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A Numerical Model for Simulating the Motions ofEllipsoidal Fibers Suspended in Low Reynolds NumberShear FlowsKevin T. Shanley a & Goodarz Ahmadi aa Department of Mechanical and Aeronautical Engineering , Clarkson University , Potsdam,New York, USAPublished online: 24 Mar 2011.

To cite this article: Kevin T. Shanley & Goodarz Ahmadi (2011) A Numerical Model for Simulating the Motions ofEllipsoidal Fibers Suspended in Low Reynolds Number Shear Flows, Aerosol Science and Technology, 45:7, 838-848, DOI:10.1080/02786826.2011.566293

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Aerosol Science and Technology, 45:838–848, 2011Copyright C© American Association for Aerosol ResearchISSN: 0278-6826 print / 1521-7388 onlineDOI: 10.1080/02786826.2011.566293

A Numerical Model for Simulating the Motions of EllipsoidalFibers Suspended in Low Reynolds Number Shear Flows

Kevin T. Shanley and Goodarz AhmadiDepartment of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, New York, USA

A computational model was developed to simulate the motionsof ellipsoidal fibers suspended in viscous shear flows. The modelincorporates drag, lift, gravity, and hydrodynamic torque acting onellipsoidal particles. Two numerical experiments were conductedto test the accuracy of the model. First, a single fiber was subjectedto a linear shear profile and allowed to rotate without translation.The model’s predictions for the fiber rotational period were eval-uated and were shown to match well with previously publishedtheoretical and numerical values. Second, fibers were injected intoa horizontal tube under different flow conditions and their motionswere tracked. Inspection of individual fiber motion showed a ten-dency to remain aligned with the direction of bulk fluid flow. Fiberaspect ratio, injection location, and flow rate were shown to playimportant roles in the details of elongated particle motions. Thepredicted sedimentation efficiencies of fibers matched well withthe previously published works. On the basis of the computationalmodel results, an empirical expression for the sedimentation rate offibers in a horizontal tube with circular cross-section was proposed.

1. INTRODUCTIONPrevalence of fibrous materials in construction and industrial

processes, coupled with their link to toxicity in humans, has ledto recent efforts being directed toward a better understandingof fiber transport mechanisms. The anisotropic shape of a fibermakes the physics of transport through fluids markedly differentfrom that of a sphere. Drag on a slender elongated body changeswith orientation that plays an important role in computing theirtrajectories.

In one of the first attempts to describe the motion of a non-spherical particle, Jeffrey (1922) examined an ellipsoidal fiber

Received 8 July 2010; accepted 18 January 2011.The financial support of the National Institute for Occupational

Safety and Health (NIOSH) through Grant R01 OH003900 and the En-vironmental Protection Agency (EPA) under grant 916716 is gratefullyacknowledged.

This work has not been subjected to either Agency’s required peeror policy review and therefore does not necessarily reflect the views ofthe agencies and no official endorsement shall be inferred.

Address correspondence to Kevin T. Shanley, Department of Me-chanical and Aeronautical Engineering, Clarkson University, 8 Clark-son Ave., Potsdam, NY 13699, USA. E-mail: ktshanley@gmail.com

subjected to a linear shear flow field. While macroscopic lin-ear shear flow is an idealization, a sufficiently small fiber sus-pended in viscous flows is exposed to velocity gradients that canbe locally approximated as linear. Additionally, Jeffrey (1922)showed that the rotations of these elongated fibers are periodicand the period of rotation is a function of their aspect ratio andthe flow shear rate, when the effects of inertia are neglected. Theexpressions derived for the coupled translational and rotationalmotions of fibers by Jeffrey (1922) have been widely acceptedand used.

Chen and Yu (1991) solved for the specific case of ellipsoidalfibers suspended in a horizontal circular cross-section, straighttube under laminar flow conditions. They presented correlationsfor fiber sedimentation with aspect ratio and flow rate. Fan andAhmadi (1995) developed a model for ellipsoidal fiber transportand deposition in the sublayer region of turbulent channel flows.They used Euler’s quaternions for describing the kinematics ofellipsoidal fibers. A deposition criterion was also developedtaking into account the orientation of the ellipsoid with respectto the wall. They suggested an expression for the deposition rateas a function of an effective dimensionless particle relaxationtime.

Asgharian and Anjilvel (1995a) developed a series of directequations for the angular acceleration of Euler’s angles. Theydemonstrated that the resulting equations could simulate theperiod predicted in Jeffrey (1922) for ellipsoidal fibers not ex-periencing significant inertial effects. As inertia became moresignificant the period first shortened before reaching a criticalinertia that slowed their rotations.

An application of these equations to track fibers in airwayswas presented by Asgharian and Anjilvel (1995b). They ex-amined a straight pulmonary branch with circular cross-sectionand length to radius ratio of 3:1. A parabolic flow profile wasassumed. After examining deposition behavior for several ori-entations of the branch, they suggested an expression for depo-sition as a function of tilt angle, fiber aspect ratio, and particlerelaxation time.

