Entry flow into blood vessels at arbitrary Reynolds number

J. Arnmcch~mrcs. Vol. 3. pp, 23-38.

Pcrgamon Press. 1970

Pnnted in Great Britain

ENTRY FLOW INTO BLOOD VESSELS ARBITRARY REYNOLDS NUMBER”

AT

H. S. LEW and Y. C. FUNG Department of AMES (Bioengineering). University of California, San Diego, LaJoIla, Calif. 92037, U.S.A. Abstract-The steady axisymmetric flow in the inlet region of a circular cylindrical tube is studied. Emphasis is laid on the entry flow at lower (but finite) Reynolds numbers, and on flow in which arbitrary axial and radial velocity distributions are prescribed at the entry section. The boundary layer approximation is not used. A solution is presented in the form of a series of two sets of properly chosen eigenfunctions. Numerical results are presented for the case of uniform entry in the Reynolds number range 0- 100. The application of the results to’ the blood flow problem is discussed. INTRODUCTION

of flow in the inlet region of a circular cylindrical tube is a classical problem with a sizable literature. A typical case is the uniform entry flow. Boussinesq was the first to make a theoretical investigation of this problem (1891). Boussinesq’s solution does not agree with the result of experiments by Nikuradse near the entrance. Schiller (1922) obtained an approximate solution by using a method analogous to that of Polhausen in his discussion of flow in a boundary layer. Schiller’s solution gave an excellent agreement with experimental measurements on velocity distribution, at least for the first third of the inlet length. Atkinson and Goldstein (1938) studied the boundary layer flow in the inlet region, by a generalization of the Blasius’s solution of the problem of the two-dimensional flat plate and matched the solution with a perturbed Poiseuille flow of the Boussinesq type at a certain section in the entry region. The corresponding parallel-plate case was solved by Schlichting (1934). Improvements on these methods are published by Siegel (1953). Shapiro, Siegel and Kline (1954), Campbell and Slattery (1963) and Collins and Schowalter (I 962). Two excellent reviews of early works on the probTHE

DEVELOPMENT

*Receiwd

2 June

1969.

lem are given in Prandtl and Tietjens’ Applied Hydro- and Aeromechanics (1934, p. 25, 26) and Goldstein’s Modern Developments in Fluid Dynamics (1938, p. 301 et seq.). The uniform entry problem was analyzed also by Han (1960), Langhaar (1942), Targ ( 1955), Bodoia and Osterlt ( 196 l), Hombeck *(I96 1). Lundgren, Sparrow and Starr ( 1965), Christiansen and Lemmon (1965) and Sparrow, Lin and Lundgren (1964) on the basis of linearized inertia terms without postulating a boundary-layer model. These authors consider only one of the three equations of the balance of momentum- the one in the direction of the mean flow, and ignore the other equations of momentum. They assume that the static pressure is uniform across each cross section. The retained equation of motion is linearized. Although there are minor differences in the methods of linearization and other assumptions made by these authors, the general features of their solutions are similar and all are more or less in agreement with the experimental results of Nikuradse (see Prandtle and Tietjens, (1934, p. 25, 26) Pfenninger (1961, in Reynolds number range 10,680-54,800) and Reshotko ( 1958, at Reynolds numbers 4100,760O and 16,000). In the present paper the radial equation of

24

H. S. LEW and Y. C. FUNG

momentum is not ignored. Our results show that at small Reynolds numbers the radial velocity in the entry region is as high as 30 per cent of the axial velocity; hence the ignoration of the radial momentum equation cannot be justified at small Reynolds numbers. Unfortunately there is no experimental data at small Reynolds numbers available for comparison. As modem computers become available, many attempts were made to solve the problem directly by solving the Navier-Stokes equation by numerical methods. Wang and Longwell (1964) studied the entry flow into a two-dimensional channel. Mills (1968) studied the low Reynolds number flow in pipe orifices, Christiansen and Lemmon (1965) worked out the uniform entry flow. These papers show that numerical analysis applied to the partial differential equations of the type considered here is not easy: considerable amount of development is required in each case, and the questions of convergence and accuracy cannot be settled easily. Recently, Lew and Fung (1969) solved the entry flow into a circular cylindrical tube at zero Reynolds number by an infinite series which is composed of two sets of eigenfunctions. Each of these eigenftmctions satisfies certain boundary conditions exactly. Their combination, therefore, leaves only one set of numerical constants to be determined by the remaining boundary conditions. Compared with the direct numerical approaches such as the methods of finite differences, relaxation, iteration, or ‘finite elements’, one might say that if the series approach solves for n unknowns, the corresponding direct numerical approach must deal with at least n2 unknowns. To obtain accurate results with large n the difference is evident. The additional labor of constructing the eigenfunctions is therefore amply justified by the final results. The object of the present study is to find a solution of the problem of entry flow with an arbitrary axisymmetric distribution in the axial and radial components of the velocity

