On the effect of nonhomogeneous flow fields on the orientation distribution and rheology of fiber...

On the Effect of Nonhomogeneous Flow Fields on the Orientation Distribution and Rheology

of Fiber Suspensions

RAVI SHANKER, J. W. GILLESPIE, JR., and S. I. GUCERI

Department of Mechanical Engineering Center for Composite Materials

University of Delaware Newark, Delaware 1971 6

The influence of nonhomogeneous flow fields on the orientation distribution and rheology of fiber suspensions is investigated. The Stokes-Burger model is used to approximate the forces and moments on the fiber for the subsequent calculation of fiber angular velocities for nonhomogeneous cases where the velocity gradients vary over the fiber length. For simple shear flows, an analytical solution based on the Stokes-Burgers model is derived. Fiber angular velocities are compared to previously published results. An iterative numerical technique is developed to calculate the otherwise indeterminate fiber angular

velocities for other nonhomogeneous flows, such as non-isothermal Hele-Shaw flows. The value of shear is derived for which the angular motion of the fiber in a quadratic velocity field is identical to that in a linear shear flow. Subsequently, numerical solutions for orientation distribution functions are presented for cases where analytical solutions for distribution functions are not obtainable. The rheological properties are then studied, using the orientation tensor description, for nonhomogeneous flow fields where the fiber motion can be approximated by that in a shear flow, the equivalent shear rate obtained from the numerical scheme.

INTRODUCTION he use of fiber suspensions in the manufacture T of reinforced composites has led to a significant

amount of theoretical and practical interest in modeling their behavior. The pioneering work in the field of suspension rheology was Einstein’s (1) investigation of the modified viscosity of a suspension of rigid spherical particles in a viscous medium at zero particle Reynolds number. There has been considerable progress since then in characterizing suspensions of varying concentrations and particle shapes and sizes in different types of fluids and under different deformation conditions. The motivation for all these efforts was to develop a comprehensive constitutive theory to model the behavior of these suspensions based on the fundamental understanding of particle motions and interactions.

Following the work of Einstein, higher order particle concentration effects were investigated by Batch- elor and Green (2, 3) by considering particle interactions. Jeffery (4) studied the motion of ellipsoidal particles of finite aspect ratios in homogeneous flow fields, and these results were further extended to arbitrarily shaped particles by Brenner (5-8) and Bretherton (9). Meanwhile, the slender body theory

was developed, through the works of Burgers (lo), Broersma (1 1). Cox (12, 131, Tillett (14), and Batche- lor (15, 16), among others, to model the motion of long slender particles in Stokes flow.

Jeffery’s model was used first by Givler et al. (17) to predict the fiber orientations in planar flows, such as those that occur during injection molding of thin section parts, and subsequently by Eduljee and Gillespie (18) to derive analytical solutions for fiber orientation in 2-D and axisymmetric flows. How- ever, the above-mentioned works are valid for dilute suspensions, since they do not account for the hydrodynamic interaction of the particles.

Folgar and Tucker (19) studied non-dilute systems and proposed a phenomenological model using a diffusion type term to model the hydrodynamic interaction between fibers and thus computed the evolution of the fiber distribution functions subjected to homogeneous flows. In a parallel effort, Dinh and Armstrong (20) developed a rheological model to characterize semi-concentrated suspensions where particle interactions were accounted for by using Batchelor’s cell model for the stresses in a non-dilute suspension. The suspension was modeled as an anisotropic fluid where the additional stress terms

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Ravi Shanker, J. W. Gillespie, Jr., and S. I . GiiCeri

due to the presence of fibers were shown to be dependent on the distribution function or, more ex- plicitly, on the fourth-order moments of the distribution function. Lipscomb et al. (21) have used a similar approach in extending the solution to the calculation of streamlines with the additional stress term in the momentum equations.

All the above-mentioned studies are valid strictly for homogeneous flows, where the components of the velocity gradient tensor are constant. Examples of such flows are simple shear, rotational, elongational, or any combination of the above. However, in a manufacturing environment, such as in injection molding, the shear rates across the gapwidth have been shown to vary by orders of magnitude (22) as depicted in Fig. 1. High injection speeds and com- plex geometries can also cause similar effects in the plane of the mold. Moreover, greater demands on the properties and performance of fiber-reinforced composites have led, inevitably, to the use of longer fibers. These effects invalidate the assumption that the velocity gradients are constant over the fiber length. Hence, the current work undertakes a rigorous treatment of the problem, for a better understanding of particle motions, orientation distributions, and subsequently the rheological behavior of a suspension in nonhomogeneous flow fields.

