On the theoretical description of the flows of dilute polymer solutions

(1)

15 SEP. 1972

Biblio[heek van

fl/I?.. _Onderafde

ee

Paris, 1972

Ya. i. Voitkounsky, Anrfilokhiev, V.A. Pavlov.k

oivß

polymer solutions.

It is known, that there exists an unusually low frictional resistance in the turbulent flows of the dilute polymer so

lutions.

This phenomenon was discovered by Thorns tre then

20 years ago and attracts

last time more and

mare attention of the scientists.

There are plenty of research rks in this field, but there does not exist pheical theory, which could be

able tç

give full explanation why such sharp dacreaBe of the frictioA

1mg takes place after uts

inffic a nt additteves of polymers

were introduced in the flow

of the water.

The flows of dilute

polymer solutions ehould De considered as anomalous from the

Of

point of vtei'Nav ru-Stokes equations because all main ph.ei' cal rhwracterietice of liquida which are maintaining in tbee

equations, such as aensity and viscosity ara practicall.y the aa as for the pure

wa;er.

The main question in tne problem of description of diu-

-lute polymer solutions flows is

tttothe

equations of the

fluid itiona . Ono of the methods to get solut..øfl of

DCUMENTAHE DocuMENTÂTI

(LeningraQ shipbuilding Instituts, USSR)

On the theoretical

cescription of the flows of dilute

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

Deift

UAI UM: _g _{oicí, 1973}

(2)

x'cble

to be cozside'ed i

to conatrct tne estea of

ua-;lona on tkie uase of molecular kinetics.

B'it i; is very ti.

f icult to use

thia way oecause of reqiremen; to

now t

l.cu1ar structur. of the ciilute 2olymer aolutn.

i

macn ire onvilent to asgtme,

that

lt is oossib.e to

con-e: the Dclyrner solution as trie isotropic continuum.

!r. ;rlic

ags it aou1d b

worth wnile to taze into ccnsiaeratlon

e11 Imown projerty of such solutions thìttheir reialatloL

time te a few orders higner then that f pure water

L i

-:t ieaas to tt. idea tO constrUct

the

local rneologicai

re-tions using the superposition principie.

!aking into account all the considerations

oove

nectix-ed the rhsological relation for ailute poimer solutions wa

-Thuria as

L-'

wnre

2 - lB

he stress tecaor,

S

is the sraii. rate tensor,

- is ;ne pressure,

- ta the iynamic viscosity cceIfìcien;,

t,t- are the tim., iiid.o..naezic var].aLe6,

anca

are rheo1ogLcaJ cnaracterietics: trie

re_a-xattoì fucc;.ori Z:

i )

cci tn. relaxaio

ibe vaiuee of both

8 a

8 snould be otainecì as

e

egulT.5

f experiments au are depending upon the sort

po

(3)

lyrzier ad solution concentration.

The

de1 of the fluid which co:respondes to the relation (1)

belors

to the mechanics of hereditary substances, because

it takes into account the time connection

between the stress

and strain-rate tensors, i.e. influeÌce of

the previous con

-dittois in tne flow on its following state.

The continuuizi

hich corres oonds to tnis mol rves

s

'is

ordinary rewtoniai f1ud when

-

1

a:;o :

flow is

dif-ferent from 1ewtonian one wren

T

Substituting the co

onents of stress censor mn;o

en-tlons of the

tion being exresse

irterrs stresses

it

is possible to obtain the following

equations, Uescribing the

movement of the viscous fluid

with relaxation

roperties

vp

+fe

wnere

V1 are the comporents of

e

elociie

vector

(i : 1, 2, 3) i

trie rec;angular Cartesian

coo:di-nate eyete,

f - are tne components

cf trie vector o: tLC niass

f:r-ccc,

- je the

irematic VisCOsity

- is trie aenety,

y - is ;te

rlamiiton opertur en

- 18 the Laplace

operdtcr.

r.e o.n ce frct tus

t:o:

ta;

r. ncri-te

13'

tiri tI'

ferec

c:weerL

re ïa:er

er...

itite xnei

aclut:ro.

