Cone rotating in newtonian liquid which has a free surface

(1)

>WÍIÏCH HAS A FREE :SÚRFACE

!.

t/;/

_I'f7

Contract No Nonr-591(2O)

Rensselaer Poljtèchnc Institute'

t J

TroyNw York

ft .51 Y j/t 1/

;stember 1966

t by I tt ( ,tS -K. Hsü 1 í.: ç. ! and--?.f J 't rt tS ç t ' ' t

Ioppit

11

j ìj

t, I' i Ii t ¡'S.. 1' / T ;S. j t/

' Office qf Naval Researc1

r. _.'

DI'STRIBIJTIÒN OF T}tISrDÖUMENT IS UNLIMITED

...,

L. ,. r '. . .' S .S I. .1' tI . -t

Lab

y. -.

Techrnscke Hogschoo

. ft .

CONE ;RQTAT 1-NG IN NEWTONIAN LIQUID

:i:--,

.,t. ''Sn ;; -t S, t . it t L S.

¿ç

( -'',L ' 't ' 't- . -,

(2)

CONE ROTATING IN NEWTONIAN LIQUID

WHICH HAS A FREE SURFACE

by

Y. K. Hsu

and

W. H. Hoppinann II

Office of Naval Research

Contract No. Nonr-59l2O)

(3)

ESTRACr

The present study represents a generalization of a preiiously studied problem of the laminar flow induced in a Newtonian liquid by

a rotating cone [ill. In the latter investigation the cone was

con-sidered to be closely fitted into a cylindrical container which held

the liquid. In such a case, there is no free surface. Further study has since been made of the nature of the flow for the case in which

the radius of the container is larger than that for the top of the

conical rotor. In fact, the flow has been studied for several ratios,

, of the radius of the container to the radius of the top of the rotor.. For the special case previously studied, was limited to the

value unity.

Using flow visualization techniques and flow timing devices,

velocity functions which very precisely define the actual flow have

been developed. The functions are such that the center line of the

vortical motion is correctly located and the flow about it is

accu-rately described. The velocity boundary conditions and the continuity equation are satisfied.

The stress field in the liquid has been determined from the

veloc-ity functions. Using these stress functions, an equation relating the total torque on the cone and its angular velocity has been developed so that it agrees with experimental data.

(4)

The nature of the pressure field is discussed in some detail.

Alio, with the aid of experimentally determined pressures and theoreti-cally dêtermined shearing stresses the total resultant thrust on the

coné was calcUlated. This thrust was then compared with that measured

with a special dynamometer.

INTRODUCflON

In 1963 the design and performance of a rotat-ional fluid flow

generator used for the study of the behavior of Newtonian and

non-Newtonian liquids was describe4 in the literatute [2]. Subsequent-ly,

the velocity functions which describe the flow generated in a. Newtonian liquid were developed [1]. In that study; it was assumed that the

radius of the top of the conical rotor which moves the liquid is equal

to the radiUs of the cylindrical cOntainer. Bcause of this assumption, an essential singularity is introduced into the kinematical

specifica-t4.on of the problem and thereby frusträtes any attempt to obtain a completely satisfactory analytical description of the flow field.

Be-cause of the fact that the generator has been very useful in studying

flÒw ôf both Newtonian and non-Newtonian liquids it was considered desirable to make a study of the flow for the case in which the radius of the cylindrical container is larger than the radius of the top of

the conical rotor. If the radius ratiO is called , it is necessary to

(5)

The purpose of the present report is to present a detailed study of the nature of the flow for Newtonian liquids for several values of

greater than unity. In such cases, the liquid has a free surface. A sketch of the cone and container is shown in Fig. 1.

EQUATIONS OF MOTION

Because of the nonlinearity of the Navier-Stokes equations, no exact

solution for the problem exists. The present report makes use of

ex-perimental observation and theoretical analysis to determine velocity

functions which accurately describe the flow. The flow is assumed to be steady, the liquid Incompressible and at constant temperature.

Based on these assumptions, the equations of motion and the continuity

equation in cylindrical coordinates are as follows [3]

equations of motion: 2

V+

....1+V...E

L_

--r

r

p

r

+

ø=_

vcc2v

r

_r

z

_r

_p

_r

ø

r2

y

v

y

rr

+ - i

+ y -

= -

+ y

y

r

p z where

V2

=À2! 11

r

Ø

(1)

(6)

and equation of continuity:

r

r 6Ø z

In order to nondimensionalize these equations the following

rela-tions are introduced:

R=--,

0=0, z=--=-- -,

= r1 z0

r0tan

Z=Ztan, r

= (r0 tan cZ)Z = z = r0hZ V V Vs. r z UR

= r00'

= h

22

= p(r,z)/(ic0/h2) , k r0 V

The. Reynolds number is proportional to k, and is defined as

RN = k/h2 = r02Ç20/v

Using these relations and the assumption of axisymmetric flow

Eqs. (1) and

(2)

are written in the following nondimnensional f orni:

)4L.

z

(7)

equations of motion: u

--+u

=-

-+

+h2[::R+1

_--i}

R R R z k L _R _R2 _R R R2 equation of continuity: u _UR _uZ = 0 (6) R R Z

Since no known solUtion for these nonlinear equations exists, it

is proposed to develop reasonable series representations for the velocity functions and then use the Navier-Stokes equations for calculation of the pressures. Since the equations are ònly of the first order in the

pressure and the velocities will be represented in approximate but

deter-minate form, the pressures can be obtained by quadratures, although the

process will be very tedious and requires a high speed digital computer

for the numerical work.