A review of the available literature reveals that the motion of particles in nonhomogeneous fiow fields has not been rigorously investigated. Altan, et al. (23, 24), addressed the nonhomogeneous flow problem by assuming that the flow is homogeneous over length scales equivalent to the particle size. How- ever, the effect of nonhomogeneous flow fields on polymer macromolecules has been investigated by several researchers. Studies by Aubert and Tirrell (25), Brunn (26). and Jhon and Freed (27) have been motivated by the need to explain the phenomenon of polymer migration and the resulting anomalous rheological behavior. An extensive review of the theoretical and experimental works and their respec- tive contributions has been published by Leal (28). In the review, weak inertial effect and the presence

of boundaries and nonlinearities in flow have been shown to be the chief causes of particle segregation.

The current work extends a previous investigation of the fiber motions and orientation distributions of fibers in nonhomogeneous flow fields by Shanker, et al. (29), to the subsequent calculation of rheological properties. The focus is primarily on the calculation of the contribution of the fiber distribution in orientation space to the bulk stress in the suspension. The method may be applied to both dilute and non- dilute systems by a suitable change in the drag coefficient as proposed by Doi and Edwards (30). The resultant hydrodynamic force and moment on a cylindrical particle in a nonhomogeneous flow field are computed from fundamental principles. To com- pare and validate the results, an analytical expression is derived for simple planar shear flow and compared to the equivalent Jeffery’s solution. Sub- sequently, the method is used for quadratic flows and other highly nonhomogeneous flows frequently observed during injection molding. Resulting distribution functions and rheological properties were calculated for the fiber suspensions in nonhomogeneous flows.

The fourth-order orientation tensor has been used to quantify the changes in the transient rheological parameters. This representation is valid only for homogeneous flow fields (20). Consequently, an equivalent homogeneous representation is used for the rheological characterization of the nonhomogeneous flows studied in this work. Results show that the equivalent shear rate obtained from our rigorous nonhomogeneous calculations differ substantially from the shear rate at the fiber center, thus considerably altering the predictions in orientation distributions and the subsequent rheological properties.

EQUATION OF MOTION FOR A FIBER

Ideally, the force on the fiber suspended in a viscous medium can be calculated by summing up the hydrodynamic stresses on its surface. Consider a fiber shown in Fig. 2 suspended in a medium. The

Test Fiber , f

2h 1

Fig. 1 . An example of a nonhomogeneous flow field as observed in the gapwidth in the injection molding offiber suspensions along with the coordinate system used in the current work.

Flow field 3$ A

Fiber center based __ coordinate system

Fig. 2. Schematic of a test fiber in an effective continuum.

162 POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1991, Yo/. 31, No. 3

Effect of Nonhomogeneous Flow Fields

total hydrodynamic force acting per unit length on the fiber can be given as

where A is the circumferential area of the fiber, n is the unit vector normal to the surface of the fiber and neff is the effective stress tensor for the medium. The calculation of the stress field is computationally quite intensive because the fiber is constantly changing orientations. Hence, a simpler means of calculating the resultant force and moments is pursued.

For the class of flows investigated, i.e., Stokes flows, where the mean Reynolds number is much less than unity, the inertial terms in the Navier- Stokes equation can be neglected. Thus, for such flows, the total hydrodynamic force is a linear function of the relative velocity of the fiber with respect to the macroscopic flow (6). The suspending medium is assumed to be a Newtonian fluid and the particles are considered to be rigid, neutrally buoyant, and inertialess. External forces such as body forces and Brownian forces are assumed to be absent. Hydro- dynamic interaction with other particles is neglected for the dilute suspension model. However, hydrodynamic interactions in semi-concentrated systems can be accounted for by modeling the suspension surrounding the fiber as an effective continuum.

Following the works of Burgers (10). Broersma (1 1). and others, the force per unit length of the fiber can be is expressed as

where rerf is an anisotropic tensorial drag coefficient and (vf- r) is the relative velocity of the fluid with respect to the test fiber, at any point r on the fiber.

