FLOW THROUGH AN ARTIFICIAL BIFURCATION WITH PARABOLIC STEADY STATE FLOW ENTRANCE

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r

ISBN 82-553-0324-3 Applied 1'1.athematics

No 1 - September 12

1977

FLOW' TEROUGH AN ARTIFICIAL BIFURCATION WITH PARABOLIC STE.ADY ST.ATE FLOW ENTBA.NCE

by

Bj~rn G. Johannessen, Hans Jensen and .A:rve Kvalheim

0 s 1 0

Supported by

the Norwegian Council on Cardiovascular Diseases.

PREPRIN¹r SERIES - Matematisk institutt., Universitetet i Oslo

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Hemodynamic .forces may be important in the patho- genesis of vascular lesions such as atherosclerosis.

The problem is complex. The major difficulty is the large number of variables involved. It seems ine- vi table that some of the many :factors involved must be ignored initially.. At least two topics are .funda- mental in the discussion~ hemodynamics and the distri-

bution o.f early plaques (and perhaps the :fatty strikes) within the bifurcations.

A number o.f authors: Fox and Hugh [ 9

J,

Caro et.al.

[ 3

J,

Flaherty et.al. [ 8

1,

Zamir and Roach. [ 22], Roach et.al. [16], Stehbens f181, Ehrlich

f

7], Fried.man el.al.

[10] and Pinchak and 0$trach L15] have done investiga~

tions on the hemodynamics iD. bifucations.. The theoreti- . cal treatment has been done only on two-dimensional

mathematical models. Most of the studies on physical models have been done on translucent models.witl!DJ?..in.ly

qualitative results.

There is doubt whether atherosclerosis is .sheardepen- dent, and whether it occurs in aeras of low shear due to diffusion problems as proposed by Caro et.al.

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. or in aeT'as of high shear due to endothelial damage as

· proposed by.Fry

l 11J_t:Ltl.LV_L~~--·----~---~--- ---

----·----~ -~-------~---·-- ~-·---.·--_"

__________ _

In general., while authors may not agree on the mechanism involved, they vill agree tbat the high incidenc.e of lesions near arterial junctions ·call .for an investigation of the flow structure.

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In symmetrical bifurcations the angle will vary within species and between species, but all o.f them

will set up secondary motion cau..sed by the turning of the .flow. Investigation of flow in curved tubes has been done, theoretically by Dean [ 5 ] and experimentally by White [ 21]

and Adler ( 1 ]. The secondary motion in a circular curved tube is shown in fig. 2a.

Our model .shown in fig. 1 consists o.f a "diverging tube" from the inlet to the apex. The pressure distribution in this section will to some extent be similar to the conditions in a two-dimensional channel. Flow in a two-dimensional channel is well described (Schlichting 1968 [17] and others). A two-dimensional channel corresponding to the mid-section (see .fig. 1) o.f our model, will at high but laminar Re-numbers have back-flow as shown in fig. 2b. In constrast to the .flow pattern in such a two- dimensional channel, the flow in the bifurcation-model will be three-dimensional. From this ract there is no reason to expect closed streamlines. The cmnbined e.ffect o.f divergence and curvature produces a rather complex three- dimensional rlow pattern~ which had to be described through ho th Yisual.iza~ion--an~ deta-iled-vel.--oci ty measurements.

---l>Ue t-0 the presence of backf'low there must exist velocity profiles with inflection point. Basicly in two-dimensional

. '

fl.ow such a velocity pro.file is unstable with a probability .for local unsteady f'low or turbulence. Measurements of the three-dimensional .flow in our model have shown neither

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any unsteady f'low nor any turbulence.

The di!f'erent hypothesis f'or relation between hemodynamics and early distribution of atheroma, given by Caro et.al. [ 3 ] and Fry et.al [ 11] has one common un- known, the shear stress distribution in the f'low.

