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Motion of a sphere oscillating at low Reynolds numbers in a viscoelastic-fluid-filled cylindrical tube
ELSEVIER
J. Non-Newtonian Fluid Mech., 66 (1996) 169-192
Motion of a sphere oscillating at low Reynolds numbers in a viscoelastic-fluid-filled cylindrical tube Renwei Mei*, Jie Xiong, Roger Tran-Son-Tay Department of Aerospace Engineering, Mechanics & Engineering Science, University of Florida, Gainesville, FL 32611-6250, USA Received 18 December 1995; in revised form 15 June 1996
Abstract The motion at low Reynolds numbers of a sphere oscillating with a small amplitude inside a cylindrical tube containing a viscous fluid (Newtonian or viscoelastic) is investigated. The linear Jeffreys model is used to describe the constitutive relation for the viscoelastic fluid. The governing equations are transformed to, and solved in, the frequency domain where the Fourier components for the flow and drag are obtained using a finite difference method. The numerical results for the Newtonian fluid velocity agree well with those given by a high frequency asymptotic solution. The effects of the Stokes number, E, on the unsteady drag, D R -b iO l, are discussed. At small •, the real component of the unsteady drag, D R, is dominated by the quasi-steady term for both Newtonian and viscoelastic fluids. For a Newtonian fluid, the acceleration-dependent terms of D R and D l are proportional to •4 and •2 respectively. Whereas D R increases monotonically with • for a Newtonian fluid, it reaches a minimum tot a Jeffreys fluid when • ~ O(1). The difference observed is caused by the shear-thinning property of the viscoelastic fluid. At large • , D R becomes proportional to •, and the imaginary component of the unsteady drag, D~, for both Newtonian and viscoelastic fluids, is dominated by the added-mass force. However, as opposed to a Newtonian fluid, D, changes sign near • ~ O(l) for a viscoelastic fluid due to shear thinning. Finally, a method for determining the rheological properties of a general viscoelastic fluid from measurements of the unsteady drag in a ball rheometer is presented.
Keywords: Oscillating sphere; Reynolds numbers; Viscoelastic fluid
1. Introduction T h e m o t i o n o f a s p h e r e falling a l o n g t h e axis o f a c y l i n d r i c a l t u b e c o n t a i n i n g a N e w t o n i a n o r v i s c o e l a s t i c fluid h a s a t t r a c t e d m u c h a t t e n t i o n in t h e last 20 y e a r s w i t h a c o n s i d e r a b l e i n c r e a s e * Corresponding author. 0377-0257/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0377-0257(96)01479-6
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in the last few years [1-9] because understanding the motion of a ball oscillating inside a cylindrical tube is critical to researchers working with a ball rheometer. The main advantage of the ball rheometer over other devices, such as cone-plate and couette rheometers, is that it requires a very small sample volume. For example, the microrheometer used by Tran-Son-Tay et al. [9] requires only 20 lal of sample, which would be impractical for other standard rheometers. This advantage is significant when one deals with fluids that can only be obtained in small quantities, e.g. mucus, sickle cell blood, or synovial fluid. Even though these studies [1-9] are of significant importance for understanding and analyzing falling ball data, our knowledge of the oscillating ball rheometer is far from being complete. Because of the importance and increased use of ball microrheometers for studying certain viscoelastic fluids, there is a need for understanding the motion of a sphere oscillating along the axis of a cylindrical tube. In comparison to the steady-state probelm, the number of studies on the motion of a sphere oscillating inside a closed container is extremely small. The effects of a closed nearby boundary on the drag of an oscillating sphere have been reported, but only for the case of a spherical container [9,10]. Rheological data using the ball microheometer exist [9], but they are of limited value since the effects of the tube wall on the unsteady drag of the fluid acting on an oscillating sphere are not known, and there is no published work on these effects. The simplest case of a particle moving at low velocity in a Newtonian fluid was solved first in 1851 by Stokes [11], but the problem of determining the near wall effect on the drag experimenced by the particle was solved much later. Analytical solutions about steady Stokes flow over a sphere in a cylinder have been reported by several investigators (e.g. Happel and Byrne [12], Haberman and Sayre [13]). Fayon and Happel [14] experimentally investigated the effect of a cylindrical boundary on a spherical particle suspended in a moving viscous liquid. Unsteady flows over a sphere in an unbounded region at finite Reynolds number have been studied by Mei and co-workers [15-18]. For the case of a sphere of radius a oscillating at a frequency ~o with a velocity amplitude 1-7oin an unbounded fluid of kinematic viscosity v, it was found that: (a) at a large Stokes number, E(~oa2/2v)1/2, the numerical result for the unsteady drag agrees well with the high frequency asymptotic solution; (b) at E ~> 1, Stokes solution is valid for finite Re; and (c) at small Strouhal number, St = coa/Uo ,~ 1, the behavior of the imaginary component of the unsteady drag is complicated by the logarithmic dependence on St and is different from an earlier result obtained [15] for an unsteady flow over stationary sphere with a small amplitude oscillation in the free-stream velocity. Numerical solutions for slightly non-Newtonian fluids over a sphere in the creeping flow limit are available [5, 6, 19]. Numerical techniques include finite difference methods, finite element methods, and boundary elements methods. The constitutive equations selected for the non-Newtonian fluids in these numerical studies are the commonly used Upper-Convected Maxwell (UCM) and Oldroyd-B fluid models. It was found that there is a limiting Weissenberg number, W e - 2 V/a, above which no steady-state solution exists. In the latter equation, 2 is the fluid relaxation time and V is the velocity of the sphere. The effects of elasticity on the drag have also been discussed. A number of studies combining falling ball experiments and analysis have been reported. Experiments with constant viscosity Boger liquids [4] show that the normalized drag can be very high for the limiting cases ( H ~ ~ and H ~ 1, where H is the tube-to-sphere radius ratio). However, there is still a gap between the available experimental data and reliable numerical simulations for simple viscoelastic models. Numerical results for a viscoelastic fluid past a sphere in the presence of a wall [20] show that the fluid elasticity and shear thinning can reduce the drag
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
171
coefficient on the sphere remarkably, up to 25% in the absence of fluid shear thinning and 40% in the presence of fluid shear thinning. A stagnation flow behind the sphere on the centerline of the cylinder in a shear-thinning viscoelastic fluid has been reported by Bush [1,2]. In this region it was found that elastic properties provide an enhanced decelerating force on the fluid far from the sphere, leading to the well-documented downstream shift in the streamlines relative to the flow of a Newtonian fluid. Tran-Son-Tay et al. [9] found that under certain conditions the effect of inertia on the force acting on a ball oscillating inside a cylindrical tube was well described, at low frequency, by the theory for a ball oscillating inside a spherical container. This finding means that when the imaginary component of the total hydrodyanamic force is negligible, a modified Stokes drag expression can be used to account for the boundary and frequency effects in the determination of the non-Newtonian viscosity with a ball rheometer. However, at higher frequencies of oscillation, the elastic componenent, G', for the non-Newtonian red blood cell suspensions is negative when the theory for a sphere oscillating without inertia inside a spherical container is used. This result is physically impossible and means that the effect of inertia in a cylindrical tube is more important at higher oscillatory frequencies than in a spherical container. This discrepancy stresses the need for a theory dealing with a cylindrical tube in order to correctly calculate the values for G' and r/' from the experimental data. The present paper focuses on the numerical simulation of the unsteady drag on a sphere oscillating inside a cylindrical tube containing a Newtonian or linear viscoelastic fluid. The motivation for this work comes from the need to understand the effect of frequency on the unsteady drag in a cylindrical tube, and the difference in behavior between Newtonian and viscoelastic fluids. This acquired knowledge will then lead to the development of a rigorous method for analyzing rheological measurements obtained with a ball rheometer over a large range of ~o, or more appropriately, E. The body of the paper is organized as follows. Section 2 formulates the problem in terms of a general Jeffreys model and subsequently presents the simplification of the governing equations, the Fourier mode representation of the solutions for the flow field and drag, and a conformal mapping for constructing an orthogonal body-fitted grid system. Section 3 presents an asymptotic analysis for the flow field at high frequency. Section 4 presents the numerical results for the unsteady flow in the tube and unsteady drag on the sphere over various frequency ranges for both Newtonian and viscoelastic fluids. The computed total unsteady drag is analyzed by decomposing it into a quasi-steady drag at zero frequency, an added-mass force which is purely inertial and dominates at high frequency, and a history force which exhibits quite complex dependence on the Stokes number for a viscoelastic fluid. It is important to note that, although most of the results presented in this paper are obtained with a linear Jeffreys model, the computational analysis developed in this study is applicable to a class of fluids more general than that which the Jeffreys model can describe. Thus, a method for determining the linear rheological properties of an unknown viscoelastic fluid from measurements of the unsteady drag is presented at the end of Section 4. Section 5 summarizes the main results.
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2. P r o b l e m f o r m u l a t i o n
The motion of a rigid sphere of radius a oscillating with a velocity Uo e x p ( - i m t ) inside a cylindrical tube of radius b = Ha containing a Newtonian or viscoelastic fluid of density p is considered. In the above complex expression, U0 and oJ are the velocity amplitude and angular frequency of the oscillation respectively, and t is time. The problem is equivalent to solving the motion of a flow oscillating with velocity u = Uo exp(-icot) over a stationary sphere with the cylindrical tube moving concurrently with the uniform flow (Fig. 1), provided that an inertial force (4/3)prca3(du/dt) is subtracted from the total force to account for the acceleration of the fluid flow. The governing equations for an incompressible, laminar, viscoelastic flow are
~ v + v. v v = _ 1 Vp + ! v-~
(1)
v. v = o
(2)
St
p
p
and
where V is the velocity vector, p is pressure, and ~ is the deviatoric stress tensor. The generalized Jeffreys model is used to describe the constituitive equation of the viscoelastic fluid [21]
"t"+ Al ~-t (1") = 2r]l D -I--~2 ~-7 (O)
(3)
D = l ( v V + V VT)
(4)
2~ - rh + a r/2
~
and
/72 22 = -~
(5)
~
b/a=H
j
__ z
V/////////////////////////////////////////////A U0 exp(-imt) Fig. 1. Sphere oscillating inside a cylindricaltube.
