Flow through gel-walled tubes

Flow Through Gel-Walled Tubes PNINA K R I N D E L AND A L E X A N D E R SILBERBERG Weizmann Institute of Science, Rehovot, Israel Received January 8, 1979; accepted March 9, 1979 The phenomenon discovered by J. Lahav, N. Eliezer, and A. Silberberg (Biorheology 10, 595, 1973) that flow through gel-coated tubes is subject to considerably increased drag was further examined. Data were taken over a much wider range of flow and it was shown that the functional dependence of the loss of throughput on the Reynolds number is similar to that in incipient turbulent flow through rigid tubes indicating a comparable mechanism. With gel-walled tubes, however, the effects are observable at much lower Reynolds numbers. The parameter a, the shear strain induced by the flow in the gel mantle, correlates with the throughput data obtained with all gel-coated systems over the entire range. The functional interdependence of all these parameters has been derived. The impression is gained that oscillations in the gel wall induce a turbulent boundary layer while core flow remains laminar. As Reynolds number increases the "boundary layer" encompasses the entire tube.

however, not been given extensive treatment. Hansen and Hunston (3), in 1974, showed that a gel-coated disk, when spun in an infinite (effectively) Newtonian medium, encountered increased drag in turbulent flow and that a traveling wave pattern appeared within the gel layer of the disk. Even earlier than that, in 1973, Lahav e t al. (4), working at low Reynolds numbers, had shown that in flow through a cylindrical tube with gel walls, the drag is considerably raised above the value expected from [1]. Lahav e t al. made a preliminary analysis of the parameter dependence of the effect. They showed that the measured throughput Q reduced by the calculated throughput Q0 (through a tube of similar geometry, but of rigid walls) was a function of a dimensionless group, ct,

INTRODUCTION

Steady laminar flow of an incompressible Navier-Stokes fluid through a rigid walled cylindrical tube is described by the well-known Poiseuille-Hagen equation Q = [IIR4/8~lVp,

[1]

where Q is the volume flow rate, ~ the Newtonian viscosity of the flow medium, R the radius of the cylinder, and Vp the absolute pressure gradient along its axis. Related flow problems have also been considered, for example, steady variations in R along the axis (1) or the flow of nonNewtonian fluids. There are only very few analyses, theoretical or experimental, of the effects which can be expected when the walls of the flow tube are not rigid, but deformable, purely elastic (1), or viscoelastic (2). Perhaps the most extensive literature relates to viscoelastic walls in connection with attempts to model the properties of the arteries in the circulation of blood. The question of very easily deformable walls, e.g., gel-coated walls, has,

a = (Rcf2G')Vp,

[2]

where Rc is the outer radius of the gel wall, G' is its shear storage modulus, and Vp is the applied pressure gradient. It is assumed 39 0021-9797/79/100039-12502.00/0

Journal of Colloid and Interface Science, Vol. 71, No. 1, August 1979

Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.

40

KRINDEL AND SILBERBERG

. ¢ - / / / / - ¢ -"~

(i)

~

~al ~G

~2R

1~'777/,. z / i

(m)~

-

(x)

FIG. 1. Gel-walled flow tube (schematic and not to scale). The rigid container and the connection to the external flow system are indicated. Pressure is measured at points (I) and (IV) although the pressure difference of interest is between points (II) and (III).

that the gel is confined within a rigid mold which keeps its outer surface fixed in space (see Fig. 1). The shear storage modulus of the gel is the real part, G', of the complex dynamic shear modulus. For the gels used, a plateau in G' exists (5) over the frequency range of interest (6). This plateau value was used in [2]. Despite this basic success, the work of Lahav et al. left a number of fundamental questions unanswered: (i) Was the difficulty of measuring the actual pressure gradient the cause of an apparently increased drag? The radius of the channel in their experiments (see Fig. 1) was only about 0.15 mm. Hence pressure

could only be measured at stations (I) and (IV) rather than at stations (II) and (III). The pressure difference due to change of radius between (II) and (I) will thus have to be recovered between (III) and (IV) if, as was assumed (4), the pressure difference between the ends of the channel Ap (III - II) is to equal the measured pressure difference Ap(IV - I). Figure 1 is not to scale and the actual length L of the channel is very much larger than the length of the entry and exit ports. (ii) Would flow through the gel tube of a fluid which is immiscible with the swelling medium of the gel give a different result? (iii) Would there be an increased drag, as compared with rigid walls, also in turbulent flow through the gel tube? The present investigation is concerned with answers to the above three questions and an investigation in particular of the turbulent regime. MATERIALS AND METHODS

