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J Magn Magn Mater. Author manuscript; available in PMC 2010 May 1.

Published in final edited form as:

J Magn Magn Mater. 2009 May 1; 321(10): 1446–1451.

doi: 10.1016/j.jmmm.2009.02.065PMCID: PMC2757298

NIHMSID: NIHMS114664

Department of Biomedical Engineering, Lerner Research Institute, Cleveland Clinic, 9500 Euclid Avenue, Cleveland, OH 44195, USA

See other articles in PMC that cite the published article.

Quadrupole magnetic field-flow fractionation (QMgFFF) is a separation and characterization technique for magnetic nanoparticles such as those used for cell labeling and for targeted drug therapy. A helical separation channel is used to efficiently exploit the quadrupole magnetic field. The fluid and sample components therefore have angular and longitudinal components to their motion in the thin annular space occupied by the helical channel. The retention ratio is defined as the ratio of the times for non retained and a retained material to pass through the channel. Equations are derived for the respective angular and longitudinal components to retention ratio.

The field-flow fractionation (FFF) techniques are for the separation and characterization of particulate and macromolecular materials suspended or dissolved in a liquid medium. They are similar to chromatography in that different components of a small sample elute from a separation channel at different times [1]. Chromatography exploits differences in partition between mobile and stationary phases to separate sample components as they are carried along a column, but FFF separation is achieved within the flowing mobile phase and does not utilize a stationary phase. The separation channel optimally has a uniform thickness and a high aspect ratio, i.e., it is much thinner than it is broad. The fluid velocity profile is therefore parabolic, or close to parabolic, with the fastest velocity close to the center and zero velocity at the walls. The small perturbation due to the viscous drag of the side-walls may be ignored for a channel of high aspect ratio. A field of some type is applied across the channel thickness, perpendicular to the direction of fluid flow. The materials to be separated (we restrict our consideration to submicron particles in this work) interact with the field and are driven toward one of the walls, known as the accumulation wall. The resulting increase in concentration adjacent to the accumulation wall is opposed by Brownian diffusion, and dynamic steady-state distributions are approached where the concentration of each species is highest adjacent to the wall and decays exponentially away from the wall. Furthermore, the rate of exponential decay is related to the strength of the interaction with the field, so that particles that interact strongly occupy thin zones close to the wall, whereas those that interact less strongly occupy more diffuse zones. It is important to point out that the approach to steady-state occurs very quickly in the thin channel because of the relatively small migration distances involved (typically 250 μm, but may be less). Because a steady-state zone occupies only a fraction of the channel thickness, it follows that there is also a very rapid exchange of particles within the thickness of a zone. This means that a collection of identical particles will be carried along the channel length as a coherent band as they all sample different streamline velocities equally. Strongly interacting particles are confined to slowly moving streamlines close to the wall and have a relatively long elution time. Particles that occupy more diffuse zones are able to sample faster moving streamlines as well as those close to the wall, and elute more quickly. Separation according to strength of interaction with the applied field is thereby obtained.