The drag coefficient, also called the translational resistance tensor (6, 7), is independent of the velocity field and incorporates only the effects of the shape and orientation of the particle.

As in the work of Goddard and Huang (31). and Dinh and Armstrong (20). the anisotropic drag coefficient is expressed as

(3)

where 6 is the unit tensor, CP and Ct are the drag coefficients parallel and transverse to the fiber, and p is the orientation vector of the particle. From the results of slender body theory we have,

rt < = - P n (4) - L

Hence, the overall drag coefficient can be expressed as

( 5 )

For investigating the motion of a single fiber, we shall currently limit ourselves to dilute solutions, thereby ignoring hydrodynamic interaction. How-

ever, as indicated earlier, the parallel and transverse drag coefficients can be suitably modified to model dilute or semi-concentrated suspensions. For the current work, the actual value of rp is not critical.

The force experienced by an element ds of the fiber can then be given as

d F = (vf- r) ds

Thus, the total force experienced by the fiber can be given as

F = S L f 2 dF - L / 2

= SLi2 ref.* (vf- r) ds - L / 2

= S L f 2 CP(26 - pp) * (vf- r) ds - L / 2

The last equality in the above equation is a consequence of the fact that the resistance tensor is not a function of s. As shown in Fig. 2, the position vector of any point r on the fiber is given by

r = r , + s p

where rc is the position vector of the center of the fiber and s is the distance from the fiber center to the point under consideration. The time derivative of Eq 8 yields

i = i , + s p (9)

The moment M, experienced by the fiber due to these forces, can then be expressed as

M = S L l 2 ro x dF - L / 2

where r,( = sp) is the moment arm length. Since the forces and moments on the fiber are represented as the sum of forces on individual elements of a fiber, the forces and moments in a nonhomogeneous field can be calculated using a constant resistance tensor approximation as shown above.

The translational and rotational velocities of the fiber in any flow field may be determined using the equilibrium conditions

F = O and M = O (11)

respectively. for axisymmetric particles, the translational and rotational motions are uncoupled and hence the calculation of one does not require knowledge of the other. Since the focus in this work is on the evolution of orientation distribution and its subsequent effects on the rheology of fiber suspensions, the calculation of equilibrium angular velocities

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Raui S h a n k e r , J . W. Gillespie, Jr., a n d S . I . GiiCeri

alone is described in the following section. However, it should be noted that in nonhomogeneous flows the non-affine motion of fiber affects the spatial distribution of particles and the rheology, some aspects of which have been addressed in a subsequent work (32).

CALCULATION OF ANGULAR VELOCITIES

The orientation of a test fiber in terms of a spherical coordinate system fixed at the center of the fiber is depicted in Fig. 1. Since the fiber is considered to be rigid, two coordinates 4 and 0 are sufficient to specify its position in orientation space. Hence, the orientation vector is given as

cos 4 sin 0 P = sin4sine (12) i cose i

The angular velocity of the particle W , can be expressed in terms of the rate of change of these angles as

i = o , (13)

6 = W Z C S C 0 (14)

where w i are the components of the angular velocity vector. The rate of change of the orientation vectors in planar flows can be calculated by setting either 4 or 0 equal to 90". Correspondingly, the rate of change of orientation vector p is given in terms of 6 or 4.

The Numerical Scheme The knowledge of the angular velocity of the fiber

for its various orientations is essential for the subsequent calculation of orientation distribution function. Analytical expressions for the angular velocity of the fiber in terms of the components of the velocity gradient tensor are available for homogeneous flow fields. However, in the present study of nonhomogeneous flow fields, the components of the velocity gradient tensor are not constant over the fiber lengths. Hence, analytical expressions for the fiber angular velocity cannot be derived (except for the case of Poiseuille flow), thus necessitating a numerical solution of the equation of motion for the fiber.

An iterative numerical scheme was devised to calculate the converged angular velocity of the fibers. Details of the numerical scheme have been given by Shanker et al. (29). The fiber was first discretized into small linear segments along its length. A numerical integration scheme was implemented to calculate the force and moment experienced by the fiber along its entire length. The converged angular velocities were obtained using underrelaxation to minimize oscillations. The angular velocities for the next orientation were then calculated using the value for the previous step as an initial guess to accelerate convergence. Influence of fiber discretization on accuracy was performed for a variety of velocity pro-

files. Results for the homogeneous case and various nonhomogeneous cases are presented and discussed in the following section.