Materials

Acrylamide--BDH Bis: (N,N'-Methylenebisacrylamide)--Eastman Kodak Temed: (N, N , N ' , N'-Tetramethylenediamine)--Eastman Kodak APS: Ammonium persulfate--BDH Potassium tetraborate (K2B406"4H20)--BDH Cyclohexane, spectroscopic grade--Fluka

Preparation of Gels As in the case of Lahav et al. permanently, in situ, crosslinked, aqueous polyacrylamide gels were used (4). These gels are transparent over the range of concentration of interest here, i.e., where the gels are strong enough to support the channel opening, attach well to the outer surfaces, and permit flow even at high rates through the channel. In this range of concentration and degree of crosslinking, moreover, the gels swell Journal o f CoUoid and Interface Science, Vol. 71, No. 1, August 1979

little so that the cylindrical channel stays open in equilibrium with water. Two stock solutions were prepared: an aqueous solution, 20% (by weight), of acrylamide and a solution of Temed and potassium tetraborate. The latter solution was prepared by mixing 0.5 ml Temed (0.25%) and 1.888 g potassium tetraborate to get 200 ml of aqueous solution (the borate is used as a buffer in this preparation). The appropriate amounts of APS and Bis were weighed and dissolved in the mixture. The exact

41

FLOW THROUGH GEL-WALLED TUBES

quantities used for the preparation are listed in Table I. The preparation procedure was: (1) Bis was mixed with acrylamide stock solution in an aspirator bottle, the bottle being immersed in a beaker full of ice water which stood on a magnetic stirrer. (2) After the Bis had dissolved, 2 ml of potassium tetraborate and Temed solution were added. (3) APS, 0.0213 g, was dissolved in cold water and added to the reaction mixture. (4) The reaction mixture was stirred while the air was sucked from the bottle. (5) The reaction mixture was poured into the cells. Gelation proceeded at room temperature. Formation of the Tube in the Gel A schematic representation of the arrangement of the gel-walled tube is shown in Fig. 1 and its details in Fig. 2. The gel was prepared in the presence of a cylindrical, smooth, and rigid former which could be withdrawn and replaced by the swelling medium after the gel had been formed. The rigid transparent outer container for the gel is a cell made by axially machining a cylindrical bore into a rectangular Perspex block (see Fig. 2a). The block was of 1 x 1-cm 2 cross section and was 5 cm long. The diameter of the bore is either 2.6 or 9.0 mm. The ends are threaded as shown in Fig. 2a, to accomodate the disposable injection-molded polyethylene end pieces (Fig. 2b). The end pieces serve both to hold the rigid former while the gelation reaction takes place, and to connect the tube to the flow system after the rigid former is removed. Each end piece has a standard Luer conical opening at one end while its other end is closed (Fig. 2b) until pierced by the former (Fig. 2c). The procedure for preparing the system thus starts by one end piece being centrally pierced from inside out using a hypodermic needle (30 gauge, 3 in. long) as former. This

TABLE I Preparation of Gels a

Designation

Total monomer (%)

Bis in monomer mixture (%)

a (ml)

b (g)

c (ml)

(6:4) (10:4)

6 10

4 4

5.1 8.5

0.0408 0.0680

9.9 6.5

a a, A m o u n t of 20% acrylamide stock solution; b, a m o u n t of Bis; c, a m o u n t of cold water.