Direet measurements of' the .shear stresses are difficult, also in models, but if the velocities are known, the

shear stresses can be calculated. by the constitutive equation. T.o ob:ta,in the wall shear stress the measured velocity p_rofiles have to be extrapolated to the wall since our techniques in the present experiments does not allow velocity measurements very close to the walls.

If' the entering flow is axi-symmetric and no transi- tion to unsteady flow or turbulence occur, then, because of the symmetric geometry, the f'low field in the bif'ur- cation must be symmetric about the planes a-a and f3-13 in fig. 1.

There is doubt wheather pressure variations in- fluence the development of' early atheroma. The magni-

tude of local pre~sure may to some extent be calculated by us·e of the Bernoulli-equation along the streamlines

---·- - - ---

and ignoring viscosity. _However~in ~orta --~:>:f man,___j;_!!~---- ____________ _ maximum "stagnation pres-sure" ·(assuming a maximum aortic

blood velocity of 2 m/s) will be about 15 mm Hg. (Caro et .. al. [ 3 1) This is less than other normal variations of the blood pressure. On this background no pressure measurements have been made in the present experiments ..

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Methods

The model used in our experiments is shown on .fig. 1.

It is made of acryl-plastics and is translucent. The

"parent" tube has a diameter oi 20 mm, and each o.f the branches bas the same diameter. The walls are smooth.

The model is symmetric in two planes, shown in .fig. 1.

It di.ffers from a model o.f a real bi.furcation with rigid walls in two ways. The aera o.f each branch is equal to the aera o.f the parent tube and the branches are tubes oi constant curvature.

The experimental set up is shown in .fig. 3. The .flow through.the model is regulated by valves and the height- dif.ference between the overflow tanks at inlet and out- lets. The inlet consists o:! a settling chamber with honeycomb, screen and contraction and a 2 m straight

tube flush to the model. The construction ensures fully developed laminar pipe .flow at testsection inlet. The rate of flow in each branch is measured by rota.meters individually calibrated for the fluid used .µi the experiments. The temperature in the SUlDJ:> is kept constant within 0,5~C by~ the~ostatie bath. The inlet tube , · . is insulated to avoid convection in the low Re-number ·

~---~----~---- - - ---~~-~----~~-~

experiments.

The'velocity measurements· is done by. laser-doppler- anemometer LDA {DISA 55L Mark II) and hence does not

. ^.

disturb the flow. References to this type o:f measurements

is

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J l

^{6 ] •}

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The stability in ti.me and the repeatability for the LDA was checked during the experiments.

The fluid consisted of 62,8% NaJ, 36,2% H2

o

and 1%

Na.2

s

2

o

_{3 •} This fluid has the same index of refraction a.s the ac.ryl-plasties. The laser light was then not refracted on the boundary between model. and fluid.

Close to the walls in a distance of about 0,5 mm we get a •dead" zone :where our LDA could not produce any results.

~he optics on our LDA. had a measurement-volume about 1 mm in the a.xis o! the laser and about 0. 1 IDJD in the other directions.

The LDA was mounted on a tree-direction traversing

unit which made it poBslble to traverse the 111easurement- volume over the whole flowfield of interest.

The visualization were produced of dye injected by canyles sited 6 diameters above the apex.

·nye

was al.so injected through small holes in model wall.s. !fo avoid buoyancy, the dye-density was within 0,5°/oo of the fluid-density.

The fluid viscosity vas measured by a J3rookf'ield

viscometer, and the density by densimetar. .The Reynolds

---~---~--~--~------~---

--- --- - - ---number-is-based on the fiow in the parent tube.

Re =

iT~

^p ⁼

t ~ ~

where Q is the f'.lov rate., d the diameter of the parent tube, P the density and Tl the dynamic viscosity.

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Measurements have -been made for Re= 1000 <;md Re""' 100 and .for some He between 1000 and 100

looking for special e.ff ects.

The velocity notation is shown in.fig. 4

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Hesults

Visualization with water.