RESULTS

Shear Flow

The case of simple linear shear, such as in a Couette flow, is considered first. Since this is a homogeneous flow problem, the velocity gradients are constant throughout the fiber length. As shown in Fig. 3, one of the coordinate systems is fixed such that an axis coincides with the centerline of the flow field, and the other is fixed at the center of the fiber but does not rotate with it. A fiber is introduced at any arbitrary point in the flow, at a distance z away from the centerline with e = 0 in the plane of the gap, i.e. the 2-3 plane as defined in Fig. I. It should be noted that for a simple shear flow, the average fluid velocity over the fiber is equal to the undis- turbed fluid velocity at the center of the fiber and that the sum of the forces acting on the fiber is identically zero.

The equilibrium condition (41 states that the sum of the moments acting on the fiber is equal to zero. As the contribution of moments from all the elements along the fiber are in the same sense, it follows that for the net moment to be zero the moment contribution from each element on the fiber must be equal to zero. Using this condition in conjunction with Eq 10, it can be shown that

I,

e = J 2 (cos2e + 1) (15)

where i, is the shear rate and 8 is the angle between the fiber and the direction normal to the flow field. This equation is valid for large aspect ratio particles and is identical to Jeffery's equation for infinite aspect ratio.

The numerical algorithm for nonhomogeneous flow fields was verified by comparing results for the simple shear flow case. In the next section the effect of a quadratic velocity field on fiber motion is investigated.

Quadratic Flows

A quadratic flow represents the simplest case of nonhomogeneous shear flow because the velocity gradients vary linearly along the fiber length. The

" f Relative v - r velocity P

Fig. 3. Schematic of a test fiber in simple shearflow.

164 POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1991, Yo/. 31, NO. 3


velocity profile for quadratic flow can be expressed as

~ f ( Z ) = urnax( 1 - kz2) (16)

where z is the distance measured from the reference plane and urnax is the maximum velocity of the fluid corresponding to the reference plane as depicted in Fig. 4.

Using Eq 10, the moment experienced by a static fiber ( i , = O ) in the direction perpendicular to the plane of the flow becomes

22 kfTuadratic = % lz1 ( z - zc) ( uf( z ) - i,) dz

For a given fiber position, an equivalent homogeneous flow field can be defined in the following two ways:

1. The velocity of the fluid at the fiber ends is calculated and then interpolated linearly, as in shear flow.

2. The velocity gradient is assumed to be equal to that at the fiber center as reported by Altan, et al. (23).

The moment experienced by a static fiber due to the linear velocity profile (as defined in method 1 above) is given as:

It can be seen from Eqs 17 and 18 that moments experienced by the fiber in the two cases are identical.

Since a quadratic flow shear rate varies linearly, effective shear rates given by the two representa- tions of the equivalent homogeneous case are identical and can be given by

where z, is the distance of the center of the fiber from the reference plane.

Hence, identical angular velocities are obtained for the quadratic flow and equivalent linear shear flow cases, as given by Eq 15 with the shear rate as defined by Eq 19.

Thus, it can be concluded that for a quadratic flow, where the shear rate varies linearly, the motion of the fiber is identical to that of a simple shear flow for an equivalent shear rate given by Eq 19. However, for higher order nonhomogeneous flows, this methodology of defining equivalent shear rates represents an improvement over the current prac- tice of approximating the fiber motion based on the fluid kinematics at the fiber center. Additionally, it can be concluded that as long as the flow over fiber can be represented exactly by a quadratic field, both equivalent homogeneous and rigorous nonhomogeneous treatments yield identical results.

Highly Nonhomogeneous Flows In injection molding, very high shear rates are

observed across the gapwidth in the mold cavity, resulting in substantial rheological changes. Conse- quently, the current study focuses on calculating the quantities of interest in the plane of the shear, i.e., across the mold gap. However, this does in not in any way restrict a full 3-D solution to the problem.