(4)

This difference

takes lac in the t'iroulent flows,

which

are non-'stationary themselv',s and in such laminar flows

wbere _,t O.

dt

To prove the last conc].ueicn a special experiment bas

been made.

In this

experiment the puls.ttng

laminar f lowe

of the pure water

and the dilute 1utiona were thveãgat&..

In the experiment the

average

discharge of the

liquid through

the round cylindrical tube was measured, The diameter of the

tube

was 118 mm. ¶here was a short rubber section in the

tube

that could be aquee.ed and

unclasped

with the def.nite

frequency

cù As the result the pressure gradIent and the

dis-charge

along th. tube iere changing as

the functions of time

t

and longitudinal coordinate z.

The experimental resulta which had

been

obtainel for the different solutions of

po]yox

were

represented in the figi.

They re plotted as the relation againat volume

con-centrations

values C

of polymer for

different Reynolde

im-oera Re. Fiere is tha average discharge of the polyox

solution and Q te the L1ircharge of the purs water i.e.,

when C z O.

lt te

eea from fig. 1 that the influence of

the pulsations takes place upon the rate of in the Q

laminar, flow of polyox solution When the concentration of

the solution is 'growing and ita relaxatiQfl properties are

changing, the ratio is growing ton, and has reached

o#w*c4

its maximum, th

locatiäTng the C-axis depends

upon the sort of polymer. To the right of the point of its

(5)

maxi-ia ratio decreases. lt te the infkuencs of the ta.

creaeir.g

'iscosity of the solution.

hta experiment confirms the theoretical conclusion, that

;iere la lift erence iii the properties between the pure water an polymer solutions

in xxnatationary laminar f lows, i.e.

as

«zien

ng equatlon.a (2) it is possible to obtain the equa

:iona cescribirig trie average turbulent motion of the fluid.

Ir the case wnen qussiatationary turbulent flow le conetde rei tesa ecuatlone can oe written as

.

,,ç

¿9i)

;:

- I

-'

i

(3)

nere i, j ,

t i seen, that in th. f lowe being considered the turbu-' ient st'isses consist of t terms. The first on. le well kn

-eynolds stress, the second repreeente the relaxation

Q

.rert1ea of the liquid

r

e

:t is necessary to take into account the.. additional eee.s

1.n the turbulent flows of polymers Th.y are the same

StreB-see the itrPlusnce f which the coneiderabis change. in the

flow structure of d ilute polymer solutions in comparison with the pur. water oue.

!he additional term in the equation (3) provides ui.oreti 'cal explanation tu the well kirn fact, that the aun of the

(6)

*primentally u1easura R

nold

and Newtoui&a mear

sre-s in the iiow of polyer.-o i

not equ1 to the measurea

full ariear airees in such flow

if

ne additoi. ehar

stress

T

taten into account the

take piace.

To siuûy trie aoditional turbulent

tsees it is

nses-gary to investigate the lomentum8

in the flow.

It would he interesting to iake attemptsto get using

th

equations (3) solution of the problem of the

turbulente flow

aiong the infinita flat wall. Such problem for ordinary vis'

cous fluid was solved by

Prandtl

1 4 1

If the y axis is cho-sen perpendicularly to hsi,a1l in

me casa being considered the

equations

) shall be tranel'

d to

C"

(4)

In the laminar sublayer of the flow,

ia. near the wail,

i.ne velocity profile is

l!biiear.

Out of that eublayer, nag

ìecting the viscous stress, we canuse

theequation

-jU'#i4''U'= cl,

where

- represents the

full friction stress on the

wall.