Typically, in an attempt to characterize the motion ofan irregular shaped body, an equivalent sphere diameter isintroduced. In the specific case of the motion of a rigid ellipsoid,

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A NUMERICAL MODEL FOR FIBER MOTION 839

a uniform distribution of the fiber’s orientation in the flow fieldis implied by averaging the mobility of the ellipsoid in the threedimensions. Many such formulations have been suggested andutilized in the literature. Stober (1972) proposed the calculationof an aerodynamic diameter (dae) for a prolate spheroid basedon a dynamic shape factor, k, that takes into account all possibleorientations of the particle with respect to the flow direction.That is, a unique dae for when the fiber is aligned with the flow,perpendicular to the flow, or randomly distributed in the flow.

Shapiro and Goldenberg (1993) proposed characterization ofan ellipsoid’s motion with a particle relaxation time dependenton the averaged mobility dyadic and the assumption of isotropicorientation in the flow. Similarly, Fan and Ahmadi (1995) sug-gested an equivalent Stokes diameter based on the averagedtranslational dyadic.

Wang et al. (2008) derived several expressions for the aerody-namic diameter of an ellipsoid when it is aligned with the flow(dae,‖), normal to the flow

(dae,⊥

), and randomly distributed

within the flow (dae,R). They used the “point force approxima-tion” technique explained in Clift et al. (1978). This techniqueuses a polynomial approximation for the distributed force alongthe ellipsoid’s axes.

Many commercial CFD software packages have the abilityto track a second phase in general flow fields. One such ex-ample is the Discrete Phase Model (DPM) in the commercialpackage ANSYS FLUENT. The DPM is a Lagrangian particletracking algorithm. Particle geometries are assumed sphericaland tracked individually with forces acting at its center. Drag isdynamic and dependent on particle Reynolds number. Contactwith a surface is assumed once the distance between the centerof mass and a boundary becomes less or equal to the radius ofthe sphere.

This work used the existing infrastructure of the FLUENTDPM as the foundation for a numerical model for tracking el-lipsoidal fibers in general laminar shear flows. A series of UserDefined Functions (UDF) were incorporated to alter the behav-ior of the particles from that of a sphere to that of an ellipsoid.The dynamic Reynolds dependent drag was expanded to includea dynamic shape factor. Equations of rotational motions weresolved to predict the instantaneous orientation of the ellipsoidat each time step.

The computational model was shown to simulate the periodtheoretically predicted by Jeffrey (1922) and simulated by As-gharian and Anjilvel (1995a) when an ellipsoidal fiber was sub-jected to a linear shear with their translation suppressed. Theflow field inside a horizontal pipe with circular cross-sectionwas evaluated numerically with ANSYS FLUENT. Sphericaland ellipsoidal fibers were tracked in the simulated flow field.Sedimentation rates for ellipsoidal fibers agreed favorably withthe experimental findings and that of equivalent sphere diameterparticles as defined by Fan and Ahmadi (1995). The sedimenta-tion rates for ellipsoidal fibers under various conditions studiedall collapsed onto a single curve and an empirical, expression

was proposed when plotted against the parameter suggested byAsgharian and Anjilvel (1995b).

2. INCOMPRESSIBLE FLOW EQUATIONSThe conservation of mass and momentum equations for

steady, laminar, incompressible flows are defined as follows.

u · ∇u = − 1

ρ∇ + ν∇2u, [1]

∇ · u = 0. [2]

In Equations (1) and (2) u is the three-dimensional (3D) fluidvelocity vector, ρ is the constant density of the fluid, and ν isthe constant kinematic viscosity.

The commercial package ANSYS FLUENT was used tonumerically solve Equations (1) and (2). ANSYS FLUENTemploys a finite volume algorithm. Discretization of the con-servation equations was performed with the upwind scheme toa second order accuracy, pressure-velocity coupling was per-formed with the SIMPLE scheme, and derivatives were com-puted via the Green-Gauss Node Based method (ANSYS FLU-ENT, Fluent User’s Manual, 2007).

3. LAGRANGIAN PARTICLE TRACKING

3.1. Spherical ParticlesA direct Lagrangian approach was used to track individual

particles in dilute concentrations. For spherical particles, po-sition and velocity were obtained by integrating the followingequations of motion.

dxdt

= v, [3]

dvdt

= fd + fl + g. [4]

Here, x represents the three coordinates of the particle posi-tion vector and v is the 3D particle velocity vector.