In the current analysis, we consider typical velocity profiles for flow across the mold gap obtained from the numerical simulation, as shown in Fig. 5, of injection of polystyrene into a 6.0cm x 2.0cm x 0.2cm rectangular mold. The velocity profiles chosen for the study are as given below (22):

Case (a) corresponding to inlet speed of 30.0cms- Case (b) corresponding to inlet speed of 50.0cms-'

The calculations for the rate of change of fiber orientation angle were performed for these nonhomogeneous velocity profiles using the numerical scheme described earlier. The robustness of the formulation was further borne out by the fact that the moments corresponding to the converged angular velocities were zero, within limits of numerical error. Comparisons were made with the corresponding homogeneous case assumption (as in method 2 of the previous section) to evaluate the difference in results obtained from the nonhomogeneous and homogeneous treatments. For the present study, the

- - - - - - t z 1

1 3 - velocity . - - - - - -

profile

Test fiber

Reference plane

Fig. 4. Schematic of a testPber in a quadraticflow.

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Ravi Shanker, J.' W. Gillespie, Jr., and S. I. Giiceri

(Polystyrene injecled into a 6crnx2crnx0.2cm mold)

- +- SO cm/a . + I 150 cm/a

-0.

b, '. x.

'. '0,

1 .- c I

I

I ID ._ : 0.2- " + i v

I N i

0.0 I " I I I I 0 100 200 300 400 500 1300 700

Velocity (crnls)

Fig. 5. Velocity profiles across the haq gapwidth [corresponding to different average injection velocities) during the injection molding of polystyrene in a rectangular mold.

fiber was taken to be 0.9 times the half gapwidth and positioned halfway between the centerline and the wall.

Results for both the homogeneous and the nonhomogeneous cases are shown in Figs. 6a and 6b. Next, the fiber angular velocities from the nonhomogeneous analysis were compared to those from an equivalent shear rate obtained by inspection of the nonhomogeneous treatment results. The fiber be- haves as if it were in a shear flow with a different shear rate, which shall be called the equivalent shear rate for the flow. The equivalent shear rate can be determined by comparing the numerically calculated fiber angular velocities with those from analytical solutions for shear flow, for different trial values of shear rates.

For example, for the case corresponding to the inlet velocity of 30cm/s and equivalent shear rate of 695.6s-' was chosen, and for the case corresponding to the inlet velocity of 50cm/s an equivalent shear rate of 1052.5s- '. The angular velocities predicted by Jeffery's equation are then compared (Fig. 7 ) to the nonhomogeneous case results shown in Figs. 6a and 6b. It should be mentioned here that, while the equivalent shear rate does not exactly represent the flow field, as is clear from Fig. 7 , it marks a definite improvement over the methods used by previous authors. Moreover, as shall be shown in a subsequent work, the error in calculating the particle-induced stress in the suspension due to the marginal inaccuracy in representing the flow field with an equivalent shear rate can be recovered.

For cases such as those mentioned above, where equivalent shear rates are obtainable, previous a n a lytical solutions can be used for calculation of distribution function and transient rheological properties.

Within the next section we outline a numerical

Inlet Velocity =30 crnls Fiber length=0.9 rnrn r.=0.5 rnrn

10 20 30 40 80 60 70 80 80

Theta (m deg)

(a)

Inlet Velocity =50 crnis Fiber length=0.9 rnrn i.=O 5 rnm

,200 1 - - - -. . . *.

- - - Homagensous

Theta (m deg)

(b)

Fig. 6. comparison of the angular Velocities of ftber (for varying orientation angles) as predicted by nonhomogeneous calculations with those based on the velocity gradient at the center of thefiber, in the

a. velocity field corresponding to 30cm / s injection ve-

b. velocity field corresponding to 50cm 1s injection velocity.

locity.

Case (a) Inlet Velocity =30 crnls, Case (b) Inlet Velocity = 5 0 cmls 1200

- - - Equiu. Horn. cis. (el

* . . - Nonhom. c a m (b)

200 -

Theta (in deg)

Fig. 7. comparison of nonhomogeneous angular velocities for the previous two cases with those based on the corresponding equivalent shear rate.

166 POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1991. Vol. 31, NO. 3


solution for the distribution function equation for the cases where analytical solutions, such as the method of characteristics, are inapplicable.

ORIENTATION DISTRIBUTION FUNCTION

The most basic and yet complete form of characterizing 2-D and 3-D orientation states is the distribution function. The distribution function can also be used in determining the rheological characteristics of fiber suspensions (19, 20).

