BULLETIN OF MATHEMATICAL BIOLOGY
VOLUME37, 1975
THREE DIMENSIONAL LAMINAR FLOW IN DISTORTING, AXISYMMETRIC, AXIALLY VARYING VESSELS
9 JULIUS MELBI~ Comparative Cardiovascular Studies Unit, School of Veterinary Medicine, University of Pennsylvania, Philadelphia, PA 191714 R. GOPALAKRISHNANand ABRAHAM NOORDERGRAAF
Department of Bioengineering, University of Pennsylvania, Philadephia, PA 191714
Three dimensional laminar, viscid flow is developed for N e w t o n i a n fluids which provides absolute values for axial, radial and tangential velocity fields everywhere if the dimensions of the vessel a r e known and two simultaneous axial velocities e.g. on and off t h e central axis in the same plane, and the central axis axial velocity gradient are measured. I n addition, normal and shear stresses are determinable. The equation set satisfies geometric and other known flow limiting conditions such as no slip at surfaces etc. and are amenable for inclusion in general, dynamic flow expressions. A l t e r n a t i v e l y t h e y m a y be used alone for certain problems involving gradients and second a r y flows. A range of illustrations axe shown for a distorting vessel with elliptic crosssection and small axial taper (analogous to the p u l m o n a r y t r u n k during ejection).
1. I~troduction. In many investigations concerning flow, simplifications are made in order to make tractable the problem under investigation (Noordergraaf, 1969). Justification is commonly based on expectation of some terms attaining low relative values. However, the interactive effect of these terms may outweigh, in significance, their relative values (Noordergraaf, 1969; Ling, 1972). Difficulty rests with the requirements of simultaneous solutions utilizing minimally specified boundary conditions usually temporal in nature. Addi489
490
J U L I U S M E L B I N , R. G O P A L A K R I S t t N A N AND A. N O O R D E R G R A A F
tional problems enter where the flows occur in vessels with moving walls and especially in those which undergo large distortions, e.g. veins. The availability of a realistic viscid flow model permits much of the difficulties to be overcome, and in some cases, solutions for a spectrum of conceivable situations are available directly from the model. In this report, we present a laminar flow model based on geometric and other known, limiting flow conditions, e.g. no slip at surfaces. With geometric considerations and quasistatic approximation, the dynamic equation of motion becomes unnecessary to specify flow kinetics. Axial, radial and tangential flows become available as functions which can be readily handled and which are amenable for inclusion in general flow expressions such as NavierStokes. They may also be used alone for certain problems involving fluid pressures, wall loading and shears. The model is not restrictive for circular shapes and is developed for any static or varying conic section with or without small variation in the axial dimension. Limitations on the analysis are discussed, where appropriate, in the text. Instead of the classic approach utilizing a simplified version of the NavierStokes equation and the continuity equation, the problem is approached from a different view. Known boundary conditions are utilized in conjunction with the continuity equation and an acceptable axial velocity function. This function defines profiles varying from fiat to parabolic, etc., is valid for both steady and putsatite flow and can vary according to the conditions encountered. Boundary conditions include zero radial and tangential velocity at the central axis, zero axial velocity (and axial velocity gradient) at the wall while tangential and radial velocities at the wall are equal to the wall velocity. Conditions include axial tethering of the wall with radial and tangential wall movements related to simultaneous stretch and distortion of the vessel wall. Radial and tangential velocities result from wall movements and axial vessel taper; the axial velocity profile varies as a function of central axial velocity, its axial gradient, wall movement and vessel taper. The vessel is taken as having an elliptic crosssection which can distort and stretch to a limiting circular shape. Presently, the flow system can be described with vessel measurements which include the variation of the major and minor semi axes (or eccentricity) with time, the axial taper and fluid measurements which track the central axis axial velocity in time and space. The measurements permit establishment of validity via experiment.
2. Geometry. Figures la, c illustrate the variables (R, 0, Z) which are defined The l~ay (Rw) is defined for an elliptic crosssection as
in the legend.