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 ¢1996) 169-192 where in the ,(2 are tensor
173
G is the elastic m o d u l u s of the spring, r/1 and r/2 are the viscosities of the two dashpots Jeffreys model in which q2 is the viscous element in parallel with the spring, and 21 and relaxation and retardation times respectively. The invariant derivative ~/~'~t for any M is defined as
~-~t (M) = 5 (M) + V " V M + M " 1 2 - 12 . M -
g(DM + MD)
(6)
w h e r e / 2 is the anti-symmetric part of the deformation tensor V V and the values g = - 1, 0 + 1 correspond to the lower, corotational, and upper models, respectively. Oldroyd's models A and B are special cases corresponding to g = 1 and - 1 respectively. When 22 = 0, Eq. (3) reduces to the Maxwell model. Newtonian fluid corresponds to 2~ = 2 2. The displacement amplitude of the ball, Zo, is very small c o m p a r e d to the radius of the ball and the dimensionless displacement, 6, is typically less than 0.1, i.e. c5 = Zo/a ~ 1. The following quantities 8~(=/~a~o), 1/a, 1~co, qlUo/a, and Uo/a are used to normalize V, V, t, r a n d D For very small c5, it can be shown that O/Ot(r)~, O(r/jco2/~) and 8 / 8 t ( D ) ~ O(to 2 c5) are the respective leading order terms for ~ / ~ t ( t ) and 9 / ~ t ( O ) . Therefore, ~ / ~ t reduces Io 8/St and a linear Jeffreys model is obtained: ~ + 2~ c~t = 2q, D + 2 2 - ~
(7)
Further, using a2p/2rl~ and pU2o to normalize 2 and p respectively, the dimensionless form of Eq. (1) becomes 8V St--~+ V'VV=
2__V -VP+Re "~
(8)
where Re = Uo2a/v, and St = e)a/Uo are the Reynolds and Strouhal numbers respectively, and v = rl~/p is the fluid kinematic viscosity. For the present problem both ~ and Re are m u c h less than one, hence the convective term V • V V is negligible, and Eq. (8) becomes linear. Since only low Reynolds n u m b e r flow will be considered, the Stokes n u m b e r E = (~oa2/2v) 1/2 = (St Re/4) 1/2
(9)
will be used for interpreting the results of the unsteady flow and drag in lieu of Re and St. Since the fluid m o t i o n is induced by a single harmonic oscillation on the b o u n d a r y and Eqs. (7) and (8) are linear, a time-domain variable f ( z , r, t) can be expressed using Fourier modes representation as f ( z , r, t) = ½If(z, r)e -~' + f * ( z , r)e +~'] = ~ R ( -7, r) cos t + ~ ( z , r) sin t
(10)
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
174
where an asterisk denotes the complex conjugate, a swung dash denotes the Fourier component, and the subscripts " R " and ' T ' denote the real and imaginary parts of a complex variable. It is noted that for a nonlinear problem, i.e. due to nonlinear convection or a nonlinear constitutive relation, more Fourier modes need to be retained in Eq. (10). The dimensionless form of the constitutive equation in the frequency domain now becomes (1 - - i E 2 2 2 ) = 2 (1 - iE22,)/~ = 2 / ? - ' / )
(1 1)
where (1
-- iE221)
(1
- (1 - i E % )
+ 2122e4) + i(22-- 21)e-z
-
(1 + E"A )
(12)
is the inverse of the dimensionless complex viscosity which reduces to one for a Newtonian fluid. The linearized m o m e n t u m equation, Eq. (8), in the frequency domain is Re 2 Vfi + 2fl-'V •/9
-2i~z P-
(13)
In cylindrical coordinates (r, ~b, z), the velocity of an axisymmetric flow is given by V = uer + we. and the vorticity ( by or
(=w:-ur
~'=W---Ur
(14)
where the subscripts denote partial derivates with respect to that variable. Using the stream function ff for an axisymmetric flow, the velocity components can be expressed as = 1 fir
and
r
a = __1 ~,,-__
(15)
y
The transport equation for ( ' c a n be easily obtained from Eq. (13) (
-
1:)
(16)
and the stream function ~ is obtained by solving the Poisson equation ¢7_-_- +
1 ~=
- r
-r~
(17)
Hereinafter, the swung dash will be dropped from the notation for simplicity. The boundary conditions are specified as follows
¢=½r
and
asz~+~
(18a)
at z = 0
(18b)
= ½r2
on the tube wall r = H = b/a
(18c)
~, = ( = 0
for z _< - 1 on the axis r = 0
(18d)
~/' = ~'n = 0
on the sphere z 2 + r 2 = 1
(180
~b: = (: = 0
(=0
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
175
where n is the outward normal of the surface of the sphere and H = b/a > 1
(19)
is the dimensionless tube radius or the tube-to-sphere radius ratio. To carry out the computation in a body-fitted coordinate system, a conformal mapping from (z, r) to (xl, x2) is first applied. Denoting Z = z + ir
and
(20)
X = Xl + ix2
the following transformation [22] X=~
W
( Z + ~ I T + ~2T 3 + ~sT 5 +" • ")
(21)
with T-- ~-~ coth
Z
(22)
maps the domain between the sphere (]Z I = 1) and the tube (Z = i l l ) into a rectangular region ( - ,~ < Xl < ~ , 0 < x2 < W / 2 ) , by choosing proper values of c~n (n = l, 2 .... ). The values of c~, and W are chosen such that the sphere IZI = 1 is mapped into a segment of line defined by - 2 _ < x t < 2 , x2= 0. To obtain an adequate spatial resolution in the r/ direction at high frequency, the following transformation Xz=
+ C2 tan -1 tan
( 2 r / - 1)
0
(23)
is applied to cluster grids near the sphere (r/=0) and the tube wall (r/= 1). Since the transformation from (xl, xz) to (z, r) is singular near xl = +_2, it is important to allocate more grids near Xl = +_2 in the x~ direction. This is accomplished through the following tranformation xt
-2-(x1_~-2) =
-
1-Cltan-
+
-
-
tan 2
for for
0<~<~10 --
-<- O . 5
(24)
where xt~ is the half-length of the strip in the (x~, x~) domain. Setting x~ = 0 (~ = 0.5), A and B are related as follows: l A-0.5-
( 2 ~lo 0.5--~1o
B)
(25)
with B < 4 / ( I - 2~o) to ensure that Xl > - 2 for ~0-< ~ <0.5. A smaller value for B corresponds to more grids near x~ = +_2. Once B is given, C~ can be obtained numerically be requiring x~ (~) to have a continuous first order derivative at xL = +_2 (or ~ = ~o). The transformed governing equations are
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R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
l[~ (~o<) , r~,,"~)
h,h2LO~\h,
o,,",,,o+~l__,( ~,--~ ~0+ r/r
+-~q t,h2 8rl)]
o~, ~) ,
~ ~o<~l+;~ r
~0)= -- r~
-~q
(27)
where hi
=
(z~+r~)l/2
and
hz=(z 2+r"2 )1/2
(28)
are the Lam6 coefficients. Finally, a finite-difference technique [23] is applied to the above equations. Spatial derivatives are respresented using central difference. The wall vorticities are evaluated as follows. On the surface of the sphere ( / = 1), the velocity is zero so that both real and imaginary components of ( are given by ~'jj-- 2(Aq)2
l,i-- ¢'j+2,,)
,
for j = 1
(29)
Since (UR, U0 = (1, 0) and (¢'R, ¢q) = ~ H 2, 0) along the tube wall ( / = Nr), one obtains
l
(~-~2)j[--7+Rj, ,i
\#'lrJj, i
_,_[-, o,~,,,~,,~,~_,} ,-or j--,,'r hlh2
~q
_JLi
,~3o,
and
'(+)~
(u.i=2(Aq)2
[80u-~,i-0u-2,i]
for j = N r
(31)
The dimensional drag on the sphere consists of the frictional part, Ff, and the pressure part, Fp. Using the steady-state Newtonian Stokes drag 6gt/l Uoa to normalize F, the dimensionless drag for the Jeffreys fluid is obtained: D = (Fp + Ff)/(6zrql Uoa) = Dp + Dr = DR + iD,
(32)
where
Dp=
1 (1 -- i~222) f0" [fo° ( -~-k-~ ~ ( ( ) ,=, d0'] sin 20 d0 6(1-iE2~.,)
(33)
Df=
1 (1 iE222) ~ 6 (1 -- i~22j) 2 ~ FIR = 1 sin z 0 dO
(34)
and -
In the above, R and 0 are the spherical coordinates, and DR and D~ are the respective real and imaginary components of the Fourier mode of the drag D. The numerical integrations are evaluated to second-order accuracy.
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R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
3. Asymptotic analysis for high frequency oscillations (~ >> 1) In this section, we present an asymptotic analysis to describe the flow structure due to a rapidly oscillating ball inside a stationary tube. As stated earlier, this is equivalent to an oscillating flow over a stationary sphere with the tube moving concurrently with the flow. At a high frequency of oscillation, the vorticity ~ is mainly confined near the surfaces of the sphere and the tube wall over a boundary layer of thickness O(e-i). Outside the boundary layers, ~ ~ 0 to the leading order. Hence, the major portion of the flow is irrotational and is governed by the Laplace equation for the Fourier mode of the velocity potential, ¢
(re)== + (rer)r
=
0
(35}
The boundary conditions in the physical domain for flow due to an oscillating sphere are
¢:=0
at : ~ - o e
(36a)
¢=0
at
z=0
(36b)
er=0
at
r=H
(36c)
¢==1
at
z2+r 2=1
(36d)
er=0
at
r=0,
- o e
l_
(36e)
Using the orthogonal transformation outlined previously, Eq. (35) becomes
rh,h
\ h,
+
\
: 0
{3V)
The solution for ¢ is obtained using the same numerical technique as described previously. For a flow oscillating rapidly over a stationary sphere, the Fourier mode for the velocity potential, q}, and the axial velocity components, Ui,v, are Op= ¢ + z
(38)
U~.v = U~n~+ 1 = ¢: + 1
(39)
and
In particular, the inviscid surface velocity on the stationary sphere is
g~(o) = Umvl:~+~: =,
(4o)
and the inviscid surface velocity on the oscillating tube is
u,(z) = u,.~]~=.
(41)
The potential flow solution not only describes the flow outside the thin Stokes layer but also provides a means to evaluate the added-mass force on the sphere inside the tube. The added-mass force, FAM(t), on a sphere moving with a velocity v ( t ) in an otherwise quiescent fluid can be expressed as
R. Mei et al. / J. Non-Newtonian FluM Mech. 66 (1996) 169-192
178
4
dv
FAM(t) = C , n ( H ) -~ ~ p a 3 d t or
4 DAM = FAM(t)/(67rtl, Uoa) = -- i-~ C~(H)e 2
(42)
where Cm(H) is the added-mass coefficient and approaches 0.5 as H--,oo. The added-mass coefficient Cm(H) can be obtained by evaluating the kinetic energy of the fluid [24] from the solution for 06 as
f
43,
C m ( H ) = - Js ¢
where S is the boundary of the domain and (4/3r0 is the volume of a sphere of unit Based on the boundary conditions, the non-zero contribution is from the surface of the For an oscillating flow, the added-mass force contributes only to D, (see Eq. (32)). The leading order solution for the Fourier mode of the tangential velocity, uo, boundary layer near the oscillating sphere can be found from the solution to Stokes' problem, i.e. the flow due to the oscillation of a plane wall:
radius. sphere. in the second
uo = UOR + iuo, = Uh(0){1 -- exp[-- e(R -- 1)1 cos[e(R - 1)1 + i exp(R - 1)] sin[e(R - 1)]} (44)
The axial velocity, u., in the boundary layer near the tube wall is of the form lg z =
$1zR ~ -
iu:l = 1 + [U, (z) - 1]{1 - e x p [ - e ( H + i [ U , ( z ) - l] e x p [ - e ( H -
r)] c o s [ e ( H - r)]}
r)] s i n [ e ( H - r)]
(45)
The vorticity on the tube wall, (w, is given by ( w = Ou: ~r r =/4 = (wR + i(w, "" --e(1 + i ) [ U , ( z ) -- 1] + O(1)
(46)
Matching between the velocities in the boundary layer and in the potential flow region can, in principle, be carried out in the entire flow field. However, a direct comparison between the results given by the asymptotic and finite difference solutions is cumbersome. The exception is along the line z = 0 or 0 = re/2 where the grid line is straight. Along this centerline, the potential flow velocity in the outer region, U,., can be evaluated numerically as o+ 1
U,.(r)= Ui,v(r,z=O)=-~z
(47)
=
In particular, we have U , ( z = O) = U,.(r = H ) and Ub(O = zr/2) = Uc(r = 1). The leading order asymptotic solution for the Fourier mode of the velocity in 1 < r < H and z --- 0 is u__(r, z = 0) ,-- 1 + [ U , . ( H ) - 1] × {1 - e x p [ - e ( H -
r)] c o s [ e ( H - r)] + i e x p [ - e ( H -
+ Uc(1){1 - e x p [ - e ( r + g,.(r)-
U,.(H)-
V,.(1)
1)] c o s [ e ( r - l ) + i e x p [ - e ( r -
r)] sin[E(H-- r)]} 1)] s i n [ e ( r - 1)]} (48)
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
179
Table 1 Comparison of DQs(H) between analytical and finite differencesolutions Solution
H 1.25
2
4
Analytical
73.555
5.970
1.960
Finite difference
74.986
5.964
1.980
As indicated by Eq. (47), the imaginary part of Uc(r) is zero to the leading order. The above asymptotic solution for u:, together with ~'w given by Eq. (46), can be used to describe the structure of the velocity field and to assess the correctness of the finite difference solution for the case of unsteady viscous flow at large E. It is noted that the high frequency asymptotic analysis is also applicable to a viscoelastic fluid whose constitutive relation is described by Eq. (11). At large ~, Eq. (12) gives [t ~ )q/;~2, so that the behavior of the viscoelastic fluid associated with viscous phenomena at high frequency corresponds to a Newtonian fluid with a Stokes number equal to (),1/22)~/2e as indicated by Eq. (ll).