at the entry section: a solution which is valid for the whole range of Reynolds number as long as the flow remains laminar. Our study was motivated by a research on the flow of blood and other fluids in animal organs. The Reynolds number of the flow involved in the living body varies from a few thousandths (lo+) in the smallest capillary blood vessels and in the bronchioles of the lung to a few hundreds (lOz) in the large arteries. The solution to the case of very small Reynolds number is presented in Lew and Fung (1969). The solution to the case of the uniform parallet entry at large Reynolds number is well known in the literature. This paper bridges the gap and is applicable to a more general type of entry flow at arbitrary Reynolds number. The applications to the blood flow problem will be discussed further in a later section. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

The non-linear convective acceleration term in the Navier-Stokes equation is the well known source of difficulty. In some cases, this non-linear term can be linearized without seriously impairing the accuracy of the analysis. For example, the boundary layer equation with a linearized inertia force provides a highly satisfactory solution to the problem of uniform entry flow at high Reynolds numbers. In the present article, the non-linear inertia force v - Vv in the Navier-Stokes equation is approximated by (v) - Vv, where (v) is the average value of the velocity over the cross section of the tube. Heuristically, such an approximation is proper when the distribution in the velocity at the inlet cross section is nearly uniform, because then v . Vv is well approximated by (v) - Vv in the inlet region where the convective inertia force is significant. Far downstream from the inlet the velocity distribution becomes parabolic and v deviates from (v) ; but then the inertia force is insignificant in such a region. It is shown that the solution based on this linearization matches the known results at

ENTRY FLOW INTO BLOOD VESSELS

both ends of the Reynolds number range; at the high Reynolds number end with the classical result of Targ, at the zero Reynolds number end with our earlier result (Lew and Fung, 1969). The following analysis is restricted to a steady axisymmetric entry flow with a nearly uniform velocity distribution at the entry section. The Reynolds number is such that the boundary layer approximation may not be valid. Under the linearization scheme named above, the equation motion becomes *

25

(5)

(6)

(ru’) = 0

g+g

where u’ and u’ denote the x and r components of v’, respectively. The no-slip condition of the fluid on the wall of the tube requires u’ t&r) I Pa = 0 0’ (XJ) 1f(l = 0

for

OCX
(7)

for

Osx
(f-4)

(1) where a is the radius of the tube. The entry condition at the inlet cross section requires where U = (v) is the average speed of flow over the cross section of the tube, v is the velocity, p is the pressure, p is the density and p is the viscosity of the fluid; V designates the gradient operator and x is the one of the cylindrical polar coordinates r, 19, x with the x-axis coinciding with the axis of the tube and the orgin of the coordinates system lying in the inlet cross section. The fluid is assumed incompressible. For an axisymmetric flow, the equation of continuity is

ff++-$(ru) =

0

(2)

where u and u denote the x and r components of the velocity, respectively. It is convenient to split v and p into two parts v = v,+v’

(3)

P=pm+pl

(4)

where vm and pm represent the Poiseuillean velocity and pressure at x = =. With (3) and (4), equations (1) and (2) become

u’(~,r>l,~~ = u(x,~)l~~-

u,

for OSrSU

u’(x,r) lzSO= r(x,r) ltPO for 0 c r G a

(9) (10)

where u(x, r) I zlo and u(x, r)lm. are the given distribution of u and tr at x = 0, respectively. Finally, as x --* m we must have u’ (x,r) lIIm = 0

for

OSrGu

(11)

r’ (x,r) IsI= = 0

for

OS

(12)

for

OS rs a. (13)

rSa

and p’ kr)

