An exact similarity solution is given for the flow of a fluid-particle suspension over an infinitely

large disk rotating at a constant velocity. Numerical solutions of the resulting ordinary differential

equations provide velocity distributions for both fluid and solid phases and density distributions

for the solid. The boundary-layer thicknesses of the particle cloud and the fluid are found to be approximately

equal. In addition to its intrinsic value as a solution to a physical problem, the results provide a convenient basis for judging the accuracy of approximate techniques

Zung, Laurence B.

English

Caltech Authors - Main

Reprinted from Printed in U. S. A .. 
THE PHYSICS OF FLUIDS VOLUME 12, NUMBER 1 JANUARY 1969 
Flow Induced in Fluid-Particle Suspension by an Infinite 
Rotating Disk * 
LAURENCE B. ZUNGt 
Daniel and Florence Guggenheim Jet Propulsion Center, Karman Laboratory of Fluid Mechanics and Jet Propulsion 
California Institute of Technology, Pasadena, California 
(Received 18 March 1968; final manuscript received 21 October 1968) 
An exact similarity solution is given for the flow of a fluid-particle suspension over an infinitely 
large disk rotating at a constant velocity. Numerical solutions of the resulting ordinary differential 
equations provide velocity distributions for both fluid and solid phases and density distributions 
for the solid. The boundary-layer thicknesses of the particle cloud and the fluid are found to be approxi-
mately equal. In addition to its intrinsic value as a solution to a physical problem, the results provide 
a convenient basis for jUdging the accuracy of approximate techniques. 
I. INTRODUCTION 
Because of the complexity of fluid-particle flow 
problems, most analyses in this field involve pertur-
bation schemes1 •2 or approximation methods.3 As a 
consequence, exact solutions of problems in two-
phase flow take on special interest. 
One example of such exact solutions is the flow 
over an infinitely large disk rotating at a constant 
velocity about an axis perpendicular to its plane. 
Von Karman 4 first used the integral method to 
investigate a similar problem of an incompressible 
viscous fluid. Using numerical integration, an exact 
solution was obtained by Cochran.5 
The layer of two-phase medium close to the 
rotating disk is carried along by the disk. Owing to 
the centrifugal forces, this layer is being transported 
outwardly towards the outer edges. This, in turn, 
is compensated for by the flow along the axial 
direction. Thus, the axial velocities of the fluid 
and the particle cloud do not vanish far away 
from the disk, but tend to a constant value. On 
the surface of the disk, the radial velocity of the 
fluid vanishes, while the radial velocity of the 
particle cloud usually attains its maximum value. 
The flow field exhibits very well the centrifugal 
separation of particulate matter from fluid as well 
as the characteristics of a secondary flow of a two-
phase medium in a three-dimensional boundary 
layer. 
* This work constitutes Chapter I of the author's Ph.D. 
thesis, California Institute of Technology (1967). 
t Present address: Dynamic Science, Monrovia, California. 
1 F. E. Marble, in Proceedings of the Fifth AGARD Com-
bustion and Propulsion Colloquium (Pergamon Press, Ltd. 
London, 1963), p. 192. 
IF. E. Marble, AIAA J. 1, 2793 (1963). 
a J. R. Kliegel, in Ninth Symposium (International) on 
Combustion (Academic Press Inc., New York, 1963), p. 811. 
4 Th. von Karman, Z. Angew. Math. Mech. 1,233 (1921). 