+

_=i....h2[

s $ R Z k L R2 R R R2

+-i-{h2I

Z+_!_.

u2uz}

R R Z h2k Z k

LR2

R R -' _Z2 J -h.

ou

+

(5) 1

R

(8)

BOUNDARY CONDITI ONS

Since there is a no-slip condition on the solid boundaries, the

velocity on the cylindrical surface and the bottom of the container is

zero while the velocity of a fluid particle at the surface of the coni-cal rotor equals the velocity of the rotor at that point.. On the free surface the pressure should be specified.

The velocity boundary conditions may be analytically expressed as

follows: at R = i (cylindrical wall) UZ = UR = = O at Z = O (bottom) UZ

= U -U0

= O at Z = i (free surface) (7) UR = F1(R,l) u0 = F2(R,i) Z

=0

at Z = R (conical surface) = R UZ = UR = O

(9)

where in general,

m=O n=O

DEVELOPMENT OF VELOCITY FIJNCrIONS

Since there are no known solutions to the equations of motion (5) for the steady flow problem under consideration, there will be developed in the following analytic expréssions for the velocity which will very

closely satisfy all of the essential requirements. The components may be broken down into two natural classes; the components UR and u which

correspond to trajectories in meridional planes andthe component

u0

which is always perpendicular to a meridional plane. First

an

analysis

will be made of the velocity components uR and u.

A. Determination of Velocity Functions uR and

It was found experimentally that in any meridional plane through

the

axis of

rotation, a Ø equal constant plane, the resultant velocity

component at a point in that plane is tangent to a trajectory which is

one of a

family

of closed curves as shown in Fig. la. The components of

that resultant velocity in the plane may be taken as UR and u. This

family may

be mathematically represented by:

r(r,z) constant (8)

mn

a r z

mn (9)

The constants a

may be determined so that the actual flow is properly

(10)

It can be shown that if the velocity functions UR and u are defined as follows: E P UR R E r R R

the continuity equation will be satisfied as in the classical case of stream functions. However, it should be noted that E is not a constant

but a function of r which in turn is a function of R and Z given by Eq. (9).

The equation of the boundary:

Z(Z-R)(l-R)(Z-1) = O (li)

suggests that it is reasonable to take r given by Eq. (9) in the form:

X X X X X

r = -(Z-R)

1 z 2(1-R) 3[P(R,Z)] + (Z -R )_C _C Z_C2(l-RC)

3[F(R,Z)1

(12)

where Z and R are the coordinates of the vortex center the X's are

C C ,

constants and F(R,Z) is a function to be determined so that UR and will describe the observed flow on the free surface. Then it can be

shown that the boundary conditions on the velocity are satisfied. It

is assumed that the surface tension and the gravitational force can be

(11)

as was determined to be approximately the case for the observed flow. Using the definitions given by Eq. (10) an4 Eq. (12) the velocity

func-tions u and u then are:

E r

uR=-;.;

=_L{(Z_R)1 zX2l_R3 F(R,Z)

+ F(R,Z)(Z-R) z 2 (1-R)

[ÏZ+2()]jL

(13) E F Uz E i 2 A.3 F(R,Z)

= - (Z-R)

z (l-R) R L, R '1 + F(R,Z)(Z-R)

l-R)3[ -X3Z+ (A.1+A.3)RA.1]}

(14)

It seems appropriate to assume:

F(R,Z) = (Z-l) f(R,Z)

= (Z-1) :i:

Rm ₍₁₅₎

m

The determination of a large number of values of a would require an mn

(12)

UZ 1 ?.

r

(Z-R)

L

_2(1R)

_(Z-l)

_{2R-Z-$+ [-)3Z +}

R L

)R_Xp1}

(20)

On the free surfacé,

Z=1,

u vanishes identically and _uR

will

bé deter-mined in accord with experimental résults.