The terms on the right-hand side of Equation (4) representthe drag, lift, and acceleration of gravity per unit mass. Dragis dynamic and dependent on particle Reynolds number anddefined as (Hinds 1982)

fd = 1

τ

CDRep24

(u − v). [5]

In Equation (5) CD is the drag coefficient, Rep is the particleReynolds number, u—v is the difference between particle andfluid velocities also known as slip velocity, and τ is the particlerelaxation time that can be defined as

τ =∫Sd2

18νCC. [6]

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840 K. T. SHANLEY AND G. AHMADI

Here S is the solid-to-fluid density ratio, d is the diameterof the particle, ν is the kinematic viscosity of the fluid, and CC

is the Cunningham slip correction factor that can be defined asfollows.

CC = 1 + 2λ

d

(1.257 + 0.4e

−1.1d/2λ

)[7]

Here λ is the mean free path of the fluid.Saffman (1965) provided an expression for the shear-induced

lift on a small sphere to be given as

fl = 2Kν1/2dij

Sd(dlkdkl)1/4(u − v). [8]

In Equation (8) K = 2.594 is the constant coefficient ofSaffman lift force and dij is the components of the deformationrate tensor.

3.2. Ellipsoidal FibersBefore discussing the motion of a fiber, it is imperative to

introduce two independent coordinate systems as shown in Fig-ure 1. The global coordinate system (x, y, z) is fixed in spacewhile the local coordinate system (x′, y′, z′) is fixed on the fiberand will translate and rotate with the fiber. Orientation of thelocal coordinate system and thereby the fiber with respect tothe global coordinate system is defined in terms of three angles(φ, θ , ψ) known as Euler’s angles. Figure 1 shows how the twocoordinate systems are related by Euler’s angles. The rotationalmotion of the fiber can now be derived from the time rate ofchange in the Euler angles.

The equations of translational and rotational motions of rigidfibers are coupled and are given as

mpdvdt

= FD + FL + (m−mp)g, [9]

FIG. 1. Relationship between local and global reference frames.

d

dt(IM · ω) = Th. [10]

Equation (9) describes the translational motion and is similarto that of a sphere. The forces on the right-hand side of Equation(9) are hydrodynamic drag (FD), shear-induced lift (FL), andgravity. Equation (10) describes the rotational motion of thefiber that is generated by the hydrodynamic torque (Th) exertedon a fiber.

The hydrodynamic drag is a dynamic function dependent onthe orientation of the fiber and its Reynolds number. It can bedefined mathematically as

FD = πµdK · (u − v) . [11]

Here d is the cross-sectional diameter or minor axis of thefiber and K is the translational dyadic, sometimes referred to asthe resistance tensor, and its elements are defined in the particlereference frame as follows (Asgharian and Anjilvel 1995a).

kx ′x ′ = 4(β2 − 1

)(2β2 − 1

) [Ln(β+√β2−1

)√β2−1

− β

] , [12a]

ky ′y ′ = kz′z′ = 8(β2 − 1

)(2β2 − 3

) [Ln(β+√β2−1

)√β2−1

+ β

] . [12b]

Here β is the major axis-to-minor axis aspect ratio, and thecomponents are placed along the diagonal of the translationaldyadic as follows.

K′ =⎛⎝kx ′x ′ 0 0

0 ky ′y ′ 00 0 kz′z′

⎞⎠ . [13]

K′ can be transposed into the global frame (i.e., K) with thebasis transformation as shown

K = A−1K′A. [14]

In Equation (14), A is the direction cosine matrix (DCM).The DCM is obtained by multiplying three matrices represent-ing three consecutive rotations about an axis. For the purposesof this formulation an x-y-x convention was assumed. The threeconsecutive rotations are sketched in Figure 2. With this conven-tion the total rotation of the ellipsoid is represented by a rotationof φ about the x-axis in the initial frame, followed by a rotationθ about an intermediate y-axis (line of nodes), finishing with arotation of ψ about the x′-axis in the final reference frame. TheDCM is then written in Equation (15) for the x-y-x convention

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A NUMERICAL MODEL FOR FIBER MOTION 841

(Goldstein 1980; Hughes 1986). That is,

A =⎛⎝ cos θ sinψ sin θ cosψ sin θ

sinφ sin θ cosψ cosφ − sinψ cos θ sinφ − (sinψ cosφ + cosψ cos θ sinφ)− sin θ cosφ cosψ cosφ + sinψ cos θ sinφ cosψ cos θ cosφ − sinψ sinφ

⎞⎠ . [15]

The DCM, A, depends on fiber orientation. To determineorientation, the rotational motion must be analyzed. The hydro-dynamic torques are given by

T hx ′ = 16πµ

3γ0

(ξ ′ − ωx ′

), [16]

T hy ′ = 16πµ

3(γ0 + β2α0

) [(β2 + 1) (η′ − ωy ′

) (β2 − 1

)dx ′z′

], [17]

T hz′ =16πµ

3(γ0 + β2α0

) [(β2 + 1) (ζ ′ − ωz′

)− (β2 − 1

)dy ′z′

].

