker andcorrespondingly the peak shear stress values. Recall thatthe bifurcation line is located at the leading edge wherethe elevated fluid region (containing the semi-torusvortex) meets the thin fluid layer in the trough (Fig. 10).The deepest extent (smallest r) of the bifurcation line,indicated by arrows in Figs. 8a and 8c corresponds to alocal minimum in sw and a local maximum in the fluc-tuation levels (Standard deviation, SD).

Biological Case Studies

In this section, it will be shown that earlier obser-vations on the orientation of endothelial cells (Dardik

et al.7) or the distribution of bacterial biofilms(Kostenko et al.15) in experiments conducted in similarculture well plates can be directly related to the wallshear stress distribution and flow topology analysed inthis study. These results help to understand theunderlying mechanisms leading to the observed dif-ferences between various locations of the culture plate.

(1) Dardik et al.7 exposed endothelial cells (EC) tosurface shear stresses inside a six-well plate (similar tothat in this study) on an orbital shaker at 210 rpm and2 mL of medium. The orientation of the EC at the wellcenter and periphery were compared. They observedthat EC in the periphery of the culture well werealigned along an angle of b = 34� ± 6� from the linetangential to the edge of the well, while the ECobserved in the center of the culture well were ran-domly aligned (Fig. 14).

The distribution of the instantaneous inclinationangle of the surface shear stress vectors, measuredrelative to a radius:

a ¼ tan�1 � sh

sr

� �� 90 � b;

where the sh and sr indicate the tangential and radialcomponents of the surface shear stress, is shown ascolored contours for an arbitrary time point inFig. 15a. Since the entire flow field rotates about thecenter of the plate with a constant rate, X, it is possibleto construct a scatter plot of a (sampled at equal timeintervals) for each radial location, r, as shown inFig. 15b. In the center region, as defined by Dardiket al.,7 a is independent of r and uniformly distributed.A histogram of a constructed for all points in thecenter region, shown in Fig. 16a, suggests that ECthroughout the center region are exposed to a rangingfrom 290� to 90� with nearly equal probability. Hence,it is expected that the EC will be randomly orientedthrough the center region as found by Dardik et al.7

In contrast, in the periphery region (9 mm< r<

16 mm), a preferential clustering in the range45� < a < 60� (30� < b < 45�) can be deduced fromthe scatter plot of Fig. 16b. The histogram of Fig. 16bshows that over the entire periphery region, the shearstress is locally oriented in the range a = 55� ± 10�with significantly greater likelihood. Dardik et al.7

estimated an EC orientation angle of a = 56� ± 6� inthe periphery region. These combined results indicate astrong correlation between the EC orientation and themost likely orientation of the local surface shear stressvector.

(2) In our earlier study (Kostenko et al.15), it wasobserved that the methicillin-resistant Staphylococcusaureus (MRSA) biofilms deposition varied radiallyacross the well and more deposition occurred in the

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FIGURE 13. Time series of z-component of wall shear stressfor 100 rpm, 4 mL at z 5 0 (a) x 5 1 mm; (b) x 5 6 mm; (c)x 5 12 mm. At this locations the z-component of wall shearstress is equal to the magnitude of tangential wall shearstress but of opposite sign. The z-component of the velocityof the well is shown by dashed line.


periphery area compared to the center regions.Figure 17a shows direct density measurements in termsof Colony Forming Units (CFU) at three radial loca-tions. Figure 17b shows the radial distribution ofoptical density (OD). The OD is directly proportionalto the biofilm density. (Additional details on method-ology are provided in Kostenko et al.15) The observeddeposition distribution is difficult to reconcile by con-sidering only the wall shear stress distribution. Forexample, as shown in Fig. 8a, at all three radial loca-tions (r = 1, 6, 12 mm) reported in Fig. 17, althoughthe mean and peak wall shear stress magnitudes arecomparable for both 2 and 4 mL cases at 100 rpm, thebiofilm deposition differs significantly between loca-tions. In contrast, for both cases at 200 rpm, while theshear stress magnitudes at r = 12 are significantlyhigher than at r = 6 mm, the biofilm deposition issimilar.

Trying to relate the amount of biofilm accumulationto the shear stress levels leads to inconsistencies whencomparing results from different studies as is discussedin Salek et al.25 It is, however, possible to relate thesebacterial biofilm deposition patterns to the flowtopology. The deposition is greatest in the region of thesemi-torus vortex, which is swept by the negative(converging) bifurcation line making the leading frontof the vortex (see Figs. 9, 10). In Fig. 17b, the deepestextent of the bifurcation line is indicated by an arrow.Thus, regions of larger radius are swept by the bifur-cation line and are characterized by higher OD (i.e.,biofilm density) than in the central regions. In the

central regions of the plate (not swept by the bifurca-tion line), the fluid layer is thin and the flow is consis-tently directed toward the bifurcation line. Thesuspended bacteria in the central region are transportedto the semi-torus vortex. The flow inside the vortex ischaracterized by a strong recirculation (e.g., Fig. 12),effectively trapping bacteria entering this region andthereby increasing the concentration of suspendedbacteria. The microorganisms can be transported to thesubstratum through transport by the flow (Abelson andDenny1) and settle. The entrainment of the free-surfacelayers into the recirculation regions can effectivelymaintain the oxygen supply, allowing bacteria to growin the nutrient-rich medium.