Rw =
b (sin 2 0 + e2 cos 2 0) lj2
(1)
THREE DIMENSIONAL LAMINAR FLOW
491
Ca) :~.
Re
. . . .
,.jm
"E (b)
I,
Re,
i
~176 \ /
, Z /N / f //~
,(~
()o~"x~
~,~6o.
I
Radiusof ~( curvolure
,~
v~ b Vew~N,
/ /
\
R,
.
VR,
Rw
I/
/
Figure 1. Segment of a vessel indicating variables. VRe is radial velocity, measured positively outward, V~ is axial velocity, R w is the distance measured from the center of s y m m e t r y to the wall, ~b is the angle of axial taper, a is t h e m a j o r semi axis; b is the minor semi axis; 0 the angle to R w measured positively counterclockwise from the m a j o r semi axis; R e is the distance to a nested oval of the same eccentricity as defined b y b/a; r is the radius of c u r v a t u r e of the wall, located by R w , with coordinates ~:, ~?; a is the angle between r and R w and ~ = O + a; V o is the tangential velocity perpendicular to Vne, and measured positively with 0; Ve~ is the tangential velocity, t a n g e n t to the actual wall of the vessel (or to a nested oval defined by Re)
492
J U L I U S MELBIN, R. G O P A L A K R I S H N A N AND A. N O O R D E R G R A A F
where R w = R a y from center o f s y m m e t r y to the wall; e = eccentricity (b/a).
Stream tubes are ellipses of constant eccentricity and crowd towards the minor semi axis (Nikuradse, 1955), a point discussed f u r t h e r in the section d e n o t e d axial velocity. The R a y ( R e ) specifying a stream t u b e is given as b  8b sin 2 0 + \ a  ~ ]
c~
where R e = R a y from center of s y m m e t r y less t h a n R w and 8b and Sa signify incremental distances measured from the wall. Since b  ~b . . a . ~a .
b a
e
(3)
(2) becomes Re =
e(a 
(4)
8a)
[sin 2 0 + e 2 cos 2 8] 112
L e t k = R e / R w = (a  $a)/a (where k is measured from the center o f symmetry), i.e. 0 < k _< 1. Figure lb illustrates relations and terms for a stream tube. A segment length o f stream t u b e is ds = p dO = [[dx~ ~
L\de]
{~]~'~ + \dO] J
dO,
(5)
where p is a length, lying on the radius of c u r v a t u r e (r) and relates a n arc (subtended b y d6) on a s t r e a m t u b e (perpendicular to its radius of curvature) to a R a y originating at the center of s y m m e t r y of the non circular crosssection. W i t h Figure lb we can show, for the same dO P ~
Re2 + \'dO] J
"
(6)
E q u a t i o n s (1)(6) therefore p e r m i t c o m p u t a t i o n of R e and p if the m a j o r and minor semi axes of the elliptic vessel is known. Also, in Figure lb, after distortion klalel Re1 = (sin 2 8 + e~ cos 2 0)1/2
(7) d R e 1 _ k l a l e l sin 0 cos 0 (e~ 
d0
1)
(sin 2 0 + e~ cos 2 0) 3/2
where the subscript 1 indicates the variable after distortion, occurring in time dt. Thus a 1  a + da and el = e + de, where d a and de represent the inere
THREE
DIMENSIONAL
LAMINAR
FLOW
493