end piece is then screwed into one end of the cell and another end piece mounted (in the reverse direction) onto the tip of the needle (Fig. 2c). The cell is placed vertically and the gelation mixture poured into the cell, the second end piece is loosely screwed into place, and the cell is totally immersed in an excess of the gelation mixture in a suitable test tube. The level of the gelation mixture above the upper end of the cell is at least 1 cm. After this, the second end piece is tightened into its place on the cell. Since oxygen is a powerful inhibitor of the polymerization of acrylamide, contact between the gelation mixture in the Perspex cell and air is thus avoided. After completion of reaction (at least 12 hr, at room temperature), the cell is removed by breaking the test tube and the excess gel is cleared away from around the cell and the parts of the former which stick out. The cell is placed under water with the tip of the needle submerged and the needle is removed with care. With the inlet under water, the space vacated by the needle is immediately filled with the flow medium. The dimensions of the tube formed in the gel are 4 cm in length and about 0.15 mm in radius. Polymerization of polyacrylamide is accompanied by a volume decrease. This is about 8% at the gel concentrations used. In order to keep the cell filled, it was fitted with two holes of diameter 0.5 mm, at each end (Fig. 2c). Through these holes contact with the gelation mixture in the test tube, outside Journal of Colloid and lnterfi~ce Science, Vol. 71, No. 1, August 1979

42

KRINDEL AND SILBERBERG (Q)

(b)

(c)

FIo. 2. Details of flow tube. (a) Perspex block representing rigid container for gel. (b) End pieces (polyethylene), unpierced. (c) Cell, half assembled.

the pressure cell, is maintained and inflow from the test tube is made possible.

Formation of the Rigid, Impermeable Flow Tube Glass tubes (Veridia, England) having a certified, uniform inner radius of 0.15 mm (_) and a length of 4.0 cm were fitted between two end pieces. A special connector (Fig. 3) was designed into which the end piece could be threaded on the one side and which attached to the glass tube on the other. Enough of the tip of the end piece was cut off to permit a direct and unobstructed connection to the cylindrical bore in the glass tube.

System for Flow Measurements and for Measuring Gel Tube Dimensions The cell (containing the gel) is mounted on a microscope stage especially modified for this purpose. The stage also accomodates the pressure transducer (a Sanborn 267 BC or a Statham UC3 + UGP4 Universal transducing cell), a thermostat reservoir for the incoming fluid, and thermostating blocks for the cell (Fig. 4).

A narrow slit below the center of the cell permits light to enter via the mirror in the bottom of the microscope. The cell is clamped between the thermostated blocks and is connected to the flow-generating system and to the pressure transducer on the one side and is open to the atmosphere on the other. It is possible to view the channel through the microscope from above. The whole stage, including the pressure transducer, can be moved in two mutually perpendicular directions in the horizontal plane. Thus it is possible to observe the entire width and length of the cell, while keeping the connection of the cell with the other parts of the system short and fixed. The radius is measured by an eyepiece scale calibrated against the dimension of the needle (before it is withdrawn). Positions along the tube axis and normal to it are read off from the scales on the stage. Fluid is supplied from an aspirator, its pressure being controlled by a mercury column. The amount of fluid passing through the tube is measured by weighing. Only a fraction of the flow tubes prepared as above is free of flaws such as cracks

FIG. 3. Details of flow tube with rigid glass walls. (1) End piece (polyethylene), as before. (2) Precision bore glass Capillary tube. (3) Bore in capillary. (4) Connector. Journal of Colloid and Interface Science,

Vol. 71, No. 1, August 1979

FLOW THROUGH GEL-WALLED TUBES

43

Th.B.

Th.B.

From flow syskPm

T

Fz~. 4. Arrangement of flow system on microscope stage. (c) Flow cell. (e) End pieces. (M) Stage micrometer. (P.T.) Pressure transducer. (Th.B.) Thermostated blocks. (Th.) Thermostat mantle for entering fluid. in the gels or imperfections in the attachment of the gel at the points of inlet or outlet. Defects most often are immediately detectable by eye. A further check consisted o f causing a solution of a high molecular weight dextran (Dextran Blue 2000, Pharmacia) to flow through the channel. Since this material cannot diffuse into the gel it v e r y clearly showed any departure from a smooth wall. This test was generally repeated at the end of each run as well. Tubes which failed this test were rejected as also were any tubes and their results which gave a different throughput when flow into them was reversed, or where the initial throughput rate, at low applied pressure differential, could not be r e c o v e r e d at the end of a run.