[18]

In Equations (16)–(18), ωi is the components of angularvelocity; η, ξ , and ζ are the components of vorticity; dij isthe components of the deformation rate tensor; and α0 and γ 0

are dimensionless constants. The vorticity and deformation ratequantities are defined in the global frame as

dxx = ∂ux

∂x, [19]

dyy = ∂uy

∂y, [20]

dzz = ∂uz

∂z, [21]

dxy = 1

2

(∂uy

∂x+ ∂ux

∂y

), [22]

dxz = 1

2

(∂uz

∂x+ ∂ux

∂z

), [23]

dyz = 1

2

(∂uz

∂y+ ∂uy

∂z

). [24]

The fluid angular velocity components are

ξ = 1

2

(∂uz

∂y− ∂uy

∂z

), [25]

η = 1

2

(∂uz

∂x− ∂ux

∂z

), [26]

ζ = 1

2

(∂uy

∂x− ∂ux

∂y

). [27]

Equations (19)–(27) can be transformed into the particleframe, as needed, by Equation (14). The angular velocity com-ponents were defined in the particle frame by Asgharian and

Anjilvel (1995a) to be

ωx ′ = dψ

dt+ dφ

dtcos θ, [28]

ωy ′ = dφ

dtsin θ sinψ + dθ

dtcosψ, [29]

ωz′ = dφ

dtsin θ cosψ − dθ

dtsinψ. [30]

The moment of inertia tensor is

IM =⎛⎝Ix ′ 0 0

0 Iy ′ 00 0 Iz′

⎞⎠ [31]

with components along the and transverse to the fiber polar axisbeing

Ix ′ = ρπβd5

60, [32]

Iy ′ = Iz′ = ρπβd5

120

(β2 + 1

). [33]

And the dimensionless parameters α0 and γ 0 were definedby Asgharian and Anjilvel (1995a) to be

α0 = 8

β(β2 − 1

)[

2 + β√β2 − 1

Ln

(β −

√β2 − 1

β +√β2 − 1

)], [34]

γ0 = 8(β2 − 1

)[β + 1

2√β2 − 1

Ln

(β −

√β2 − 1

β +√β2 − 1

)]. [35]

After substituting Equations (19)–(35) into Equations(16)–(18) and considerable simplification, the angular accel-eration components can be expressed as follows.

d2φ

dt2= 2(

β2 + 1)

sin θ

dθ

dt

[dψ

dt− β2 cos θ

dφ

dt

]

+ µ

ρd2

20

β(γ0 + β2α0

)•{[

(ζ cosφ − η sinφ) cot θ + ξ − dφ

dt

]

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842 K. T. SHANLEY AND G. AHMADI

FIG. 2. Intermediate rotations that make a single fiber rotation with the x-y-x convention.

+ β2 − 1

β2 + 1

[(dxy cosφ + dxz sinφ

)cot θ − dyz cos 2φ

+ 1

2

(dyy − dzz

)sin 2φ

]}, [36]

d2θ

dt2= − 2(

β2 + 1) dφdt

[dψ

dtsin θ + 1

4

(β2 − 3

)sin 2θ

dφ

dt

]

+ µ

ρd2

20

β(γ0 + β2α0

)•{[η cosφ + ζ sinφ − dθ

dt

]− β2 − 1

β2 + 1

• [(dyz cosφ − dxy sinφ

)cos 2θ

+ 1

2sin 2θ

(dxx+dyz sin 2φ−dyy sin2 θ−dzz cos2 θ

)]},

[37]

d2ψ

dt2= cot θ(

β2 + 1) dθdt

[(cos θ +

(β2 + 1

)(β2 − 1

))dφ

dt− 2

dψ

dt

]

+ µ

ρd2

20 cos θ

β(γ0 + β2α0

)•{[

(ζ cosφ − η sinφ) cot θ + ξ − dφ

dt

]+ β2 − 1

β2 + 1

• [(dxy cosφ + dxz sinφ)

cot θ − dyz cos 2φ

+ 1

2

(dyy − dzz

)sin 2φ

]}

+ µ

ρd2

20 cos θ

βγ0

{(ξ − dφ

dt

)cos θ

+ (ζ sinφ − η cosφ) sin θ − dψ

dt

}. [38]

For an ellipsoid of revolution suspended in a simple shearflow, the shear-induced lift can be expressed as follows.

FL = π2µd2

4√ν

∂u/∂y∣∣∂u

/∂y∣∣ 1

2

(K · L · K) · (u − v) . [39]

In Equation (39), L is the lift coefficient matrix described byFan and Ahmadi (1995) as

L =⎛⎝0.0501 0.0329 0.0000

0.0182 0.0173 0.00000.0000 0.0000 0.0373

⎞⎠ . [40]

Determining interception between an ellipsoid and a planeis both a function of the ellipsoid’s orientation and the distancefrom the plane. The distance is defined by the radius vector inthe particle frame. Figure 3 shows a two-dimensional sketchdefining the radius vector. To determine the radius vector first,define a unit vector pointing out from the center of mass of thefiber. This vector will sit on a unit sphere about the center ofmass. If this vector is scaled by the semiminor and semimajoraxes of the ellipsoid, the vector will point from the center ofmass to a point on the ellipsoid surface. The point at which thevector meets the ellipsoid surface is the interception point. Ifthe unit vector is taken to be the inverse of the unit normal of theplane and the distance between the plane and the center of mass

FIG. 3. 2D illustration of the location of the interception point on an ellipsoid.

