Slow Steady Viscous Flow of
Newtonian Fluids in Parallel-Disk
Viscometer With Wall Slip
Y. Leong Yeow1
Department of Chemical
and Biomolecular Engineering,
The University of Melbourne,
Victoria, Australia 3010
e-mail: yly@unimelb.edu.au
Yee-Kwong Leong
School of Mechanical Engineering,
The University of Western Australia,
Crawley, Western Australia, Australia 6009
Ash Khan
Department of Civil
and Chemical Engineering,
RMIT University,
Victoria, Australia 3000
The parallel-disk viscometer is a widely used instrument for measuring the rheological
properties of Newtonian and non-Newtonian fluids. The torque-rotational speed data
from the viscometer are converted into viscosity and other rheological properties of the
fluid under test. The classical no-slip boundary condition is usually assumed at the
disk-fluid interface. This leads to a simple azimuthal flow in the disk gap with the azimuthal velocity linearly varying in the radial and normal directions of the disk surfaces.
For some complex fluids, the no-slip boundary condition may not be valid. The present
investigation considers the flow field when the fluid under test exhibits wall slip. The
equation for slow steady azimuthal flow of Newtonian fluids in parallel-disk viscometer in
the presence of wall slip is solved by the method of separation of variables. Both linear
and nonlinear slip functions are considered. The solution takes the form of a Bessel
series. It shows that, in general, as a result of wall slip the azimuthal velocity no longer
linearly varies in the radial direction. However, under conditions pertinent to paralleldisk viscometry, it approximately remains linear in the normal direction. The implications
of these observations on the processing of parallel-disk viscometry data are discussed.
They indicate that the method of Yoshimura and Prud’homme (1988, “Wall Slip Corrections for Couette and Parallel-Disk Viscometers,” J. Rheol., 32(1), pp. 53–67) for the
determination of the wall slip function remains valid but the simple and popular procedure for converting the measured torque into rim shear stress is likely to incur significant
error as a result of the nonlinearity in the radial direction. 关DOI: 10.1115/1.2910901兴
Keywords: parallel-disk viscometer, wall slip, slow viscous flow, Navier slip law, Bessel
series
1
Introduction
The parallel-disk viscometer is employed by rheologists to
measure the shear properties of a wide range of fluids. In a typical
steady-shear measurement, the gap between the two parallel disks
is filled with the fluid under test. The upper disk is rotated at a
steady angular speed while the lower disk is held stationary.
The torque ⌫ is recorded for a series of rotational speeds. These
⌫- data are then converted into material properties of the fluid.
The no-slip boundary condition is assumed at the disk-fluid
boundaries. Under the normal conditions of parallel-disk viscometry, the inertia terms in the equation of motion are small and can
be ignored. It is also observed that the stress-free fluid surface at
the rim of the disks essentially remains flat in the vertical direction. As a consequence, the flow field inside the gap is approximately unidirectional and the cylindrical polar velocity components in the r and z directions can be ignored. The azimuthal
component v also takes on a particularly simple form. The v that
satisfies the equation of motion and the no-slip boundary condition is
v共r,z兲 =
冉 冊
1 z
+ r,
2 h
0 艋 r 艋 R and − h/2 艋 z 艋 h/2 共1兲
where R is the disk radius and h 共ⰆR兲 is the disk gap. In Eq. 共1兲,
z = 0 is the midplane between the disks. This v共r , z兲 is linear in r
and z. At the midplane, it is exactly half the speed of the upper
disk. v共r , z兲 is antisymmetric about the midplane velocity. Equa1
Corresponding author.
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received April 3, 2006; final manuscript
received March 3, 2008; published online May 9, 2008. Review conducted by Bassam A. Younis.
Journal of Applied Mechanics
tion 共1兲 is assumed by many of the commercial software that
accompany the current generation of parallel-disk viscometers for
converting the measured ⌫- data into rheological properties.