Evaluation of Data It was found that when a pressure gradient was applied the dimensions o f the gel tube changed slightly. The tube tended to widen where the fluid entered and to narrow near its end. A taper developed which was small

(angle less than 0.1 rad) in all cases, but was of importance in the calculation of Q0. The tube profile was thus determined in each case, i.e., the radius R was measured at various positions x along the tube and Q0 calculated from the function R(x), so determined, by the use of the integrated form of [1],

1/Qo = [8~/IIAp] I~ R-'(x)dx,

[3]

where L is the total length of the tube. Equation [3] is only valid if the taper angle is as small as was observed in our cases. On the same basis an effective (average) Reynolds n u m b e r was calculated according to Re = [2pQ/II~71

R-!(x)dx/L,

[4]

where P is the density and ~0 the viscosity of the flow medium. As already pointed out, it was assumed that the average pressure gradient Vp was correctly given by dividing the pressure difference Ap between stations (I) and (IV) in Fig. 1 by the length L. Journal of Colloid and Interface Science, Vol. 71, No, 1, August 1979

44

KRINDEL AND SILBERBERG

Check for Turbulence The occurrence of turbulence could be monitored using the dye stream method. This involves introducing a hypodermic needle through a bent Tygon connection upstream of the tap in Fig. 4. The exit of the hypodermic could be located in the center line of flow and a drop of dye introduced. In the region of the narrow tube these drops appeared as long drawn-out threads under the microscope. In laminar flow the edges were sharp and well defined. In turbulence either the whole channel lumen was dyed or the sides of the thread had a frayed, diffuse appearance. It was not always possible to define the Reynolds number for transition with very high precision but we could definitely establish a Reynolds number below which the flow was laminar and another Reynolds number above which it was clearly turbulent. It was thus possible to define the transition to within about 10%. The method, however, gives information only about the nature of the flow in the core of the tube. Near the wall a turbulent layer could still exist even if the core flow is laminar. RESULTS Five systems were investigated which all had similar entrance and exit flow ports. All tubes were 40 mm long and hadR -- 0.15 mm (nominally). (i) Flow of water in tubes with rigid walls. (ii) Flow of water in tubes with gel (6:4) of thickness 1.3 ram. (iii) Flow of water in tubes with gel (10:4) of thickness 1.3 mm. (iv) Flow of water in tubes with gel (10:4) of thickness 4.5 mm. (v) Flow of water-saturated cyclohexane in tubes with gel (10:4) of thickness 1.3 mm. All these results are reported in the form of plots of [Q/Qo] versus log Re in Figs. 5 and 6, with Q0 calculated from [3] using the relevant R(x) versus x curve in each case. For water V = 0.936 cP, p = 0.998 Journal of Colloid and Interface Science, Vol. 71, No. 1, August 1979

g/ml; for water-saturated cyclohexane = 0.916 cP, p = 0.776 g/ml all at 23°C. It is seen that the flow in the gel-walled tubes deviates considerably from the resuits obtained with rigid walls. Typical results for the profile R(x) are shown in Fig. 7. The transition from laminar to turbulent core flow was not determined for all systems. Where the dye stream test was performed the following data were obtained: (ii) Gel (6:4); wall thickness, 1.3 mm; R e t u r b u l e a c e = 570. (iii) Gel (10:4); wall thickness, 1.3 mm; R e t u r b u l e n c e - - 870. DISCUSSION The fact that Q0 4: Q in most cases shows that the observed departure from a straight cylindrical geometry does not account for the deviations. The present case is thus very different from the problem considered by Rubinow and Keller (1). In their case the elasticity of the walls produced distortions which affected throughput. This occurs in our case as well, but the correction due to that is already embodied in Q0 if flow is laminar. In our case, therefore, the deviation of Q/Qo from unity must be the result of some additional pressure loss. Such a loss might occur if the kinetic energy in the flow leaving the tube at station (III) in Fig. 1 is not converted into a pressure head (as required by the Bernoulli equation), but is dissipated. Since the average kinetic energy per unit volume of the fluid is V2p~2, where ~ = Q/IIR z, the pressure should have risen by ½p~2 to compensate for the pressure drop (of this magnitude) which occurred between stations (I) and (II). The true pressure drop in the tube is thus Ap [measured between (I) and (IV)] less 1/2pb2 and it is this corrected pressure drop which should have been used in [3] to determine Q0. If, therefore, this is the only reason why Q does not equal Q0 (as calculated according to [3]) we should have

FLOW THROUGH GEL-WALLED TUBES I

Foe

I

I

I

I

•

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I

I.(3

I

.

45

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.

.

I .

.

I

.

.

.

'

.

0.9 0.8

"%_

0.7 0.6 Q "Q'.0,5 ..,.