4. Results and discussion
4.1. Validathm o f the numerical results
The computation is carried out in the (~, v/) domain using 401 grid points in the ~ direction and either 65 (for E < 1) or 129 (for E > 1) grid points in the v/direction for both Newtonian and viscoelastic fluids, and 65 grid points in the r/ direction for the potential flow for all values of ~. The parameter B in Eq. (24) is chosen to be two which gives A = 24 and C1 = 0.677. The parameter C2 used for stretching the grids in the r/direction is set to 0.38. The stagnation point in front of the oscillating sphere in the (~, r/) coordinate is located at ~ = .~0 = 0.25, A coarser grid system (201 × 65) was used to examine the dependence of the drag on the grid resolution; it gives a dimensionless drag D ( H = 2) = 5.9632 at steady state. Since the difference in D ( H = 2) between the coarse grid and the fine grid (401 × 65) is only 0.001, it is concluded that the fine grid system is sufficient for obtaining accurate, low frequency flow field and drag. In the limit E--, 0, the effect of the unsteadiness vanishes and the drag is determined by the quasi-steady value DQs(H): DQs(H)=DR(H,E)
as
E~0
(49)
The analytical solution for the Newtonian steady-state creeping flow has been reported by Haberman and Sayre [13]. It can be used in this study since the convection terms are neglected. The comparision between the analytical and the present finite difference results for Newtonian fluids is shown in Table 1. It is seen the finite difference solution performs satisfactorily for Stokes flows at a low frequency of oscillation.
180
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
An analytical result for Cm was reported by Cai and Wallis [25] for an inviscid fluid over a row of identical spheres in a tube. Fig. 2 compares the finite difference solution for Cm with that of Cai and Wallis for the case of a single sphere moving in a tube. Excellent agreement is observed, which indicates that the finite difference algorithm and grid system work very well for the potential flow. Fig. 3 compares the values for u:(r, z = 0) given by the finite difference solution and those given by the high frequency asymptotic solution, Eq. (48), for a Newtonian fluid at E = 64 and H = 2. It is seen that there exist two boundary layers: one near r = 1 and a weaker one near r = H. The velocity profiles in the boundary layers match smoothly those given by the inviscid solution. Excellent agreement between the finite difference and asymptotic solutions is observed. This indicates that the finite difference algorithm in the body-fitted orthogonal coordinate system, with the present set of computational parameters, gives accurate results even in the presence of thin Stokes layers. Fig. 4a shows the imaginary part of u:(r, z = 0), u:~, for E = 16, 32 and 64. A boundary layer of thickness O(E- ') on the surfaces of both the sphere and tube can be identified. Fig. 4b shows the enlarged profiles of u:, near the sphere at E = 16, 32 and 64 together with the asymptotic solution for • = 64 given by Eq. (48). A good agreement between the two solutions is observed within the Stokes layers. Outside the Stokes layers, it is found from the numerical solution that the values for u:,(r) are inversely proportional to • as can be seen from Fig. 4a. The behavior of u:~(r) outside the Stokes region is similar to that for u:R(r) shown in Fig. 3. This behavior is not surprising since Mei [18] has shown, for a rapidly oscillation flow over a stationary sphere in an unbounded Newtonian medium, that: (i) the leading order term of u__~outside the thin Stokes layer is zero and that of u:R is given by U~nv(r,z) (see Eq. (39)); (ii) the next-order term in the solution for (u-R + iu:,) is of O(• - ' ) and is induced by the viscous displacement effect 2
,
I
,
f
,
l
,
I
,
I
,
0
o Finitedifference + Cai & Wallis (1992) M Unbounded ÷
C
1
o
m
÷
0
0
0
0
I
I
i
I
i
l
2
3
4
5
H Fig. 2. Comparison of the added-mass coefficients, Cm, of the present finite difference solution and the analytical result of Cai and Wallis [25].
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
2.0
~
I
~
I
~
I
181
t
~=64 Finite difference
1.8
................... Asymptoticsolution 1.6
r 1.4
1.2"
1.0 0.0
~
~
,
0.5
1.0
1.5
U
2.0
zR
Fig. 3. C o m p a r i s o n b e t w e e n n u m e r i c a l a n d a s y m p t o t i c s o l u t i o n s for the real c o m p o n e n t o f the N e w t o n i a n fluid velocity o n the centerline, z = 0, for H - = 2 a n d E = 64.
within the boundary layers; and (iii) the curvature effect of the sphere contributes to (U_-R+ iu=1) only in higher order terms. The present numerical result for u=i(r, z = 0) shows qualitatively the same feature. The fact that u__~(r) has the same shape as U-R(r) means that the mechanism for generating u=~(r) in the outer region is of inviscid type. It was also observed that the numerical and asymptotic solutions for the real component of the Newtonian flow vorticity on the tube, ~wR, agree well at H = 2 and E = 64. The maximum values of numerical and asymptotic solutions for (wR are 16.99 and 15.95 respectively, and this O(1) difference quickly vanishes for Izl > 1. At a high frequency of oscillation, the good agreement between the asymptotic and numerical solutions provides convincing evidence that the numerical results are reliable in this domain. The fact that the numerical results also agree with the analytical solution at a low frequency of oscillation establishes confidence in the numerical solution. The behavior of the unsteady drag in Newtonian and viscoelastic fluids over a wide range of frequencies is discussed in the next section.
4.2. Unsteady drag in the Newtonian fluid As E increases, DR(H, E) increases and deviates gradually from DQs(H). Since DQs(H) does not depend on the acceleration, the difference between DR(H, E) and Dos(H) is denoted as the real part of the acceleration-dependent drag, i.e. D R A c ( H , e') :
DR(H, ~) --
DQs(H)
(50)
182
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
Fig. 5 shows DRAC for H = 1.25 and 2 for the Newtonian fluid case. It is clearly seen that DRAC(E) ~ O(E 4) for small E and finite H. As H increases, the region where DRAC(E) ~ O(E 4) decreases. As H--. ~ , the unbounded Stokes flow case is recovered and DRAC(6) ~ 0(62). The behavior DRAC(6) "~ 0(64) at finite H is caused by the limited diffusion in the r direction by the tube wall which forces the vorticity field generated on the surface of the ball to decay exponentially at a large distance away from the ball.
2.0
,
I
,
1.8
,
I
,
g ........ 16 32 ................... 64
1.6
r
I
i
1.4'
1.2'
~ - " "
10
.... i ..............
-0.6
-0.4
i
.......
-0,2
-0.0
0.2
UzI 1.25
,
t
,
i
,
)
I
O I
1.20
---o--- Asymptotic solution E=64
o 1
9, V 1.15
r
e=l 6 1.10
~
E
=32 ~
/ o~
1.05
l.O0
,
-0.6
-0.4
8
",
-0.2
-0.0
Uzl Fig. 4. (a) Numerical solutions for the imaginary component of the Newtonian fluid velocity on the centerline, z = 0, for H = 2 and e = 16, 32, and 64. (b) Asymptotic solution and numerical solution within the Stokes layer.
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
183
I0 2
10°
_
10 -2 c) 10 -4 I0 -6 10 -8
H = 1.25 10-10
.
10-2
.
.
.
.
.
.
.
I
10-1
.
.
.
.
.
.
.
.
]
10 0
.
.
.
.
.
.
.
.
i
10 1
.
.
.
.
.
.
[l
10"~
E Fig. 5. Effect o f ~ on the real part of the acceleration-dependent drag DR~,c in a N e w t o n i a n fluid for H = 1.25 and
2.
At large E, it is expected that DRAc(H, E) is proportional to E based on the following considerations: (i) DRAc(H--~ oo, E) is about equal to e even at finite Re; and (ii) the Stokes layer thickness is of O(E -~) and the surface vorticity on the sphere is of O(E). This asymptotic behavior is not established until E reaches a value of about 20. It must be pointed out that at much higher values of E the grid resolution near the sphere becomes a major concern. However, for practical purposes, E = 64 is beyond the normal operating range of the ball rheometer. Hence the grid resolution for the present computation is sufficient. The imaginary component of the history force, D~H, is obtained by subtracting the addedmass force DAM(H, ¢)= i(4/9)Cm(H)E 2 from Dr,
DIH(H, E) = DI(H, ¢) - DAM(H, E)
(51)
It should be noted that the Stokes solution gives D.u(E)~:E for H--,wJ. Fig. 6 shows D~H(H= 2, E) over a range of E. In the limit of large E, both DRAc(H , E) and D~H(H, ~) are linearly proportional to E because the wall vorticity is of O(E) on the surface of the ball which results from the Stokes layer of thickness O(E-~). Experimental results for the unsteady drag on a Newtonian fluid in a ball rheometer have been reported for E up to 0.25 at H = 2 by Tran-Son-Tay et al. [9]. It is found that DRA c (E = 0.25) ~ 5.5 × 10 -5 (Fig. 5), which is negligibly small in comparison with DQs = 5.97. The experimental results of Tran-Son-Tay et al. [9] are presented in the form of a correction factor KD for the Stokes solution, D R ( H ~ ~ ) = 1 + ~;
KD(H, ~)
DR(H, ~) I+E
(52)
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
184
10 3
. . . . . . . .
I
. . . . . . . .
,
. . . . . . . .
l
101
,
,
,
.....
,.-
10-1
I
-
. . . . . . . .
1
i
l
~.,-~
I0 -3
2
10"5
~
H =2
•
10-7
'
•
0-3
•
'
''''I
•
'
10"2
'
" 1 ' " I
'
. . . . . . .
I
10-1
"
H = 1.25 . . . . .
100
''I
•
'
'
' ' ' ' '
10 1
102
Fig. 6. Effect of • on the imaginary part of the history force D m in a Newtonian fluid.
Fig. 7 shows that the present numerical results for DR(H, E) compare very well with the measured data of Tran-Son-Tay et al [9]. However, values for the imaginary component of the drag, Dr, could not be compared. At E = 0.25, the numerical solution gives D~ ~ -0.0417, which represents only 0.7% of the values for the real component. This is a rather small value to be determined accurately with an instrument that has a much larger inherent phase error. ,
7.0
I
,
I
,
I
~
I
Finite difference
6.5 o
Measurement by Tran-Son-Tay et al. (1990)
6.0
5.5
5,0
~
4.5
4.0
0.00
|
i
i
i
0.05
0.10
0.15
0.Z0
0.2!
E Fig. 7. Comparison between the numerical solution and the experimental data o f Tran-Son-Tay et al. [9] for D R at H = 2.0 in a Newtonian fluid.
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
185
................... Newtonian l O2 .:
Viscoelastic
..".....
10 1
100
.
. . . . . . .
10 -3
i
10 "2
•
.
,
, . . . . i
•
,
. . . .
10 "1
..i
. . . . . . . .
10 0
i
10 1
. . . . . . . .
i
10 2
. . . . . . . .
10
E Fig. 8. Effect o f E o n D R f o r a N e w t o n i a n fluid a n d a v i s c o e l a s t i c fluid at H = 2 a n d 22/2~ = 0.5.
4.3. Unsteadydragin the viscoelasticfluid For small E, fl given by Eq. (12) is close to one and the viscoelastic fluid behaves just like a Newtonian fluid. The drag is determined by the "quasi-steady" value of the Newtonian flow. The effect of viscoelasticity is too weak to be noticed at small E. It is instructive to compare first the values of DR for viscoelastic (22/21 = 0.5) and Newtonian fluids at a given H. Fig. 8 shows that DR of the viscoelastic fluid is noticeably smaller than that of the Newtonian fluid for E > 0.1 and H = 2. While the Newtonian DR increases monotonically with E, the viscoelastic DR reaches a minimum at E ~ O(1). At large E, DR ~ O(E) in all three cases. The drag reduction in the viscoelastic fluid case for E > 0.1 is associated with an unsteady shear-thinning effect. This is clearly seen from the dependence of fl on E (Eq. (12)). The fact that DR for the viscoelastic fluid possesses a minimum indicates a competition between the decreasing effective viscosity and increasing wall vorticity associated with a thinner Stokes layer as E increases. To further illustrate the drag reduction in a Jeffreys fluid and the competition between the two mechanisms affecting DR, a sphere oscillating in an unbounded viscoelastic medium is considered. The dimensionless drag D can be easily found for the Jeffreys fluid: D-
l+2')tzE4+i(2'--22)Ez2E2{ 2+2,2~E-4 2 - - i ~1 - i
1
q_ 212~.4 [(22- 21)E2-F i(1
O1-2122{:'4)]
I' 1/2
(53)
The small E limit and large E limit of D are thus given by D=I+E--iE and
forE<~l
(54)
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
186
D=(22~/ze--i~(22y/2E+~¢2]
L\ZI
for e>> 1
(55)
For the Jeffreys fluid, 2~/2~ < 1, the real part of the high frequency drag, (22/2~)~/2E, is clearly smaller than the value of E for the Newtonian fluid. The presence of the tube does not qualitatively affect this conclusion. Fig. 9 shows the values for DR given by the analytical (H-o oo) and finite difference ( H = 2) solutions for 22/21 = 0.5. A small dip for the unbounded case is still observed near E = 0.6. It is interesting to note that DR reaches a local maximum first near E = 0.2 for the unbounded case before it starts decreasing. This behavior is not observed for the bounded case, H = 2. The lack of an obvious local maximum in DR in the bounded case is due to the suppression of the effect of acceleration by the cylindrical wall at H = 2. The extent of shear thinning strongly depends on the value of 22/2~ as seen from Eq. (55). Fig. I0 shows that the value of the local minimum for DR(H, E) decreases with decreasing 2z/2L. The Newtonian case is recovered when 22/2~ = 1 and the local minimum disappears. Fig. 11 shows the effect of 21 on DR(H, E) at e = 1 for H = 2 and various values of 22/21. The drag decreases as 2~ increases for a given value of 22/21. The largest decrease is observed for the smallest value of 22/21, which is consistent with the results shown in Fig. 10. At large 21, DR approaches a constant which indicates that the effective viscosity has reached its minimum. The imaginary part of the drag, DI, is in general very small compared to DR for small E. When increases, a local minimum for D~ is also observed for the sphere oscillating in the unbounded viscoelastic medium with 22/21= 0.5 (Fig. 12). The behavior of DI for a finite H is further complicated, as compared with that of DR, by the change of its sign as ¢ increases. The change in the sign of D~ indicates that the effect of shear thinning overwhelms the effect of inertia for intermediate values of E. At large E, the added-mass force DAM dominates D~, meaning that D~ ~ O(E2). Since the added-mass force is purely inertial, it does not depend on the constitutive model. Hence the expression of DAM for viscoelastic fluids is the same as for Newtonian fluids. 10 2
~ .......