Furthermore, observation shows that UR and u must vanish at the

the function F was chosen as:

F(R,Z) = (Z-R)(1-R)(Z-1) (16)

Differentiating F(R,Z) with respect to Z and regarding R as constant, then differentiating F(R,Z) with respect to R and regarding Z as con-stant, we obtain

= (i-R)(2Z-R-i) (17)

= (Z-1)(2R-Z-)

(18)

Substituting the valves of F/Z and F/R into Eqs. (13) and (14)

respectively yields,

UR = -

(Z)1

z2 (lR)3

{z2zR_l+ (Z_l)[1Z+?2(Z-R)i}

(13)

(Z-l)[X1Z + X2(Z-R)] + (2Z-R-i)Z

= O

and

+ (2R-Z-) = O

Solving these equations for X2 and X3 in terms of X1 yields,

Z (2Z -R. -l)Z

(Z-R)

(Z-l)(ZR)

(R -1)

(2R -Z -)

X

X,+'

C

(Z-R)

1

(Z_R)

All X's should be such that there is no singularity in the velocity; that is:

X1-1 > O ,

X2l > O

, X3-1 > O

It should be recalled that UR and vanish on the solid portion of the boundary.

Furthermore, all X's should'be such that the theoretical streamlines

agree with the experimental findings.

B. Determination of the Velocity Component

u0,

It is considered that the velocity function u0 may be represented in series form as fôliows:

(14)

where the coefficients A are to be determined from known conditions Inn

of the problem.

It can be shown that all of the boundary conditions except at the

free surface are satisfied. That is, u is zero on the cylindrical surface and on the bottom of the container. Also u matches the cone velocity at the surface of the cone where Z equals R.

As a first approximation, to satisfy the known flow conditions at the free surface, only the linear terms in the series expansion are used and they are defined as follows:

A10 = a1 , A01 = a2 , A00 = a3 , A11 = O

Thus Z(l-R) + Z(Z-R)(l-R) ( -Z) 1 1

YA RmZn=aR+aZa

¿ ,' mn 1 2 3 m=O n=0

m

n A

RmZn

Inn (23)

(15)

METHODS OF DETERMINATION OF TI-lE UNKNOWN PARAMETERS

IN THE

VELOCITY FUNCtIONS

1. The Determination of the

)t5

The values of ? may be determined from a knowledge of the position

of the vortex center. The follòwi:ng is a description of the experimental method used to locate the center.

Small water particles were injected into the flow domain with a hypodermic needle. Using two micrometer screws, the coordinates of the

particle with reference to the top and side of the cylindrical container

were determined.. Water particles moving in a circular path about the

axis of rotation of the cone are at the vortex center. For castor oil as the liquid and two values of , the coordinates and angular velocity ratio íI are given in Table I. The Ç' is' the ratio of angular velocity

of a particle at the vortex center to the angular velocity of the cone.

Table I

Coordinates of Vortex Center and Angular Velocity Ratio

r Z _R Z Z

mm

nmi

mm

1 16.74 44.12 53.11 19.38 0.835 0.436 0.303 0.438

(16)

It may be noted that the coordinates of the vortex center for

equal to 2.3 are different from those for equal to 1. The vertical position z is changed little, but the radial distance is shifted about

18 m.Tn. farther away from the axis of rotation for the larger . The

angular velocity at the vortex center for equal to 2.3 is 0.188 revo-lution per second, and Ç' , the ratio of angular velocity of vortex

center to angular velocity of conical rotor, was found to be 0.282. The ratio was determined for an angular velocity of cone of 40 r.p.m.

With theaid of Eqs. (19) and (20) and using the relation uR = u O at the vortex center, we have from Eqs. (21) and (22)

Z (2Z -R -1)Z

=

(Z-R)

1 -

(Z-1)(ZR)

(R -1)

(2R -z -)

c c

(Zc_Rc)

(Z-R)

For castor oil as the liquid the values of R and Z are shown in

c c

Table I. Assuming a suitable value of ?, to obtain the best agreement

of the theoretical velocity trajectories with experimental results the

corresponding values of and X3 can be obtained.

(24)

(17)

x2 = O.6443X1 + 1.4238 = l.739X1 + 0.739

and it turns out that for best fit to expérimental data

X1 = 6

x2 = 0.6443 x 6 + 1.4238 = 5.2896

and X3 = 1.7396 x 6 + 0.739 = 11.173

For convenience, in subsequent analysis, assume = 6, X2 = , X. = 11.

The developed velocity trajectories are shown in Fig. la and except for scale factor they agree with the actual flow pattern as shown in

Fig. lb.

2. Determination of E

The unknown E can be determined from tle following consideration.

The time required for a liquid particle to travel around a vortical

surface as i rotates about the cone axis can be considered as the circu-lation time about the vortex center an it is given by the relation:

P

-

ds

Li

where ds is the differential path length on a r equal to a constant

curve. The velocity y is the magnitude of the vector sum of the velocity

components y and y of a particle which moves on that toroidal surface

r z

of which the r equal to a constant curve is a meridional section.

(18)

The quantity E varies with P and the angular velocity of the cone. Its determination depends on the fact that a given particle of liquid always moves on a particular toroidal surface. Consequently, the parti-cle moves not only around the fixed circular axis which coincides with the vortex centér in the liquid but also around the axis of rotation of the cone.