Important aspects of this optimal implementation of the FFF separation mechanism are the thin channel of uniform thickness across its breadth, and the transverse field applied uniformly across the channel breadth. Quadrupole magnetic field-flow fractionation (QMgFFF) uses a thin helical channel mounted in an axisymmetric magnetic field. Inside the aperture of the quadrupole, the field strength increases radially from the axis. The gradient in field strength is constant and directed radially outward from the axis. Magnetization of the particulate sample is induced by the applied magnetic field and the particles are driven toward the outer channel wall by their interaction with the field gradient. The helical channel has a number of advantages over a full annular channel. An annular channel requires the carrier fluid and sample be distributed uniformly around the annulus at the inlet and collected uniformly around the annulus at the outlet. This is not easily achieved in practice. Multiple inlets and outlets, distributed around the annulus are required. Partial or complete blockage of any of the multiple inlets or outlets would disrupt flow patterns and contribute to band-spreading and loss of resolution. Any variation in field gradient around the circumference of the annular channel would result in variation of elution times, again leading to loss of resolution. It is also difficult to fabricate and maintain a perfectly uniform, thin annular channel. Variations in thickness around the circumference would offer variations in resistance to flow; in fact, mean velocity varies with the square of the channel thickness for a given pressure drop. Again, the net result would be a loss of resolution. On the other hand, a helical channel has a single inlet and a single outlet, the helical path forces all particles to sample any circumferential variations in field gradient, and the approach taken for channel assembly tends to reduce variations in channel thickness. The helical channel is machined into the surface of a polyoxymethylene rod (Delrin™ from DuPont) that fits tightly into an internally polished stainless steel tube [2–4]. It is therefore self-centering throughout the length of the tube. The helical nature of the QMgFFF channel does, however, require a careful consideration of the fluid velocity profile, as well as the influence of any secondary flow effects due to channel curvature and torsion. This work is concerned with only the effect of fluid velocity profile on sample elution.

QMgFFF has been used to characterize magnetic nanoparticles such as those used for cell labeling and for targeted drug therapy. These particles consist of a magnetic component with biocompatible coatings carrying antibodies or therapeutic drugs. The coatings stabilize the materials in suspension and also reduce the magnetic dipole-dipole interactions between particles in a magnetic field. The force exerted on a single magnetic particle due to its interaction with a magnetic field gradient is given by

$${\mathit{F}}_{m}={V}_{m}M\nabla B$$

(1)

where *V _{m}* is the volume of the magnetized component of the particle,

$$u=\frac{{F}_{m}}{f}=\frac{{F}_{m}}{3\pi \eta {d}_{p}}$$

(2)

in which *f* is the friction coefficient given by 3*πηd _{p}* where

$${J}_{r}=uc-D\phantom{\rule{0.16667em}{0ex}}dc/dr$$

(3)

In this equation *D* is the particle diffusion coefficient, *c* is the local concentration, and *dc/dr* is the local gradient in concentration in the radial direction. In the thin channel, a steady state distribution is quickly approached and *J _{r}* is then effectively zero at every point across the channel thickness, so that

$$\frac{dc}{dr}=\frac{uc}{D}=\frac{{F}_{m}c}{kT}$$

(4)

The second form on the right of eq (4) is obtained by the substitution of *D* by *kT*/*f* = *kT*/3*πηd _{p}*. The friction coefficient cancels and the steady state concentration profile is therefore independent of both the hydrodynamic diameter of the particles and the viscosity of the suspending fluid. It is therefore only the force on the particles due to the interaction of their magnetic component with the field gradient that determines the concentration profile. The steady state profile is obtained by solving the integral equation

$$\underset{{c}_{0}}{\overset{c}{\int}}\frac{dc}{c}=\frac{1}{kT}\underset{{r}_{o}}{\overset{r}{\int}}{F}_{m}\phantom{\rule{0.16667em}{0ex}}dr=\frac{{r}_{o}}{kT}\underset{1}{\overset{\rho}{\int}}{F}_{m}\phantom{\rule{0.16667em}{0ex}}d\rho $$

(5)

in which *c*_{0} is the particle concentration adjacent to the accumulation wall, *r _{o}* is the radius of the (outer) accumulation wall, and the new variable of integration

$$c={c}_{0}exp\left(-\frac{1}{\lambda}\frac{(1-\rho )}{(1-{\rho}_{i})}\right)={c}_{0}exp\left(-\frac{\xi}{\lambda}\right)$$

(6)

where

$$\lambda =\frac{kT}{{W}_{o}}=\frac{kT}{{F}_{m}{r}_{o}(1-{\rho}_{i})}=\frac{kT}{{F}_{m}w}$$

(7)

so that *λ* is the ratio of thermal energy *kT* to the work *W _{o}* required to drive a particle a distance equal to the channel thickness