04 •

0.3

:

o : ~ l (io:4) RG=I.3 •

0.2

o~o

Rigid w a l l s - w a t e r

5 \

•

..... ter

~o~

•

: Gel (10:4) R G = 1 3 m m - c y c l o h e x a n e

•

o v

-

~

,

~

4~

0.1 0

I 1.4

I 1.6

I 1.8

I 20

l 2.2

l 2.4

~

.6

I 2.8

J 30

I 3.2

I 3.4

I 36

Ioglo Re

FIG. 5. Reduced throughput as a function of effective Reynolds number. Curve 1, Eq. [8] with R ° = 8533. Curve 2, Eq. [11] withR* = 3950. Curve 3, Eq. [11] withR* = 706. Curve 4, Eq. [11] with R* = 552. Curve 5, Eqs. [15] and [16]. Q/Qo = (Ap - 1/2p~Z)/Ap.

[5]

we find, introducing [6] and [7] into [5] and rearranging,

At higher Reynolds numbers, one can presume that instability, i.e., turbulence, sets in and that due to vortex formation additional kinetic energy is lost. It is as though the effect occurring b e t w e e n stations (III) and (IV) in Fig. 1 is repeated m a n y times along the length of the flow tube. One should hence be able to allow for this effect by modifying [5] as follows:

[1 - (Q/Qo)]/(Q/Qo) = R J R °,

[8]

Q/Qo = ( A p - yl/2pb2)/Ap,

R o = 32L/R.

[9]

where the p a r a m e t e r , y, is a function of the g e o m e t r y of the flow tube and indicates the way turbulence modifies the effective pressure loss. Instead of [8] we thus find

Since we m a y write (approximating [3] and [4]), Ap = 8~LQo/FIR 4

[6]

and Re = 2 p Q / H ~ R ,

[7]

where F o r the tubes in question here with L = 40 m m a n d R = 0.15 m m , we h a v e R ° = 8533. The line predicted by [8] and [9] has b e e n entered in Figs. 5 and 6 as c u r v e 1. It is seen that this model indeed well accounts for the deviation f r o m Q/Qo = 1 which characterizes the low to intermediate Reynolds n u m b e r portion of the results obtained with rigid tubes where a distortion of tube dimensions is out of question and Q0 is given b y [1]. On the other hand, it is clear that a " k i n e t i c energy end e f f e c t " could only v e r y partially account for the deviation of Q f r o m Q0 in the gel tubes.

[1 - (Q/Qo)]/(Q/Qo) = Re~R*,

[10]

[11]

where R* numerically equals R ° / T but should on this model be conceived as a new parameter. Turbulence, in the rigid tube in our case, starts at a Reynolds n u m b e r of about 700 and c u r v e 2 is the best-fit c u r v e to our turbulent flow data. The value of R* (based on curve 2) is 3950 but our inability with the present setup to m a k e measurements at very large Re m a y m e a n that the good fit is not necessarily significant. Journal of Colloid and lnterj?tce Science, Vol. 71, No. 1, August 1979

46

KRINDEL AND SILBERBERG I

IOO

I

~1

. . . . . .

•

I

•

I

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•

•

o91-...

[

""

Q °6

I

I

I

"3""-.

°

Q oO5

~

i ~ %0 o o

~^

0.4

~o~ ~

.,%o,

• : Rigid wolls:woter o : Gel (6:4) R6=l.Srnm-woter

0.3 0.2

o

oo~_ =° o

~o o~

& : Gel (10:4) RG=4.5rnm-woter

-

0.1

o° o

0

I 1.4

I 1.6

I 18

I 20

I 2.2

I

•

I 24

r 26

o

I 2.8

I 3.0

I 3.2

I 54

I 56

IOglO R e

FIG. 6. Reduced throughput as a function of effective Reynolds number. Curves 1 and 2 as in Fig. 5. Curve 3: Eq. [11] with R* = 200.

It will be noted, however (Fig. 5), that the data obtained for the gel (10:4) system with the 1.3-mm-thick walls both in the case of water and in the case of cyclohexane (curves 3 and 4, respectively) can also be represented reasonably well by an equation of the type [11], but with a different, much lower R*. 40

The same conclusion can also be drawn for the two other gel systems studied (see Fig. 6), Here, too, the results within their scatter conform to [11], but with a still smaller R*. (The reason why the two seemingly disparate systems in Fig. 6 agree with each other will become apparent a little later.)