mental change of the major semi axis and eccentricity respectively due to the distortion so that, e.g. e 1 becomes the new eccentricity applicable for the stream tubes, i.e. de = ~e/~t dt. The stream tube defined by pz(O) is downstream of the tube defined by p(O) except at the wall, where we assume no slip and tethering in the z direction, i.e. at k = 1, V~ = ~V~/~z = O. The error introduced b y projecting Pl back onto the original R  O plane is small as long as geometric taper, if it exists, is small and the axial fluid velocity realistic. T h a t is, in time dt, the Pz stream line is close to the original R  O plane. We assume that, with distension, wall stretch is uniformly distributed between the semi axes. Also with stream line flow, the stream tubes do not intersect and the equivalent statement in the fluid is that, with distension, the fluid on a stream tube is uniformly dispersed between the semi axes. The lengths of a stream tube to the angle O, prior to and after distortion are given by
and SOz =
l
' O + 50
Pl dO ~
f~
(8) Pl dO + Pz 80,
0
where 30 is small. With uniform dispersion, the fractional change in perimeter (per unit of original) is (St  S ) / S where S and S 1 are known lengths of initial and subsequent stream tube lengths respectively in a subtended angle, e.g. length over a quadrant or total perimeter etc. Therefore, we can write, using (8),
,o: p~, f: p dO  f; p~ dO],
(9)
where in terms of R w and R w l (the respective wall dimensions)
[
[~
p = k Rw 2 + \ 7
f j
(lo) p, = k,
+ kbg/J
Let ;~ and ;~1, equal the respective bracketed terms of (10). Since S = k S w and Sz = k l S w z , where S w and S w 1 are the lengths of wall perimeter [i.e. k = k 1 = 1 in (lO)] equation (9) becomes
, r[8w w, j.:, )t dO 
30 = ~z
f:o ]
~z dO 9
(11)
494
J U L I U S M E L B I N , R. G O P A L A K R I S H N A N A N D A. N O O R D E R G R A A F
Since two reference systems exist (r, Rw) a f u r t h e r geometric relationship m u s t be considered which relates two t a n g e n t i a l velocities, i.e. perpendicular to r and Rw. The relationship between these velocities m u s t be defined. I f R w is specified as a R a y from the center of s y m m e t r y to the wall, t h e n the tangential velocity (Vs), derived utilizing Rw, is perpendicular to the radial velocity (Vne is defined on the R a y , e.g. VRw = ~Rw/~t a t the wall) and n o t t a n g e n t to the vessel wall, e x c e p t for the circular vessel. I n this l a t t e r case R w becomes the radius of curvature. The tangential velocity, t a n g e n t to a stream t u b e (Ve~) or to the wall (Vow) is related to V0 b y the angle ( a ) f o r m e d between the R a y (Rw) and the radius of c u r v a t u r e (r), i.e. with reference to Figure lc. Ve~ = Ve(cos a )  1
(12)
The coordinates for the radius of c u r v a t u r e (r) are =
x 
r~/(~ 2 +
~)2).2
(13)
where
= ~Rw/~O cos t?  R w sin ~) = ~Rw/~O sin 0 + R w cos
(14)
therefore cos r =
R w cos 0 + ~Rw/~O sin O ;~
(15)
R w sin ~  ~Rw/~O cos 0 sin r = )I Since a = r 0 and, b y identity, cos a = cosScos~b + s i n C s i n S , with relations corresponding to (6) and k 1 = k = 1 (da = de = 0) we find sin 2 ~ + e 2 cos 2 cos a = (sin 2 ~ + e4 cos2 ~)112
(16)
SO t h a t for a n y condition or point in the t u b e the relationships between the velocities m a y be determined from (12).