In FFF, the retention time, or alternatively, the elution time, may be given in terms of the retention ratio *R* which is defined as the ratio of the time *t*^{0} for a non-retained material to pass through the channel to that of the retained material *t _{r}*. Alternatively,

$$R=\frac{{v}_{p}}{<v>}=\frac{<cv>}{<c><v>}$$

(8)

where the angle-brackets indicate the mean of the enclosed quantity over the channel cross section. The velocity of the non-retained material is therefore assumed to be equal to the mean fluid velocity in a cross section of the channel, and the retained material has a velocity that weights local velocity by local particle concentration, so that *v _{p}* = <

$$<c>=\frac{\underset{0}{\overset{w}{\int}}c\phantom{\rule{0.16667em}{0ex}}dx}{\underset{0}{\overset{w}{\int}}dx}=\frac{1}{w}\underset{0}{\overset{w}{\int}}c\phantom{\rule{0.16667em}{0ex}}dx$$

(9)

The fluid velocity for a helical channel may be considered to be, to a first approximation, the summation of two orthogonal components: the longitudinal and the angular velocity components. The longitudinal component is in the direction parallel to the axis of the annulus, and the angular component circles around the axis. The two components of sample velocity may be given in terms of the two fluid velocity components and the two respective retention ratios. The retention ratio *R _{z}* for longitudinal flow in an annular channel is given by

$${R}_{z}=\frac{{v}_{zp}}{<{v}_{z}>}=\frac{<c{v}_{z}>}{<c><{v}_{z}>}$$

(10)

where *v _{zp}* is the sample velocity along the length of the annulus and <

$$<c>=\frac{2\pi \underset{{r}_{o}}{\overset{{r}_{i}}{\int}}r\phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}dr}{2\pi \underset{{r}_{o}}{\overset{{r}_{i}}{\int}}r\phantom{\rule{0.16667em}{0ex}}dr}=\frac{\underset{1}{\overset{{\rho}_{i}}{\int}}\rho \phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}d\rho}{\underset{1}{\overset{{\rho}_{i}}{\int}}\rho \phantom{\rule{0.16667em}{0ex}}d\rho}$$

(11)

Solving the integral in the denominator and changing the variable of integration in the numerator to *ξ*, results in

$$<c>=\frac{2\underset{0}{\overset{1}{\int}}(1-\xi (1-{\rho}_{i}))\phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}d\xi}{(1+{\rho}_{i})}$$

(12)

The retention ratio for the angular component to flow is given by

$${R}_{\omega}=\frac{{\omega}_{p}}{<\omega >}=\frac{<c\omega >}{<c><\omega >}$$

(13)

where *ω _{p}* is the angular component to sample velocity and <

Analytical solutions for the longitudinal *v _{z}*, angular

$$\frac{\partial p}{\partial z}=\eta \left(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial {v}_{z}}{\partial r}\right)\right)$$

(14)

in which *p/z* is the pressure drop in the *z*-direction, and *η* is the fluid viscosity. Double integration, and imposing the no-slip boundary conditions at the walls, results in

$${v}_{z}=\frac{\mathrm{\Delta}p{{r}_{o}}^{2}}{4\eta L}\left(1-{\rho}^{2}+\frac{({{\rho}_{i}}^{2}-1)}{ln{\rho}_{i}}ln\rho \right)$$

(15)

where Δ*p* is the pressure drop along the annulus and *L* is the length of the annulus. Determination of <*v _{z}*> by integration of

$$<{v}_{z}>=\frac{\mathrm{\Delta}p{{r}_{o}}^{2}}{8\eta L}\left(1+{{\rho}_{i}}^{2}-\frac{({{\rho}_{i}}^{2}-1)}{ln{\rho}_{i}}\right)$$

(16)