IFDo= 0

METHOD OF ANALYSIS

The equation of continuity (6) is satisfied if v’ is derived from an arbitrary f(x,r) as follows:

where _?and i are unit base vectors of x and r axis, respectively. Substitution of (14) into (5) yields

*Equation (I) is formally identical with Oseen’s equation (Oseen. 1910). the reason by which this equation is obtained here however, is entirely different from Oseen’s.

26

H. S. LEW and Y. C. FUNG

-vb’-P(C2-~&)gl (15)

-np(v2-~.$v2f=0

ently general to generate the solution to our problem. In order to satisfy the boundary conditions (7) and (8), we write (19) in the following form

where

is the Reynolds satisfied if

number.

Equation

(15) is

pf =++-L)~

(17)

(v+-$v2f

= 0.

(18)

By direct substitution, it can be verified easily that a general solution of (18), which has bounded value in the whole region *

(~~10 c x < CO,Os f c aj is

f(w) = We -AZ+

Be-(v4An+R,Z-Re)$

] Jo

+[Ccos(q$+Dsin(q$)]

5 >

Z,(4)

+E{

cos(~~)Z,[ei~(l+~)“‘~~]~R

-sin

(7:)

(21)

Z,[eiI(l+~)“‘~~]]Z}

+F { cos(r~$ $-sin (q:)

* (

where pn and qn denote the nth root of the equations J,(x) = 0 and J,(x) = 0, respectively; J, is the first order Bessel function of the first kind, CJis the mean velocity of the flow, and

Z,[ei~(l+$Q1’4~~]lZ

I, [eiF(*+Z$)li”qi]IR}

C19J

(22) where A, B, C, D. E, F, A and 71are constants, J,, is a Bessel function and Z0 a modified Bessel function, both of the first kind and order zero. I& and loI1 denote the real and imaginary part of I,,, respectively, and R

(Y= tan-l”.

77

From this to (14L

solution

we obtain,

according

(20)

We shall show that the solution (19) is suffici-

A

<4pn2 i- Rc2 - R, e

-(ti4p.‘+R.2-RJ

&

1

ENTRY

FLOW INTO BLOOD VESSELS

27

_ e-‘dit+Rd~

(24)

where tR and 5, denote the real and imaginary part of 5. When the boundary condition (7) and (8) are applied to (23) and (24), the following results are obtained:

aR2-&2)10(4dlR -YRtJoW,l

C,(q) =-E,(r))

-F,(q)

[(bR*--‘if?)~O(~~)

tI

(25)

+25,&&&)(~1

Ql(17)

=E,(7))[5R~1(~7))11+~l~l(~rl)lRl

-F,(1)[~~Z,(511)lR-5,1,(571)lIl

(23)

(26)

=

-(” (27)

28

H. S. LEW and Y. C. FUNG

from the physics of the flow as well as from the mathematical nature of the governing equations that an alteration of the flow at a given point does not influence the flow at a (28) great distance away from that point. If L We can determine C,,(v), D,(q), E,,(T) and designates a distance within which the inF,(v) in terms of A,, and B, by solving (25), fluence of the perturbation of the flow is (26), (27) and (28) simultaneously. Thus, all effectively limited, the condition (1 l), (12) constants involved in (23) and (24) are deter- and (13) can be approximated as mined except A,, and B,, which can be obtainv'=O and p ’ = 0 forL < x. (30) ed by applying the boundary conditions (9) and (10). Since the sequence of terms involved in these equations is not orthogonal, the deter- This suggests that the solution in the inlet mination of A,, and B, is rather tedious. One region given by (23), (24) and (29) can be way to do it is the method of collocation: one approximated by replacing the Fourier insolves for A,, B, from a set of linear simul- tegral in these solutions by Fouries series. In taneous equations which are obtained by so doing, the region where the solutions are applying the boundary conditions (9) and (10) approximated by the Fourier series has to be at least 0 < x < 2L so that the flow at x < L to a set of specific values of r. is not influenced by the flow downstream at The pressure can be obtained from ( 17): x > 2L. The degree of approximation can be checked by varying L and examining the asymptotic convergence of the result with respect to L. The numerical calculations become a great deal simpler by such a replacement. If one denotes the ratio of 2L to the radius of the tube by N, the solution can be obtained from (23), (24), (25), (26) and (29) by replacing the Fourier integral by the Fourier series:

This theoretically completes the solution of the problem.