3. Velocities. Axial Velocity (V~): Since the crosssection is elliptical, an initial assumption must be m a d e concerning the distribution of stream tubes. F o r example, one m a y consider the b o u n d a r y layer constant, in which case flow at the center is a ribbon whose width is d e p e n d e n t on the relative dimensions of the semi axes. Alternatively, one m a y p e r m i t stream tubes to crowd a t the minor semi axis and flow on the central axis is t h e n a point on the crosssection. Since the former case becomes intractable b y virtue of discontinuities at the
THREE DIMENSIONAL LAMINAR FLOW
495
center, it is fortunate t h a t there is evidence for the latter (Nikuradse, 1955) in non circular shapes and precedence for the form of the initial assumption. Thus we assume for stream line flow W z = (1 
]~N)Vr
(17)
where Vet = center line axial velocity; N = a number defining the profile. Thus V. is not a function of 0 and at N = 2, V~ takes on the parabolic form and is the solution for axial velocity with the PoiseuiUe assumptions. As N increases in value V~ becomes flatter and approaches plug flow. Since the axial profile can v a r y with vascular distortion, N can be a function of time. The boundary conditions satisfied by (1) are: (a) At the wall (k = 1) V~  OV~/~z  O. (b) At the central axis (k = 0) V~ = Vcz. In Figure la the variables are illustrated and defined in the legend. Figure 2a shows V~ for various values of N. Figure 2b shows stream tubes in a crosssection of an elliptic vessel having an eccentricity (e) of 0.5. The stream tubes are all of e = 0.5 and are functions of 0. Here the constant velocities are defined by the dimensionless q u a n t i t y ,.. = V~/Vcz. Each quadrant depicts v~ according to N corresponding with Figure 2a. Figure 2c illustrates the distribution of v~ for e = 0.9. As the vessel becomes circular vz becomes less dependent on 8. F r o m (17) we derive c~V~ OVr _ kU ON Oz  (1  k N ) ~ Oz Vczlnk"
(18)
Equations (17) and (18) express the relationships between axial velocity and the center line velocity and their spatial derivatives. Since the axial velocity profile can develop in z, N is a function of z. Tangential velocity (V0): The boundary conditions are: (a) At the central axis (k = 0) Vo = O. (b) At the wall (k = 1) Vo equals the wall velocity in the direction perpendicular to Rw. (c) In the circular vessel (e = 1) Vo = OVo/~O = 0, everywhere. With reference to Figure lb and c
~Rwi38)
Re'l = k i R w i + ~
(19)
496
J U L I U S MELBIN, R. G O P A L A K R I S H N A N AND A. N O O R D E R G R A A F (a}
R
(b)
b N=4
t N=2 4~=09 J ~ "~z:0.75 ~ _ l _ l ~ / " ~'z=0.6^ ,~ yZ=U.=,{.;~ " vz:O.3
~
e=O.~
N=8 t
N=50
b
(c) N=4
~~Vz=0.9 ~.,.,,.,."".,~Vz =0.75
~ ' ~ : o . 6
F~,~ ,7,,,,X/X"~ Vz=O.5
L../"% \ \ '~kV,'z=o.~5
e=0.9
U
F i g u r e 2. Axial velocity profiles and s t r e a m tubes. (a) V2 for various values of z~T f r o m parabolic (N = 2) to " p l u g " (57 = 50). (b) S t r e a m tubes defined in a n elliptic vessel of eccentricity (e) of 0.5 as f u n c t i o n s of 0 in t e r m s of vz = V z / V c ~ . E a c h q u a d r a n t depicts the profile of v z according to N corresponding to Figure 2a. (e) S t r e a m tubes defined in elliptic vessel of eccentricity, e  0.9 as functions of O, in t e r m s of v z
THREE DIMENSIONAL LAMINAR FLOW
497
SO that from Figure lb
Vo dt = k 1 Rwl + ~
(20)
80 sin 88
if dt is small, then 80 is small and sin 80 _~ 80, thus
o,.o 0o
at I,L ao
+ ,so ~
+
a~
aoj
+ gg
[
(21) where, with (1 l) ~80
8w1
o# = awaS a
1 0~
al o0
AdO
+
~ Tg
t~ d O 
1.