Making the appropriate substitution into eq (15) gives the final result

$${v}_{z}=\frac{2<{v}_{z}>}{{A}_{1}}\left(1-{\rho}^{2}+{A}_{2}ln\rho \right)$$

(17)

in which

$${A}_{1}=1+{{\rho}_{i}}^{2}-\frac{({{\rho}_{i}}^{2}-1)}{ln{\rho}_{i}};\phantom{\rule{0.38889em}{0ex}}{A}_{2}=\frac{({{\rho}_{i}}^{2}-1)}{ln{\rho}_{i}}$$

(18)

This result has been presented earlier by us [6–9] and by Davis and Giddings [10]. For purely azimuthal, steady fluid flow, the relevant Navier-Stokes equation may be written as

$$\frac{1}{r}\frac{\partial p}{\partial \theta}=\eta \left(\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial (r{v}_{\theta})}{\partial r}\right)\right)$$

(19)

and since *v _{θ}* =

$$\frac{1}{r}\frac{\partial p}{\partial \theta}=\eta \left(\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial ({r}^{2}\omega )}{\partial r}\right)\right)$$

(20)

Following the same approach as for *v _{z}*, we obtain

$$\omega =\frac{2<\omega >}{{B}_{1}}\left(1-\frac{1}{{\rho}^{2}}+{B}_{2}ln\rho \right)$$

(21)

in which

$${B}_{1}=\frac{(4-{{\rho}_{i}}^{2})ln{\rho}_{i}}{(1-{{\rho}_{i}}^{2})};\phantom{\rule{0.38889em}{0ex}}{B}_{2}=\frac{(1-{{\rho}_{i}}^{2})}{{{\rho}_{i}}^{2}ln{\rho}_{i}}$$

(22)

The azimuthal velocity is obtained in the same manner from the starting point of eq (19):

$${v}_{\theta}=\frac{2<{v}_{\theta}>}{{C}_{1}}\rho \left(1-\frac{1}{{\rho}^{2}}+{C}_{2}ln\rho \right)$$

(23)

in which

$${C}_{1}=\frac{4({{\rho}_{i}}^{3}-1)}{9{{\rho}_{i}}^{2}ln{\rho}_{i}}-\frac{8}{3(1+{\rho}_{i})};\phantom{\rule{0.38889em}{0ex}}{C}_{2}={B}_{2}=\frac{(1-{{\rho}_{i}}^{2})}{{{\rho}_{i}}^{2}ln{\rho}_{i}}$$

(24)

This result may be shown to be equivalent to that derived by Davis [11]. Figure 1 shows a comparison of the three normalized velocity profiles *v _{z}*/<

Comparison of the normalized velocity profiles *v*_{z}/<*v*_{z}>, *v*_{θ}/<*v*_{θ}>, and *ω*/<*ω*> for the annular channel with the ratio *ρ*_{i} of inner to outer radii of 0.5, and *v*/< **...**

The forms of eqs (17) and (21) preclude the derivation of analytical solutions for *R _{z}* and

$$v=6<v>\left((1+\upsilon )\xi -(1+3\upsilon ){\xi}^{2}+2\upsilon {\xi}^{3}\right)$$

(25)

In this equation, *υ* is an adjustable parameter that allows for deviation of the shear rate at the accumulation wall from that of the isoviscous, parabolic profile. The equation also has the required property of predicting zero velocity at each of the walls. The region close to the accumulation wall is important for predicting particle elution and the parameter is adjusted so that 6<*v*>(1+*υ*)/*w* is a good fit to the true shear rate at the wall. This approach may be adapted to the velocity profiles for an annular channel. In this case, a second parameter *μ* must be included that is a function of the ratio *ρ _{i}* of the inner to the outer radii. The cubic equation has the general form:

$$v=6<v>\left((1+\upsilon )\xi -(1+\mu \upsilon ){\xi}^{2}+(\mu -1)\upsilon {\xi}^{3}\right)$$

(26)