When the shear stress at the disk surfaces is sufficiently high,
some fluids may no longer adhere to the disk and begin to exhibit
wall slip. When this happens, the rheological properties derived
from the ⌫- data becomes gap dependent and are no longer
genuine properties of the fluid under test. This gap dependence is,
in fact, used as an indicator of wall slip 关1兴. In fluids with suspended particles or droplets, shear-induced segregation may result
in the formation of a thin layer of low viscosity fluid next to the
disk surfaces. This low viscosity layer then acts as a lubricating
film resulting in high velocity gradient next to the disk surfaces.
While there is no genuine breakdown of the no-slip boundary
condition, this shear-induced inhomogeneity will again result in
gap-dependent rheological properties. This is often referred to as
apparent wall slip. In either real or apparent slip, an additional
material property function, the slip function S共vslip兲 is often introduced to relate the slip velocity vslip to the wall shear stress w
= S共vslip兲. The slip velocity is defined by vslip ⬅ vdisk − vfluid at the
disk surface. Yoshimura and Prud’homme 关1兴 have developed a
procedure in which two sets of ⌫- data for two different gaps are
used to obtain S共vslip兲. This procedure is now routinely used to
obtain this additional material property function 关2–4兴.
When wall slip is present, it is not immediately clear that the
simple kinematics represented by Eq. 共1兲 is still valid within the
disk gap. There have been a number of investigations into this
issue—both for genuine and for apparent slip 关5,6兴. The general
consensus is that while the analysis of Yoshimura and
Prud’homme 关1兴 is not exact, it is an acceptable approximation—
particularly when the h : R ratio is small. In this investigation, a
series solution for v共r , z兲 is obtained for Newtonian fluids that
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exhibit wall slip. The solution is applicable to linear and nonlinear
S共vslip兲. This series solution is used to investigate the nature of the
flow field within the disk gap and to verify some of the assumptions inherent in the procedure of Yoshimura and Prud’homme
关1兴.
considerations require v共r , z兲 to be antisymmetric.
The admissible values of are determined by Eq. 共4兲. In terms
of J1共r兲, this takes the form
2
From the relationships between Bessel functions of different orders, this boundary condition reduces to 关8兴
Equation of Motion
For a Newtonian fluid under the normal conditions of paralleldisk viscometery, the azimuthal equation of motion reduces to 关7兴
2v 1 v v 2v
− +
+
=0
r2 r r r2 z2
共2兲
The inertia term has been ignored and the flow is assumed to be
axisymmetric so that the azimuthal velocity is a function of r and
z only, i.e., v共r , z兲. It is further assumed that wall slip occurs to
the same extent at the upper and lower disks and consequently
v共r , z兲 remains antisymmetric about the midplane. This more
general form of v共r , z兲 replaces that given by Eq. 共1兲.
At the disk surfaces, the no-slip boundary condition is replaced
by
d共J1共r兲/r兲
=0
dr
J2共R兲 = 0
⬁
v共r,z兲 = r/2 + ␣0rz +
These expressions equate the viscous shear stress w generated by
the steady shear of the fluid next to the disk surfaces with that
generated by slippage. In addition to Eq. 共3兲, v共r , z兲 also has to
satisfy the stress-free condition at the rim of the disk, i.e., r = 0.
In terms of v共r , z兲, this takes the form 关7兴
4
To cope with wall slip, instead of Eq. 共1兲, v共r , z兲 will be assumed to be of the form
共5兲
where ␣0 is a constant and V共r , z兲 is a function to be determined.
The first two terms in this representation of v共r , z兲 automatically
satisfy Eq. 共2兲. Equation 共5兲 reduces to Eq. 共1兲, i.e., ␣0 → / h and
V共r , z兲 → 0, when the no-slip boundary condition applies. In the
solution scheme to be developed, it is further assumed that the r
and z in V共r , z兲 are separable so that V共r , z兲 = F共r兲G共z兲. Substituting Eq. 共5兲 into Eq. 共2兲 leads to
冋
2
册
再 冎
冋
⬁
d2G共z兲
= 2G共z兲
dz2
共7b兲
where −2 is the separation constant and is taken to be real and
positive. Equation 共7a兲 is a first order Bessel equation and its
solution is J1共r兲—Bessel function of the first kind of order 1.