Fig. 5a b c d e f g h show photographs of streamlines at

Re 1000. Dye is injected through a canyle 6 diameters upstream from apex, at different positions along the z-axis.

The stream.lines with high entrance velocity (central.

region of inlet tube) will paas through the bifurcation forming a double helix in each branch, much like the flow in a regular curved pipe. An example of this is shown in fig. 5a where the dye is released at the center of the inlet tube. The dyed streamline attach at the apex and then £ollow the wall of ~he branch tube with a slow ^swirl~·

Streamlines closer to the wall (fig. 5b) show a similar deflection by the secondary now.

Frg. 5c1 however, snow streamlines enterfilg a region of very low velocity, and some of the dye is caught in backflow.

Figures 5d to Sh all show streamlines entering the low velocity region and with backflow of ·different , magnitude_.

-- - - --~------

-- - ----- ---~

--- ---·- --- - - - -- - - - -- - - -

The regions of low velocity and back.flow are sketched in figure 6.

Measurements.

Figures ?a., b., c show velocity profiles for vq:i

·-·----~---

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at Re

=

^·1QOO ^andfigure 8 corresponding profile at Re = 100 • In figures 9a, b. c~ d are shown the related velocity pro.files for vt at Re = 1000 •

From the velocity profiles at Re = 1000 we see that the magnitude of v~ in the backfl01r1 region is small oompared to the velocity in the main .flow. The mass transport through this area is small compared with

the mass .flow in the main .flow. The .lateral velocity component _v~ is small compared to vcp ⁱⁿthe main .flow. In the back.flow region and in the low velocity region shown in .figures 6a, b are the two velocity com- ponents and o.f the same order of magnitude.

As seen .from the JDeasured velocity pro.files .for Re = 100

there is no back.flow at this Reynolds number. The .flow- .field resembles .flow-in a regular curved pipe •

. M~asurements in the back.flow area shows that signi- ficant back.!low _s_tart_a be_tveen Re : 25-0 and Be = 300 and increases -with Reynolds-number.

~e rluid flow through the testsection has been computed on the basis of the measured velocity pro.files (parabolic profiles} at testsection entrance •

. 1?.hes_e computations .sho~TS agreem.ent-wi th the. f'l-0w

measured by the rotameters -within O, 5~- -- -^{- - -}^{- - - -} ^~----~--^-- ^{- - - -} ^~^--- From the velocity profiles we have estimated the

magnitude of shear stresses at the walls. The shear stresses has a maximum near the apex.

In

^fig. 10 we have sketched four zones, 0 to 25% of

max.

stress,

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25 t.o 50% of max .. stress-. 50 to 75% of max. stress and 75 to L100% of max. stress.

-~= ---~ ~ -·---' - - - = - - - - - ---

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•

... 10 -

Conclusions.

Ii' endothelial damage due to high shear stresses is important for initiating the atheroschrotic process, then ^theearly lessiOll.B will be loca.J..ized to ^ZOJle 4 on fig. 10. On the other band, il' low mass diffusion is a governing £actor in the formation oi' early athero- sclerotic plaques., one would expect tba. t early lessions will be sited in zone 1 on i'i,g .• 10.

The pathological description :for the localization o:f plaques is not uniform,. [ ; ] [ 4 ] [ 14]

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23]

This lack of precise pathological .information ^isconfu- sing as it ^isnot possible i'rom the excisting material

to find where the majority oi' early lessions are loca- lized. We will therefore suggest that some type of standard pathological description is introduced since it is important tbat the localization together with the degree of lession is described precisely and uniformly.

I .

At present it is possible tn select difi'e:rent

pathological descriptions supporting both the high shear stress hypothesis (zone 4) and th~ mass di.fi'usion hypothesis (zone 1).