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A NUMERICAL MODEL FOR FIBER MOTION 843

FIG. 4. Flow chart of the process by which fiber trajectories were computed.

of the fiber is less or equal to the magnitude of the radius vector,the fiber will intercept the plane (i.e., deposition has occurred).

4. MODEL IMPLEMENTATIONA series of three UDF’s were developed for ANSYS FLU-

ENT to perform the calculations of fiber motion. Figure 4 showsa flow chart of the process from flow solution to fiber trajec-tory. Because of the one-way coupling assumption fiber trackingcould be treated as a postprocessing step. The flow field neednot to be updated after trajectory iterations.

The ANSYS FLUENT default drag coefficient assumes thatthe particle is spherical. This was set to zero so a nonscalardrag could be applied. This was done by returning 0.0 with theDEFINE DPM DRAG macro. This macro was called once periteration.

Drag was applied as a 3D body force. The results ofEquation (11) were returned componentwise from the DE-FINE DPM BODY FORCE macro. This macro was calledonce per component per iteration.

The DEFINE DPM SCALAR UPDATE macro was used fora number of bookkeeping items. Equations (36)–(38) were inte-grated to update the Euler angles at every iteration. This updatewas necessary to provide the appropriate orientation for Equa-tion (11). Within this same macro the distance from a wallboundary was checked if the fiber was located in a cell thattouched a wall boundary and position, velocity, and orientationwere written to an output file. This macro was called once periteration.

5. ROTATIONAL MOTIONTo validate the computation of rotational motion in this

model, a series of simulations were performed that subjectedan ellipsoidal fiber to a simple shear flow. Particular attentionwas given to the examination of the predicted rotational periodsof the ellipsoids and their comparison against the theoreticalmodel of Jeffrey (1922) and the simulations of Asgharian andAnjilvel (1995a). The results of these simulations are presentedin this section.

FIG. 5. Translating flat plate over a stationary flat plate to induce a linearvelocity profile in the interspatial air.

To establish a linear shear flow, a 3D configuration betweentwo parallel plates shown schematically in Figure 5 was preparedin GAMBIT for use in ANSYS FLUENT. The domain was 2 mmtall (y-direction), 2 mm wide (x-direction), and 5 mm long (z-direction). The sides of the channel that are in the yz-plane weremodeled with a specified shear of zero. Velocity values wereextrapolated from within the domain to the exit plane that washeld at a constant static pressure. A linear shear velocity profilewas applied to the inlet surface. The top wall was set to translatewith constant velocity equal to the maximum value along theinlet profile, while the bottom wall remains stationary. The no-slip boundary condition was imposed on the top moving walland stationary bottom wall. Steady state flow fields for differenttop wall velocities were evaluated.

The velocity of the top wall (U0) was varied from 0.02 to2 cm/s to examine the effects of shear rate on fiber rotationalperiod. A single ellipsoidal fiber was placed at the center of thechannel for different generated steady simple shear flow fields.The fiber was allowed to rotate, but the drag, lift, and gravi-tational sedimentation were set to zero to prevent translation.Three aspect ratios of 5, 10, and 15 were examined. The massof the fiber was varied by examining a range of diameters from1 to 100 µm and densities from 1 to 10 g/cm3.

Jeffrey (1922) analyzed the motion of an ellipsoidal fiberunder a constant velocity gradient condition. He showed thatunder this condition the rotational period was scaled with thevelocity gradient. This resulted in a dimensionless period (T∗)that was shown to vary with the fiber aspect ratio. This periodwas presented as

T ∗ = 2π(β2 + 1

)β

. [41]

Using a computational model, Asgharian and Anjilvel(1995a) showed fibers with negligible inertia to rotate with theperiod predicted by Equation (41). They defined a rotationalReynolds number to account for the particle cross-sectionaldiameter (d), density (ρ), fluid dynamic viscosity (µ), and

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844 K. T. SHANLEY AND G. AHMADI

FIG. 6. Simulated rotational period as a function of Rotational Reynolds num-ber at 3 aspect ratios. (Color figure available online).

magnitude of the constant velocity gradient (G). That is,

ReR = ρfGd2

µ. [42]

Figure 6 shows the present simulated rotational period of theparticle as a function of the rotational Reynolds number. Alsoshown are the predicted period given by Equation (41) and theearlier simulations of Asgharian and Anjilvel (1995a). Simu-lations performed with the ANSYS-FLUENT UDF developedhere agreed quite well with the independent code developed byAsgharian and Anjilvel (1995a) and the theoretical values ofJeffrey (1922) for the range of interest. Both sets of simulateddata show a decrease in rotational period at the higher end ofrotational Reynolds number. Asgharian and Anjilvel (1995a)explained this to be an effect of inertia. They also showed thatif the rotational Reynolds number were increased further theperiod would reach a minimum then increase in the limit thatthe rotational Reynolds number becomes very large.