The other linearly independent solution is the corresponding
Bessel function of the second kind Y 1共r兲. Since Y 1共r兲 is singular at r = 0, it is discarded. The solution to Eq. 共7b兲 is sinh共r兲. The
second solution is cosh共r兲. This second solution also has to be
discarded as it is symmetric about the midplane while physical
i 1
i
i=1
From Eq. 共3兲, w at the upper disk is
冉
⬁
w = S r/2 − ␣0rh/2 −
兺 ␣ J 共 r兲sinh共 h/2兲
i 1
i
i
i=1
册
冊
共11兲
Because of the anti-symmetric nature of Eq. 共9兲, Eq. 共11兲 also
applies at the lower disk. At the upper disk, for a Newtonian fluid
with viscosity , w is also given by
w =
冏
v共r,z兲
z
冏
⬁
= ␣ 0r +
h/2
兺 ␣ J 共 r兲cosh 共 h/2兲
i i 1
i
i
i=1
共12兲
Again this expression also applies at the lower disk. Equating the
two expressions for w gives
⬁
␣ 0r +
兺 ␣ J 共 r兲cosh 共 h/2兲
i i 1
i
i
i=1
冉
= S r/2 − ␣0rh/2 −
共7a兲
兺 ␣ J 共 r兲sinh共h/2兲
共10兲
2
1
d2F共r兲 1 dF共r兲
+ 2 − 2 F共r兲 = 0
2 +
dr
r dr
r
041001-2 / Vol. 75, JULY 2008
共9兲
i
Matching Slip Boundary Condition
vslip共r,h/2兲 = r − r/2 + ␣0rh/2 +
1 d G共z兲
1 d F共r兲 1 dF共r兲 F共r兲
+
= − 2 共6兲
− 2 =−
F共r兲 dr2
r dr
r
G共z兲 dz2
or
i
In terms of Eq. 共9兲, the slip velocity at the upper disk is
Separable Solution
r
r
+ ␣0rz + V共r,z兲 =
+ ␣0rz + F共r兲G共z兲
v共r,z兲 =
2
2
i 1
where ␣0 , ␣1 , ␣2 , ␣3 , . . . are the coefficients yet to be determined.
共4兲
The method of separation of variables will be used to construct a
solution for Eq. 共2兲 subject to Eqs. 共3兲 and 共4兲.
3
兺 ␣ J 共 r兲sinh共 z兲
i=1
共3兲
at r = R
共8b兲
This equation has infinitely many solutions. The first ten of these
are
R = 兵5.13562, 8.41724, 11.6198, 14.796, 17.9598, 21.117,
24.2701, 27.4206, 30.5692, 33.7165其. These were obtained by numerically solving Eq. 共8b兲. The ␣0rz term in Eq. 共5兲 can be regarded as the solution associated with R = 0. This term together
with the J1共ir兲 terms forms an orthogonal basis that can be used
to construct a generalized Fourier series to represent most well
behaved functions 关8兴. The series solution for v共r , z兲 can thus be
written as
w共r,h/2兲 = w共r,− h/2兲 = S共r − v共r,h/2兲兲 = S共v共r,− h/2兲兲
共v/r兲
=0
r
共8a兲
at r = R
⬁
兺 ␣ J 共 r兲sinh共 h/2兲
i 1
i=1
i
i
冊
共13兲
The coefficients ␣0 , ␣1 , ␣2 , ␣3 , . . . are determined so that this condition is met for 0 艋 r 艋 R or approximately satisfied at a large
number of collocation points for r in this range. The values of
␣0 , ␣1 , ␣2 , ␣3. . . and the way they are obtained depend on the form
of S共vslip兲, as will be demonstrated in the next section.