I

I

t

o 35 °

•~

o

o

o o •

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o

o

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•

o

30

go • - A P = ZOB rnmHg o - Z~P= 1074mmHg

25

o I 0

I0

20

I

I

3o

4o

X (ram)

FIG. 7. Diameter of gel tube at various locations x along axis for different applied pressure differences Ap. The reading on the ordinate gives 2R(x) in scale divisions. Note that the changes in R are much enlarged in this representation. Journal of Colloid and Interface Science, Vol. 71, No. 1, August 1979

FLOW THROUGH GEL-WALLED TUBES '

'

I

I ....

I

'

'

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I ....

I

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47 '

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1.0-

0.8

o

0.6

0

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,

~ I,,,~I

,

,

I 0 -i

, I ,,,,I

, I

A

~ , l,,,t,l I0

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C/

FIG. 8. Reduced throughput as a function of shear strain ct in gel. Points: @, Gel (6:4), thickness, 1.3 mm; flow medium, water. +, Gel (10:4), thickness, 1.3 mm; flow medium, water. A, Gel (10:4), thickness, 4.5 mm; flow medium, water. ×, Gel (10:4), thickness, 1.3 mm; flow medium, cyclohexane saturated with water. Curve: Equation [13] with factor in square brackets put equal to 0.5. a is given by Eq, [2]. One can easily convince oneself that no reasonable model of energy dissipation in the gel layer could a c c o u n t for the m a s s i v e pressure losses to which the r e d u c e d throughput has to be attributed. The m o s t simple and direct explanation is the existence of additional velocity gradients in the fluid flowing through the channel as would arise, for example, in turbulence. T h e r e is no question but that the flow is turbulent at the higher Reynolds n u m b e r s tested. The direct dye s t r e a m e x p e r i m e n t s mentioned h a v e p r o v e d this. The transition m o r e o v e r s e e m s s m o o t h and the data s e e m r e a s o n a b l y consistent with Eq. [11]. H e n c e it is quite feasible that turbulence accounts for the entire effect and that interaction with the gel walls p r o d u c e s a flow-destabilizing dist u r b a n c e which creates a turbulent wallb o u n d layer. At low Reynolds n u m b e r this layer is thin and core flow is laminar but as Reynolds n u m b e r is increased m o r e and m o r e of the flow b e c o m e s turbulent. The dye stream m e t h o d detects turbulence w h e n the b o u n d a r y b e t w e e n the dyed and u n d y e d

surrounding fluid c o m e s to overlap the turbulent region. Oscillations of the gel surface are k n o w n to be induced b y the flow [3] and destabilization can be viewed to occur in this case, therefore, not by r a n d o m noise but by a w a v e of specific frequency and amplitude. A direct i n v o l v e m e n t of the gel walls is also brought out by the fact that plotted against the p a r a m e t e r ~ defined in [2] the difference b e t w e e n all the gel systems disappear (see Fig. 8). This confirms the observations of L a h a v et al. (4). Our data, h o w e v e r , c o v e r a m u c h wider range than those of L a h a v et al. The G ' values for gels (6:4) and (10:4) needed for the calculation of a were kindly made available to us by Dr. N. Weiss. T h e y are 3.2 × 103 and 12 x 10z dyn/cm 2, respectively. It should be pointed out that replotting the data according to a in Fig. 8 depends on Eq. [2] only and has nothing to do with our a b o v e discussion in terms of Reynolds number. Figure 8 represents an independent look at the results. It will be noted in parJournal of Colloid and lnterjace Science, Vol. 71, No. 1, August 1979