Given the close agreement of the simulated rotational periodto the theoretical period and the similar trends published inAsgharian and Anjilvel (1995a), it can be concluded that thedeveloped model can accurately predict the rotational motion ofan ellipsoid suspended in linear shear flows.

6. MOTIONS OF A FIBER IN A VISCOUS FLUIDThe anisotropic shape of a fiber will cause motions that

markedly depart from that of a sphere when released in a movingviscous fluid. Fluid vorticity acts on the elongated body caus-ing the fiber to tumble as it translates. This section discussesthe motions of a fiber when released in the center plane of ahorizontal tube with circular cross-section.

A horizontal tube geometry was prepared and meshed inGAMBIT for use in ANSYS FLUENT. The tube was 792 mmlong with an inner diameter of 4.05 mm. The walls of the tube

FIG. 7. Horizontal tube with circular cross-section; parabolic velocity profileimposed at the inlet surface.

were taken to be rigid and smooth with a no-slip boundarycondition. Two flow rates corresponding to Reynolds numbersof 150 and 450 were examined where Re = U0d/ν. A schematicdiagram of the computational domain is shown in Figure 7.

A parabolic velocity profile was imposed at the inlet surfaceand the outlet was maintained at a constant 0 Pa static gagepressure. Flow field variables were solved numerically withinthe domain using the ANSYS FLUENT code (Fluent User’sManual, 2007). Figure 8 shows contours of axial velocity alongthe center plane of the tube. Consistent with typical pipe flow,

FIG. 8. Axial velocity contours along the center plane of the horizontal tube;Re = 150 (top) and Re = 450 (bottom).

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A NUMERICAL MODEL FOR FIBER MOTION 845

FIG. 9. Direction cosines as a function of time for a fiber with aspect ratioof 5 released a half radius above and a half radius below the center line in thecenter plane of the tube with Re = 150. (Color figure available online).

the axial velocity reaches a peak at the center and decays withthe square of the radius to zero at the walls.

Three numerical experiments were conducted to examine themotions of individual fibers. The results of these experimentsare shown in Figures 9–11. In each experiment, two fibers werereleased in the center plane of the tube. The two fibers wereused to vary in injection location, aspect ratio, and flow rate.Direction cosines, as functions of time, were used to record themotions of the fibers (Figures 9–11). The three direction cosinesfor the fiber are the cosine of the angle made between the fiber’ssemimajor axis and the x-axis, y-axis, and z-axis, respectively.

FIG. 10. Direction cosines as a function of time for a fiber with aspect ratioof 5 and aspect ratio 15 released a half radius above the center line in the centerplane of the tube with Re = 150. (Color figure available online).

FIG. 11. Direction cosines as a function of time for a fiber with aspect ratioof 5 released a half radius above the center line in the center plane of the tubewith Re = 150 and Re = 450. (Color figure available online).

A cosine of zero indicated that the fiber was perpendicular to theaxis, while a cosine of one indicated that the fiber was parallelwith the axis.

In general, the fibers tended to remain parallel with the di-rection of flow (i.e., x and y direction cosines were nearly zerowith z-direction cosines nearly ±1 in Figures 9–11). Rotationshappened suddenly. This can be observed in Figures 9–11 by thesudden switch from 1 to −1 (or −1 to 1) in z-direction cosinesand the sharp spikes to ±1 in y-direction cosines. This indicatesthat once enough torque was present to perturb the fiber off ofthe initially assumed equilibrium state, the fiber was destabi-lized, rotated 180◦, and a new quasi-equilibrium was achieved.Since the center plane of the tube was normal to the x-axis, thex-direction cosines remained zero for each of the experimentsconducted.

Figure 9 shows the direction cosines of two fibers releasedin the center plane of the tube. One was released a half radiusabove the center and the other was released a half radius belowthe center. The fibers are initially rotated in opposite directionsbecause the direction of the shear gradients are opposite aboveand below the center of the tube. Time between rotations wasvery similar for the two fibers. As time progressed, however,the effect of gravity was observed. The fiber released above thecenter was pulled toward the center by gravity, into smaller sheargradients. This caused the time between rotations to lengthen.Conversely, the fiber released below the center was pulled awayfrom the center by gravity, into larger velocity gradients. Thiscaused the time between rotations to shorten. Both y and zdirection cosines show a small phase shift in Figure 9 as timeprogresses.