5
Results
5.1 Linear Slip Law. In the boundary condition proposed by
Navier in 1823, the wall shear stress is assumed to be a linear
function of the slip velocity, i.e., w = k1vslip 关9兴. Substituting this
into Eq. 共13兲 leads to
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⬁
␣ 0r +
兺 ␣ J 共 r兲cosh 共 h/2兲
i i 1
i
i
i=1
冋
⬁
= k1 r/2 − ␣0rh/2 −
兺 ␣ J 共 r兲sinh共 h/2兲
i 1
i
i
i=1
册
0艋r艋R
,
共14兲
Linearity of the slip function means that all the coefficients of the
Bessel series are identically zero and the unknown ␣0 is given by
R␣0
=1
冒冉
2 h
+
k 1R R
冊
共15兲
Equation 共15兲 gives the dimensionless parameter R␣0 / in terms
of the dimensionless group / k1R and the gap ratio h / R. / k1R is
a measure of the relative importance of the viscous shear stress
and the slippage shear stress. h / R is the key geometric parameter
of the viscometer. The resulting v共r , z兲 is
冉 冊 冉 冊冉 冊 冒 冉
r
v共r,z兲 1 r
+
=
2 R
R
R
z
R
2 h
+
k 1R R
冊
(a)
共16兲
Equation 共16兲 clearly shows that, in the presence of linear slip, the
velocity remains linear in r and in z. When k1 Ⰷ / h, corresponding to negligible wall slip, Eq. 共16兲 reduces to Eq. 共1兲.
Based on Eq. 共16兲, the wall shear stress is given by
冉 冊冒冋
w
r
=
R
2 h
+
k 1R R
册
共17兲
Thus, for a Newtonian fluid that follows the linear slip law, the
wall shear stress remains a linear function of r as is observed
when the no-slip boundary condition applies. The shear stress
within the disk gap is independent of z—a condition observed for
all fluids when the no-slip boundary condition applies 关1兴.
Typical v共r , z兲 profiles for the linear slip law are shown in Fig.
1. These results are for k1R / = 5 and h / R = 0.15. U, M, and L in
Fig. 1共a兲 refer to the upper disk, the midplane, and the lower disk,
respectively. These curves reveal very significant wall slip. For
example, at the rim of the disks, vslip is as high as 36.4% of the
rim speed of the upper disk. Figure 1共b兲 shows the velocity profiles across the gap at different radial positions. Wall slip has
greatly reduced the slope of these curves. The maximum shear
rate is only 共100− 2 ⫻ 36.4兲 = 27.2% of that given by Eq. 共1兲 when
there is no slip.
The linear relationship between shear stress and r, as described
by Eq. 共17兲, is shown in Fig. 2 by the straight line “ln.” This stress
is independent of z. The line “ns” on the same plot is the corresponding stress for a Newtonian fluid with the same viscosity but
does not exhibit wall slip. Because of wall slip, the stress on ln is
only 27.2% of that on ns. For ease of comparison, the stress
curves for all the S共vslip兲 subsequently investigated are summarized in Fig. 2.
(b)
Fig. 1 Linear slip law. „a… Variation of azimuthal velocity with
radius at the upper disk „U…, midplane „M…, and lower disk „L….
„b… Variation of azimuthal velocity with axial coordinate at different radial positions.
coefficients ␣0 , ␣1 , ␣2 , ␣3 , . . .. Instead, they are determined by
least-squares minimization of the difference between the left hand
side 共LHS兲 and right hand side 共RHS兲 of Eq. 共18兲 at a set of
preselected collocation points. Typically, 150–200 collocation
points uniformly spaced between 0.02艋 r / R 艋 1 are included in
the minimization process. The region 0 艋 r / R 艋 0.02 has been arbitrarily excluded as v共r , z兲 and w there are too small to be
accurately evaluated. In the least-squares minimization process,
the series representation of v共r , z兲 is terminated after N terms,
typically N = 10– 20. N is adjusted so that the average difference
5.2 Square Dependence on Slip Velocity. The linear slip
function considered in Sec. 5.1 does not test the numerical performance of the Bessel series solution. In this example, the linear slip
2
. Substituting this into
law of Navier is generalized to w = k2vslip
Eq. 共13兲 yields
⬁
␣ 0r +
兺 ␣ J 共 r兲cosh 共 h/2兲
i i 1
i
i
i=1
冋
= k2 r/2 − ␣0rh/2 −
⬁
兺
i=1
0艋r艋R
␣iJ1共ir兲sinh共ih/2兲
册
2
,
共18兲
The nonlinear nature of Eq. 共18兲 does not permit the exploitation
of the orthogonal properties of J1共ir兲 to determine the unknown
Journal of Applied Mechanics
Fig. 2 Radial variation of shear stress for no slip „ns…, linear
slip „ln…, square dependence „sq…, square-root dependence „rt…,
and linear slip with critical wall shear stress „cr…. The darker
curves are for the stress at the disk surfaces and the lighter
curves are for the stress at the midplane.