Changes in biofilm morphology are often of interestas these correlate with susceptibility.15 Whereas dif-ferences in biofilm accumulation can be expanded interms of the flow topology, the shear stress distributionalso plays an important role in determining biofilmmorphology. Figure 18 shows the typical morphotypesobserved, and their location on the plates are sum-marized in Table 1.

The biofilms forming in the central region differfrom those formed in the semi-torus region (i.e., regionswept by the bifurcation line). At the lower rotationalrate (X = 100 rpm), the biofilms form low density,randomly distributed clumps in the central regions,while in the region swept by the bifurcation lines,biofilms form higher density, monolayer webs. ForX = 200 rpm, multilayered webs are observed in thecentral and distal regions. In the semi-torus region, the

FIGURE 14. (a) Diagram of differential cell seeding (b) Morphology of ECs after 5 days of culture, ECs exposed to orbital motion atthe center and periphery of culture well are marked by arrow. Figure from Dardik et al.7 with some changes.

SALEK et al.

biofilm is compact and structure more complex (cup).Upon comparison of Table 1 and Fig. 8, these changesin morphology are not simply related to the density or

accumulation, but appear related to the fluctuationlevels (or maximum value) of the shear stress magni-tude. For example, the web morphology is observed inthe semi-torus region at 100 rpm and in the centralregion at 200 rpm, for which the fluctuation andmaximum shear stress levels are comparable. Note the

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work is being pursued with the aim to distinguishbetween the influence of fluctuation and maximumshear stress levels.

CONCLUSION

The present study has shown that the fluid motionand wall shear stress distribution within a cylindricalwell undergoing orbital translation in a horizontalplane is controlled by the liquid medium volume, V,the orbital radius of gyration, Rg, and angular speed,X. For operating conditions typical of microbiological

experiments using orbital shakers, the fluid undergoesa solid body rotation, at the rotational speed X, aboutthe center of the well. Consequently, for a fixed pointon the well surface, the wall shear stress, sw, fluctuatesperiodically at the frequency X and the mean andstandard deviation of sw are axisymmetric. The peri-odic fluctuations are generally not sinusoidal due to thepresence of secondary flows, which arise in regions ofhigher liquid (free surface) elevation and are respon-sible for the observed radial distribution of the meanwall shear stress as well as the standard deviation. Thelocal liquid elevation, h, appears to be governed byinertial effects and can be approximated based on a

FIGURE 18. Representative samples of MRSA morphotypes found for (a) 100 rpm in central region; (b) 100 rpm in peripheralregion; (c) 200 rpm in peripheral region; (d) 200 rpm in central region.

TABLE 1. Summary of the typical biofilm morphotypes and their location on the plates.

Experimental conditions

Radial distance from the center of the plate

2 mm 4 mm 6 mm 8 mm 10 mm 12 mm 14 mm 16 mm

100 rpm 4 mL Clumps Clumps Clumps Clumps Clumps Web Web Web

100 rpm 2 mL Clumps Clumps Web Web Web Web Web Web

200 rpm 4 mL Multilayer web Cup Cup Cup Cup Cup Cup Multilayer web

200 rpm 2 mL Multilayer web Cup Cup Cup Cup Cup Multilayer web Multilayer web

SALEK et al.

simplified analytical model as a function of V, X,and Rg.

This study is beneficial for interpreting results frombiological experiments where the mean or fluctuatingshear stress levels are important and also allows forsystematic comparisons between different experiments.As examples, the orientation of endothelial cells anddeposition patterns of bacterial biofilms have been re-lated to the flow topology andwall shear stress behavior.The results suggest that the shear stress is not a sufficientfactor to interpret the observations occurring in differ-ent experiments, but when it is linked to the topology offlow field, it helps to understand the underlying mech-anisms of different biological processes.

ACKNOWLEDGMENTS

The authors recognize valuable discussions withDr. Victoria Kostenko. M. Mehdi Salek would like toacknowledge Alberta Ingenuity Fund (AIF), now partof Alberta Innovates Technology Futures for theirsupport through a scholarship. This research has beenenabled by the use of computing resources provided byWestGrid and Compute/Calcul Canada.

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