The circulation time t given by Eq. (25) can also be related to the average angular velocity, w , of the particle as it rotates once about

the fixed circular axis. The subscript

a.

refers to a particular surface defined by the parameter P. The time corresponding to thïs single

revolution is: where

t

(J V i 2

21/2

2 2

21/2

y

= (Vr

+

y

) , or

y

= (uR h u r0 1/2 ds = (dr2

+

dz2) , or ds = (dR2

+

a2h2dZ

a=

r1

Nondimensionalizing Eq. (26) yields

2t

_,rds

(J

J

1/2 r1

(19)

Le t 1/2 (dR2

+

a2h2dZ2) = Ti 1/2

+ h2u2)

r0 UR UR , uz E

Substituting Eq. (28) in Eq. (27,) gives

f

(dRl±ah4Z)

2 2_2hh/2

E(UR ± h u

)

For a particular surface defined by

r,

E is a constant, and hence we

ob tain E 1

i

Ç (dR2

+

a2h2dZ2) rl 2ic 2

2 2_2112

O

_{(UR +hu)}

r0 (29)

It was found experimentally that the average angular velocity of the particle about the cone axis,

a

, is related to the average

angu-lar v1ocity cu of the particle about the fixed ciru1ar axis as

follows,:

U)a.

(20)

where R is a parameter depending only upon r. The maximum angular

velocity of the cone for the velocity range studied is Also it was

experimentally found that is related to the angular velocity of the cone as follows:

= G

(31)

i

where G is a function of given as follows:

T(R,Z) r(Rc,zc)

The above two parameters Ra and G can be determined experimentally by

measuring the time required for the particle to travel once about a

trajectory defined by r, and by counting the number of times the parti-cles move abòut the cone axis while moving once about the plane trajectory. For equal to 2.3, the experimental results are as shown in Fig. 2 and

Fig. 3.

Using Eqs.. (29), (30) and (31,) the parameter E is given as follows:

Ra (1)G(F)

E=

2t E 2

22 21/2

R + a h dz

1/2 r

2 2-2

O

[u.n +hu]

The above integral can be numerically evaluated by using the derived velocity functiOns UR and u. For each

F,

there is a corresponding velocity trajectory. The function E can be found from Eq. (32).

(21)

Function E is given approximately as fo11s:

ç0

103.77

Therefore t-le velocity components UR and are as follows:

u = -103.17 (-) R Ç2 R u 103.77

(2)

(Z-R)4Z4(l-R)8(Z-1)

_12z+19R_7}

(33) Ç R s

3. Determination of the a's

There are three undetermined coefficients in the approximate velocity

function u0, and these can be determined by satisfying three flow

condi-tions. These conditions are that u0 match the measured velocity on the

free surface, at least at ne point., the measured velocity at the vortex center and in addition such that the torque equation is satisfied. The determination of the vortex center was previously mentionEd. The torque and the velocity on free surface cOnditions will now be developed.

First, in order to calculate the torque on the conical surface of

the rotor it is only necessary to use the shearing stress component In spherical coordinates this function is given as follows [3]:

1 e 1

j

cot9

- V r-sjn 9 $ r* Z-R)4Z3 (l-R)9 {(Z_l)(llZ_5R)+ 2Z2-(R.+ l)Z} (34)

E=

Z-R)2Z(l-R)3

r0 (See Fig. 4)

(22)

where r* is

because the dimensional

used as the symbol for the radial spherical coordinate usual symbol R is used in this paper to designate the

non-ratio r. r1

where

ds = sec a dr (38)

Integration of Eq. (37) gives the torque

For the present investigation of steady axisyetr.ical flow, which

is independent of

_5,

the stress equation becomes simply:

T L(

cote

e$ _r* _e

Making the transformations

r = r* sin S , z = r* cos S ,

5 = 0

=

IL

-2 T es _r _r v,)

r*

" èos *J , (3.5) (36)

where * is an angle and not to be confused with the radius ratio .

On the cone surface equals the angle a. The torque element is given as follows:

dM = [Teø]

z=r tan a

(23)

Since

u Z(1-R) + Z(Z-R)(l-R)(a1R + a2Z + a3)

0

_(p-Z)

the final form of torque becomes:

M sin a cos a

sii2a { l a2 + a + loge(l_a)_ 2[(a1+ a2)( -2tpÇ0r1 r1

+ a3( - )]} =

2a{1

-L.

a2 a - log )]

M sin a cos a 27LQ0r13

(_ -

)sin2- cos2a

r1 R R 5 6

+

[(al + a2)(

- -) + a

5 6 5 5 2 dR (39)

where is the angular velocity of the conical rotor, 14 is the

coef-ficient of

viscosity,

r0 is the radius of the top of conical rotor, is the radius of the cylindrical container, a is the reciprocal of ,

which is the same as previously defined.

For three different values of the torque was measured by a

sensitive transducer, described elsewhere [2]. it is statically

(24)

The torque-angular velocity relations based on measurement are

shown in Fig'. 5. Now the velocity on the free surface will be defined.