JULY 2008, Vol. 75 / 041001-3
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between the LHS and RHS of Eq. 共18兲 is less than about 1%. As
expected in the neighborhood of r = 0, the difference can be as
large as 10–15%. However, this does not appear to have a great
influence on the behavior of v共r , z兲.
The search of the ␣0 , ␣1 , ␣2 , . . ., ␣N that minimize the leastsquares difference was performed using the commercial software
®
MATHEMATICA 关10兴. The computation starts with a relatively
small N, typically N = 5 – 6, and this is progressively increased
using the previously obtained coefficients as the starting point. To
avoid being trapped by local minima, the minimization computation was repeated with slightly modified starting values of the
coefficients and also by using different minimization strategies
provided by the commercial software. No numerical difficulties
were encountered in this and in the subsequent examples.
2
Results for the slip law w = k2vslip
with k2R2 / = 5 and h / R
= 0.15 are summarized in Figure 3. Figure 3共a兲 shows that, except
at the midplane, v共r , z兲 is clearly no longer linear in r. The three
velocity profiles are also much closer to one another compared to
that in Fig. 1共a兲, indicating increased wall slip. For example, the
velocity of the fluid at the rim of the upper disk is only 56.8% of
the rim speed giving a vslip of 43.2%. With this vslip, the maximum
shear rate attained is only 共100− 2 ⫻ 43.2兲 % = 13.6% of that given
by Eq. 共1兲 when the no-slip boundary condition applies. The velocity profiles in Fig. 3共b兲 show that v共r , z兲 approximately remains linear in z for all r. The slight deviation from linearity can
only be observed by plotting 共v / z兲r against z at different radial
positions, see Fig. 3共c兲. In general, for a fixed r, 共v / z兲r varied
by less than 3.5% for −1 / 2 ⬍ z / h 艋 1 / 2. The two curves marked
“sq” in Fig. 2 show the nonlinear variation of the shear stress with
r as a consequence of the square dependence on vslip. The shear
stress at the disk surfaces w 共darker curve兲 and that at the midplane m 共lighter curve兲 are very close together, reflecting the
small variation of shear rate with z. These curves also show the
very significant reduction in shear stress brought about by wall
slip.
(a)
(b)
5.3 Square-Root Dependence on Slip Velocity. The compu1/2
as an example where
tation in Sec. 5.2 is repeated for w = k1/2vslip
the wall shear stress increases less than linearly with vslip. Substituting this into Eq. 共13兲 resulted in
⬁
␣ 0r +
兺 ␣ J 共 r兲cosh 共 h/2兲
i i 1
i
i
i=1
冋
= k1/2 r/2 − ␣0rzw −
⬁
兺
␣iJ1共ir兲sinh共ih/2兲
i=1
0艋r艋R
册
1/2
(c)
,
共19兲
Least-squares minimization is again used to determine ␣0 , ␣2 , . . .,
␣N. The same set of collocation points in Sec. 5.2 was used. It was
found that, for this slip law, a slightly larger number of terms,
25⬍ N ⬍ 30, of the Bessel series was required to keep the average
difference between the LHS and RHS of Eq. 共19兲 to around 1%.