~
........
~
........
~
........
I
........
j
. . . . . . . .
i
........
~
........
i
................... HU~2 (finite d i f f e r e n c e )
1o 1
DR
f .F" ......................................... ,,.....-o"" 10
0
.
I0"3
.
; .....
j
. . . . . . . .
10"2
,
10 °1
. . . . . . . .
,
100
. . . . . . . .
I0 1
10 2
103
£ Fig. 9. C o m p a r i s o n
o f D R f o r a v i s c o e l a s t i c fluid b e t w e e n H - - , ~
and H=
2.
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
102
........
,
........
,
........
I
.......
~.2/~, i 0.50
~
..........
. ' d " ' ,s ."/,~ .~/,f~
] ] 1
.,.t,
1.00
10 1
....................... -2"/7
DR
~
. . . . . . . .
100 0 -3
I
10 "2
"
'
........
,,." ~./
0.75
' ' ' ' " I
/
. . . . . . . .
10 "1
I
100
. . . . .
!
'''I
187
I
. . . . . . . .
101
10
E Fig. 10. Effect o f ~ o n D e at H = 2 for v a r i o u s values o f 22/2~.
Fig. 13 shows that DIH-----(=D~-DAM) at large E for H = 2 and H---, oo, and 22/2~ =0.5. It is seen that for the Jeffreys fluid Dm ~ O(e). Fig. 14 shows the decrease of D~ with increasing 2~ for e = 1, H = 2, and various values of 22/2~. The change in the sign of D~ can be clearly seen as 2~ increases. The behavior of D~ at large 2, is quite similar to that of DR.
~,2 / ~. 1
i
0.7
5
..........
0.5 0.3
4 I\ I; | ",
D
R
i
3
,
I
\ \ 2
~
]
0
. . . . . . . . . . . . . . . . . . .
I
I
I
I
10
20
30
40
Fig. 11. Effect o f 2~ o n D R at H = 2 for v a r i o u s values o f 22/2~.
50
188
R. Meiet al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192 10
4
,
.
.
. . . . .
i
,
.
. . . . . .
I
.
.
.
.
.
.
.
.
i
,
,
,
,
.
,..i
,
e
. . . .
10 3
102 10 1
~
~
2
~
I
100 -DI 10-1 10-2
I
10-3 10
-4 10-2
.
.
.
.
.
.
.
.
I
'
"
''
. . . . .
I0-1
!
.
.
.
.
.
.
.
.
100
I
•
,
I0 I
10 2
E Fig. 12. Behavior of the imaginary component of the unsteady drag, D~, on a sphere oscillating in an unbounded Jeffreys fluid with 22/21 = 0.5. 4.4. Determination o f the rheological properties from measured drag data
The foregoing numerical analyses pertain to a linear Jeffreys fluid, but can easily be extended to a class of more general viscoelastic fluids. One of the main purposes of this computational study is to provide a means for characterizing the rheological behavior of viscoelastic fluids using the ball rheometer. In this section, a method to obtain the rheological properties from 102
~
,
,
,
,
,
, , I
,
.
Analytical solution (F_-~ o . ) .
.
.
.
.
.
.
. ~
.......... Finite difference ( H = 21~ . . . . . ~
-DIIJ1°1
100 10°
I
......
1
i 101
,
.......
102 E Fig. 13. Effect of E on DIH for the Jeffreys fluid at large c, H = 2, and H ~ ~ with 2~/21 = 0.5.
189
R. Meiet al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
2
,
I
,
I
,
I
,
I
i
~-2 /~.1 1
'" -'ira
0
0.7 0.5
.....
-
.................. 0.3
\ ~" ......... ~'"""'"'"'"'""2:'"'""'-_"
-1
-2
0
.....................................................
I
I
!
I
10
20
30
40
k 1
50
Fig. 14. Effect of 2~ on DT for different values of ).2/2, at • = 1, and H = 2.
measured unsteady drag data and the numerical simulation is presented. It is emphasized that the assumption of a Jefreys model is not necessary in order to determine the unsteady drag from the ball rheometer data. The general constitutive equation for a linear viscoelastic fluid in the frequency domain can be written in dimensional form as = 2q*/5
(56)
where the complex viscosity, r/*, consists of a dynamics viscosity, q', and an elastic modulus, G': q* = q'(a)) + iG'(~o)/co
(57)
Experimentally, it is possible to measure the steady viscosity, r/o, and the c o m p o n e n t s of the unsteady drag, DR and D~, at a given co [9]. Using rio and Dlov/a 2 respectivel3) to normalize q' and G', and keeping the same notation to represent the dimensionless variables, it follows that the reciprocal dimensionless viscosity fl-I (see Eq. (12)) is given by E2
/] -- r/'E 2 -t- iG'
(58)
One can start searching for/~ by guessing a pair of values for r/' and G'. For small E, r/' = 1 and G ' = 0 are good initial starting values. As E increases, one makes initial guesses for r/' and G' based on the values already found at the nearest E. By solving Eq. (16), one obtains numerical values for the vorticity, and unsteady drag can be obtained by intergrating the stress along the surface of the sphere. The real and imaginary c o m p o n e n t s of the unsteady drag are given by
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
190
\ - ~ - + - R - ) R =, d0'3 sin 20 d0
1G' fo~ [fo° (eC' + ~
R/.=, d0'3 sin 20 d0
1 r/' 3
e 2 Jo (,IR sin 2 0 dO =~ fo(R[R= ,sin2 0 dO + 3 £(" 1 d0']
Dl=-6~'fo~r[fo°(~(l+~)R=~-R
\0R
6 e"2 -lr/'3
(59)
sin 20 d0
R JR= 1 dO' sin 20 dO
(IIR = lsin2 0 d0 - 5 ~5
(RIR =, sin2 0 d0
(60)
If the numerical results for DR and D~ agree with the measured values, the guessed r/' and G' values are accepted. If the calculated values are too high or too low, a new pair is selected to correct for the difference. This iterative process continues until the values for the calculated and measured drags agree. The development of a linear viscoelastic model is then possible after r/'(o~) and G'(~o) are obtained over a range of o9. To this end, it is necessary to examine the sensitivity of the drag (DR + iDa) on r/' and G'. Fig. 15 shows the dependence of DR and D~ on G' with H = 2 and q ' = 0 . 5 . Fig. 16 shows the 10 1 I 00
........
~
........
t
........
t
........
'.......................o ..........o-o .................... - . - . - o - . . . - . . . - o . - - . . . . - . - . - . - . - - . . . . . - . ~ .
10 -1
D . . . . .
103
/
-
rl = 0.5, e = 0.01 H=2.0
10 -4
.
10"8
.
.
.
.
.
.
.
I
10"7
.
.
.
.
.
.
.
.
I
'
10"6
G
"
"
'
'
' ' ' l
10"5
i
Fig. 15. Effect o f G' on the unsteady drag at e = 0 . 0 1 .
.
.
.
.
'
" ' '
10-4
R . M e i e t a l . / J. N o n - N e w t o n i a n
6
,
I
n
5
D 21
,
I
,
/ ~ ~ " J l ~
D I
[
=
•
0.0
I
191
66 (1996) 169-192
G'= 1.0e-8, H = 2 /
3
-I
i
0-'- D R ........° .......
4
o
I
Fluid Mech.
...............................• ................................• .................................................• ..............• ...............
i
I
i
I
0.2
0.4
0.6
0.8
1.0
Fig. 16. Effect of ~/' on the unsteady drag at E= 0.01. dependence of DR and DI on r/' with H = 2 and G' = 10 -~. In both cases, e = 0.01. It can clearly be seen that DR depends mainly on I/' and is insenstive to G', while DI depends mainly on G' and is insensitive to r/'. Similar behavior is observed at E = 1. Thus, 1/' and G' can easily be related to the measured DR and DI through numerical simulations. The principal difficulty lies in the accurate measurement of the unsteady drag (DR + iDi).
5. Conclusion
Numerical analyses for a sphere oscillating along the centerline of a cylinder at small Reynolds numbers are presented for Newtonian and Jeffreys fluids over a wide range of frequencies. The unsteady drag and the corresponding flow field are examined based on the finite difference and asymptotic solutions. The effect of the cylindrical tube wall on the unsteady drag is elucidated. At small Stokes number ~, the drag is dominated by the quasi-steady value for both Newtonian and viscoelastic fluids. At intermediate E, the effect of shear thinning on reducing the drag is observed and the dependence of the drag on the frequency becomes complicated. At large E, the added-mass force, which is the same for both Newtonian and viscoelastic fluids, dominates the imaginary component of the drag. The history force is proportional to E for large e. The presence of the wall reduces the low frequency history force and the real and imaginary componenets change from O(E) in the unbounded domain to 0(~"4) and O(e 2) in the bounded case respectively. A method for obtaining the viscoelastic properties of a fluid from measurements and numerical simulations of the unsteady drag is proposed. It is found that the real component of the drag depends strongly and monotonically on the dynamic viscosity ~/' while the imaginary component depends strongly and monotonically on the elastic modulus G'.
192
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
Acknowledgement R.M. acknowledges the support of the Engineering Research Center (ERC) for Particle Science & Technology at the University of Florida, the National Science Foundation (EEC9402989), and industrial partners of ERC. J.X. and R.T.S acknowledge the support of the National Institutes of Health through Grant 4 RO1 HL49060.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
M.B. Bush, J. Non-Newtonian Fluid Mech., 49 (1993) 103. M.B. Bush, J. Non-Newtonian Fluid Mech., 55 (1994) 229. D.D. Joseph, Y.J. Liu, M. Poletto and J, Feng, J. Non-Newtonian Fluid Mech., 54 (1994) 45. W.M. Jones, A.H. Price and K. Waiters, J. Non-Newtonian Fluid Mech., 53 (1994) 175. C. Bodart and M.J. Crochet, J. Non-Newtonian Fluid Mech., 54 (1994) 303. J.V. Satrape and M.J. Crochet, J. Non-Newtonian Fluid Mech., 55 (1994) 91. R.J. Lutz, M. Litt and L.W. Chakrin, in H.L. Gabelnick and M. Litt (Eds.), Rheology of Biological Systems, Thomas, Springfield, IL, 1973. R. Tran-Son-Tay, B.B. Beaty, D.N. Acker and R.M. Hochmuth, Rev. Sci. Instrum., 59(8) (1988) 1399. R. Tran-Son-Tay, B.E. Coffey and R.M. Rochmuth, J. Rheol., 34(2) (1990) 169. H.S. Sellers, W.H. Schwartz, M. Sato and T. Pollard, J. Non-Newtonian Fluid Mech., 26 (1987) 43. G.G. Stokes, Trans. Cambridge Philos. Soc., 9 (1851) 8. J. Happel and B.J. Byrne, Ind. Eng. Chem., 46 (1954) 1181. W.L. Haberman and R.M. Sayre, David Taylor Model Basin Report No. 1143, U.S. Navy Department, Washington, DC, 1958. A.M. Fayon and J. Happel, AIChE J., 6 (1960) 55. R. Mei, C.J. Lawrence and R.J. Adrian, J. Fluid Mech., 233 (1991) 613. R. Mei and J.F. Kausner, Phys. Fluids A, 4(1) (1992) 63. R. Mei and R.J. Adrian, J. Fluid Mech., 237 (1992) 323. R. Mei, J. Fluid Mech., 270 (1994) 133. R. Zheng, N. Phan-Thien and R.J. Tanner, J. Non-Newtonian Fluid Mech., 36 (1990) 27. F. Sugeng and R.I. Tanner, J. Non-Newtonian Fluid Mech., 20 (1986) 281. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, Berlin, 1990. B. Fornberg, J. Fluid Mech., 225 (1991) 655. R. Mei and A. Plotkin, Gomput. Fluids, 14 (1986) 239. C. Yih, Fluid Mechanics,West River, Ann Arbor, MI, 1979. X. Cai and G.B. Wallis, Phys. Fluids A, 4(5) (1992) 904.