The distribution function C(r,,p, t ) is defined as the probability of having a test particle under consideration with its center of mass at position rc and orientation p at time t. Hence, Wr,, p, t) dr, dp is the number of particles with their center of mass in the range dr, about r,, and orientation in the range dp about p. For a homogeneous flow field, Wr,, p, t ) is assumed to be separable into two mutually independent quantities

p(r, ,p. t ) = n(r,, t )$(p. t ) (20)

where C(p, t ) is an internal distribution function which is independent of the spatial location of the particle and n(r,, t) is a spatial location dependent number density.

While the above assumption is not valid for a nonhomogeneous velocity field, the present calculations are for distribution functions at different spatial locations: therefore, the solution of the distribution function is dependent only on the local values of the velocity in the flow field. This is in contrast to homogeneous flow fields, which can be defined in terms of a macroscopic velocity gradient tensor, thus permitting the distribution function for the entire flow field to be expressed in terms of the velocity gradient tensor components. Thus,

$ = $ ( P * t ) , represents the distribution function at any particu- lar point in the flow domain.

For the 2-D case, i.e. when all the fibers are in a plane, the distribution function has a period of T ,

$(e l = $40 + T )

$ ( e , + ) = $ ( T - e , + + .)

(21)

(22)

whereas for the 3-D case we have

As shown in Fig. 2, 0 and + represent the orientation angles in the spherical coordinate system.

The normalization condition of the distribution function requires that the sum of the distribution function over all possible orientations be unity. This can be stated for the 2-D and 3-D cases as

2 0 : L T $ ( 0 ) d0 = 1 (23)

The governing equation for the distribution function, also known as the Fokker-Planck equation, is

based on the principle of conservation. It is given as

125)

In Eq 25, p is the rate of change of the orientation vector p for a given set of particles in the flow domain.

Given the equation of motion of the fiber in the flow field, the distribution function can be calculated from E q 25. In terms of the spherical coordinate system, it can be expressed as

For our 2-D case, where & = O , the above expression simplifies to

Analytical solutions for the distribution function exist for homogeneous flow fields, subject to ran- dom initial distribution (20, 33). For simple 2-D shear flow, the distribution function for a suspension subjected to a total shear y has been derived analytically by Altan, et al. (33), and can be expressed as

$ ( e , t ) = - ( 1 - y ~ i n 2 0 + y ~ c o s ~ 1 3 ) - ~ (28)

Note that the above expression is based on Jeffery’s equation of motion for infinite aspect ratio particles.

Numerical Calculation of Distribution Function

For the nonhomogeneous case, there are two op- tions that can be pursued. If the nonhomogeneous flow is such that the angular velocities calculated using an effective shear rate are comparable to the exact nonhomogeneous calculation, then the analytical results described above may be used. If, however, the results differ significantly from the effective homogeneous treatment, i.e. if no analytical solution for p is available, then the numerical calculation of distribution function is necessitated.

It is possible to express E q 27 as a first-order linear equation of the form

1 7r

au au - +p- + qu = 0 a t ax (29)

This can then be transformed to a first-order wave equation

a U a U ax

- a t +p- = o

by making the substitution u = ueqt. Explicit or implicit methods can be used to solve

the resulting wave equation (34, 35). In this work, both methods, having second-order accuracy in time and space, were used and identical results were obtained. As a result of the change of variables, the errors due to the presence of large gradients caused

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Ravi Shanker, J . W. Gillespie, Jr., and S . I. Giiceri

lar positions in space. The time scales for our calculations were chosen so as to depict the transient distribution functions for a total shear of the order of unity. Beyond this, the results show high alignment of fibers, and as a consequence the numerical results become inaccurate.

by the exponential behavior in time were significantly reduced. Numerical results for the distribution function in homogeneous flow fields resulting from Eq 15 and those obtained analytically using Jeffery’s equation with infinite aspect ratio are found to be comparable for total shears not exceeding 2.5. The analysis was then extended to nonhomogeneous flow fields such as those discussed in the previous section. Subsequently, distribution functions derived by the equivalent shear rate assumption (nonhomogeneous treatment) were compared to those obtained from a simplistic homogeneous treatment, as shown in Fig. 8.