The effect of aspect ratio was examined by releasing fiberswith aspect ratio of 5 and 15 a half radius above the center intothe flow field with Re = 150. The direction cosines of these fibers

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846 K. T. SHANLEY AND G. AHMADI

as they translate along the center plane of the tube are shown inFigure 10. More torque was required to rotate the larger fiber.This is why Figure 10 shows a longer time for the fiber withaspect ratio 15 to rotate than the fiber with aspect ratio 5.

Figure 11 shows the direction cosines of a fiber with aspectratio 5 released a half radius above the center under flow con-ditions of Re = 150 and Re = 450. The fiber released into theflow field with Re = 450 rotated more frequently. This is due tothe larger shear gradients present in the faster moving flow.

7. DEPOSITION IN A HORIZONTAL TUBEWhen injected to a moving, bounded flow field in the pres-

ence of gravity, a small ellipsoidal fiber will translate and eitherescape the domain or deposit on the tube wall. To validate theaccuracy of the developed model under these conditions, sed-imentation rate in the horizontal tube domain was computed.Rates of gravitational sedimentation were compared with theearlier work of Asgharian and Anjilvel (1995b). These resultsare presented in this section.

To examine the deposition rate of fibers in the horizontal tubeconfiguration discussed above a lattice of equal-spaced fibrousparticles was released at the inlet surface with random orienta-tions. This resulted in the release of 6375 particles per trial. Eachparticle was tracked individually, and trials were conducted fordiameter 0.5, 1, 1.5, and 2 µm with aspect ratio ranging from4 to 24. The same procedure was repeated for spheres with theequivalent diameter computed with the suggested approach ofFan and Ahmadi (1995). They suggested the equivalent diameterof a fiber to be

dF−A = 3d

√2β

kxx + kyy + kzz. [43]

Figure 12 shows the sedimentation rate as a function of as-pect ratio for ellipsoidal fibers released in the horizontal tubesetup. Also shown in Figure 12 are sedimentation efficienciesfor equivalent sphere particles, the diameter of which are com-puted by Equation (43). Favorable agreement was observed forfibers and equivalent spheres. Sedimentation increased with as-pect ratio. This was due to the fact that fiber mass increasedwith aspect ratio. Minor differences were observed betweenfibers and equivalent spheres at higher aspect ratios. This couldindicate that Equation (43) is appropriate for certain sized fibersbut not for all. Sedimentation was also seen to decrease with in-creasing Reynolds number. This is due to the reduced residencetime with higher flow rate.

Asgharian and Anjilvel (1995b) showed fiber sedimentationin a straight circular airway to vary with dimensionless relax-ation, fiber aspect ratio, and inclination angle of the airway.

FIG. 12. Sedimentation efficiency as a function of aspect ratio for ellipsoidalfibers and equivalent sphere diameter particles as computed by Equation (43).(Color figure available online).

They proposed the following relationship

Eff = τ1β−0.774

5∑n=0

cnαn. [44]

In Equation (44), cn are constants defined in Table 1, α isthe inclination angle of the tube, and τ 1 is the dimensionless

TABLE 1Coefficients for Equation (23) (Asgharian and Anjilvel 1995b)

N cn

0 −0.8441 14.4002 −22.4093 16.5864 −5.6695 0.726

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A NUMERICAL MODEL FOR FIBER MOTION 847

FIG. 13. Sedimentation efficiency as a function of aspect ratio and particlerelaxation time. (Color figure available online).

relaxation defined as

τ1 = d2Sβ

18ν

g

U. [45]

Here U is a characteristic velocity. All of the simulations con-ducted here were done with the tube in a horizontal alignment(α = π /2).

Figure 13 shows the simulated sedimentation efficiency offibers in the horizontal tube across all aspect ratios and fiberdiameters under flow rates with Re = 150 and Re = 450. Thesimulation data collapse approximately onto a single curve thatcan be curve fitted with the following simple expression:

Eff = 4.30τ1β−0.774. [46]

Close agreement with the earlier prediction of Asgharian andAnjilvel (1995b) indicates that the proposed augmentation to theANSYS FLUENT DPM is capable of accurately predicting thetransport and sedimentation of ellipsoidal fibers.

8. CONCLUSIONSA computational model for simulating the rotational and

translational motions of an ellipsoidal fiber was developed. Twoshear flow conditions were simulated to validate the model.First, a single particle was subjected to a constant linear shearflow and allowed to rotate without translation. The model wasshown to simulate a rotational period similar to the predictedperiod of Jeffrey (1922) and the simulations of Asgharian andAnjilvel (1995a). Next, flow through a horizontal straight pipewas numerically simulated. Ellipsoidal fibers were injected intothe resulting steady flow fields, and the motions and sedimenta-tion were evaluated. The following conclusions may be drawnfrom the results presented here:

• Normalized rotational period of an ellipsoidal fiber wasa function of its aspect ratio and the developed modelwas able to accurately predict the period.

• Gravity was able to shift the path of fibers enough to al-ter the period with which they rotated while translatingthrough the horizontal tube.