J. Non-Newtonian Fluid Mech., 66 (1996) 169-192
Motion of a sphere oscillating at low Reynolds numbers in a viscoelastic-fluid-filled cylindrical tube Renwei Mei*, Jie Xiong, Roger Tran-Son-Tay Department of Aerospace Engineering, Mechanics & Engineering Science, University of Florida, Gainesville, FL 32611-6250, USA Received 18 December 1995; in revised form 15 June 1996
Abstract The motion at low Reynolds numbers of a sphere oscillating with a small amplitude inside a cylindrical tube containing a viscous fluid (Newtonian or viscoelastic) is investigated. The linear Jeffreys model is used to describe the constitutive relation for the viscoelastic fluid. The governing equations are transformed to, and solved in, the frequency domain where the Fourier components for the flow and drag are obtained using a finite difference method. The numerical results for the Newtonian fluid velocity agree well with those given by a high frequency asymptotic solution. The effects of the Stokes number, E, on the unsteady drag, D R -b iO l, are discussed. At small •, the real component of the unsteady drag, D R, is dominated by the quasi-steady term for both Newtonian and viscoelastic fluids. For a Newtonian fluid, the acceleration-dependent terms of D R and D l are proportional to •4 and •2 respectively. Whereas D R increases monotonically with • for a Newtonian fluid, it reaches a minimum tot a Jeffreys fluid when • ~ O(1). The difference observed is caused by the shear-thinning property of the viscoelastic fluid. At large • , D R becomes proportional to •, and the imaginary component of the unsteady drag, D~, for both Newtonian and viscoelastic fluids, is dominated by the added-mass force. However, as opposed to a Newtonian fluid, D, changes sign near • ~ O(l) for a viscoelastic fluid due to shear thinning. Finally, a method for determining the rheological properties of a general viscoelastic fluid from measurements of the unsteady drag in a ball rheometer is presented.
Keywords: Oscillating sphere; Reynolds numbers; Viscoelastic fluid
1. Introduction T h e m o t i o n o f a s p h e r e falling a l o n g t h e axis o f a c y l i n d r i c a l t u b e c o n t a i n i n g a N e w t o n i a n o r v i s c o e l a s t i c fluid h a s a t t r a c t e d m u c h a t t e n t i o n in t h e last 20 y e a r s w i t h a c o n s i d e r a b l e i n c r e a s e * Corresponding author. 0377-0257/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0377-0257(96)01479-6
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in the last few years [1-9] because understanding the motion of a ball oscillating inside a cylindrical tube is critical to researchers working with a ball rheometer. The main advantage of the ball rheometer over other devices, such as cone-plate and couette rheometers, is that it requires a very small sample volume. For example, the microrheometer used by Tran-Son-Tay et al. [9] requires only 20 lal of sample, which would be impractical for other standard rheometers. This advantage is significant when one deals with fluids that can only be obtained in small quantities, e.g. mucus, sickle cell blood, or synovial fluid. Even though these studies [1-9] are of significant importance for understanding and analyzing falling ball data, our knowledge of the oscillating ball rheometer is far from being complete. Because of the importance and increased use of ball microrheometers for studying certain viscoelastic fluids, there is a need for understanding the motion of a sphere oscillating along the axis of a cylindrical tube. In comparison to the steady-state probelm, the number of studies on the motion of a sphere oscillating inside a closed container is extremely small. The effects of a closed nearby boundary on the drag of an oscillating sphere have been reported, but only for the case of a spherical container [9,10]. Rheological data using the ball microheometer exist [9], but they are of limited value since the effects of the tube wall on the unsteady drag of the fluid acting on an oscillating sphere are not known, and there is no published work on these effects. The simplest case of a particle moving at low velocity in a Newtonian fluid was solved first in 1851 by Stokes [11], but the problem of determining the near wall effect on the drag experimenced by the particle was solved much later. Analytical solutions about steady Stokes flow over a sphere in a cylinder have been reported by several investigators (e.g. Happel and Byrne [12], Haberman and Sayre [13]). Fayon and Happel [14] experimentally investigated the effect of a cylindrical boundary on a spherical particle suspended in a moving viscous liquid. Unsteady flows over a sphere in an unbounded region at finite Reynolds number have been studied by Mei and co-workers [15-18]. For the case of a sphere of radius a oscillating at a frequency ~o with a velocity amplitude 1-7oin an unbounded fluid of kinematic viscosity v, it was found that: (a) at a large Stokes number, E(~oa2/2v)1/2, the numerical result for the unsteady drag agrees well with the high frequency asymptotic solution; (b) at E ~> 1, Stokes solution is valid for finite Re; and (c) at small Strouhal number, St = coa/Uo ,~ 1, the behavior of the imaginary component of the unsteady drag is complicated by the logarithmic dependence on St and is different from an earlier result obtained [15] for an unsteady flow over stationary sphere with a small amplitude oscillation in the free-stream velocity. Numerical solutions for slightly non-Newtonian fluids over a sphere in the creeping flow limit are available [5, 6, 19]. Numerical techniques include finite difference methods, finite element methods, and boundary elements methods. The constitutive equations selected for the non-Newtonian fluids in these numerical studies are the commonly used Upper-Convected Maxwell (UCM) and Oldroyd-B fluid models. It was found that there is a limiting Weissenberg number, W e - 2 V/a, above which no steady-state solution exists. In the latter equation, 2 is the fluid relaxation time and V is the velocity of the sphere. The effects of elasticity on the drag have also been discussed. A number of studies combining falling ball experiments and analysis have been reported. Experiments with constant viscosity Boger liquids [4] show that the normalized drag can be very high for the limiting cases ( H ~ ~ and H ~ 1, where H is the tube-to-sphere radius ratio). However, there is still a gap between the available experimental data and reliable numerical simulations for simple viscoelastic models. Numerical results for a viscoelastic fluid past a sphere in the presence of a wall [20] show that the fluid elasticity and shear thinning can reduce the drag
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
171
coefficient on the sphere remarkably, up to 25% in the absence of fluid shear thinning and 40% in the presence of fluid shear thinning. A stagnation flow behind the sphere on the centerline of the cylinder in a shear-thinning viscoelastic fluid has been reported by Bush [1,2]. In this region it was found that elastic properties provide an enhanced decelerating force on the fluid far from the sphere, leading to the well-documented downstream shift in the streamlines relative to the flow of a Newtonian fluid. Tran-Son-Tay et al. [9] found that under certain conditions the effect of inertia on the force acting on a ball oscillating inside a cylindrical tube was well described, at low frequency, by the theory for a ball oscillating inside a spherical container. This finding means that when the imaginary component of the total hydrodyanamic force is negligible, a modified Stokes drag expression can be used to account for the boundary and frequency effects in the determination of the non-Newtonian viscosity with a ball rheometer. However, at higher frequencies of oscillation, the elastic componenent, G', for the non-Newtonian red blood cell suspensions is negative when the theory for a sphere oscillating without inertia inside a spherical container is used. This result is physically impossible and means that the effect of inertia in a cylindrical tube is more important at higher oscillatory frequencies than in a spherical container. This discrepancy stresses the need for a theory dealing with a cylindrical tube in order to correctly calculate the values for G' and r/' from the experimental data. The present paper focuses on the numerical simulation of the unsteady drag on a sphere oscillating inside a cylindrical tube containing a Newtonian or linear viscoelastic fluid. The motivation for this work comes from the need to understand the effect of frequency on the unsteady drag in a cylindrical tube, and the difference in behavior between Newtonian and viscoelastic fluids. This acquired knowledge will then lead to the development of a rigorous method for analyzing rheological measurements obtained with a ball rheometer over a large range of ~o, or more appropriately, E. The body of the paper is organized as follows. Section 2 formulates the problem in terms of a general Jeffreys model and subsequently presents the simplification of the governing equations, the Fourier mode representation of the solutions for the flow field and drag, and a conformal mapping for constructing an orthogonal body-fitted grid system. Section 3 presents an asymptotic analysis for the flow field at high frequency. Section 4 presents the numerical results for the unsteady flow in the tube and unsteady drag on the sphere over various frequency ranges for both Newtonian and viscoelastic fluids. The computed total unsteady drag is analyzed by decomposing it into a quasi-steady drag at zero frequency, an added-mass force which is purely inertial and dominates at high frequency, and a history force which exhibits quite complex dependence on the Stokes number for a viscoelastic fluid. It is important to note that, although most of the results presented in this paper are obtained with a linear Jeffreys model, the computational analysis developed in this study is applicable to a class of fluids more general than that which the Jeffreys model can describe. Thus, a method for determining the linear rheological properties of an unknown viscoelastic fluid from measurements of the unsteady drag is presented at the end of Section 4. Section 5 summarizes the main results.
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2. P r o b l e m f o r m u l a t i o n
The motion of a rigid sphere of radius a oscillating with a velocity Uo e x p ( - i m t ) inside a cylindrical tube of radius b = Ha containing a Newtonian or viscoelastic fluid of density p is considered. In the above complex expression, U0 and oJ are the velocity amplitude and angular frequency of the oscillation respectively, and t is time. The problem is equivalent to solving the motion of a flow oscillating with velocity u = Uo exp(-icot) over a stationary sphere with the cylindrical tube moving concurrently with the uniform flow (Fig. 1), provided that an inertial force (4/3)prca3(du/dt) is subtracted from the total force to account for the acceleration of the fluid flow. The governing equations for an incompressible, laminar, viscoelastic flow are
~ v + v. v v = _ 1 Vp + ! v-~
(1)
v. v = o
(2)
St
p
p
and
where V is the velocity vector, p is pressure, and ~ is the deviatoric stress tensor. The generalized Jeffreys model is used to describe the constituitive equation of the viscoelastic fluid [21]
"t"+ Al ~-t (1") = 2r]l D -I--~2 ~-7 (O)
(3)
D = l ( v V + V VT)
(4)
2~ - rh + a r/2
~
and
/72 22 = -~
(5)
~
b/a=H
j
__ z
V/////////////////////////////////////////////A U0 exp(-imt) Fig. 1. Sphere oscillating inside a cylindricaltube.