(22)
Equations (20) and (21) express the relationships between tangential velocity and geometric distortions. Radial v d o c i t y (VRe): VRe is defined on the ray Rw (which becomes confluent with the radius of curvature at the axes or if the vessel is circular). The boundary conditions are: (a) At the central axis (k = 0) VRe = 0. (b) At the wall (k = l) Vne = VRw the wall velocity in the direction of Rw. (c) In the stiff walled, straight tube VRe = 0, everywhere. With reference to Figure lb tan 86 = V8 dt/(Re + VRe dt)
(23)
so that
k~( ORw~ ) VRe = ~ Rwl + ~ 80 cos 8 0 
kRw d7"
(24)
With 80 small, cos 80 _~ 1 and we note, from (11), that 080/~k = 0 since A and Sw are not functions of k, so that
~R7 = R w OTo= Rw d'tL Ok
Rw~ + ~
80
 Rw 9
(25)
Equations (24) and (25) express the relationships between radial velocity and geometric distortions. Of interest, as discussed in the introduction, is the determination of the dynamic three dimensional flow field. In this context, the previous sections express the axial, tangential and radial velocities in terms of the center line axial velocity, variation of the axial velocity profile in time and axial distance and vessel shape and wall distortion. For this development NavierStokes relations have not been used and we have thus far not introduced the continuity equation.
498
JULIUS MELBIN, R. GOPALAKRISHNAN AND A. NOORDERGRAAF
The Continuity Expression: c o n t i n u i t y relationships m u s t be satisfied and m a y be invoked to yield ]q in terms of the other variables. W e utilize the cont i n u i t y expression at points as expressed for the circular cylindrical coordinate system. F o r an incompressible fluid we have  8VR~ ~V~ 8VO VRe "} k   ~  + k R w  ~ + 80 = 0
(26)
which embodies the set of equations (1), (11), (18), (21), (22), (24) and (25). Since (26) applies at a n y point, we can evaluate at 0 = 0, here 8Rw 8Rw~ 8)t I ~0 8080 = 0"0= Jo ~ dO = O,
80
R w = A = a, 8bO ~0

(27)
R w l = AI = a + da,
aSwl  1 (a + da)Sw
so t h a t (26) becomes
8kl kl 8k + e  ~ =
a
a + da
( 2 _ 8 ~Vd.t )
(2s)
a linear differential equation of first order, where aZw 1 (a + da)Sw
(29)
The solution to (28) is =
a
~z
dt
+ c}
(30)
with (18)
2 kl = ~ k a a + da k'e + l
[ 1 ( ~
1
(~l)~N]ev~,
Yjiu
N + e + 1 In/c
8z
N + ~ + 1  ~ Vcz dt ,
where the constant of integration (c) is zero, since at k = 0, kl = 0. we also h a v e kl = 1 at k = 1. This constraint, therefore, implies
]
~Vczdt=(N+~+ l) 2 _[
E+ 1
N ( N + E + I) 8Vc~ ,, dt E+
1
8z
(31)
However,
(32)
THREE DIMENSIONAL L.~2~IINARFLOW
499
SO that (31) becomes kl = (a + da)(eka + 1 ) { [ ( a + d a ) ( e + a + 2  [1
1)
_ 2][kU_ ( N + e + 1)kNlnk]
 kN + NkNlnk] OV~zdt t 9
(33)
Thus with the continuity equation, k~ is expressed in terms of variables relating to wall dimensions and axial velocity variation. This information is utilized in (20) and (24) for determination of radial and tangential velocities.