The distribution functions exhibit some interest- ing features. For example, they do not vary much when the fibers are oriented at small angles to the flow direction. This is a direct result of both the angular velocity and its rate of change with orientation being very small corresponding to those angu-

40 -80 -40 -20 0 20 40 60 80

Theta (m deg)

(a)

Fig. 8. Comparison of distribution functions predictions based on thefiber center gradient with those based on the equivalent shear rate.

a. velocity field corresponding to 30cm J s injection velocity.

b. velocity field corresponding to 50cm J s injection veloci ty.

RHEOLOGICAL CHARACTERIZATION

In flow processes, such as in injection molding, accurate knowledge of the rheological behavior of suspensions helps determine flow characteristics and plays an important role in the overall design of the mold and part performance. The flow analysis process is coupled with fiber orientation, since the flow-dependent orientation states determine the rheology, which in turn determines the flow characteristics. Moreover, the performance of the finished product is a function of the fiber orientation states, which are determined by the flow processes. Rheol- ogy thus determines the flow characteristics and plays an important role in the overall design of the mold and part performance.

The suspension is assumed to be a uniform homogeneous fluid (not to be confused with a homogeneous flow) with effective rheological properties different from the ambient fluid in which the particles are suspended. The most important rheological property of suspensions is the bulk induced stress which is caused by the presence of particles.

Batchelor (16) developed a general constitutive equation for suspensions of arbitrary concentrations with particles of arbitrary shapes given as

7 = 7, + r p

where r is the non-equilibrium stress, rs and rP are the contributions to the stress from the ambient fluid and the particles respectively, + is the bulk shear rate and r, is surface fluid stress. As mentioned earlier, the evaluation of these surface integrals in a dynamic system is almost impossible due to the computational effort involved. The problem was simplified by Batchelor (15) in a subsequent work, where he developed the cell model for suspensions of slender particles in elongational flows, modeling the suspension as an effective continuum. One possible representation of the stresses due to the presence of the particles is by using the rheological model developed by Dinh and Armstrong [20], which can be expressed as

where C is a constant depending on the concentration of the suspension, the particle aspect ratio, and other parameters of the system, and Vv is the velocity gradient tensor. It should be noted that Eq 32 has been obtained using Jeffery’s equation for infinite aspect ratio particles and is valid strictly for homogeneous flow fields.

168 POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1991, VOl, 31, NO. 3


POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1991, Vol. 31, NO. 3 169

The distribution function can be calculated in either of two ways. If the corresponding equivalent shear rates are known, one can use analytical results for $ based on Jeffery’s equation or use E q 15 for the numerical calculation of $. However, if the fiber angular velocity profile indicates that the motion of the fiber in the flow field differs considerably from that in an equivalent homogeneous flow situa- tion, this technique cannot be used.

However, as we have demonstrated earlier, based on the motion of rigid fibers in nonhomogeneous flow fields studied here, these fields can be expressed in terms of an equivalent homogeneous flow field. Consequently, the distribution function and the rheological properties can be expressed using the equivalent homogeneous flow field.

Following the work of Advani and Tucker (36) and Altan (24, 331, we shall express the rheological properties of the suspension in terms of the components of the fourth-order orientation tensor. The second- and fourth-order orientation tensors are defined as

(33)

(34)

These tensors are invariant under orthogonal trans- formations, are completely symmetric, and can be used for both two- and three-dimensional description of the orientation.

The transient shear viscosity q+ and the first normal stress difference *: are the rheological quantities of interest for shear flows. During injection molding, shear predominantly occurs across the gapwidth of the mold represented by the 2-3 plane (Fig . 1 ) . The above-mentioned rheological quantities can be expressed as

(35)

*: s2223 - s2333 = c-

D

Figure 9 shows predictions of transient rheological parameters (in terms of the various orientation tensor components) based on the instantaneous orientation states. These were obtained by evaluating the integrals given in E q s 33 and 34 using Simp- son’s rule for different total elapsed times corresponding to a range of total shears. Since we are dealing with Newtonian flows where the orientation state is a function of the total deformation, the rheological properties are the same for equal total shears. However, it is obvious from the figures that there is a significant difference in the evolution of the orientation distribution and hence the transient rheological properties.