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 ¢1996) 169-192 where in the ,(2 are tensor
173
G is the elastic m o d u l u s of the spring, r/1 and r/2 are the viscosities of the two dashpots Jeffreys model in which q2 is the viscous element in parallel with the spring, and 21 and relaxation and retardation times respectively. The invariant derivative ~/~'~t for any M is defined as
~-~t (M) = 5 (M) + V " V M + M " 1 2 - 12 . M -
g(DM + MD)
(6)
w h e r e / 2 is the anti-symmetric part of the deformation tensor V V and the values g = - 1, 0 + 1 correspond to the lower, corotational, and upper models, respectively. Oldroyd's models A and B are special cases corresponding to g = 1 and - 1 respectively. When 22 = 0, Eq. (3) reduces to the Maxwell model. Newtonian fluid corresponds to 2~ = 2 2. The displacement amplitude of the ball, Zo, is very small c o m p a r e d to the radius of the ball and the dimensionless displacement, 6, is typically less than 0.1, i.e. c5 = Zo/a ~ 1. The following quantities 8~(=/~a~o), 1/a, 1~co, qlUo/a, and Uo/a are used to normalize V, V, t, r a n d D For very small c5, it can be shown that O/Ot(r)~, O(r/jco2/~) and 8 / 8 t ( D ) ~ O(to 2 c5) are the respective leading order terms for ~ / ~ t ( t ) and 9 / ~ t ( O ) . Therefore, ~ / ~ t reduces Io 8/St and a linear Jeffreys model is obtained: ~ + 2~ c~t = 2q, D + 2 2 - ~
(7)
Further, using a2p/2rl~ and pU2o to normalize 2 and p respectively, the dimensionless form of Eq. (1) becomes 8V St--~+ V'VV=
2__V -VP+Re "~
(8)
where Re = Uo2a/v, and St = e)a/Uo are the Reynolds and Strouhal numbers respectively, and v = rl~/p is the fluid kinematic viscosity. For the present problem both ~ and Re are m u c h less than one, hence the convective term V • V V is negligible, and Eq. (8) becomes linear. Since only low Reynolds n u m b e r flow will be considered, the Stokes n u m b e r E = (~oa2/2v) 1/2 = (St Re/4) 1/2
(9)
will be used for interpreting the results of the unsteady flow and drag in lieu of Re and St. Since the fluid m o t i o n is induced by a single harmonic oscillation on the b o u n d a r y and Eqs. (7) and (8) are linear, a time-domain variable f ( z , r, t) can be expressed using Fourier modes representation as f ( z , r, t) = ½If(z, r)e -~' + f * ( z , r)e +~'] = ~ R ( -7, r) cos t + ~ ( z , r) sin t
(10)
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
174
where an asterisk denotes the complex conjugate, a swung dash denotes the Fourier component, and the subscripts " R " and ' T ' denote the real and imaginary parts of a complex variable. It is noted that for a nonlinear problem, i.e. due to nonlinear convection or a nonlinear constitutive relation, more Fourier modes need to be retained in Eq. (10). The dimensionless form of the constitutive equation in the frequency domain now becomes (1 - - i E 2 2 2 ) = 2 (1 - iE22,)/~ = 2 / ? - ' / )
(1 1)
where (1
-- iE221)
(1
- (1 - i E % )
+ 2122e4) + i(22-- 21)e-z
-
(1 + E"A )
(12)
is the inverse of the dimensionless complex viscosity which reduces to one for a Newtonian fluid. The linearized m o m e n t u m equation, Eq. (8), in the frequency domain is Re 2 Vfi + 2fl-'V •/9
-2i~z P-
(13)
In cylindrical coordinates (r, ~b, z), the velocity of an axisymmetric flow is given by V = uer + we. and the vorticity ( by or
(=w:-ur
~'=W---Ur
(14)
where the subscripts denote partial derivates with respect to that variable. Using the stream function ff for an axisymmetric flow, the velocity components can be expressed as = 1 fir
and
r
a = __1 ~,,-__
(15)
y
The transport equation for ( ' c a n be easily obtained from Eq. (13) (
-
1:)
(16)
and the stream function ~ is obtained by solving the Poisson equation ¢7_-_- +
1 ~=
- r
-r~
(17)
Hereinafter, the swung dash will be dropped from the notation for simplicity. The boundary conditions are specified as follows
¢=½r
and
asz~+~
(18a)
at z = 0
(18b)
= ½r2
on the tube wall r = H = b/a
(18c)
~, = ( = 0
for z _< - 1 on the axis r = 0
(18d)
~/' = ~'n = 0
on the sphere z 2 + r 2 = 1
(180
~b: = (: = 0
(=0
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
175
where n is the outward normal of the surface of the sphere and H = b/a > 1
(19)
is the dimensionless tube radius or the tube-to-sphere radius ratio. To carry out the computation in a body-fitted coordinate system, a conformal mapping from (z, r) to (xl, x2) is first applied. Denoting Z = z + ir
and
(20)
X = Xl + ix2
the following transformation [22] X=~
W
( Z + ~ I T + ~2T 3 + ~sT 5 +" • ")
(21)
with T-- ~-~ coth
Z
(22)
maps the domain between the sphere (]Z I = 1) and the tube (Z = i l l ) into a rectangular region ( - ,~ < Xl < ~ , 0 < x2 < W / 2 ) , by choosing proper values of c~n (n = l, 2 .... ). The values of c~, and W are chosen such that the sphere IZI = 1 is mapped into a segment of line defined by - 2 _ < x t < 2 , x2= 0. To obtain an adequate spatial resolution in the r/ direction at high frequency, the following transformation Xz=
+ C2 tan -1 tan
( 2 r / - 1)
0
(23)
is applied to cluster grids near the sphere (r/=0) and the tube wall (r/= 1). Since the transformation from (xl, xz) to (z, r) is singular near xl = +_2, it is important to allocate more grids near Xl = +_2 in the x~ direction. This is accomplished through the following tranformation xt
-2-(x1_~-2) =
-
1-Cltan-
+
-
-
tan 2
for for
0<~<~10 --
-<- O . 5
(24)
where xt~ is the half-length of the strip in the (x~, x~) domain. Setting x~ = 0 (~ = 0.5), A and B are related as follows: l A-0.5-
( 2 ~lo 0.5--~1o
B)
(25)
with B < 4 / ( I - 2~o) to ensure that Xl > - 2 for ~0-< ~ <0.5. A smaller value for B corresponds to more grids near x~ = +_2. Once B is given, C~ can be obtained numerically be requiring x~ (~) to have a continuous first order derivative at xL = +_2 (or ~ = ~o). The transformed governing equations are
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R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
l[~ (~o<) , r~,,"~)
h,h2LO~\h,
o,,",,,o+~l__,( ~,--~ ~0+ r/r
+-~q t,h2 8rl)]
o~, ~) ,
~ ~o<~l+;~ r
~0)= -- r~
-~q
(27)
where hi
=
(z~+r~)l/2
and
hz=(z 2+r"2 )1/2
(28)
are the Lam6 coefficients. Finally, a finite-difference technique [23] is applied to the above equations. Spatial derivatives are respresented using central difference. The wall vorticities are evaluated as follows. On the surface of the sphere ( / = 1), the velocity is zero so that both real and imaginary components of ( are given by ~'jj-- 2(Aq)2
l,i-- ¢'j+2,,)
,
for j = 1
(29)
Since (UR, U0 = (1, 0) and (¢'R, ¢q) = ~ H 2, 0) along the tube wall ( / = Nr), one obtains
l
(~-~2)j[--7+Rj, ,i
\#'lrJj, i
_,_[-, o,~,,,~,,~,~_,} ,-or j--,,'r hlh2
~q
_JLi
,~3o,
and
'(+)~
(u.i=2(Aq)2
[80u-~,i-0u-2,i]
for j = N r
(31)
The dimensional drag on the sphere consists of the frictional part, Ff, and the pressure part, Fp. Using the steady-state Newtonian Stokes drag 6gt/l Uoa to normalize F, the dimensionless drag for the Jeffreys fluid is obtained: D = (Fp + Ff)/(6zrql Uoa) = Dp + Dr = DR + iD,
(32)
where
Dp=
1 (1 -- i~222) f0" [fo° ( -~-k-~ ~ ( ( ) ,=, d0'] sin 20 d0 6(1-iE2~.,)
(33)
Df=
1 (1 iE222) ~ 6 (1 -- i~22j) 2 ~ FIR = 1 sin z 0 dO
(34)
and -
In the above, R and 0 are the spherical coordinates, and DR and D~ are the respective real and imaginary components of the Fourier mode of the drag D. The numerical integrations are evaluated to second-order accuracy.
177
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
3. Asymptotic analysis for high frequency oscillations (~ >> 1) In this section, we present an asymptotic analysis to describe the flow structure due to a rapidly oscillating ball inside a stationary tube. As stated earlier, this is equivalent to an oscillating flow over a stationary sphere with the tube moving concurrently with the flow. At a high frequency of oscillation, the vorticity ~ is mainly confined near the surfaces of the sphere and the tube wall over a boundary layer of thickness O(e-i). Outside the boundary layers, ~ ~ 0 to the leading order. Hence, the major portion of the flow is irrotational and is governed by the Laplace equation for the Fourier mode of the velocity potential, ¢
(re)== + (rer)r
=
0
(35}
The boundary conditions in the physical domain for flow due to an oscillating sphere are
¢:=0
at : ~ - o e
(36a)
¢=0
at
z=0
(36b)
er=0
at
r=H
(36c)
¢==1
at
z2+r 2=1
(36d)
er=0
at
r=0,
- o e
l_
(36e)
Using the orthogonal transformation outlined previously, Eq. (35) becomes
rh,h
\ h,
+
\
: 0
{3V)
The solution for ¢ is obtained using the same numerical technique as described previously. For a flow oscillating rapidly over a stationary sphere, the Fourier mode for the velocity potential, q}, and the axial velocity components, Ui,v, are Op= ¢ + z
(38)
U~.v = U~n~+ 1 = ¢: + 1
(39)
and
In particular, the inviscid surface velocity on the stationary sphere is
g~(o) = Umvl:~+~: =,
(4o)
and the inviscid surface velocity on the oscillating tube is
u,(z) = u,.~]~=.
(41)
The potential flow solution not only describes the flow outside the thin Stokes layer but also provides a means to evaluate the added-mass force on the sphere inside the tube. The added-mass force, FAM(t), on a sphere moving with a velocity v ( t ) in an otherwise quiescent fluid can be expressed as
R. Mei et al. / J. Non-Newtonian FluM Mech. 66 (1996) 169-192
178
4
dv
FAM(t) = C , n ( H ) -~ ~ p a 3 d t or
4 DAM = FAM(t)/(67rtl, Uoa) = -- i-~ C~(H)e 2
(42)
where Cm(H) is the added-mass coefficient and approaches 0.5 as H--,oo. The added-mass coefficient Cm(H) can be obtained by evaluating the kinetic energy of the fluid [24] from the solution for 06 as
f
43,
C m ( H ) = - Js ¢
where S is the boundary of the domain and (4/3r0 is the volume of a sphere of unit Based on the boundary conditions, the non-zero contribution is from the surface of the For an oscillating flow, the added-mass force contributes only to D, (see Eq. (32)). The leading order solution for the Fourier mode of the tangential velocity, uo, boundary layer near the oscillating sphere can be found from the solution to Stokes' problem, i.e. the flow due to the oscillation of a plane wall:
radius. sphere. in the second
uo = UOR + iuo, = Uh(0){1 -- exp[-- e(R -- 1)1 cos[e(R - 1)1 + i exp(R - 1)] sin[e(R - 1)]} (44)
The axial velocity, u., in the boundary layer near the tube wall is of the form lg z =
$1zR ~ -
iu:l = 1 + [U, (z) - 1]{1 - e x p [ - e ( H + i [ U , ( z ) - l] e x p [ - e ( H -
r)] c o s [ e ( H - r)]}
r)] s i n [ e ( H - r)]
(45)
The vorticity on the tube wall, (w, is given by ( w = Ou: ~r r =/4 = (wR + i(w, "" --e(1 + i ) [ U , ( z ) -- 1] + O(1)
(46)
Matching between the velocities in the boundary layer and in the potential flow region can, in principle, be carried out in the entire flow field. However, a direct comparison between the results given by the asymptotic and finite difference solutions is cumbersome. The exception is along the line z = 0 or 0 = re/2 where the grid line is straight. Along this centerline, the potential flow velocity in the outer region, U,., can be evaluated numerically as o+ 1
U,.(r)= Ui,v(r,z=O)=-~z
(47)
=
In particular, we have U , ( z = O) = U,.(r = H ) and Ub(O = zr/2) = Uc(r = 1). The leading order asymptotic solution for the Fourier mode of the velocity in 1 < r < H and z --- 0 is u__(r, z = 0) ,-- 1 + [ U , . ( H ) - 1] × {1 - e x p [ - e ( H -
r)] c o s [ e ( H - r)] + i e x p [ - e ( H -
+ Uc(1){1 - e x p [ - e ( r + g,.(r)-
U,.(H)-
V,.(1)
1)] c o s [ e ( r - l ) + i e x p [ - e ( r -
r)] sin[E(H-- r)]} 1)] s i n [ e ( r - 1)]} (48)
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
179
Table 1 Comparison of DQs(H) between analytical and finite differencesolutions Solution
H 1.25
2
4
Analytical
73.555
5.970
1.960
Finite difference
74.986
5.964
1.980
As indicated by Eq. (47), the imaginary part of Uc(r) is zero to the leading order. The above asymptotic solution for u:, together with ~'w given by Eq. (46), can be used to describe the structure of the velocity field and to assess the correctness of the finite difference solution for the case of unsteady viscous flow at large E. It is noted that the high frequency asymptotic analysis is also applicable to a viscoelastic fluid whose constitutive relation is described by Eq. (11). At large ~, Eq. (12) gives [t ~ )q/;~2, so that the behavior of the viscoelastic fluid associated with viscous phenomena at high frequency corresponds to a Newtonian fluid with a Stokes number equal to (),1/22)~/2e as indicated by Eq. (ll).