4. Genesis of Variation of N with Other Variables. Equations (18) and (32) relate the axial change in velocity to the velocity profile which, in turn, may be expected to be a function of wall movements, vessel taper and independent variations of V~z. Although at present OVez/Ozdt in (33) is expected to be a measured variable, it is of interest to evaluate the form of contributors to dN/dz. Consider a circular, straight, compliant tube. For this condition Vo = OVo/O0 = 0, and with (18) the continuity equation is
O(RVR) OR
= 
R OVez Rlq+IOVet RN+IO~z ( R) ~
+ RcNOz + RcN
In Rc
where R e is the wall dimension {radius of curvature of the wall). with respect to R we find
Vet
(34)
Integrating
VR = R{[ ('~/Rc)N ~10Vc~~z+ 'N~2Vcz + (R)N ~ON lnRcR N + 2 1 ) }
(35)
at the wall VR = VRc, R = R e, thus Re VRo =
N + 2
0 ez ~
Vez + N + 2 7z
(36)
V.c (N + 2)~ VctRc
(37)
therefore
ON 0z
[ ~NRcOVc' [2(N + 2) Oz
for the circular, straight, compliant tube. Consider a circular, tapered, stiff tube. At two transverse planes a distance dz apart, the radii are R c and Re1 respectively, the center line velocities are Vcz and Ve~l respectively and the velocity profile indices are N and N1. Utilizing (17) and integrating over the upstream plane with respect to k we can determine the flow, which when equated to the flow downstream results (with neglect of higher order terms) in the continuity equation
500
JULIUS MELBIN, R. GOPALAKRISHNANAND A. NOORDERGRAAF
however Re1 = Rc + ~Rc/~z dz N1 = N + ~NIOz dz Vczl = Vcl + ~Vcz/~z dz.
(39)
From (38) and (39) we find
0zr ~
 N ( N + 2 ) [ R Vcz =
2ReVel
2" 0R2
[ c ~z +
[ct~z]"
(40)
Adding (37) and (40), with the assumption of superposition and the variables remaining unchanged, we have, in general, for the circular, tapered, compliant tube ON  N ( N + 2) OV~ N + 2 [(N + 2) 0R~] (41)
Oz =
2V~
0z
~R~ [
V~
VRo + N  ~ z j.
With examination of (32), (33) and (41)i it can be seen that secondary flows can be expressed in terms of variations in dimensions, velocity profile and axial velocity. 5. A n Approximation for Constant Wall Perimeter. Changes in perimeter lengths are embodied in (29). To pursue the phenomena of wall stretch to the fullest extent, the elastic properties of the wall must be introduced. However, we can proceed more readily to examine the simpler case where the vessel distorts without stretch. This will be the major effect in the variation of vessel shape until the elliptic vessel approaches circularity. Utilizing the mensuration expression for the elliptic perimeter for the case where only distortions exist (stretch insignificant) we have S w = era[2(1 + e2)]1/2.
(42)
Since ~Sw/~t = 0, we find, with da = ~a/~t dt, de = ~e/~t dt (1 + e2) da =  a e de.
(43)
With S w = S w 1 and e = a/(a + da), (33) becomes ka f[2a+da kl  2a + da a
+ 2
2][k N 
(a + (N + 1)(a + d a ) ) k N l n k ] a + da
[1  k N + N k N l n k ] ~ Vcl dt'~ j"
(44)
6. Restraints on $fl. For sin 30 and cos 30 to be approximated by 3fl and 1 respectively, we limit 3fl < 0.1. For our worst case (e ~ 0.5, fl ~ 30 ~ at the
THREE DIMENSIONAL LAMINAR FLOW
501
wall, utilizing (11) we note that the condition 80 < 0.1 is met if ldat < ]0.05a 1 and [de[ < [0.05e I. However, with (43), the condition for constant perimeter, if de = 0.0he, da is limited to  0.01a.