The ef:fective

mass

diffusion in the backfl.ow and low velocity region in fig. 6 is far less than the

diffusion in the main flow. But the backflow region is not part of a separation bubble with closed streamlines. '

It is rather a "'ventilated bubble" with continuous (but low) supply of ttfresh⁹ fluid from the main flow. The

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ef'Jccttve mass exchange between vessel walls and fluid should therefore be much more effecient than in a corresponding region with closed streamlines. As mentioned earlier will two dimensional models of bifurcations have separated regions with closed streamlines at Reynolds numbers of interest. The study of two dimensional models are therefore of little interest in light of

the mass diffusion hypothesis.

Symmetrical bifurcations in the human circulation will differ from our model.

If the ratio is defined as the sum of the cross- sectional area of the two branching tubes divided by the cross-sectional area of the pa.rent tube, then our model has an area ratio of 2, whereas the aorta-bifurcation of man has an area ration of O. 77 to 1. 25

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J.

Compared to our model this smaller area ration will resiiTt in

a

change

In

pressure distribu:tion in the bifurcation. For the same Reynolds number the .separation will occur further from the inlet in a bifurcation with a smaller area ratio, and the region of backflow and the ventilated bubble are probably smaller. Our experi-

mel'l:~-~ were d_()ne at Reynolds number up to 1000 which

is a fairly typical Reynold-s---nu:mbeT-in human---ei.--reul--a-t-i--en-.--- As a part of further investigations in this field we will

study the flow field in a cast o:r human aorta-bifurcation.

Acknowledgment-:

We would like to thank La.rs \Vallee for valuable discussions.

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R E F E R E N C E S •

[ 1

l

M~ Adler, Stromung in gekrUmmten R0hren ( 1934) . UMM 14. 257-275.

(2] Beales, J.S.M. and Steiner, R.E. Radiological

assessment of arterial branching coeffients (1972) Cardiovare. Res. 6. 181-186.

[3] Caro, C.G.,Ritz-Gerald, J.M., and Schooter, R.C.

[4]

Atheroma and arterial wall shear. Observation, correlation and proposal of a shear dependent mass·

transfer mechanism :for atherogenesis. (1971) Proc. Roy. Soc. Lond. B. 177 109-159.

Clarkson, T.B. Atherosclerosis-spontanions and , induced. (1963). In Advances in lipid research

(ed. R. J>aoletti and D. Xritchevsky). New York and London: Academic Press.

f,5] Dean, W.R. The stramline motion of a i'luid in a curved pipe. (1927) Phil. Mag. (7) 4. 208 and (1928)

- -- - ---

Phil. Mag. 5. 673.

[6] Durst, F., Melling A. and Whitelaw, J .H. Priciples and practice or Laser-doppler anemometry (1976) Academic press. London, New York, San Francisco.

[7] Ehrlich, L.W. Digital simulation of periodic flow - in a bifurcation.; { 1-~74} Co-input. Fluids~ 2. 237;..:247.

- I

- - - - ---~--- -~ - - - ---~~---·--·--- ·------ --- ---·--·--- -~ ---- - - -----

Tsl

^Flaherty, ^J.'I'., ^Pierce, ^J.E., Ferrans, V.J., Patel D.J ••

Tucker, W.K., Fry, D.L. Endothelial nuclear patterns.

in the canine arterial tree with particular reference to hemodynamic events. (1972) Circ. Res. 30 23-33.

f9l Fox, J.A. and Hugh, A.E. Localization of atheroma:

A theory Based on Boundary Layer Separation {1966) Br. Heart. J. 28. 388.

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f

^10] :B'riedman. M.H., O'Brian, V. and Ehrlich, L.W.

Calculations of pui.satic flow through a bianch.

(1975) Circ. Res. j6. 277-285

(11] Fry, D.L. Acute vascular endothelial changes

associated with increased blood velocity gradients.

(1968) Circ. Res. 22. 165-197.