6 G. W. Cochran, Proc. Cambridge Phil. Soc. 30, 365, 
(1934). 
18 
II. BASIC EQUATIONS 
A half-infinite space, z > 0, is filled with an in-
compressible fluid containing small particles of a 
single size with radius IT. The boundary plane, 
z = 0, is rotating at a constant angular velocity w. 
Using cylindrical coordinates and taking account of 
the rotational symmetry, we easily observe that the 
flow quantities are independent of the angle 9. The 
governing equations describing the flow field of a 
two-phase medium are discussed in detail in Ref. 1. 
Denote the radial, angular, and axial velocities of 
the fluid by u, v, and w; the mass density by Pi 
and designate quantities associated with the particle 
cloud by subscript p. The continuity equations for 
the two phases are 
o 0 
- (ru) + - (rw) = 0 
or oz ' 
The corresponding momentum equations are 
ou v2 ou 
u---+w-
or r OZ 
UOV+UV+WOV 
or r OZ 
uow+w ow 
or OZ 
(1) 
(2) 
(4) 
= _! op + 1I(02!!!, + ! oW + 02W) + ! F (5) 
P OZ or2 r or OZ2 p" 
FLOW INDUCED BY ROT ATING DISK 19 
(6) u. = u" - u, ' 
v, = v" - v, (15) 
u av" + u"v" + w av" = _1. Fo, 
" ar r " az p" 
(7) w. = w" - w. 
The momentum equations for the particle cloud 
(8) can be expressed as 
where p is the local gas pressure and F" Fo, and F. 
are the forces in the radial, angular, and axial 
directions exerted on a unit volume of fluid by the 
particles. 
Using a linear drag law as a first approximation 
to describe the particle-fluid interaction Fr is 
(9) 
where T has the dimension of time. In the case when 
Stokes drag law is a valid approximation, T assumes 
a simple form 
T = m/67rf./.u, (10) 
where m is the mass of a single particle. Similarly, 
Fo and F. are 
(11) 
(12) 
The boundary conditions for the fluid properties 
are 
u = 0, 
u = 0, 
p = p",. 
v = rw, 
v = 0, 
w = ° for z = 0, 
for z = CX), (13) 
As was mentioned previously w does not vanish as 
z -7 CX), but tends towards a constant value. The 
boundary conditions for the quantities pertaining 
to the particle cloud are 
p" = Kp 
w" = w for z = CX), (14) 
for z = CX), 
where K is the density ratio for z = CX) and is equal 
to a constant. 
ill. SOLUTION FOR THE EQUILmRIUM FLOW 
From the definition of T, Eq. (10), the limiting 
case when T -7 ° corresponds to either very small 
particles or the viscosity of the fluid f./. is very large. 
Under such circumstances, the velocity difference 
between the particle cloud and the fluid is expected 
to be very small. This is the case when we examine 
Eqs. (6), (7), and (8). Define 
au v2 au u 
u - - - + w - = --'- + o (u.) , 
ar r az T 
(16) 
u av + uv + w av = -~ + O(v.), 
ar r az T 
(17) 
aw aw w. 
u - + w - = -- + O(w ). 
ar az T 8 (18) 
The continuity equation is 
(19) 
Since both the velocities and their spatial derivatives 
are finite, the limiting case when T -7 ° requires 
u. -7 0, V. -7 0, and w. -7 O. Thus, Eqs. (16)-(19) 
are reduced to 
au v2 au u. 
_u" - u (20) u---+w-= --"= 
ar r az T T 
uav+uv+wav= v. v - v (21) -- = -~
ar r ar T T 
aw aw w. w" - w (22) u-+w-= --= 
ar az T T 
From the continuity equation we observe that p" = 
const = Kp. 
The momentum equations for the whole system, 
in the limiting case when T -7 0, can be expressed as 
au v2 au 
u---+w-
ar r az 
(23) 
(24) 
(25) 
where 
p* = p + Pv( CX) = p(l + K), (26) 
p* = fJ./[p + Pv( CX)] = f././(1 + K)p. 
20 LAURENCE B. ZUNG 
The solution for T ~ ° is, therefore, identical with 
the flow without particles with an effective density 
p* = p(1 + K). This limiting case is usually referred 