• Sedimentation rate of fibers increased with aspect ratioand relaxation time but decreased with flow rate.

• The developed empirical curve fit model given byEquation (46) is suitable for predicting the sedimenta-tion rate of fibers in a circular tube.

• The developed model was able to accurately predict thetranslation, rotation, and sedimentation of ellipsoidalfibers in laminar viscous shear flows.

NomenclatureLatin Symbolsa Semiminor axis of an ellipsoida+ Dimensionless semiminor axis of an ellipsoidA Direction cosine matrixb Semimajor axis of an ellipsoidb+ Dimensionless semimajor axis of an ellipsoidcn Constant coefficients in Equation (23)CD Drag coefficientdae Aerodynamic diameter (m)dev Diameter of an equivalent volume sphere (m)df Fiber cross-sectional diameter (m)dij Deformation rate tensorEff Sedimentation efficiencyfd Drag force on a sphere per unit massfl Lift force on a sphere per unit massFD Drag force on an ellipsoidFL Lift force on an ellipsoidG Magnitude of linear velocity gradientg Gravitational acceleration (m/s2)IM Moment of inertia tensorK Saffman lift force = 2.594K Translational dyadicm Mass of fluid (kg)mp Mass of particle (kg)Re Fluid Reynolds numberRep Particle Reynolds numberReR Rotational Reynolds numberS Solid-to-gas density ratioT∗ Normalized rotational periodTh Hydrodynamic torquet Time (s)u Local fluid velocity (m/s)U Characteristic velocity (m/s)U0 Velocity of translating plate (m/s)v Instantaneous particle velocity (m/s)x, y, z Position coordinates of particle center of mass

(m)

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848 K. T. SHANLEY AND G. AHMADI

Greek Symbolsα Inclination angleα0 Dimensionless parameterβ Aspect ratioδ Distance of fiber center of mass from a wall

(m)ϕ, θ , ψ Euler angles (rad)γ 0 Dimensionless parameterη, ξ , ζ Vorticity componentsκ Dynamic shape factorκ‖ parallel to a wallκ⊥ normal to a wallκR randomly orientedµ Dynamic viscosity (kg/m/s)ν Kinematic viscosity (m2/s)ρ Fluid densityρp Particle densityρ f Particle densityρH2O Density of waterτ Particle relaxation time (s)τ+ Dimensionless particle relaxation timeτ 1 Ratio of particle relaxation timeω Angular velocity (rad/s)

REFERENCESANSYS FLUENT. (2007). Fluent User’s Manual. ANSYS, Canonsburg, PA.Asgharian, B., and Anjilvel, S. (1995a). The Effect of Fiber Inertia on Its

Orientation in a Shear Flow with Application to Lung Dosimetry. AerosolSci. Technol. 22:261–270.

Asgharian, B., and Anjilvel, S. (1995b). Movement and Deposition of Fibersin an Airway with Steady Viscous Flow. Aerosol Sci. Technol. 23:282–290.

Chen, Y. K., and Yu, C. P. (1991). Sedimentation of Fibers from Lami-nar Flows in a Horizontal Circular Duct. Aerosol Sci. Technol. 14:343–347.

Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles, Drops, and Particles.Academic Press, New York.

Fan, F. G., and Ahmadi, G. (1995). A Sublayer Model for Wall Depositionof Ellipsoidal Particles in Turbulent Streams. J. Aerosol. Sci. 26(5):813–840.

Goldstein, H. (1980). Classical Mechanics, 2nd ed. Addison-Wesley, Reading,MA.

Hinds, W. C. (1982). Aerosol Technology: Properties, Behavior, and Measure-ment of Airborne Particles. John Wiley and Sons, New York.

Hughes, P. C. (1986). Spacecraft Attitude Dynamics. John Wiley and Sons, NewYork.

Jeffrey, G. B. (1922). The Motion of Ellipsoidal Particles Immersed in a ViscousFluid. Proceedings of the Royal Society of London, Series A. London, UK,102:161–179.

Saffman, P. G. (1965). The Lift on a Small Sphere in Slow Shear Flow (LiftForce on Small Sphere in Slow Shear Flow Through Highly Viscous Liquid).JFM 22:385–400.

Shapiro, M., and Goldenberg, M. (1993). Deposition of Glass Fiber Particlesfrom Turbulent Air Flow in a Pipe. J. Aerosol. Sci. 24:65–87.

Stober, W. (1972). Dynamic Shape Factors of Nonspherical Aerosol Particles. inAssessment of Airborne Particles, T. Mercer et al., ed., Charles C. Thomas,Springfield, IL, pp. 249–289.

Wang, Z., Hopke, P. K., Ahmadi, G., Cheng, Y. S., and Baron, P. A. (2008).Fibrous Particle Deposition in Human Nasal Passage: The Influence ofParticle Length, Flow Rate, and Geometry of Nasal Airway. J. Aerosol. Sci.39:1040–1054.

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