NUMRRICAL CALCULATIONS FOR UNIFORM ENTRY FLOW

The numerical calculations of the velocity given by (23) and (24) and of the pressure given by (29) are rather complicated due to the Fourier integrals involved. It can be seen

etc. We mention that the terms corresponding to m = 0 is omitted because the sine series expansion employed to satisfy v’ = 0 on the wall generates the cosine series. Of course, we have to make the following changes to (20) and (22):

ENTRY

FLOW

INTO

BLOOD

VESSELS

29

After making the above changes and replaceing q by mr/N and C,(r)) by C,,, etc. we can apply the boundary conditions (7) and (8) to (23) and (24) to obtain four equations similar to those of (25), (26), (27) and (28), from which C,,, D,,, E,,, and F,, can be determined in terms of A, and B,. To determine A,, and B, by method of collocation, we apply the boundary conditions (9) and (lo), which, in case of a uniform entry flow, are

numerical calculation is carried out for six different values of Reynolds number: R, = 1,5, 10,30,50 and 100. We should take larger numbers for N if the calculation were carried out for larger Reynolds numbers. Of course, the asymptotic convergence of the solution with respect to the values of N can be checked by comparing the result of calculations for various values of N. The values of A, and B, listed in Table 1 show that the magnitude of A, and B, defor 0 S rd a u’(x.~)[~~= U 2$-l creases rapidly as n increases. With all the ( ) (31) constants involved in the solution determined, one can calculate the axial component of the for 0 G r G a. v’(x,r)j&j = 0 non-dimensionalized velocity ul U by adding (32) NojU = 2(1- (r2/a2)) to u’/U. The radial In the calculations to be presented below, the component of the dimensionless velocity v/U collocation is applied to the section x = 0 at is simply equal to v’/U because v,,= 0. 19 different values of r, which are uniformly Omitting the lengthy formulas for u and v, we distributed in the closed interval [O.0.91, present the formula for the average pressure after retaining the first 120 terms in the sum- (P). mation over m for each value of n. This provides 38 algebraic equations for A,, and B,, from which the values of the first 19 A,‘s and 19B,,‘s are determined, after retaining the first 19 terms in the summation over n. Here the value of N is chosen equal to 30 and the Table l(a). Value of&

R,= 1 n

1 2 3 4 5 6 7 8 9 10 11 12 13 ;: 16 17 18 19

A” 1.30389 1 168730 0 -8.59735-l -1.77772 0 -2.11686 0 -2*17%7 0 -2+8842 0 -1+0609 0 -1.67145 0 -1.41131 0 -1*14567 0 -8+JOO81-1 -6.56747 - 1 -4.54720 - 1 -2@38611 -164775-l -7.88394 -2 -2.80893 -2 -5.39761-3

and B.

R,=5

R c= 10

42

A.

&I

-240547 - 1 -3.09617-2 -1.96935-3 446539 -3 6.07384 -3 6.06188 -3 5.41387 -3 4.59866-3 3.81550-3 3.11034-3 248708 -3 196629-3 1.56632-3 1.28210-3 1.07080-3 8.90529 -4 6.95074 -4 4.80552 -4 2.06200 -4

1.37486 0 -9.39487-2 -4.27938 - 1 -5.25836 - 1 -5*41630-l -5.20250-l -4.79118-l -4.26716-l -3.68283 - 1 -3.07698 - 1 -248056-l -1.91859-l -1.41143-I -9.75141-2 -6.20630 -2 -3-52534 -2 -168718-2 -6.01789-3 -1.15800-3