Plots of the variation of v共r , z兲 with r for this slip law, for
k1/2共R / 兲1/2 / = 5 and h / R = 0.15, are shown in Fig. 4共a兲. v共r , z兲,
for z ⫽ 0, is now even more nonlinear in r. This may explain the
increase in N required to represent v共r , z兲 to the same degree of
accuracy as before. A reduction in wall slip, compared to the
linear slip law, is observed. The maximum vslip is reduced from
36.4% to 29.3% of the rim speed, see Fig. 4共a兲. Consequently, the
maximum shear rate experienced by the Newtonian fluid is now
about 41.4% of that when there is no slip. As in the previous
examples, Fig. 4共b兲 shows that v共r , z兲 approximately remains linear in z. The slight deviation from linearity is shown in the plots
of 共v / z兲r against z in Fig. 4共c兲. The maximum variation, at
r / R = 1, is about 2.0%.
The general lowering of vslip, compared to Secs. 5.1 and 5.2, for
041001-4 / Vol. 75, JULY 2008
Fig. 3 Square dependence wall slip. „a… Variation of azimuthal
velocity with radius at the upper disk „U…, mid plane „M…, and
lower disk „L…. „b… Variation of azimuthal velocity with axial coordinate at different radial positions. „c… Variation of shear rate
with axial coordinate at different radial positions.
the square-root dependent S共vslip兲 means that the shear stress is
correspondingly higher. This can be observed from the stress
curves “rt” in Fig. 2. The stress at the disk surfaces 共darker curve兲
and that for the midplane 共lighter curve兲 are again very close
together confirming that the variation of shear rate with z can be
ignored. The shear stress is again nonlinear in r.
5.4 Fluid With Critical Wall Shear Stress. Materials, such
as polymer melts, do not exhibit wall slip at low wall shear stress.
Wall slip often only becomes noticeable when w has exceeded
some threshold value. To model this observation, the linear slip
law is modified to
vslip = 0
for w 艋 wcrit
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⬁
␣ 0r +
兺 ␣ J 共 r兲cosh 共 h/2兲
i i 1
i=1
冋
i
i
⬁
= wcrit + kY r/2 − ␣0rh/2 −
兺 ␣ J 共 r兲sinh共 h/2兲
i 1
i=1
i
i
册
共21b兲
for w ⬎ wcrit
Because of the antisymmetric nature of v共r , z兲, Eqs. 共21a兲 and
共21b兲 ensure that Eq. 共20兲 is automatically met at the stationary
disk. As in the last two examples, the coefficients ␣0 , ␣1 , ␣2 , . . .,
␣N are determined by least-squares minimization of the difference
between the LHS and RHS of Eqs. 共21a兲 and 共21b兲 over a set of
regularly spaced collocation points. In computing the leastsquares deviation, Eq. 共21a兲 applies for 0 ⬍ r 艋 rcrit and Eq. 共21b兲
for rcrit ⬍ r 艋 R, where rcrit is the radial position for the onset of
wall slip. For a Newtonian fluid with viscosity , it is given by
(a)
rcrit = wcrith/共兲
(b)
(c)
Fig. 4 Square-root dependence wall slip. „a… Variation of azimuthal velocity with radius at the upper disk „U…, midplane „M…,
and lower disk „L…. „b… Variation of azimuthal velocity with axial
coordinate at different radial positions. „c… Variation of shear
rate with axial coordinate at different radial positions.
共22兲
This expression is based on the assumption that the no-slip boundary condition for r 艋 rcrit is sufficient to ensure that the velocity is
linear in z and hence the wall shear stress is given by r / h. If
the velocity profiles given by the Bessel series indicate significant
deviation from linearity, then an iterative procedure will have to
be adopted to evaluate rcrit. In the least-squares minimization process, typically, the span 0 ⬍ r 艋 rcrit is divided into 100–150 collocation points and that for rcrit ⬍ r 艋 R into 200–300 points.