The experimental determination of the velocity on the free surface

was made with a special reference frame and a stop watch. Very small solid particles of low density were placed on the free surface and their motions were carefully observed and timed. It is considered that the

techniqué was hgh1.y successful for the purpose. Óbservations show that

the velocity trajectory is a spiral curve which may be represented very

closely as follows:

-k(Ø-Ø0)

r = r0 + (r1-r0)e [1 + k(Ø-Ø0)]

where r0 radius of the top of conical rotor, r1 = radius of the

cylindrical container, k and

0 are constants.

The component of velocity y is taken zero at the free surface, 'in accord with observation. The remaining two components y0 and Vr should

be such as to agree with the experimentally determined velocity of a

particle on the free surface.

In order to determine experimentally the velocity of 'a particle on the free surface it is necessary to have an equation for distance

travelled along curve (40). The differential arc length ds is given ás follows [4]:

(25)

(45)

v=

hrn Ls

ds

(42)

t-4Ot

dt where d 2 -k

_(Ø-

_{øû)(ni_ r)e}

(43) dØ

and ds/dt is the magnitude of the instantaneoùs velocity of a fluid particle on the free surface. It may be approximated as follows:

s =FL

(44)

2 1/2

F = (r2

+ ()

d

Ø may be chosen as small as we please. For a given Ø, F can be

calcu-lated and thus ¿s may be obtained. The time t may be measured by a stop watch, the ratio ¿s/it would give the average velocity of the fluid

¿s ds

particle on the free surface, and the

um

= the instantaneous

velocity.

(46)

where r is the angle which the radius vector makes with the tangent, r* is the angle which the polar axis makes with the tangent, and $ is

a given angle. On the free surface, the radiál component of the velocity

Vr

equals y cos ir, and the tangential component of the velocity v equals

(26)

It should be noted that the angle fl* is given by the following relation [4]:

tan P* (dy/d$) f($)cos $ + f' ($)sin $

(dx/dØ) -f($)sin $ + f'($)cos Ø

f($) + f'($)tan $ -f($)tan 0 + f'($)

tan r* = -f'($)/f($) if cos = O

Solving for f'(Ø) gives

f($)(1 + tan

r*

tan $) = f(Ø)cot(r*-0) tan fl* - tan Ø Using Eq. (46): = f($)cot _iV thus tan ill = f($)/f'(Ø) (48)

The velocity on the free surface has been determined for two dif-ferent values of B at the same angular velocity of conical rotor. The

if cos Ø O

(27)

Therefore, corresponding to the three flow conditions the following

three equations for the a1, and a3 are:

u

Z(l-R)

()

= C C

+ --

Z (Z -R )(l-R )(a1R + a Z + a3)

RR=R

C

R(-Z)

R

CC

c

C C

2c

C C C C Z=Z C u (l-R1) (-i)

=çì=

R R=R1

R1(-1)

R1 Z=Zl (l-R1) (l-R1) (a1R1 + a (49)

and the torque equation. The values of the a's will be calculated for

the case in which equals 2.3, equals 40 r.p.m., and the liquid is castor oil, having a viscosity coefficient, , equal 8.70 poises. The

vortex center is given by R equal 0.486 and Z equal 0.438. The value of Ç' is 0.282 and that of is 0.03896. For the cone under study r0

is 2.5 inches and the internairadius ofthe container, r1, is 5.75 inches. The radius in the free surface at which the velocity is

con-sidered, R1, is 0.826. The torque, 14, is 1.05 ounce-inches.

Using these numerical values, Eqs. (39) and (49) give the following three linear equations för the determination of a1, a2, a3:

0.486a1 + 0.438a2 + a3 = -0. 105479

0.826a1 + a2 + a3 = 0.606027

(28)

Solving the above set of simultaneous equations:

a1 = -5.5704

a2 = 4.6360

and a3 0.57120

Thus, the velocity component u becomes

Z(l-R)

+ z(z-2.3R)(l-R)(-5.5704R + 4.6360Z + 05712)

(2.3-Z)

(50)

THEORETICAL DETERNINATION 0F PRESSURE

As previously mentioned, once the velocity functions have been

determined the pressure function may be obtained from Eq. (5) by

quadra-tures. The first and third equations are the only ones that contain P. Since the velocity functions which have been developed are only

approxi-mate, the p obtained from the two different equations will be probably

somewhat different from each other. For convenience, they may be desig-nated and

2

Therefore, after integration the functions and

(29)

and p2

=j.Lç

+ h(

2

-UR _i 0U UR R R R 2 2 =

Jth4(

+

h2 R dZ+ F2(R)

(51)

where F1 and F2 are arbitrary functions of integration.

The integrations, while obviously straightforward, are long and very

time consuming. With a digital computer program set up the calculations could obviously be carried out to any reasonable degree of accuracy. For

the pressure on the bOttom of the cylindrical container the integrals are very simple to evaluate because of the vanishing of most of the terms for

Z equal to zero. For equal 2.3, curves showing p1, p2 and the average of p1 and p2 are given in Fig. 8. Measured pressures are shown for the purpose of comparison.