-7MO58 -2 6.84258 -3 8.72147-3 5*12811-3 2.67818 -3 l-43835 -3 8.07683 -4 4.27990-4 1.99891-4 9.54856 -5 5.07137 -5 -2.72179-6 -7*19837-5 -1.12861-4 -9.76803 -5 -5.27400-5 -2.11626-5 -1.45184-5 -9.26262 -6

A. 3.95514- 1 -1.35369-l -2.53211-l -2.83249-i -2.81748-l -2.65935 - 1 -2.42569-l -2.14867-l -1+34846-l -1.54111-l -1.24058-l -9.58685 -2 -7.05098 -2 -4.87290 -2 -3.10296-2 -1.76300-2 -8.43449-3 -3.00571-3 -5.77794-4

&I -4.35285 -3 1.93720-2 1.13164-2 4.76366 -3 1.35349 -3 -9.29106 -5 -6.49790 -4 -8.63078 -4 -9.15980-4 -8.72926 -4 -7.85107-4 -6.86515 -4 -5.85302 -4 -4.83964-4 -3.86771-4 -294243 -4 -2U436 -4 -1.13726-4 -4.06767 -5

30

H. S. LEW and Y. C. FUNG Table 1(b). Value of A ,, and E,

R c= 50

R,=30 n

A.

B.

A.

1 2 3 4 5 6

4.63041-2 -6.37976 -2 -844257 -2 -8.85282 -2 -8.61074-2 -8.05980 -2 -7.34479-2 -6.52417 -2 -5.63393 -2 -4.71392-2 -3.80623 -2 -2+iS17-2 -2~17110-2 -1.50064-2 -9.54842 -3 -5.41832 -3 -2.58909-3 -9.22135 -4 -1.77351-4

2.20718-2 1.77465 -2 1.06458-2 5.49567 -3 2.49076 -3 9.63628 -4 2.10761-4 -1.85642-4 -3.93446 -4 -4.83162 -4 -5.05234-4 -4.91472 -4 -4.51796-4 -3.90312 -4 -3.13471-4 -2.32883 -4 -1.57081-4 -9.43331-5 -3.70253 -5

1.73916-2 -3.89113 -2 -4.79399 -2 -4.92796-2 -4.76762 -2 -4.45663 -2 -4.06121-2 -360961-2 -3.12083 -2 -2.61634-2 -2.11841-2 -164654-2 -1.21744-2 -8.45325 -3 -5.40493 -3 -3.08110-3 - 1.47752 -3 -5.27458 -4 --1*01598-4

il3 9 10 11 12 13 14 15 16 17 18 19

.o II

F

mrrx

( > ( >

m7r m7r NZ” yiy-

cos

--g-a

R,= &

l-71389-2 1.23492 -2 790424 - 3 4.70641-3 264561-3 I.40869 - 3 6.74319-4 2.31543 -4 -2.53717 -5 -1.59113-4 -2.19407-4 -2.39363 -4 -2.34312-4 -2.12850-4 -1.80726-4 -1*43342-4 -1.03162-4 -6.52441-5 -2.62757 -5

N -8;.

(33) NUMERICAL RESULTS

The result of the calculations are presented in the figures at the end of this article. De-

A” 5.39957 -3 -1~91511-2 -2-22596 -2 -2.23608 -2 -2.13988-2 -1+9055-2 -1w901-2 -160502-2 -1.38640-2 -1.16231-2 -9.42100-3 -7.33956 -3 -544885 -3 -3.80726 -3 -2.45610 -3 -1.41607-3 -6.87858 -4 -248712-4 4*84311_r5

103318-2 6.585 14 -3 4.36681-3 2.8942 1 - 3 1.91518-3 1.25878 -3 8.05385 -4 4.86336 -4 2.67111-4 1.23997 -4 3.53946 -5 -1.62085 -5 -4.29276 -5 -5.41953 -5 -5.61553 -5 -5.25060-5 -4.36567 -5 -3.12023-5 -1.34848-5