7. Restraints on dt and ~a/~z. To limit the error imposed on the geometric projection, when small axial taper is the case, we examine the limiting condition of V, (N> oo). We require, at worst, that ?a/~z dz be at least an order of magnitude less than ~a/~t dt at the RO plane of interest so that the stream tube projection, after a time dr, back into this plane is a reasonable approximation. For a taper of angle ~b, tan~b = ~a/~z and  t a n ~ b d z = tan~bVcz dt = ~a/~z dz. We require ~a/~zdz < 0.1 ~a/~tdt = 0.1 da. For a maximum angle, ~b = 2 ~ and V~z of the order 100 cm/sec and maximum Ida[ = [0.05a[ the quasi static condition can be approximated with time increments (dt) of about 1.5 msec. The additional condition of constant perimeter, if [de[ = [0.05e[, limits dt to about 0.3 msec. 8. Discussion and Results for a Case of Distortion at Constant Perimeter. Computations would proceed by first determing da in time dt with the existing profile (N) and center line velocity spatial gradient (~Vcd~Z). These are utilized in (33) or (44) to determine k 1 as k is traversed from 0 to 1. For each k and k 1 (18) and (24) determine Ve and VRe for each ~, where (1) and (11) provide the required variables Rw and 86 etc. In these illustrations the contribution of ~Vcz/~z is examined by inserting a value. In reality this and da would be obtained b y measurement or, if amenable for evaluation, calculated by NavierStokes relations utilizing known pressure gradients and vessel wall properties. The flow model provides a convenient means of determining velocities everywhere, in terms of the profile index (N), if the dimensional variations of the vessel are known. The index (N) itself is available from two simultaneous axial velocity measurements. Evaluation of phenomena under quasi static conditions requires that the values of a, ~a/~t, e, ~e/~t and N at time t = 0, are appropriate for the entire period dt. Thus to evaluate over large distortions, all variables must be newly established after dt, at which time a new perturbation can be considered. This requires the additional information concerning ~N/?t which, within the scope of this paper, is a measured variable. In the illustrations, a very small perturbation is applied, to comply with the restraints for the quasistatic evaluation. Thus, all variables remain close to original values and small non linearities (especially in Vo) are obscured in the plots.
502
J U L I U S MELBIN, R. G O P A L A K R I S H N A N A N D A. N O O R D E R G R A A F
tR"N=2 o oo4z
/ai~
8:75 8=90 _ 8=60
/
~'N=2 00043 /div
(b) : I
~
k
0 0043}
(C)
8=0
]
~
=2 00045tldiN ]
0
8=0 . . . . . . . . . . . . . . . . .
YRe
8=90 8=75
k 0.0043 i /div
0.0043 /div. l
, ~ /
I
~
N=50 e=0.5 8Vr do=O  ~  dt=O.025
N=50  ~ ' ~
N=50
e=0.5 i)Va da =O.OI ~ dt =0 v.,
e=0.5
~__~z~dt =0.025
da=O.OI
%~
%,
'~ N=2 0 0043i/div,
N=2
o.oo43 i/divN =2
8=75 8:90
0 0045 /div.
t ~ 8 = 4 5 (e)
t ....
i ........
1 N=50 e=0.9 DVcl da=O.OI ~ dt=O
~
0
k
t 0.004:
0 0 0 4 3 1/6iv.
i
,

/div.

(f)
......... 0
I
0,0043
iv [
o e=0.9 ' ~ z j da=O.OI dt =0.025
' N=50 e=0.9 aVd da:O ~ dt =0.025
Figure 3. Illustrations of radial velocity (VRe). Each quadrant demonstrates curves corresponding to the axial velocity profile (defined by N) as a parameter. Other parameters, eccentricity (e), OVcz/Oz and Vae dt at the major semi axis (da), for a = 1, are shown at the lower lefthand corner, k is the radial coordinate where k = 0 at R e = 0 and k  1 at R e = R w . Curves of VRe for different rays are marked by the angles Distortions cause secondazy velocities t h r o u g h the factor k 1 which, however, differs from k only a t the third or f o u r t h decimal place for these v e r y small perturbations. This close correspondence between k a n d kl is obvious from e x a m i n a t i o n of (44). Illustrations are presented with geometric p a r a m e t e r s ehosen in the range o f measurements m a d e for the p u l m o n a r y trunk. Figure 3 illustrates the radial velocity for a v a r i e t y o f circumstances. Similar
THREE DIMENSIONAL LAMINAR FLOW
503
to Figures 2b a n d c each q u a d r a n t defines VR for the N values 2 a n d 50 as noted. T h e p a r a m e t e r s for each g r a p h are shown in the lower lefthand corner. Figures 3a a n d b are illustrations for u n t a p e r e d a n d t a p e r e d vessels respectively, with eccentricity of 0.5, where da =  0.01 in t i m e dr. The negative sign indicates t h a t this point m o v e s t o w a r d s the center of s y m m e t r y as the vessel distorts t o w a r d s a circular shape. Curves for different r a y s are m a r k e d b y the angles 8. These curves indicate the radial variations of the r a y s as a function of k, and it should be n o t e d t h a t the m o v e m e n t s a t different angles do not change u n i f o r m l y in magnitude. This is so, because for e = 0.5 the ellipse flattens as the m i n o r semi axis is a p p r o a c h e d a n d r a y s at m a x i m u m c u r v a t u r e are longer and their changes, therefore, m o r e rapid. I t should also be n o t e d t h a t with distortion of a non circular vessel m a x i m u m VRe is not always realized a t the wall a n d is d e p e n d e n t on the velocity profile. W h e r e the profile is constrained as plug flow (large N values), this p h e n o m e n o n persists even as the vessel achieves a m o r e circular shape. I n addition the direction o f VRe along a r a y m a y a l t e r n a t e in k.