To get solution of the problem it is

usefül to apply

the Prandtl's idea, which allows to connect the pulsating

(

i ,

'

) and average

(l/ ) velocities in the turbulent

(7)

¿I"

Zft'

t'

rìer

t

a co at ant, i t

nitLaie wa f od in

i's

emirical theory

f turbulence a

Trvin in the zone of fu1' develop-es turbulent flow the

ro file ccn be obtained

frodi the

.. ¿1/_I ¿1,_e,

/

, ,a,,

2 i,-.

dq3'Y

"-e boufliary cr'1tiris

re

s fo'ìow:

13

.ne

ontant whicn valua

the 3.mi

eirical tneory of turbulence shouLd o.

ta-a-w

., =i -;:;--

_q

y -'

integral of

qua;ion (7) satisfying bu.ndary conitions

' cn De dritten in non-uinaionl form ea

T

obtain uon-dimensiorja], runction

to solve the equation

J-Jp3

_:

_L'/;

_V'"

(')

(7)

(9)

wriere

ÇP

arid

7

are zne non-dimenelonal

ve-13c1ty and distance fc

trie waLl eccoriingly

weIkoe

(8)

where

a

anu

i

are

the e3Ipirical constante,

can be called as "the dynamical lengtb

The for.ilae (9) and (10) give us the profile of velocity in the turbulent zone of the flow takir. into aecou.nt the existence of time connection between the stress and strajn rate ten8ars.

The analysis of equations (9) nd (10) shows us, that tb.

profile should be considered se logaritb.mical st o dia-tance from the wall.

rt

coula be seen that the velocities distribution ('7) aepende on both the

rel*zatioafunc-tion and the iynaLtcal length ¿

It gives na opportunity to explain the so called diame-ter effect" which takes place when theflowe of polymer so-lutions having the same conc4rations are inve.tigst.d in

the tubes of different diameters. Using th. dynaaical

l.igth

it is possible to get a new nOn-dimensional nit.hsr

This number might be considered as the additional

similari-ty parameter whicfl is necessary to use or the explanation

of "diameter effect." The structure of this parameter could re obtained by means of s.na1ysie of the last term in equa

-tion (2). in the fig. 2 ere indicated the resulte of

calw-lationa in wuicb expressions (9) and

(10) w used.

The velocity distributions were calculat.d : 11,5 and some different values ox parsmetera s, D

ana *

The

theoretical curves of this figure are in agreement with the experimental r.sulte by different

authors.

(9)

1x te f.. 2

rrer.s

t

ca

of the

TD

rcve tI1 nuiercal a-eernerit of trie

oricai rui;s i

ill

e :ecesry n fw

vajuç

D

i

._..i

_::

S,

tat trie

DO

mer a

itior :io

:

'zer

_{the incmaEz ot the time ecale iz the}

k4

ace

osica1 c: racte'iti.cs ana1 ir pa

ie faxdton rcrti

e

resui

t iriu

-cea

the

rture o

trie flo

The ezeer presectt

Wnet

tie v-iocjtjeE of th

rzie are

_ot

ecìe%-o

the

Tht

t.ica1 case wrer

tfla secii proeertee of the

d-t10

iciuìe upofl both 'trie s;r'cure

fT.ow arad ii.

f'íct1n.. reci.;aLc

trie

uroient

io

;eretica1iy.

(10)

lo

References

Ya. I. Voitkounsky, LB. Amflokhiev, V.A. Pavlovelq.

The ecuations of the motion of the fluid taking into cone

3ideration

its relaxation properties. froces1ings nf the

Leningrad Shipbuilding Institute. Vl IJIX, Leningrad,

197C.

Ya. I.

Voi.kouneky, LB.

Amfilokhlev, V.A. Eaviovaky.

Account

for relaxation properties

in

rbeological relation

and viacOt4a fluid motion equations. Internatiobal Semina

kieat

and aee Transfer

in Itheologically Complex

Fluide

1970 aerceg-novi. Jugoeiavia.

, VA. Pa'vlovsky. To the problem of th. theoretical

descii-tion of dilute

polymer solutione.

Academy of Science of th& USSR. Reporte. Vol 200, No 4, Moscow, 1971.

K.K. Pedyaeveky, Ya, I.

Voitkou.neky, Ju. I. Faddeev.

Rydro-iynamice. Leningrad. Sudostroenie Ed. 1968, (in Russian). M.K. Gupta, A.B. Metzner, S.t. Hartnell. Turbulent heat-tran8fer characteristics of Visco-elastic fluid.