(30)

EXPERIMENTAL DETERMINATION OF PRESSURE

The pressures on the surface of the cone were measured by special

pressure transducers which were embedded in the cone. The electric signals from the transducers were conducted from the cone, along wires attáched to the rotating shaft, through terminals immersed in circular troughs of mercury which were fixed to the frame of the flow generator.

The measurement of these steady state signals was made with a very

sensitive wheatstone bridge. I-n addition, pressures were measured also over the fixed surface of the different containers. There were nine stat-ions for pressure measurements for equals 1.1 and twelve stations

for equals 2.3. The pressures on the surface of each container were measured by means of wall taps and inclined manometers. Pressures on the surfaces of each container and cone were measured, at nearly constant temperature, around 24°C., and at the same angular velocity, 40 r.p.m. The pressure results are plotte4 in Fig. 9 and Fig. 10. It can

be seen that the pressure on the surface of thé cone for equal to 1. 1

rises markedly in the inward direction away from the periphery of the

cone to about R = 0.75, then drops to zero and further decreases toward the axis of rotation. The pressure on the cylindrical wall has a similar

distribution to that on the cone surface but the pressures are of small

magnitude. However,, on the bottom of the container, which corresponds to the plate in the cone-plate viscometer., the pressure rises much less

rapidly as the axis of rotation is approached. The influence of the container .size can be readily seen. For equal to 2.3, the pressure on the cylindrical wall drops rapidly at Z equal to about .9, then

(31)

in-bottom, the pressure drops rapidly toward the axis of rotation. The

pressure on the conic surface was measured at equals 1.1, 1.55 and

2.3. As increases the pressure decreases. For equal to 2.3, the pressure is lower than atmospheric pressure at each of the three measured points, and it has approximately a parabolic distribution, which can be

represented as follows:

r2r2

p=-b(x-)

(J:)

r1 r0

THRUST ON THE CONE

On the cone, the thrust consists of the vectorial sum of shearing force along cone generator ánd normal force perpendicular to cone

gener-ator. The shearing stress acting along cone generator is given by:

V

1 r*

e

*

= L( -

+ - - - )

r* 9 r* r*

where r* is the radial vector of the.spherical coordinates as previously

defined.

The Vr*

ve, y0

are the velocity components of a fluid particle in

spherical coordinates;

r' v

are the velocity components in cylindrical coordinates.

Since y0 and Ve are perpendicular to the cone generator, they do not give any contribution to the shearing stress along the cone generator.

(32)

The magnitude of ve/r* and of Vr*/9 are obtained as follOws. Making the transformation:

r=rsine, z=rcose, ø=$

r* r*

r*cose

+r*sine

Ve

- =sine+cose

The velocity components Ve and Vr* are expressed in terms of and

y as follows: z y

=v sina- y cosa

e r z y

=v cosa+v sina

r* r z

and the corresponding nondimensional form of y8 and are:

U8

_{=uRsinauZcosa}

U

=U

cos+u sina

r* R Z

The corresponding nôndimensional form of Vr*/8 and v8/r* are:

r* r* R . Z r* . .R Z

-

= - cos e(cos a - + sin a -) + - sn e(cos a - + sin a -)

(33)

r* r1

Ue j 1

sin 9(sin a - cos a-) + - cos 9(sin - -. còs

z0

From the previous results:

n -103.77(--.) ì R s

$R)4Z3(lR)9 {(Z!.l)(1iZ-) + 2z2

- + i 1O3.77(-)

(zR)4Z3(1-R)8(Z-1) {.lZ + l9R -

_{Th }}

R

It can be seen that since UR and u have the factor

(Z-R)4

Ve

=0 at Z=R

8 r*

As a result

8r* equals zero along the cone generator. If the shearing stress vanishes along the cone. generator, the thrust will be determined solely by the àxial compOnent of. the integrated pressures. The. thrust was

measured by means of a precision thrust transducer, which is described

elséwhere [5]. It was developed at Rensselaer for the purpose of con-ducting experiments of the type under discussion. It is statically

calibrated for each experiment and is foúnd to be highly reliable in

performance. The small electric signals from the transducer are con-ducted through a mercury trough to a precision potentiometer. The

electrical arrangement for thrust measurements is the same as that used for pressure measurement. The results are plotted in Fig. il for three

(34)

Now the experimentally measured thrust may be compared with that obtained from the integratedpressures. The differentialeletnent of

thrust is:

dT = 23trp sin

a

ds

where p is the pressure either obtained from the integration of the Navier-Stokes equation or established from the experimental

investiga-tion.