This has the same adjustable parameter *υ* for the shear rate at the accumulation wall. The equation also retains the required property of giving zero velocity at each of the walls. The value for *μ* is obtained from the required identity (recalling eq (12)):

$$<v>=\frac{2\underset{0}{\overset{1}{\int}}(1-\xi (1-{\rho}_{i}))v\phantom{\rule{0.16667em}{0ex}}d\xi}{(1+{\rho}_{i})}$$

(27)

Solution of this equation reveals that, for all *υ* ≠ 0, *μ* is given by

$$\mu =\frac{7+8{\rho}_{i}}{2+3{\rho}_{i}}$$

(28)

The parallel-plate system corresponds to *ρ _{i}* = 1 in which case

We can obtain cubic approximations to each of the velocity profiles in the annulus as follows. For the longitudinal velocity profile, we have

$${v}_{z}=6<{v}_{z}>\left((1+{\upsilon}_{z})\xi -(1+\mu {\upsilon}_{z}){\xi}^{2}+(\mu -1){\upsilon}_{z}{\xi}^{3}\right)$$

(29)

The parameter *υ _{z}* must be adjusted to equate 6<

$${\upsilon}_{z}=\frac{({\rho}_{i}-1)({A}_{2}-2)}{3{A}_{1}}-1$$

(30)

The same exercise may be carried out for the cubic approximations for angular and azimuthal velocity profiles. These results are given by

$$\omega =6<\omega >\left((1+{\upsilon}_{\omega})\xi -(1+\mu {\upsilon}_{\omega}){\xi}^{2}+(\mu -1){\upsilon}_{\omega}{\xi}^{3}\right)$$

(31)

in which

$${\upsilon}_{\omega}=\frac{({\rho}_{i}-1)({B}_{2}+2)}{3{B}_{1}}-1$$

(32)

and

$${v}_{\theta}=6<{v}_{\theta}>\left((1+{\upsilon}_{\theta})\xi -(1+\mu {\upsilon}_{\theta}){\xi}^{2}+(\mu -1){\upsilon}_{\theta}{\xi}^{3}\right)$$

(33)

in which

$${\upsilon}_{\theta}=\frac{({\rho}_{i}-1)({C}_{2}+2)}{3{C}_{1}}-1$$

(34)

Figures 2a, b, and c show the plots of the first differentials of the normalized profiles with respect to *ξ* for the true profiles and the cubic approximations to the profiles. Also shown are the % errors of the first differentials of the cubic approximations compared to thoseof the true profiles. A ratio *ρ _{i}* of 0.5 was chosen, as for Figure 1, in order to show significant deviation of the approximate equations from the true values. It may be seen that the first derivative of the cubic approximation for angular velocity is significantly worse than those for longitudinal and azimuthal velocities. This is simply because the angular velocity profile deviates further from parabolic than the other two, as shown in Figure 1.

The first derivative of the normalized fluid velocity profiles with respect to *ξ* in the region of the accumulation (outer) wall. The approximate curves are based on the respective cubic approximations to velocity profiles. The curves of % errors **...**

We are now in a position to derive approximate expressions for the longitudinal and angular retention ratios. For the longitudinal retention ratio

$${R}_{z}=\frac{\underset{1}{\overset{{\rho}_{i}}{\int}}\rho \phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}{v}_{z}\phantom{\rule{0.16667em}{0ex}}d\rho}{<{v}_{z}>\underset{1}{\overset{{\rho}_{i}}{\int}}\rho \phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}d\rho}=\frac{\underset{0}{\overset{1}{\int}}(1-\xi (1-{\rho}_{i}))c\phantom{\rule{0.16667em}{0ex}}{v}_{z}\phantom{\rule{0.16667em}{0ex}}d\xi}{<{v}_{z}>\underset{0}{\overset{1}{\int}}(1-\xi (1-{\rho}_{i}))c\phantom{\rule{0.16667em}{0ex}}d\xi}$$

(35)