tailed numerical tables are presented in Lew and Fung (1968). Although the figures are self-explanatory, a few remarks are necessary. Notice that the speed U is the axial component of velocity at x = 0,O 6 r =S 0.9. In the region of x = 0, O-9 < r/a < 1.0, the velocity component u decreases from u = U at r/a = 0.9 to u=O at r/a= 1 in an unspecified manner. Consequently, the mean speed of the flow is slightly less than U. This can be easily seen by observing the maximum value of the developed flow which is slightly less than 2U. The small variation of the velocity distribution in 0.9 < r/a < 1-O for each set of calculations is also responsible for the slightly different values of volume flow for the different sets of calculations with various Reynolds numbers. As a consequence of the truncation of the infinite series in the summation over m, the velocity does not vanish exactly on the wall of the tube, but they are small enough and the error due to the truncation of the series in the summation over m is negligible. We can see clearly from the data presented that the effect of the entry flow is well limited to the region 0 s x/a < 30 for the Reynolds numbers considered. The far-

+~;zl(~)lR]-F~_[&l,(~)l~ -MI(~)]] f--+sinr+i)}

100

ENTRY FLOW INTO BLOOD VESSELS

downstream conditions (1 l), (12) and (13) simply require that the flow develops into the Poiseuille flow as x -+ 03. The numerical resuits check this condition very well. The distributions of velocity in the inlet region are shown in Figs. 1(a-c). It is interest-

0

02 09 06

08

IB M

ing to note that the maximum value of radial component of velocity takes place about at x/a = O-2 and r/a = O-7. The ratio of the maximum value of the radial component of velocity to the mean speed to flow is about O-3 for small value of Reynolds number. This

1% I% 20 2.2 24 2-6 2% 30

Fig. l(a). Change of velocity vs. the dimensionless downstream distance, x/a (R, = 0, 1 and 5).

B.M. Vol. 3 No. 1 -C

31

X/O

32

H. S. LEW and Y. C. FUNG

0

02

09 06 06

OQ

w

I-0 I4

l-6 I.6 20

2.2 214 96 2-6 30

x/a

IQ 06 06 O-4 02 =O OS? 04 06 0% 16 -0

06

34 ;90 36

08

la

36 40

I.2

4-2 44 46

x/a

4i3 50

56 S* W 56

60

x/a

Fig. l(b). Change of velocity vs. the dimensionless downstream distance. x/n(R, = 10and 30).

ratio becomes about O-1 for the case of Reynolds number equal to 50. Therefore, we conclude that the radial component of the velocity is not negligibly small compared to

the axial component of velocity in the intermediate range of Reynolds numbers and, consequently, the boundary layer theory should not be used in such cases.

ENTRY

FLOW INTO BLOOD

33

VESSELS

. . ,

04 -

o-6-

0

-U2 -04 -06 .OB 40

i2

94 16 *IS -20

I.0 0% u U

06 04 02 +=q 02

10.4 u 06 0.6 I.0 x/O% a)

-

\

1

-TR, =I00

Fig. I(c). Change of velocity vs. the dimensionless downstream distance divided by Reynolds number, X/R& (R, = 30.50 and 100).

INLET LENGTH

For an entry flow into a circular cylinder with a uniform axial velocity at the entry section, it is common to define an ‘inlet length’ as a distance through which the velocity has

redistributed itself approximately into a Poiseuille profile. Since the approach to Poiseuille profile is asymptotic, there is no unique definition of the inlet length. Boussinesq (1891) defined an inlet length as the dis-

34

H. S. LEW and Y. C. FUNG

tance from the entry section to a point where the deviation from Poiseuille profile is less than 1 per cent. He obtained the result that the entry length is equal to O-26R,a. With the same definition, Schiller (1922) obtained an inlet length O-115 Rp, whereas Targ (1954) obtained the result

k= 0.16 R,.