to as the "equilibrium flow." 
In another limiting case when T. ~ 00, the presence 
of the particle cloud does not change the fluid 
quantities. The particles are moving with constant 
velocities u" = 0, v" = 0, w.(r, (0), and p" = Kp. 
This case is usually referred to as the "frozen flow." 
IV. METHOD OF SOLUTION 
To investigate the problem for finite values of T, 
it is convenient to introduce a dimensionless distance 
variable 71. 71 is defined as 
(27) 
dG" dQ 
H"Q d;; - 2QG"F" + H"G" d71 
- (3(Q2G" - Q3G) = 0, (36) 
H! ~~ + 2H"QF" - (3(Q2H" - Q3H) = 0. (37) 
The boundary conditions can be obtained from Eqs. 
(13) and (14). They are 
F(71) = 0, G(71) = 1, H(71) = ° for 71 = 0, 
F(71) = 0, G(71) = 0, P(71) = ° for 71 = 00, 
F,,(71) = 0, G,,(71) = 0, H,,(71) = KH(71) (38) 
for 71 = 00 
Q(71) = K for 71 = 00 
By expressing the fluid velocity components and (3 is defined as 
pressure in the following form: 
u(r, z) = rwF(71), 
vCr, z) = rwG(71), 
w(r, z) = (wv)!H(71) , 
per, z) = per, (0) = pwvP(71) , 
(28) 
defining a relative density Pr = pp/ p, and expressing 
the particle velocities and density as 
u,,(r, z) = p~lrwF,,(71), 
v,,(r, z) = p~lrwG,,(71), 
w,,(r, z) = p~1(vw)!Hp(71)' 
Pr(r, z) = Q(71), 
(29) 
the governing equations (1)-(8) are reduced to the 
following form: 
a:: + 2F = 0, (30) 
~2~ _ F2 _ H ~: + G2 + (3(Fp - QF) = 0, (31) 
d2G dG 
d712 - 2FG - H d71 + (3(G" - QG) = 0, (32) 
a:: - 2HF + 2 ~: + (3(-H" + QH) = 0, (33) 
dH" + 2F = ° (34) d71 " , 
(3 = l/wT. (39) 
Asymptotic solutions which are valid for large 
values of 71 were first obtained. Functions F(71), 
F,,(71) , etc., were evaluated for some large value 
of 71 = 710 using the asymptotic solutions. Numerical 
integration was carried out from the point 710 to 
71 = 0. The asymptotic solutions contain three 
parameters. Using an iteration procedure based on 
Newton's method, the boundary conditions on the 
disk, namely, F(O) = 0, G(O) = 1, and H(O) = 0, 
were satisfied by varying the three parameters. 
Numerical integration was performed with an IBM 
7094 computer using the DEQ subroutine. 
Two sets of solutions were obtained by varying, 
respectively, the particle loading K and the particle-
fluid interaction parameter (3. 
v. DISCUSSION 
The fluid velocitv profiles are shown in Figs. 1 
and 2 for various particle loading K. Figures 3 and 
4 show the corresponding particle velocity profiles. 
Several observations can be made based on these 
solutions. The flow field is very much like Prandtl's 
boundary-layer type. The boundary-layer thick-
nesses 8 and 8" for the fluid and the particle cloud 
are approximately equal. The distances from the 
wall over which the angular velocities of the particle 
cloud and the fluid are reduced to half of their disk 
values differ by less than 10%. Their values are 
close to (v / w)!. For small values of 8, the radial and 
angular velocities of both phases have appreciable 
values only within a layer of thickness of the order 
(v/w)t. Close to the disk, Up usually attains its 
maximum value while u approaches zero. This is 
FLOW INDUCED BY ROTATING DISK 21 
1.0 
.8 
v rw (1("02.5) 
.6 
.4 
.2 
o 2 3 4 5 
~WIV z 
FIG. 1. Dependence of fluid velocities in tangential and axial 
directions upon particle loading ({3 = 0.5). 
simply because of tbe large centrifugal force exerted 
on the particle cloud. The value of Up very close to 
the disk depends on the interaction parameter 
{3. Figures 5 and 6 show the particle velocity profiles 
for various particle-fluid interaction parameters {3. 