Substitution for *ρ _{i}* using a rearrangement of eq (28), and for

$$\begin{array}{l}{R}_{z}=6\lambda [(3\mu -8)(1+{\upsilon}_{z})+2\lambda (23-8\mu +3(5-(\mu -1)\mu ){\upsilon}_{z})\\ +6{\lambda}^{2}(5(\mu -3)+2(4+(4\mu -13)\mu ){\upsilon}_{z})-120{\lambda}^{3}(\mu -3)(\mu -1){\upsilon}_{z}\\ +exp(-1/\lambda )\{(7-2\mu )(1+(2-\mu ){\upsilon}_{z})+2\lambda (22-7\mu +3(17+3(\mu -5)\mu ){\upsilon}_{z})\\ +6{\lambda}^{2}(5(3-\mu )+2(26-3(9-2\mu )\mu ){\upsilon}_{z})+120{\lambda}^{3}(\mu -3)(\mu -1){\upsilon}_{z}\}]/\\ [3\mu -8-5\lambda (\mu -3)-exp(-1/\lambda )(7-2\mu -5\lambda (\mu -3))]\end{array}$$

(36)

For material that is acceptably well retained where *λ* is less than about 0.15, the terms in exp(−1/*λ*) are insignificant, and the expression reduces to

$$\begin{array}{l}{R}_{z}=6\lambda [(3\mu -8)(1+{\upsilon}_{z})+2\lambda (23-8\mu +3(5-(\mu -1)\mu ){\upsilon}_{z})\\ +6{\lambda}^{2}(5(\mu -3)+2(4+(4\mu -13)\mu ){\upsilon}_{z})-120{\lambda}^{3}(\mu -3)(\mu -1){\upsilon}_{z}]/\\ [3\mu -8-5\lambda (\mu -3)]\end{array}$$

(37)

Figure 3a shows the comparison of the true longitudinal retention ratio calculated by numerical integrations involving the true velocity profile given by eq (17) (full line) with that given by eq (36) (short-dashed line). A ratio of inner to outer channel radii *ρ _{i}* of 0.5 was selected to show a visible error. The percentage difference is shown by the longer-dashed line and this refers to the right hand axis, and is seen to be less than 0.6% in the range of

As a check on consistency, it may be shown that if *μ* is set to 3 in eq (36), the equation does indeed reduce to the expected parallel-plate solution with first order correction for deviation of the velocity profile from parabolic, given by [12,13]

$$R=6\lambda \upsilon (1-{R}_{2})+{R}_{2}$$

(38)

in which *R*_{2} is the retention ratio for an ideal parabolic velocity profile:

$${R}_{2}=6\lambda (coth(1/2\lambda )-2\lambda )$$

(39)

Similarly, when *μ* in eq (37) is set to 3, the equation reduces to the form given by eq (38), but with *R*_{2} given by the approximate parallel-plate solution for retention ratio, accurate to better than 0.4% for *λ* < 0.15:

$${R}_{2}=6\lambda (1-2\lambda )$$

(40)

In the case of the angular retention ratio, we take a similar initial approach:

$${R}_{\omega}=\frac{<c\omega >}{<c><\omega >}=\frac{\underset{1}{\overset{{\rho}_{i}}{\int}}\rho \phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}\omega \phantom{\rule{0.16667em}{0ex}}d\rho}{<\omega >\underset{1}{\overset{{\rho}_{i}}{\int}}\rho \phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}d\rho}$$

(41)