re,
N=2 '/~v
000085
Y" N=2
845
845>//
0.0008[ /div.
@.60~~
//
=75
(o)
830
(b) ~
O
O
k
k
I N=5O
0.00085~ _ _
] t
e=o.5
da=O.OI
N=50 0.00085
e=0.9
da: 0.01
Figure4. Illustrationsofnormalizedtangentialveloeity(Vs~). Each quadrant demonstrates curves for the axial velocity profile (defined by N) as a parameter. Other parameters, eccentricity (e), avcdaz and VRe dt at the major semi axis (da), for a = 1, are shown at the lower lefthand corner, k is the radial coordinate wherek = 0 a t R e = 0 a n d k = l a t R e = Rw. Curves of Vew for different rays are marked by the angles 8
504
JULIUS MELBIN, R. GOPALAKR!SHNAI~ AND A. NOORDERGRAAF
Figure 3b shows conditions where ~Vcz/~zdt is given a value of 0.025. This may be due to vessel taper or other variation of Vc~ in the compliant tube. The value is held constant, in these figures, a condition not necessarily true for the real case where, for example, taper m a y vary with e or with z. In general, increasing ~Vcz/~z m a y be expected to generate negative components to VRe. Figure 30 indicates the contribution of ~Vcz/~zdt to VRe. Figures 3d, e and f are similar to Figures 3a, b and c, however in these former illustrations e = 0.9. It can be seen that as the vessel becomes circular, secondary flows are reduced. Figure 4 illustrates the tangential velocity (tangent to the stream tubes and wall but not perpendicular to VRe) Although the values of tangential velocity perpendicular to VRe are different with respect to 0, qualitatively the results are similar and the discussion can apply to both. Two sets are shown for two different eccentricities. Variations due to velocity profile (N value) is indiscernible and this is true also for small variations of ~V=J~z. Thus, these serve as representative illustrations for all Ve~ at the level of these small perturbations. Vow appears as a linear function of k in the plots at this scaling since k 1 _ k and the factor (k Rw) which emphasizes the difference between ]c1 and k, in VR~, is not present in V0. At the axes (0 = 0, ~ = ~r/2)F0~ is zero and reaches a maximum at about 0 = 45 ~ t o 60 ~ depending on existing eccentricity. I t should be noted that the scale of Figure 4 is expanded to five times that for VRe so that Ve~ flows are to be seen considerably smaller. This project was supported b y USPHHL4885 and H L 10,330. LITERATURE Ling, S. C. and H. S. Atabek. 1972. "Nonlinear Analysis of Pulsatile Flow in Arteries." J. Fluid Mech., 55, 493511. Nikuradse, J. 1955. I n Boundary Layer Theory, Schlichting, H., ed., Chapter 20. New York: McGrawHill. l~oordergraaf, A. i969. I n Biological Engineering, Schwan, H. P., ed. New York: McGrawHill. RECEIVED 92074