[12] Fry, D.L. Certain histological and chemical res- ponses of the vascular interface to acutely induced mechanical stress in' the aorta of the dog. (1969)

Circ. Res. 24. 93-108.

f

13] Fry, D.L. Certain chemorheological considerations regarding the blood vascular interface with particular reference to coronary artery disease. Circu- lation 39. 40 Suppl. IV, 38-59.

(14] Hugh, A.E., and Fox J.A. The precise localization of atheroma and its association with stasis at

the origin of the internal carotial artery a radio graphic investigation. (1970) Brit. J. Radiol.

43. 377-383.

[15] Pinchak, A.C. and O~trach, S. Blood flow in

branching vessels. (1976) J. Appl. Physiol. 41. 646-658.

f16] Roach, M.R., Scott, S. and Fergxnon, G.G. The hemodynamic importance of the geometry o.f bifur·ca- tions in the circle of Willis. (Glass model studies)

· (1'972) -storks 3. · 255-267~

---·---·--- ---~·· .--- --- - - -

- ----- - - ---~-~---- - -~ ----~-~ ---- - - -- -~ - -

I

^{17] .}Schlichting, H. Boundary-Layer Theory ( 1968) McGraw-Hill Book Campany., .blew York. ·

[18] Stehbens,. W.E. Flow in glass models oi arterial bifurcations and Berry aneurysms at low Reynolds number (1975) Quart. J. Exper. Physiol. 60. 181-192.

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f

19] Thompson, H.D. and Stevenson, W.H. Proceedings o.f the second international workshops on laser veloci- metry (1974) Pendue University.

r20] Tomm, D. Determination of velocity distributions in models for aterial bifucations by laser-doppler- anemometry. (1975) Proc • .5th Over the -water Meet.

Biol. Eng. Soc. London in Aachen June 18-20 1975 Biol. Eng. (.Special ed.)

(21] White, C.M. Stramline f'low through a curved pipe.

(1929) Proc. Roy~ Soc. London A 123. 645.

(22] Zamir, M. and Roach, M.R. Blood flow downstram oi a two-dimensional bifurcation. (1973) J. theor.

Biol. 42. 33-48.

[23] Zemplenyi, T., Lojda, Z. and Mrhova, O. Enzymes of the vascular wall in experimental atherosclerosis in the rabbit. (1963) In Atherosclerosis and its origin (ed. M. Sandler and G.H. Boune) New York and London Academic Press.

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Fig. 1

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..

^-

./--]

//

Fig. 2 a

Fig. 2 b

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3 3

4 4

1

6

Fig .. ' 3

1.. Constant head tank

' '

2. Setling chamber 3 .. Rotametre

.4. Valve

5 .. ~est

--'6-~

-sump- - --

7. Pump

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z

Fig. 4

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Fig. 5 a

-

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S I

- _lt:'i:?·

•

Fig. 5 ^b

1

. ·r.

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-

Fig. 5 e

i' i:

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^a··~:).

~ .

Fig. 5 f

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®

^...

Fig. 6

Centre ·section 1 Backf'low zone 2 Low velocity zone

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r

I

. . 1 . . ~ ^•

1 O cm!\~

Fig. 7 a Velocity profile

- Centre Section

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C)

>::: H ^~ 0 0 ..-f .p ⁽⁾_Q) ~

ti) (Yf

, ..

.0 I"-

tlO

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rx,.

. ----~---·----··-·----··-·-·~

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r+-1 0 H 0.

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.µ ..-t 0 0 .--I

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::>

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- - - --

- - - -- - - -

<1>

s:..

µ c

llJ 0

s:: Ci-l ~

0 0

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0 e

a>

Cfl l.O

__ Ill_~--------~---~----- - - -

...

e 0 0 ....-

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::> 9 co

0 r l

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0 k

M Pi

"l'"i

µ:.

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0 0 C""I

u 9

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Fig. 10

Projection of shear stress zones

1 . 0-25 % max shear stress 2. 25-50 % ¹¹

3. 50-75 % ^Tf 4 .. 75-100% ¹¹