From

experimental results, at = 2.3, _p can be expressed approximately as follows: r

2r

2 p = -b(x -

(J)

r1 r0 where b = 625(dynes/cm2) r1 r1

r0 and r1 are the same as previo.usly definéd. Substituting p into the thrust equation and integrating gives

r 2

r02

T = -2tb tan a

f

(i)

(x -

r dr

LO

r0 r1 it 2

=--br

tana

6 0

(35)

tan a = 0.7 , b = 625(dynes/cm2) , r0 = 6.35 cm

T = - X 625 X (6.35)2 X 0.7 = -9160 (dynes)

6

at

o

equal to 40 r.p.m , the measured thrust is -7800 dynas, the calcu-lated thrust T is larger in magnitude than the measured one. If we take T to be 7800 dynes as reference, the calculated one is about 17 percent higher.

It may be interesting to consider the average shearing stress that

would be required to obtain perfect agreement between the measured thrust

and the one calculated from the measured pressures. It was shown that the shearing stress component along the cone generator calculated from the analytically developed velocity function is zero.

From the previous results we have

T +T =-7800

p r

where T is the calculated thrust T is the thrust that would be

con-p , ¶

tributed by the shear, thus T = 9160 - 7800 = 1360 (dynes) but:

f

sin a ro

m

sin sec a 2 r dr o 2

= it

o tan a . r0

(36)

where r0 = 6.35 cm , tan a = 0.7 , and hence,

1360

2 = 15.45 (dynes/cm2)

3.1416 X 0.7 X (6.35)

In a similar way, the average pressure on the cone surface is found to be approximately 100 dynes/cm2. Thus it is seen that even if the

whole difference between the thrust calculated from the experimental pressure data and the measured thrust were caused by the existence of a

shearing stress in the direction of cone generator, the magnitude of the average shearing stress. is only about l57 of the average pressure acting on the conic surface of the rotor and hence a relatively small quantity.

It should be noted that the vanishing of the shearing, stress r8

along the cone generator is a consequence of the particular form chosen

for the velocity functions. While the velocity functions were obtained so as to satisfy fundamental physical facts and therefore approximately

describe the velocity field, the stress field obtained therefrom may not

have the same degree of accuracy. Thus the vanishing of shearing stress along the cone generator as obtained in this paper is open to some

ques-tion. However, the evidence is considerable thàt even if it does not

(37)

DISCUSSION AND CONCLUSIONS

The kinematic, difficulty which arises from the ssupti9n in the

previous studies [1] that the cone generators intersect the cylindrical

wall of the container was overcome by taking the ratio of radius of the cylindrical container to radius of rotor, greater than unity. The study of such free surface conditions showed that velocity functions could be developed which describe reasonably well the major physical features observed in the fluid. The velocity functions which are ob-tained in this paper are considered to be a close approximation to the

required solution of the Navier-Stokes equations because of the fact that they describe satisfactorily the major physical features of the

f1 ow.

The approximate solutions of the pressure equation can be seen to

contain no singularity as exists when equals unity, the case for which there is no free surface. The calculations of pressure everywhere pn

the boundazy except on the bottom of the container require an extensive high speed computer program. It is considered this study should be

carried out in the future. However, it is considered that to obtain

good agreement between theoretical values and measured values more

terms must be assumed in the infinite series expansion for the velocity

function u0. However, it can be seen that for the bottom of the

cylin-drical container, the pressure function reduces to a function of R

while reduces to a constant, therefore there is a discrepancy between

and p.

The ieason for this is that the velocity functions are not

(38)

the exact solutions to the Navier-Stokes equations. Furthermore, it can

be seen that the average of the two pressure functions can be made to

agree as closely as desired with the experimentally determined pressures by the proper choice of the function of integration F(R).

The torque-angular velocity relation is also free from the

singu-larity which occurred in the original derivation [1] for the case of equal to unity. The measured torque for three different values of

shows that the torque varies linearly with angular velocity, and de-creases as increases.

The shearing stress calculated from the velocity functions is zero on the surface of the cone so that in suàh a case the total thrust is determined solely-by the integrated normal pressure. The discrepancy between the measured thrust and the integrated experimental pressure was

shown tobe only i77. Probably, the sheàring stress along cOne genera-tor is very small but not exactly equal to zero. It should again be emphasized that the vanishing of the shearing st-ress along the cone

generator is a consequence of the particular form chosen for the velocity

functions. The average shearing stress calculated so as to obtain

agree-ment between the measured thrust and that calculated from the measured pressures shows that the magnitude of shearing stress along the cone

generator is small. The thrust angular velocity relation is approxi-mately parabolic.

(39)

that it agrees with measured velocity at various points in the liquid. The theoretical velocity components tiR and u developed in this paper describe the Observed flow in meridional planes very satisfactorily. The developed velocity functions accurately describe the shape of the observed velocity trajectories and properly locate the vortex center.

It was found experimentally that when increases, the pressure, torque and thrust decrease. In the limiting case for which equals

unity [1], the theoretical values of pressure, torque and thrust become

unbounded. Also, it can be seen that from the measured values that the pressure builds up rapidly on the cone near the free surface as

ap-proaches unity. It is considered that this is a very significant factor if the rotor is to be used in a viscometer.