48

KRINDEL AND SILBERBERG

ticular that the data for the different gel systems truly overlap. Although there is considerable scatter, the scatter is about the same mean. The full line curve is not drawn to represent this mean but is semitheoretical as will be explained below. The correlation on an axis of a explains why the two seemingly very different cases compared in Fig. 6 agree so well with each other at the same Re. By coincidence the R J G ' for these two gel systems are approximately the same. They thus have the same a value for the same Re value and correlate on the Re axis for this reason. It is o f importance to point out that a has an exact meaning. Let us consider the system shown in Fig. 1. If P(II) is the pressure inside the flow tube at station (II), it is also approximately the pressure inside the gel at this location along the axis. Similarly for the pressure P(III) = Pox) - Ap at location (III). In particular the difference in these pressures, the pressure drop Ap between (II) and (III), will be the same in flow across the entire gel block of length L and of outer radius Re. H e n c e the force on the gel, due to this pressure difference (from left to right), is IIR~Ap. This is resisted by the shear stress exerted by the gel on the cylindrical rigid shell encompassing the gel. If the gel has a shear modulus G', this force will be aG'2HRGL where a is the shear strain induced by the gel at its outer periphery in contact with the rigid tube. Equating gives

a G ' 2 I I R c L = IIR~Ap

[12]

which reduces to the definition [2] for a. The parameter a is thus a measure of the degree of shear deformation induced in the gel by the applied pressure gradient. Using Eqs. [6], [7], and [11] in [2] we can rewrite a as

a = [2RG~2R*/G'R3p] × [1 - (Q/Qo)]/(Q/Qo) 2,

[13]

where R* is the characteristic Reynolds number which locates curves 3 and 4 in Journal of Colloid and Interface Science, Vol. 71, No. 1, August 1979

Figs. 5 and 6. It is that value of R* in [11] which will link the Q/Qo values with the tube Reynolds n u m b e r Re in the case of each of the gel-walled systems. The good correlation between the data and the line in Fig. 8 drawn according to [13] indicates that the factor in square brackets in [13] is a constant for all systems with deformable walls. Since our data (Fig. 8) show that Q/Qo = 0.5 when a = 1 its value according to [13] should be 0.5, leading to the following equation for R* :

R* = 0.25[ G'R3p/RG'rl2].

[14]

The line in Fig. 8 is drawn using 0.5 for the constant in [13]. The good correlation o f the data by the line in Fig. 8 thus supports representation of the data by Eq. [11] in Figs. 5 and 6. With R* calculated from [14], Eq. [11] can be used to predict behavior for unknown gelwalled systems, as a function o f tube Reynolds number. While at first glance the close agreement between the cases of water and cyclohexane seems to prove that permeation flow through the gel is unimportant, this is in fact not necessarily clear-cut. In the case of water the amount of fluid passing through the gel is a minute fraction (about 10-4) o f the total flux (8). [In (8) the relative flux was actually calculated to be 10-5, but the polyacrylamide gels used here are some 10 times more permeable than there assumed (5).] Since water saturated cyclohexane contains 0.01% water (9), there is in fact a supply o f water through the tube of the same order of magnitude as the needed flux of water through the gel. It is thus not at all inconceivable that a transfer o f water occurs from cyclohexane to the gel which is fast and copious enough to provide the same water flow in the gel as when pure water is flowing in the channel. Since the deformation of the gel is characterized by a and a reflects the pressure gradient in the gel, the flow in the gel should be some function of a as well. H e n c e a different a and a different case would arise

FLOW

THROUGH

GEL-WALLED

were flow in the gel prevented. In consequence the good comparison noted in Fig. 8, which takes in the cyclohexane case as well, would probably not hold. What the experiment with water-saturated cyclohexane has thus proved is that the flow reduction is a phenomenon linked to the properties and state of the wall, but that flow in the tube itself is properly characterized by its Reynolds number. Since turbulence is involved, it is of interest to discuss our results also in terms of the frequently applied (7, 10) semiempirical equation f-1/2 =

--0.8

+

log10 (Reef)

[15]

based on the theories of von Karman and Prandtl (7). While the form of [15] is dictated by these theories, its numerical parameters are in fact adjusted to represent data at very high Re in smooth-walled tubes. The parame t e r f in [15] is the so-called friction factor which in terms of Re and (Q/Qo) can be expressed as follows (7): f = 64/Re(Q/Qo).

[16]

Combining [15] and [16] gives a relationship (curve 5 in Fig. 5) between Q/Qo and Re. The results (10) to which [15] was fitted do not in fact lie on curve 5 in the region of Fig. 5. The fit is matched to data of very much higher Re. In the region of Re values in Fig. 5, the data (10) lie atRe values to the right of curve 5. This is in disagreement with our results which lie mainly to the left of curve 5. It is, moreover, obvious that curve 2 in Fig. 5 fits our data much better than curve 5. When ideally rough- rather than smoothwalled tubes are used, the constant on the right-hand side [15] is -3.3 rather than -0.8 (10). This predicts a transition curve which corresponds approximately to a horizontal line at (Q/Qo) ~ O. 17 over the range of Fig. 5. Most of our results with gels fall in the region between this line and curve 5.