4. Results and discussion
4.1. Validathm o f the numerical results
The computation is carried out in the (~, v/) domain using 401 grid points in the ~ direction and either 65 (for E < 1) or 129 (for E > 1) grid points in the v/direction for both Newtonian and viscoelastic fluids, and 65 grid points in the r/ direction for the potential flow for all values of ~. The parameter B in Eq. (24) is chosen to be two which gives A = 24 and C1 = 0.677. The parameter C2 used for stretching the grids in the r/direction is set to 0.38. The stagnation point in front of the oscillating sphere in the (~, r/) coordinate is located at ~ = .~0 = 0.25, A coarser grid system (201 × 65) was used to examine the dependence of the drag on the grid resolution; it gives a dimensionless drag D ( H = 2) = 5.9632 at steady state. Since the difference in D ( H = 2) between the coarse grid and the fine grid (401 × 65) is only 0.001, it is concluded that the fine grid system is sufficient for obtaining accurate, low frequency flow field and drag. In the limit E--, 0, the effect of the unsteadiness vanishes and the drag is determined by the quasi-steady value DQs(H): DQs(H)=DR(H,E)
as
E~0
(49)
The analytical solution for the Newtonian steady-state creeping flow has been reported by Haberman and Sayre [13]. It can be used in this study since the convection terms are neglected. The comparision between the analytical and the present finite difference results for Newtonian fluids is shown in Table 1. It is seen the finite difference solution performs satisfactorily for Stokes flows at a low frequency of oscillation.
180
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
An analytical result for Cm was reported by Cai and Wallis [25] for an inviscid fluid over a row of identical spheres in a tube. Fig. 2 compares the finite difference solution for Cm with that of Cai and Wallis for the case of a single sphere moving in a tube. Excellent agreement is observed, which indicates that the finite difference algorithm and grid system work very well for the potential flow. Fig. 3 compares the values for u:(r, z = 0) given by the finite difference solution and those given by the high frequency asymptotic solution, Eq. (48), for a Newtonian fluid at E = 64 and H = 2. It is seen that there exist two boundary layers: one near r = 1 and a weaker one near r = H. The velocity profiles in the boundary layers match smoothly those given by the inviscid solution. Excellent agreement between the finite difference and asymptotic solutions is observed. This indicates that the finite difference algorithm in the body-fitted orthogonal coordinate system, with the present set of computational parameters, gives accurate results even in the presence of thin Stokes layers. Fig. 4a shows the imaginary part of u:(r, z = 0), u:~, for E = 16, 32 and 64. A boundary layer of thickness O(E- ') on the surfaces of both the sphere and tube can be identified. Fig. 4b shows the enlarged profiles of u:, near the sphere at E = 16, 32 and 64 together with the asymptotic solution for • = 64 given by Eq. (48). A good agreement between the two solutions is observed within the Stokes layers. Outside the Stokes layers, it is found from the numerical solution that the values for u:,(r) are inversely proportional to • as can be seen from Fig. 4a. The behavior of u:~(r) outside the Stokes region is similar to that for u:R(r) shown in Fig. 3. This behavior is not surprising since Mei [18] has shown, for a rapidly oscillation flow over a stationary sphere in an unbounded Newtonian medium, that: (i) the leading order term of u__~outside the thin Stokes layer is zero and that of u:R is given by U~nv(r,z) (see Eq. (39)); (ii) the next-order term in the solution for (u-R + iu:,) is of O(• - ' ) and is induced by the viscous displacement effect 2
,
I
,
f
,
l
,
I
,
I
,
0
o Finitedifference + Cai & Wallis (1992) M Unbounded ÷
C
1
o
m
÷
0
0
0
0
I
I
i
I
i
l
2
3
4
5
H Fig. 2. Comparison of the added-mass coefficients, Cm, of the present finite difference solution and the analytical result of Cai and Wallis [25].
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
2.0
~
I
~
I
~
I
181
t
~=64 Finite difference
1.8
................... Asymptoticsolution 1.6
r 1.4
1.2"
1.0 0.0
~
~
,
0.5
1.0
1.5
U
2.0
zR
Fig. 3. C o m p a r i s o n b e t w e e n n u m e r i c a l a n d a s y m p t o t i c s o l u t i o n s for the real c o m p o n e n t o f the N e w t o n i a n fluid velocity o n the centerline, z = 0, for H - = 2 a n d E = 64.
within the boundary layers; and (iii) the curvature effect of the sphere contributes to (U_-R+ iu=1) only in higher order terms. The present numerical result for u=i(r, z = 0) shows qualitatively the same feature. The fact that u__~(r) has the same shape as U-R(r) means that the mechanism for generating u=~(r) in the outer region is of inviscid type. It was also observed that the numerical and asymptotic solutions for the real component of the Newtonian flow vorticity on the tube, ~wR, agree well at H = 2 and E = 64. The maximum values of numerical and asymptotic solutions for (wR are 16.99 and 15.95 respectively, and this O(1) difference quickly vanishes for Izl > 1. At a high frequency of oscillation, the good agreement between the asymptotic and numerical solutions provides convincing evidence that the numerical results are reliable in this domain. The fact that the numerical results also agree with the analytical solution at a low frequency of oscillation establishes confidence in the numerical solution. The behavior of the unsteady drag in Newtonian and viscoelastic fluids over a wide range of frequencies is discussed in the next section.
4.2. Unsteady drag in the Newtonian fluid As E increases, DR(H, E) increases and deviates gradually from DQs(H). Since DQs(H) does not depend on the acceleration, the difference between DR(H, E) and Dos(H) is denoted as the real part of the acceleration-dependent drag, i.e. D R A c ( H , e') :
DR(H, ~) --
DQs(H)
(50)
182
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
Fig. 5 shows DRAC for H = 1.25 and 2 for the Newtonian fluid case. It is clearly seen that DRAC(E) ~ O(E 4) for small E and finite H. As H increases, the region where DRAC(E) ~ O(E 4) decreases. As H--. ~ , the unbounded Stokes flow case is recovered and DRAC(6) ~ 0(62). The behavior DRAC(6) "~ 0(64) at finite H is caused by the limited diffusion in the r direction by the tube wall which forces the vorticity field generated on the surface of the ball to decay exponentially at a large distance away from the ball.
2.0
,
I
,
1.8
,
I
,
g ........ 16 32 ................... 64
1.6
r
I
i
1.4'
1.2'
~ - " "
10
.... i ..............
-0.6
-0.4
i
.......
-0,2
-0.0
0.2
UzI 1.25
,
t
,
i
,
)
I
O I
1.20
---o--- Asymptotic solution E=64
o 1
9, V 1.15
r
e=l 6 1.10
~
E
=32 ~
/ o~
1.05
l.O0
,
-0.6
-0.4
8
",
-0.2
-0.0
Uzl Fig. 4. (a) Numerical solutions for the imaginary component of the Newtonian fluid velocity on the centerline, z = 0, for H = 2 and e = 16, 32, and 64. (b) Asymptotic solution and numerical solution within the Stokes layer.
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
183
I0 2
10°
_
10 -2 c) 10 -4 I0 -6 10 -8
H = 1.25 10-10
.
10-2
.
.
.
.
.
.
.
I
10-1
.
.
.
.
.
.
.
.
]
10 0
.
.
.
.
.
.
.
.
i
10 1
.
.
.
.
.
.
[l
10"~
E Fig. 5. Effect o f ~ on the real part of the acceleration-dependent drag DR~,c in a N e w t o n i a n fluid for H = 1.25 and
2.
At large E, it is expected that DRAc(H, E) is proportional to E based on the following considerations: (i) DRAc(H--~ oo, E) is about equal to e even at finite Re; and (ii) the Stokes layer thickness is of O(E -~) and the surface vorticity on the sphere is of O(E). This asymptotic behavior is not established until E reaches a value of about 20. It must be pointed out that at much higher values of E the grid resolution near the sphere becomes a major concern. However, for practical purposes, E = 64 is beyond the normal operating range of the ball rheometer. Hence the grid resolution for the present computation is sufficient. The imaginary component of the history force, D~H, is obtained by subtracting the addedmass force DAM(H, ¢)= i(4/9)Cm(H)E 2 from Dr,
DIH(H, E) = DI(H, ¢) - DAM(H, E)
(51)
It should be noted that the Stokes solution gives D.u(E)~:E for H--,wJ. Fig. 6 shows D~H(H= 2, E) over a range of E. In the limit of large E, both DRAc(H , E) and D~H(H, ~) are linearly proportional to E because the wall vorticity is of O(E) on the surface of the ball which results from the Stokes layer of thickness O(E-~). Experimental results for the unsteady drag on a Newtonian fluid in a ball rheometer have been reported for E up to 0.25 at H = 2 by Tran-Son-Tay et al. [9]. It is found that DRA c (E = 0.25) ~ 5.5 × 10 -5 (Fig. 5), which is negligibly small in comparison with DQs = 5.97. The experimental results of Tran-Son-Tay et al. [9] are presented in the form of a correction factor KD for the Stokes solution, D R ( H ~ ~ ) = 1 + ~;
KD(H, ~)
DR(H, ~) I+E
(52)
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
184
10 3
. . . . . . . .
I
. . . . . . . .
,
. . . . . . . .
l
101
,
,
,
.....
,.-
10-1
I
-
. . . . . . . .
1
i
l
~.,-~
I0 -3
2
10"5
~
H =2
•
10-7
'
•
0-3
•
'
''''I
•
'
10"2
'
" 1 ' " I
'
. . . . . . .
I
10-1
"
H = 1.25 . . . . .
100
''I
•
'
'
' ' ' ' '
10 1
102
Fig. 6. Effect of • on the imaginary part of the history force D m in a Newtonian fluid.
Fig. 7 shows that the present numerical results for DR(H, E) compare very well with the measured data of Tran-Son-Tay et al [9]. However, values for the imaginary component of the drag, Dr, could not be compared. At E = 0.25, the numerical solution gives D~ ~ -0.0417, which represents only 0.7% of the values for the real component. This is a rather small value to be determined accurately with an instrument that has a much larger inherent phase error. ,
7.0
I
,
I
,
I
~
I
Finite difference
6.5 o
Measurement by Tran-Son-Tay et al. (1990)
6.0
5.5
5,0
~
4.5
4.0
0.00
|
i
i
i
0.05
0.10
0.15
0.Z0
0.2!
E Fig. 7. Comparison between the numerical solution and the experimental data o f Tran-Son-Tay et al. [9] for D R at H = 2.0 in a Newtonian fluid.
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
185
................... Newtonian l O2 .:
Viscoelastic
..".....
10 1
100
.
. . . . . . .
10 -3
i
10 "2
•
.
,
, . . . . i
•
,
. . . .
10 "1
..i
. . . . . . . .
10 0
i
10 1
. . . . . . . .
i
10 2
. . . . . . . .
10
E Fig. 8. Effect o f E o n D R f o r a N e w t o n i a n fluid a n d a v i s c o e l a s t i c fluid at H = 2 a n d 22/2~ = 0.5.