Typical results for a Newtonian fluid with wcrit are shown in
Fig. 5. These results are for wcrit / = 2, kY R / = 5 and h / R
= 0.15. The corresponding rcrit / R = 0.3. Figure 5共a兲 shows the distinct change in the slope of the v共r , z兲 curves at r = rcrit as the fluid
begins to slip at the disk surfaces. As expected, at the lower disk,
v共r , −h / 2兲 = 0 for r 艋 rcrit. The v共r , −h / 2兲 given by the series
solution shows small fluctuations about 0 for r 艋 rcrit but these
fluctuations do not show up in the scales of Fig. 5共a兲. As in the
previous examples, v共r , z兲 again appears to be approximately linear in z, see Fig. 5共b兲. The 共v / z兲r plots in Fig. 5共c兲 reveal the
small deviation from linearity in z and, in particular, they show
that the most significant deviation is in the neighborhood of r
= rcrit. There the maximum difference in 共v / z兲r is around
7.65%. At r = R, the maximum difference is reduced to 2.34% 共in
the opposite direction兲. These were not considered as sufficiently
large to warrant the iterative recalculation of rcrit. The wcrit has
imparted a very distinctive shape on the shear stress versus r
plots. See the “cr” curves for w 共darker curve兲 and m 共lighter
curve兲 in Fig. 2. These curves are highly nonlinear. As expected,
they closely follow the no-slip line ns for r ⬍ rcrit. As wall slip sets
in at r = rcrit, these curves sharply change slope. As in the previous
examples, the close proximity of the w and m curves confirms
that the variation of shear rate with z can be ignored.
Discussion
w = wcrit + kY vslip for w ⬎ wcrit
共20兲
where wcrit is the critical wall shear stress below which the classical no-slip boundary condition is observed. kY plays the same
role as k1 in Sec. 5.1. For this generalized S共vslip兲, the matching of
boundary condition at the rotating disk takes the form
⬁
r/2 + ␣0rh/2 +
兺 ␣ J 共 r兲sinh共 h/2兲 = r
i 1
i
i
for w 艋 wcrit
i=1
共21a兲
Journal of Applied Mechanics
For the S共vslip兲 investigated, the numerical performance of the
Bessel series representation of v共r , z兲 appears to be satisfactory.
The method of separation of variables adopted here can, in principle, be extended to even more general slip behaviors. For example, if the slip behavior of a real fluid can be approximated by
a low order polynomial of the form
2
3
+ 3vslip
w = 1vslip + 2vslip
共23兲
where 1 , 2. . . are known empirical coefficients, then the procedure described above can be applied to obtain the series representation of v共r , z兲.
Similarly, the treatment of S共vslip兲 with a wcrit can be extended
to a more general form of the slip function. However, for such
materials, it is probably more convenient to inverse the relationship between vslip and w and to express it in the form
JULY 2008, Vol. 75 / 041001-5
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gation is nonlinear and the standard method of separation of variables becomes inapplicable. For this class of more general problems, it is more efficient to resort to finite element computation.
This is currently under progress. The Newtonian results reported
here can then be used as a special case in the validation of the
finite element method 共FEM兲 results.
As mentioned above, the method of Yoshimura and
Prud’homme 关1兴 is now widely used in the determination of
S共vslip兲. One of the key assumptions in this method is that the
shear rate in the gap is function of r only and is independent of z.
The shear rate plots in Figs. 3共c兲, 4共c兲, and 5共c兲 show that, for the
S共vslip兲 investigated, the shear rate is not exactly independent of z.
However, as the variation is generally less than 10%, this small
variation can most probably be ignored. Similarly, only small
variations in shear rate were observed for h : R as large as 0.2 共not
shown兲. Since in most parallel-disk viscometry measurements
h : R ⬍ 0.2, one is therefore justified to apply the method of
Yoshimura and Prud’homme to determine wall slip.
For the case of linear wall slip, the shear stress remains linear in
r as is expected of a Newtonian fluid. From the shear stress plots
in Fig. 2, it is clear that, even for a Newtonian fluid, this stress is
no longer linear in r for a general S共vslip兲. This has an important
practical implication on the processing of the ⌫- data of paralleldisk viscometry. The measured torque ⌫ is often converted into
shear stress R at the rim of the disks using the simple expression
(a)
R = 2⌫/共R3兲
共25兲
This is the expression implemented in many of the software that
accompany the current generation of parallel-disk viscometers.