DISCUSSION AND CONCLUSIONS

The current work investigates the effect of nonhomogeneous velocity fields on the orientation distri-

bution and rheology of fiber suspensions. While in the current investigation the ambient fluid viscosity has been assumed to be constant, varying viscosi- ties can be easily incorporated into the formulation. Wall effects, while of considerable importance in the cases discussed, are neglected, as the object of the paper is to develop a methodology for quantifying the effect of nonhomogeneous flows on the rotational motion of fibers and the rheology of fiber suspensions.

A rigorous analysis was performed to analyze the motion of long slender particles in creeping flow. The angular velocities of a test fiber were determined using the Stokes-Burger model to calculate the force and moment on the fiber. The use of this model yields, for the simple shear flow, Jeffery’s equation for infinite aspect ratio.

The technique was then used for nonhomogeneous flows where quadratic and some highly nonhomogeneous profiles were considered. It was shown that the motion of fibers in a quadratic flow is identical to that of a homogeneous flow case for an equivalent shear rate as given by E q 19. When the combination of flow field and fiber lengths is such that the flow over the fibers can be represented by a quadratic flow, a homogeneous consideration is sufficient for accurate representation of fiber motion in orientation space. If this is not the case, then a nonhomogeneous treatment is required. This is a very important result, in that it furnishes us with a criterion with which to evaluate whether or not, in a given problem, nonhomogeneous flow considerations are necessary.

For the case of the highly nonhomogeneous flows investigated in this study, it is notable that even for fiber lengths of only 0.9mm, a rigorous nonhomogeneous flow solution yields results for fiber motion that are significantly different from a simple homogeneous consideration. The transient fiber distribution obtained from a rigorous nonhomogeneous treatment departs significantly from that obtained by a simplistic homogeneous treatment. Conse- quenty, the transient rheological properties of the suspension due to particle rotation, which are expressed in terms of the fourth-order orientation tensor, subjected to a nonhomogeneous flow, differ from those predicted on the basis of homogeneous flow field considerations. The stresses due to relative translation motion between the particle and the fluid in nonhomogeneous flow fields shall be addressed in future work.

The current study has been motivated by the need to simulate shear-dominated manufacturing processes more realistically and accurately. This technique is valid for both dilute suspensions and semi- concentrated suspensions, as the coefficient C in E q 32 incorporates the structure of the suspension, thus accounting for the interactions. While it is known that the Dinh-Armstrong model is incomplete in its description of semi-concentrated suspensions, it is used merely as a tool to demonstrate the effect of a rigorous treatment of nonhomogeneous flow fields

Ravi Shanker, J . W. Gillespie, Jr., and S . I . GiiCeri

z 2 3 vs T i m e

-.. caqe (b) N o n h o m o g e n e o u s

0 050 001 002 003 004 005 006 (;I): ( l i lR

T i m e (secs)

(a)

S33 vs T i m e

o o n 1 , I 001 002 001 004 003 006 n o 7 rmq

T i m e (secs)

(el

S,, vs Time

0 5 ' , 1 001 002 003 004 005 006 007 008

T i m e (secs)

(b)

S z Z z 2 vs T i m e

n 4 -

n a -

- - I c u s e ( u j t lanhomogeneous . . . cose(b) Homogeneous -.- case(b) N o n h o m o g e n e o u s I 1-

001 002 005 004 005 006 007 008

Time ( s e c s )

T i m e (secs)

(fl

Fig. 9. Comparison of predictions of various second-order and fourth-order tensor components based on equivalent shear rates with those based onftber center velocity gradients corresponding to the two velocity proftles under consideration.

170 POLYMER ENGINEERING AND SCIENCE, MID-FEBRUARY 1991, YO/. 31, NO. 3


5 2 2 2 3 V S T i m e

o i n 0 4 :II

0025 ’ , I I COl 0@7 003 004 005 006 007 008

Time (secs)

(€9

Fig. 9. Continued

on the rheology. Moreover, the method outlined, namely, calculating the equivalent shear rates, can be used in conjunction with any rheological model.

ACKNOWLEDGMENT

The authors are indebted to Prof. C. L. Tucker, University of Illinois, Urbana-Champaign, for his in- valuable comments and suggestions. The authors gratefully acknowledge the support of the Univer- sity-Industry Consortium, “Application of Compos- ite Materials to Industrial Products,” at the Univer- sity of Delaware Center for Composite Materials.

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