Substitution using eqs (6), (28), and (33), yields the following solution:

$$\begin{array}{l}{R}_{\omega}=6\lambda [(3\mu -8)(1+{\upsilon}_{\omega})+2\lambda (23-8\mu +3(5-(\mu -1)\mu ){\upsilon}_{\omega})\\ +6{\lambda}^{2}(5(\mu -3)+2(4+(4\mu -13)\mu ){\upsilon}_{\omega})-120{\lambda}^{3}(\mu -3)(\mu -1){\upsilon}_{\omega}\\ +exp(-1/\lambda )\{(7-2\mu )(1+(2-\mu ){\upsilon}_{\omega})+2\lambda (22-7\mu +3(17+3(\mu -5)\mu ){\upsilon}_{\omega})\\ +6{\lambda}^{2}(5(3-\mu )+2(26-3(9-2\mu )\mu ){\upsilon}_{\omega})+120{\lambda}^{3}(\mu -3)(\mu -1){\upsilon}_{\omega}\}]/\\ [3\mu -8-5\lambda (\mu -3)-exp(-1/\lambda )(7-2\mu -5\lambda (\mu -3))]\end{array}$$

(42)

This has exactly the same form as eq (36), but incorporates the parameter *υ _{ω}* rather than

$${v}_{\theta}=r\omega ={r}_{o}\rho \omega $$

(43)

and we can advantageously substitute for *ρω* in eq (41). It is advantageous because the cubic approximation for *v _{θ}* was found to be better than that for

$$\frac{1}{<\omega >}=\frac{{C}_{1}}{{B}_{1}}\frac{{r}_{o}}{<{v}_{\theta}>}$$

(44)

Making these two substitutions into eq (41), changing the variable of integration to *ξ*, and substituting for *ρ _{i}* using a rearrangement of eq (28) results in

$${R}_{\omega}=\frac{{C}_{1}}{{B}_{1}<{v}_{\theta}>}\frac{\underset{0}{\overset{1}{\int}}c\phantom{\rule{0.16667em}{0ex}}{v}_{\theta}\phantom{\rule{0.16667em}{0ex}}d\xi}{\underset{0}{\overset{1}{\int}}(1-5\xi (\mu -3)/(3\mu -8))\phantom{\rule{0.16667em}{0ex}}c\phantom{\rule{0.16667em}{0ex}}d\xi}$$

(45)

and on substituting eq (6) for *c* and eq (33) for *v _{θ}* gives the solution

$$\begin{array}{l}{R}_{\omega}=\frac{6{C}_{1}\lambda (3\mu -8)}{{B}_{1}}[1+{\upsilon}_{\theta}-2\lambda (1+\mu {\upsilon}_{\theta})+6{\lambda}^{2}(\mu -1){\upsilon}_{\theta}\\ +exp(-1/\lambda )\left\{1+(2-\mu ){\upsilon}_{\theta}+2\lambda (1+(3-2\mu ){\upsilon}_{\theta})-6{\lambda}^{2}(\mu -1){\upsilon}_{\theta})\right\}]/\\ [3\mu -8-5\lambda (\mu -3)-exp(-1/\lambda )\{7-2\mu -5\lambda (\mu -3)\left\}\right]\end{array}$$

(46)

Again, for relatively well retained material where *λ* is less than about 0.15, this reduces to

$${R}_{\omega}=\frac{6{C}_{1}\lambda (3\mu -8)}{{B}_{1}}\frac{\left[1+{\upsilon}_{\theta}-2\lambda (1+\mu {\upsilon}_{\theta})+6{\lambda}^{2}(\mu -1){\upsilon}_{\theta}\right]}{[3\mu -8-5\lambda (\mu -3)]}$$

(47)

Figures 3b and c show the comparison of the results based on eqs (42) and (46), respectively, with the true angular retention ratio based on numerical integrations involving the true angular velocity profile given by eq (21). As expected, the result given by eq (46), obtained by consideration of the better cubic approximation to the azimuthal velocity profile, is a better match to the true retention ratio. The result given by eq (42) deviates from the true retention ratio by over 5% at *λ* around 0.1, whereas eq (46) deviates by only about 1.2%.

The eqs (36) and (46) have been shown to be good approximations for *R _{z}* and

This work was supported by grants CTS-0125657 from the National Science Foundation and R01 CA62349 from the National Institutes of Health.

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