Equation 共25兲 is an approximation and is strictly valid only when
the shear stress is linear in r as is in the case of a Newtonian fluid
that does not exhibit wall slip. The exact expression relating R to
⌫ is 关7兴
(b)
R =
冋
d loge ⌫
⌫
3+
2R3
d loge ␥˙ R
册
共26兲
where ␥˙ R = R / h is the shear rate at the rim of the disks. The main
reason for adopting the approximate equation 共25兲 instead of the
exact equation 共26兲 is that the evaluation of the derivative term on
the RHS of Eq. 共26兲 is difficult and is likely to amplify the noise
in the ⌫- data. For Newtonian fluid with nonlinear wall slip and
certainly for non-Newtonian fluids with or without wall slip, the
use of Eq. 共25兲 is likely to lead to significant error and is therefore
not recommended 关11兴.
(c)
Conclusion
Fig. 5 Linear slip with critical wall shear stress. „a… Variation of
azimuthal velocity with radius at the upper disk „U…, midplane
„M…, and lower disk „L…. „b… Variation of azimuthal velocity with
axial coordinate at different radial positions. „c… Variation of
shear rate with axial coordinate at different radial positions.
vslip = 0
vslip = T共w兲
for w 艋 wcrit
for w ⬎ wcrit
共24兲
where T共w兲 is a general function of the wall shear stress. For this
type of slip function, a boundary velocity matching condition,
similar to the stress matching condition in Eq. 共13兲, can be developed. The main issue here is the numerical performance of this
new matching condition. This has not been investigated.
The present investigation only dealt with Newtonian fluids exhibiting wall slip. Most real fluids that exhibit wall slip are likely
to simultaneously exhibit one or more non-Newtonian behaviors.
It is therefore of practical interest to extend this slip investigation
to non-Newtonian fluids. Unfortunately, the non-Newtonian
equivalent of Eq. 共2兲 that forms the starting point of this investi041001-6 / Vol. 75, JULY 2008
The method of separation of variables gave a reliable series
representation of the azimuthal velocity for Newtonian fluids in
steady parallel-disk flow in the presence of wall slip. The solution
shows that the azimuthal velocity approximately remains linear in
z and hence justifies the use of the procedure of Yoshimura and
Prud’homme for the determination of the wall slip function. As a
consequence of wall slip, the wall shear stress, even for a Newtonian fluid, is no longer linear in r. This is not consistent with the
key assumption of the simple but very popular method for converting the measured torque into rim shear stress.
References
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Couette and Parallel-Disk Viscometers,” J. Rheol., 32共1兲, pp. 53–67.
关2兴 Yilmazer, U., and Kalyon, D. H., 1989, “Slip Effects in Capillary and Parallel
Disk Torsional Flows of Highly Filled Suspensions,” J. Rheol., 33共8兲, pp.
1197–1212.
关3兴 Hartman Kok, P. J. A., Kazrian, S. G., Lawrence, C. J., and Briscoe, B. J.,
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关6兴 Wein, O., 2005, “Viscometric Flow under Apparent Wall Slip in Parallel-Plate
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关7兴 Bird., R. B., Armstrong, R. C., and Hassager, O., 1987, Dynamics of Polymeric Liquids, 2nd ed., Wiley-Interscience, New York, Vol. 1, Chap. 10.
关8兴 Andrews, L. C., 1986, Special Functions for Engineers and Applied Mathematicians, Macmillan, New York, Chap. 6.
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Concentric Cylinder Viscometry” 关J. Rheol. 49, 807–818 共2005兲兴. The Relevance of the Early Days of Viscosity, Slip at the Wall, and Stability in Concentric Cylinder Viscometry,” J. Cryst. Growth, 49共6兲, pp. 1539–1550.
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