For (3 equal to 0.2, the nondimensionalized velocity 
up/(wr) has a value of 0.12, while for (3 equal to 
1.0, up/(wr) is close to 0.3. This is attributed to the 
difference in angular velocity of the particle cloud; 
v.Jewr) is 0.24 and 0.62, respectively, for {3 equal to 
0.2 and 1.0. For large values of {3, Up on the wall 
.2 
u 
«rwl .I 
o 2 3 4 5 
'iWIV Z 
FIG. 2. DependencQ of fluid velocities in radial directions upon 
particle loading ({3 = 0.5). 
1.0 
0.8 
Wp 
~w v (1(01.0) 
o 2 ;, 4 
~WIV z 
FIG. 3. Dependence of particle velocities in tangential and 
axial directions upon particle loading ({3 = 0.5). 
again decreases, and for infinitely large values of {3 
when there is equilibrium flow, Up approaches zero 
on the wall. 
The change of particle density and fluid pressure 
for various particle loading K is shown in Figs. 7 
.3 
.2 
.1 
2 ;, 4 5 
\fW7i1a 
FIG. 4. Dependence of particle velocities in radial direction 
upon particle loading (fl = 0.5). 
22 LAURENCE B. ZUNG 
1.0 
Wp 
\fW""V (13-0.2) 
.8 
o 2 3 4 5 
\J WIll z 
FIG. 5. Dependence of particle velocity in tangential and axial 
directions upon interaction parameter fJ (K = 0.5). 
• 3 
o 2 3 4 5 
~ WIll z 
FIG. 6. Dependence of particle velocity in radial direction 
upon interaction parameter fJ (K = 0.5). 
.4 
.2 
o .1 .2 .3 .4 .5 
\jWllI z 
FIG. 7. Dependence of particle density upon particle loading (fJ = 0.5) . 
.5 
\lWllI Z 
FIG. 8. Dependence of pressure upon particle loading 
(p = 0.5). 
FLOW INDUCED BY ROTATING DISK 23 
1.2 
1.1 
1.0 
c~ 
M7T 
.9 
.8 
.7 
.6 
.5L---~----~----~--~~--~ 
.5 1.0 1.5 2.0 2.5 
K 
FIG. 9. Dependence of coefficient of turning moment upon 
particle loading for different interaction parameter /3. 
and 8. The pressure attains its maximum value on 
the disk. The two phase mixture is thus flowing in 
a region with an adverse pressure gradient. This is 
contrary to the solution obtained by Cochran,s since 
he has a sign error in the pressure distribution. 
It is noteworthy that w" does not vanish on the 
disk. The problem is solved with the boundary condi-
tion that the particles stick to the disk or that the 
amount of particles reflected from the disk is neg-
ligible. This latter assumption is well justified, since 
from the solution we observe that w" on the disk is 
indeed small except for very small values of {3. 
Furthermore, the axial velocities are of the order 
(VW)l and at any rate are small. 
When the radius R of the disk is large compared 
with the boundary-layer thickness, the boundary 
conditions along the edge of the disk can be neg-
lected. The solution for the infinite disk can be used 
in this case to evaluate the turning moment 
M = iR 27rr2Toz dr 
of the disk. Define eM = M / (!pw2 R5) as the coeffi-
cient of the turning moment of the disk; eM can 
be easily calculated and is equal to 
eM = - ddG I IRe, (40) 
'I) ~~o 
where Re is the Reynolds number defined as Re = 
R 2w/v. eM increases for increasing values of {3 and K 
as shown in Fig. 9. 
ACKNOWLEDGMENTS 
The author is indebted to Professor Frank E. 
Marble for his guidance and helpful suggestions. 
This work was supported by Aerospace Research 
Laboratories, Office of Aerospace Research, under 
Contract AF33(615)-3785. 


Flow Induced in Fluid-Particle Suspension by an Infinite

Rotating Disk

http://authors.library.caltech.edu/21482/1/195_Zung_LB_1969.pdf

Flow Induced in Fluid-Particle Suspension by an Infinite Rotating Disk

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