TUBES

49

The effect can thus also be regarded as a consequence of an artificially created (by oscillations) surface roughness. The question may be asked why such a major effect on tube flow has not been noted before. The answer in part is given by expression [14] for R*. Only when G' is low enough and Rc is large enough will Rg lie in a range of tube Reynolds numbers which are below the normal transition to turbulence. The effect is thus confined to very soft gel-like walls as is often the case with biological systems and is of no consequence in the case of rubber tubes. For the blood circulation the problem was discussed by Lahav et al. (4). In other biological situations perhaps the most apposite observations are those of Clarke (11) on flow of air through glass tubes coated with sputum or with polymer solutions of similar viscoelasticity. He observed the appearance of waves in the coating which was accompanied by an increased resistance to flow. Since G' for sputum is of the order of 102 dyn/cm2 (12), the conditions under which Clarke performed his experiments correspond to c~ values of the order of 0.1. There are probably also other cases, biological and nonbiological, where the effect has been observed, but attributed to other or unknown causes. While loss of throughput is generally to be regarded as a disadvantage the presence of a turbulent boundary layer may help to dissipate concentration polarization in filtration and membrane transfer problems. This may, in fact, be an important contribution to transfer problems in biological systems. CONCLUSIONS

It was shown that the loss of throughput as a function of Reynolds number which marks incipient turbulence in flow through rigidly walled tubes can be represented by an equation of the form of [11]. A tube Reynolds number R* characterizes the loss phenomenon. In rigidly walled tubes it is Journal of Colloid and Interface Science, Vol. 71, No. 1, August 1979

50

KRINDEL AND SILBERBERG

about 4000. When gel-walled tubes are employed [11] again represents the data. This similarity between the loss curves suggests that a similar mechanism (i.e., turbulencelike disturbances) is the cause. R* in the gel case, however, is smaller in value and turns out to be a function of the gel system (Eq. [14]). The expression for R*, which results, derives from the fact that the loss phenomenon in gel-walled tubes can be shown to be correlated to the shear strains c~ in the gel. At the same strain a the same loss of throughput is recorded. Oscillations in the gel systems [discovered by Hansen and Hunston (3), but not seen directly by us] may possibly be the cause of the additional instability. Unlike with flow through rigid tubes, turbulence seems to develop along a smooth transition indicating that a boundary layer may be turbulent and may grow monotonously at the expense of a laminar flow core as the tube Reynolds number is increased.

Journal of Colloid and lnterj2tce Science, Vol. 71, No. 1, August 1979

REFERENCES 1. Rubinow, S. I., and Keller, J. B., J. Theor. Biol. 35, 299 (1972). 2. Schultz, D. L., in "Cardiovascular Fluid Dynamics" (D. M. Bergel, Ed.), Vol. 1, p. 287. Academic Press, London/New York, 1972. 3. Hansen, R. J., and Hunston, D. L., J. Sound Vibration 34, 297 (1974). 4. Lahav, J., Eliezer, N., and Silberberg, A., Biorheology 10, 595 (1973). 5. Weiss, N., and Silberberg, A., Brit. Polymer J., 144 (June 1977). 6. Silberberg, A.,Ann. N. Y. Acad. Sci. 275, 2 (1976). 7. Landau, L. D., and Lifschitz, E. M., "Fluid Mechanics," pp. 159-167. Pergamon, Oxford, 1959. 8. Apelblat, A., Katzir-Katchalsky, A., and Silberberg, A., Biorheology 11, 1 (1974). 9. Seidell, A., "Solubilities of Organic Compounds," 3rd Ed., Vol. 2, p. 432. Van Nostrand, New York, 1941. I0. Van Driest, E. R., J. Aeronaut. Sci. 23, 1007 (1956). 11. Clarke, S. W., Bull. Physio-pathol. Resp. 9, 359 (1973). 12. King, M., and Macklem, P, T., J. Appl. Physiol. 42, 797 (1977).