4.3. Unsteadydragin the viscoelasticfluid For small E, fl given by Eq. (12) is close to one and the viscoelastic fluid behaves just like a Newtonian fluid. The drag is determined by the "quasi-steady" value of the Newtonian flow. The effect of viscoelasticity is too weak to be noticed at small E. It is instructive to compare first the values of DR for viscoelastic (22/21 = 0.5) and Newtonian fluids at a given H. Fig. 8 shows that DR of the viscoelastic fluid is noticeably smaller than that of the Newtonian fluid for E > 0.1 and H = 2. While the Newtonian DR increases monotonically with E, the viscoelastic DR reaches a minimum at E ~ O(1). At large E, DR ~ O(E) in all three cases. The drag reduction in the viscoelastic fluid case for E > 0.1 is associated with an unsteady shear-thinning effect. This is clearly seen from the dependence of fl on E (Eq. (12)). The fact that DR for the viscoelastic fluid possesses a minimum indicates a competition between the decreasing effective viscosity and increasing wall vorticity associated with a thinner Stokes layer as E increases. To further illustrate the drag reduction in a Jeffreys fluid and the competition between the two mechanisms affecting DR, a sphere oscillating in an unbounded viscoelastic medium is considered. The dimensionless drag D can be easily found for the Jeffreys fluid: D-
l+2')tzE4+i(2'--22)Ez2E2{ 2+2,2~E-4 2 - - i ~1 - i
1
q_ 212~.4 [(22- 21)E2-F i(1
O1-2122{:'4)]
I' 1/2
(53)
The small E limit and large E limit of D are thus given by D=I+E--iE and
forE<~l
(54)
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
186
D=(22~/ze--i~(22y/2E+~¢2]
L\ZI
for e>> 1
(55)
For the Jeffreys fluid, 2~/2~ < 1, the real part of the high frequency drag, (22/2~)~/2E, is clearly smaller than the value of E for the Newtonian fluid. The presence of the tube does not qualitatively affect this conclusion. Fig. 9 shows the values for DR given by the analytical (H-o oo) and finite difference ( H = 2) solutions for 22/21 = 0.5. A small dip for the unbounded case is still observed near E = 0.6. It is interesting to note that DR reaches a local maximum first near E = 0.2 for the unbounded case before it starts decreasing. This behavior is not observed for the bounded case, H = 2. The lack of an obvious local maximum in DR in the bounded case is due to the suppression of the effect of acceleration by the cylindrical wall at H = 2. The extent of shear thinning strongly depends on the value of 22/2~ as seen from Eq. (55). Fig. I0 shows that the value of the local minimum for DR(H, E) decreases with decreasing 2z/2L. The Newtonian case is recovered when 22/2~ = 1 and the local minimum disappears. Fig. 11 shows the effect of 21 on DR(H, E) at e = 1 for H = 2 and various values of 22/21. The drag decreases as 2~ increases for a given value of 22/21. The largest decrease is observed for the smallest value of 22/21, which is consistent with the results shown in Fig. 10. At large 21, DR approaches a constant which indicates that the effective viscosity has reached its minimum. The imaginary part of the drag, DI, is in general very small compared to DR for small E. When increases, a local minimum for D~ is also observed for the sphere oscillating in the unbounded viscoelastic medium with 22/21= 0.5 (Fig. 12). The behavior of DI for a finite H is further complicated, as compared with that of DR, by the change of its sign as ¢ increases. The change in the sign of D~ indicates that the effect of shear thinning overwhelms the effect of inertia for intermediate values of E. At large E, the added-mass force DAM dominates D~, meaning that D~ ~ O(E2). Since the added-mass force is purely inertial, it does not depend on the constitutive model. Hence the expression of DAM for viscoelastic fluids is the same as for Newtonian fluids. 10 2
~ .......
~
........
~
........
~
........
I
........
j
. . . . . . . .
i
........
~
........
i
................... HU~2 (finite d i f f e r e n c e )
1o 1
DR
f .F" ......................................... ,,.....-o"" 10
0
.
I0"3
.
; .....
j
. . . . . . . .
10"2
,
10 °1
. . . . . . . .
,
100
. . . . . . . .
I0 1
10 2
103
£ Fig. 9. C o m p a r i s o n
o f D R f o r a v i s c o e l a s t i c fluid b e t w e e n H - - , ~
and H=
2.
R. Mei et al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
102
........
,
........
,
........
I
.......
~.2/~, i 0.50
~
..........
. ' d " ' ,s ."/,~ .~/,f~
] ] 1
.,.t,
1.00
10 1
....................... -2"/7
DR
~
. . . . . . . .
100 0 -3
I
10 "2
"
'
........
,,." ~./
0.75
' ' ' ' " I
/
. . . . . . . .
10 "1
I
100
. . . . .
!
'''I
187
I
. . . . . . . .
101
10
E Fig. 10. Effect o f ~ o n D e at H = 2 for v a r i o u s values o f 22/2~.
Fig. 13 shows that DIH-----(=D~-DAM) at large E for H = 2 and H---, oo, and 22/2~ =0.5. It is seen that for the Jeffreys fluid Dm ~ O(e). Fig. 14 shows the decrease of D~ with increasing 2~ for e = 1, H = 2, and various values of 22/2~. The change in the sign of D~ can be clearly seen as 2~ increases. The behavior of D~ at large 2, is quite similar to that of DR.
~,2 / ~. 1
i
0.7
5
..........
0.5 0.3
4 I\ I; | ",
D
R
i
3
,
I
\ \ 2
~
]
0
. . . . . . . . . . . . . . . . . . .
I
I
I
I
10
20
30
40
Fig. 11. Effect o f 2~ o n D R at H = 2 for v a r i o u s values o f 22/2~.
50
188
R. Meiet al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192 10
4
,
.
.
. . . . .
i
,
.
. . . . . .
I
.
.
.
.
.
.
.
.
i
,
,
,
,
.
,..i
,
e
. . . .
10 3
102 10 1
~
~
2
~
I
100 -DI 10-1 10-2
I
10-3 10
-4 10-2
.
.
.
.
.
.
.
.
I
'
"
''
. . . . .
I0-1
!
.
.
.
.
.
.
.
.
100
I
•
,
I0 I
10 2
E Fig. 12. Behavior of the imaginary component of the unsteady drag, D~, on a sphere oscillating in an unbounded Jeffreys fluid with 22/21 = 0.5. 4.4. Determination o f the rheological properties from measured drag data
The foregoing numerical analyses pertain to a linear Jeffreys fluid, but can easily be extended to a class of more general viscoelastic fluids. One of the main purposes of this computational study is to provide a means for characterizing the rheological behavior of viscoelastic fluids using the ball rheometer. In this section, a method to obtain the rheological properties from 102
~
,
,
,
,
,
, , I
,
.
Analytical solution (F_-~ o . ) .
.
.
.
.
.
.
. ~
.......... Finite difference ( H = 21~ . . . . . ~
-DIIJ1°1
100 10°
I
......
1
i 101
,
.......
102 E Fig. 13. Effect of E on DIH for the Jeffreys fluid at large c, H = 2, and H ~ ~ with 2~/21 = 0.5.
189
R. Meiet al./ J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
2
,
I
,
I
,
I
,
I
i
~-2 /~.1 1
'" -'ira
0
0.7 0.5
.....
-
.................. 0.3
\ ~" ......... ~'"""'"'"'"'""2:'"'""'-_"
-1
-2
0
.....................................................
I
I
!
I
10
20
30
40
k 1
50
Fig. 14. Effect of 2~ on DT for different values of ).2/2, at • = 1, and H = 2.
measured unsteady drag data and the numerical simulation is presented. It is emphasized that the assumption of a Jefreys model is not necessary in order to determine the unsteady drag from the ball rheometer data. The general constitutive equation for a linear viscoelastic fluid in the frequency domain can be written in dimensional form as = 2q*/5
(56)
where the complex viscosity, r/*, consists of a dynamics viscosity, q', and an elastic modulus, G': q* = q'(a)) + iG'(~o)/co
(57)
Experimentally, it is possible to measure the steady viscosity, r/o, and the c o m p o n e n t s of the unsteady drag, DR and D~, at a given co [9]. Using rio and Dlov/a 2 respectivel3) to normalize q' and G', and keeping the same notation to represent the dimensionless variables, it follows that the reciprocal dimensionless viscosity fl-I (see Eq. (12)) is given by E2
/] -- r/'E 2 -t- iG'
(58)
One can start searching for/~ by guessing a pair of values for r/' and G'. For small E, r/' = 1 and G ' = 0 are good initial starting values. As E increases, one makes initial guesses for r/' and G' based on the values already found at the nearest E. By solving Eq. (16), one obtains numerical values for the vorticity, and unsteady drag can be obtained by intergrating the stress along the surface of the sphere. The real and imaginary c o m p o n e n t s of the unsteady drag are given by
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
190
\ - ~ - + - R - ) R =, d0'3 sin 20 d0
1G' fo~ [fo° (eC' + ~
R/.=, d0'3 sin 20 d0
1 r/' 3
e 2 Jo (,IR sin 2 0 dO =~ fo(R[R= ,sin2 0 dO + 3 £(" 1 d0']
Dl=-6~'fo~r[fo°(~(l+~)R=~-R
\0R
6 e"2 -lr/'3
(59)
sin 20 d0
R JR= 1 dO' sin 20 dO
(IIR = lsin2 0 d0 - 5 ~5
(RIR =, sin2 0 d0
(60)
If the numerical results for DR and D~ agree with the measured values, the guessed r/' and G' values are accepted. If the calculated values are too high or too low, a new pair is selected to correct for the difference. This iterative process continues until the values for the calculated and measured drags agree. The development of a linear viscoelastic model is then possible after r/'(o~) and G'(~o) are obtained over a range of o9. To this end, it is necessary to examine the sensitivity of the drag (DR + iDa) on r/' and G'. Fig. 15 shows the dependence of DR and D~ on G' with H = 2 and q ' = 0 . 5 . Fig. 16 shows the 10 1 I 00
........
~
........
t
........
t
........
'.......................o ..........o-o .................... - . - . - o - . . . - . . . - o . - - . . . . - . - . - . - . - - . . . . . - . ~ .
10 -1
D . . . . .
103
/
-
rl = 0.5, e = 0.01 H=2.0
10 -4
.
10"8
.
.
.
.
.
.
.
I
10"7
.
.
.
.
.
.
.
.
I
'
10"6
G
"
"
'
'
' ' ' l
10"5
i
Fig. 15. Effect o f G' on the unsteady drag at e = 0 . 0 1 .
.
.
.
.
'
" ' '
10-4
R . M e i e t a l . / J. N o n - N e w t o n i a n
6
,
I
n
5
D 21
,
I
,
/ ~ ~ " J l ~
D I
[
=
•
0.0
I
191
66 (1996) 169-192
G'= 1.0e-8, H = 2 /
3
-I
i
0-'- D R ........° .......
4
o
I
Fluid Mech.
...............................• ................................• .................................................• ..............• ...............
i
I
i
I
0.2
0.4
0.6
0.8
1.0
Fig. 16. Effect of ~/' on the unsteady drag at E= 0.01. dependence of DR and DI on r/' with H = 2 and G' = 10 -~. In both cases, e = 0.01. It can clearly be seen that DR depends mainly on I/' and is insenstive to G', while DI depends mainly on G' and is insensitive to r/'. Similar behavior is observed at E = 1. Thus, 1/' and G' can easily be related to the measured DR and DI through numerical simulations. The principal difficulty lies in the accurate measurement of the unsteady drag (DR + iDi).
5. Conclusion
Numerical analyses for a sphere oscillating along the centerline of a cylinder at small Reynolds numbers are presented for Newtonian and Jeffreys fluids over a wide range of frequencies. The unsteady drag and the corresponding flow field are examined based on the finite difference and asymptotic solutions. The effect of the cylindrical tube wall on the unsteady drag is elucidated. At small Stokes number ~, the drag is dominated by the quasi-steady value for both Newtonian and viscoelastic fluids. At intermediate E, the effect of shear thinning on reducing the drag is observed and the dependence of the drag on the frequency becomes complicated. At large E, the added-mass force, which is the same for both Newtonian and viscoelastic fluids, dominates the imaginary component of the drag. The history force is proportional to E for large e. The presence of the wall reduces the low frequency history force and the real and imaginary componenets change from O(E) in the unbounded domain to 0(~"4) and O(e 2) in the bounded case respectively. A method for obtaining the viscoelastic properties of a fluid from measurements and numerical simulations of the unsteady drag is proposed. It is found that the real component of the drag depends strongly and monotonically on the dynamic viscosity ~/' while the imaginary component depends strongly and monotonically on the elastic modulus G'.
192
R. Mei et al. / J. Non-Newtonian Fluid Mech. 66 (1996) 169-192
Acknowledgement R.M. acknowledges the support of the Engineering Research Center (ERC) for Particle Science & Technology at the University of Florida, the National Science Foundation (EEC9402989), and industrial partners of ERC. J.X. and R.T.S acknowledge the support of the National Institutes of Health through Grant 4 RO1 HL49060.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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