(40)

REFERENCE S

Miller, C. E. and Hoppmann, W. H. II, "Velocity Field Induced in a

Liquid by a Rotating Cone," Proceedings of the Fourth International Congress on Rheology, Part 2, pp. 619-635 (.1965).

Hoppmann, W. H. II and Miller C. E., "Rotational Fluid Flow Generator

for Studies in Rheology," Transactions ofthe.Societyof Rheology,

Vol. VII, pp. 181-193 (1963).

3.. Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Pergamon Press,

Addison-Wesley Publishing Company, Inc., pp. 51-52 (195.9). Courant, R.,. Differential and Integral Calculus, Interscience Publishers, Inc., pp. 262-266 (1953).

Hoppmann, W. H. II and Baronet, C. N., "Apparatus for Measuring Thrust Produced on Rotating Body Immersed in a Liquid," Journal of Applied Mechanics, March, 1965.

(41)

CYLINDRICAL

CONTAINER

CONE

I I

"2

3"

5'

FIG.I

DIMENSIONAL CROSS SECTION OF FLOW

CHAMBER WITH A

350 ANGLE CONE

(42)

0.7 0.6 0.5 0.4

NO3

0.2 0.1

LUID FREE SURFACE

(Z-ßR)Z(I-R)F(R.z) r.i (ZR)z(I-RrF(R,z) 0.1 0.2 0.3 0.4 OES

06

0.7 0.8

R+

FIG. la. VELOCITY TRAJECTORIES IN

A ØNCONSTANT PLANE FOR

(43)

Fig. lb.

Photograph o.f Velocity

Trajectories in a

Ø Equal Constant Plane

for

a =

(44)

r

FIG. 2 R,(r) AS A FUNCTION OF

r

o

0.2

0.4

0.6

0.8 LO

(45)

ii

o

I I I I I

i

î

S2ai = GÇ20

I

î

I

0.2

0.4 o

0.6

0.8

I.0

04 I

I-I

I

0.2 FIG. 3

G(r) AS A FUNCTION OF r

(46)

8

6

4

2 DOTTED CURVE

: FITTED CURVE

SOLID CURVE

: ACTUAL CURVE

o

-0.2

0.4 r

0.6

0.8

1.0

FIG.4

E

versus r

(47)

3.00.

I I I I I I I I I I

20

40

60

80 ANGULAR VELOCITY IN

R.PM.

FIG. 5

TORQUE AS FUNCTION OF ANGULAR

VELOCITY

(48)

24

6

8 lo

2 r

(cm)

FIG. 6 TANGENTIAL COMPONENT OF THE

(49)

FIG.7 RADIAL COMPONENT OF THE VELOCITY

ON FREE SURFACE AT .2c, EQUAL TO

4Orp.rn.

(50)

20-o

('J

E

0

-20-CI) cl)

>

D

40-LU

J

60-(J)

C/)

u

80 /3=2.3

P2

_100A--'

0:2

o.4

o.8

;

.Li.O

/3=2.3 -.(

/3=2.3

F

O

EXPERIMENTAL VALUE OF P

CALCULATED VALUE OF P

CALCULATED VALUE OF P1

D

AVERAGE VALUE OF 9 AND P2

/3=2.3

F)

X-FIG.8

PRESSURE ON THE BOTTOM OF

THE CYLINDRICAL CONTAINER AS A

FUNCTION OF R FOR £L EQUAL TO

40 rp.rn. AND FOR TWO DIFFERENT

(51)

400-FIG. 9

PRESSURE ON CONIC SURFACE

AS A FUNCTION OF R FOR £?. EQUAL IO

40 rp.m.AND FOR THREE

DIFFERENT

(52)

FIG.IO PRESSURE ON CYLINDRICAL

WALL OF THE CONTAINER AS A

FUNCTION OF Z FOR QIOEQUAL TO

40 r.p.m. FOR TWO DIFFERENT

(53)

16

U)

C

>

24 H

(J,

D

a:

I

H

32

40 ANGULAR VELOCITY20 .(r.p.m)

20

40

60

80 lOO

T ¡ I

:1

I

o /3=2.3

¿S /3 = 1.55

-

x /3=1.10

I I i

FIG.II THRUST AS A

FUNCTION OF

(54)

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Palo Alto, California

Dr. W. J. Jacobs Lockheed-Georgia Co.

Marietta, Georgia

Dr. J. Harkness

LTV Research Center

Ling-Temco-VoUght Aerospace Corp. P0 Box 5907

Dallas, Texas 75222 Library

-The Marquardt Corporation

16555 Saticoy

Van Nuys, California 91409 Dr. B. Sternlicht

Mechanical Technology Inc. 968 Albany-Shakér Road

Lathàm, New York 12110

Nr. P. S. Francis

North Star R & D Institute 3100 Thirty-Eighth Ave. South Minneapolis., Minnesota 55.408

Ocean Systems

North xnerican Aviation, Inc. 12214 Lakewood Blvd. Downey, California 90241 Dr. B. N. Pridinore Brown Northrop Corp. Norair-Div. Hawthorne, California 90250