(34)

We shall adopt a similar definition. However, since a numerical solution inevitably involves truncation errors, it is rather difficult to pin point the cross section where the tIow profile differs from the Poiseuille profile by exactly 1 per cent. In the present study, the inlet length is determined as a down-stream distance from the entry section where the maximum velocity (axial component of velocity at the axis of the tube) reaches the value of

99 per cent of the maximum velocity of the fully developed flow. In Fig. 2, the inlet length based on the present analysis is plotted vs. the Reynolds number as a solid line. A dotted straight line representing Targ’s solution (34) of the linearized boundary layer equation is also plotted. It is seen that Targ’s formula L/a= O-16R, is a good approximation for inlet length for Reynolds number greater than 50. For smaller Reynolds numbers, however, equation (34) does not apply. When the Reynolds number tends to zero, the inlet length tends to a constant 1.3 II. MEAN PRESWRE

DI!STlUBLJTION

The mean pressure over the cross section of the tube is plotted vs. the downstream distance in Fig. 3. We adjusted the average pressure in such a way that it vanishes at the entry section. The pressure gradient is seen to be accentuated in the entry region.

REYNOLDS

NUMBER, R,

Fig. 2. Change of inlet length vs. Reynolds number.

ENTRY

-25

-60

-65

FLOW INTO BLOOD

VESSELS

-

c

0

I

2 DISTANCE

3 FROM

I

6

I

I

4

5

6

7

ENTRY

4

8

9

SECflON,xh

Fig. 3. Change of mean pressure in the inlet region.

DISCUSSION: APPLICATIONS TO BLOOD FLOW

The boundary conditions of the ‘uniform entry’ problem, (U = const., v = 0 at x = O), are the same if a rigid piston is placed at x = 0. With this realization, it is seen at once that our analysis is applicable to the flow of a viscous fluid in a tube pushed by a rigid body. In particular, it is applicable to the flow of the plasma in a capillary blood vessel pushed

by a red blood cell. We know that the plasma is a Newtonian viscous fluid. At first sight the application seems extremely limited, because it is necessary to have an infinitely long column of plasma in front of a red cell which is a very unlikely situation. However, it is fortunate that the ‘inlet length’ turns out to be quite small. Consequantly, the results of preceding analysis are applicable

36

H. S. LEW and Y. C. FUNG

to a finite column of plasma in front of a red cell. It is seen from Fig. 2 that the inlet length is a continuous function of the Reynolds number. When R,= 0 the inlet length is equal to L a=

1.3.

(35)

l-3 times the radius of the tube a. For all Reynolds numbers c 1 the inlet length is no more than 1.4~. This rapid adjustment of a uniform entry to a Poiseuille flow makes the analysis realistic to microcirculation. In a capillary blood vessel the red cells are separated by segments of plasma whose length are of the order of the diameter of the tube. The flow of a plasma between the red cells is called the ‘bolus flow’. Lew and Fung (1968) presented an analysis of the bolus flow at zero Reynolds number. It was found that if the faces of the consecutive red cells were separated by a distance equal to one tube radius or larger, the cells have very little interaction, and the flow is essentially a superposition of two independent entry flows one at each red cell. When the red cells are closer to each other, they interact to reduce the required pressure drop to a value smaller than the sum of two entry flows. These features were derived at zero Reynolds number. Now the present paper shows that the inlet length varies slowly with Reynolds number. We conclude therefore that the earlier results on bolus flow must prevail to all small but finite Reynolds numbers (say, R, < I). At the point where a capillary blood vessel branches off from an arteriole, there is a segment of sphincter of length equal to about 3 diameters. This is long enough for the plasma flow to be described by the results of the present paper. A classical difficulty with the entry problem of a tube opening into a reservoir is the uncertainty as to what the velocity distribution should be at the inlet section. For an inviscid fluid the solution of Borda mouth piece clearly

shows the nonuniformity of the velocity profile at the inlet section. For a viscous fluid Wang and Longwell (1964) tried to determine the inlet condition under certain idealized upstream conditions. This question is not investigated in the present paper, but we offer a formal solution which is applicable to an arbitrary axisymmetric distribution of velocities u and z, at the inlet section, thus providing a theoretical scheme with which the full inlet problem can be solved if a corresponding analysis of the reservoir is adjointed to the tube at the inlet section. In larger blood vessels whose diameters are much larger than the diameter of red cells the blood may be treated as a homogeneous continuum. The flow in these vessels is pulsatile. The unsteady entry flow in these vessels has been investigated by Chang and Atabek (1961), Atabek and Chang (1961), Atabek, Chang and Fingerson (1964) and Atabek (1962,1964) for flows at large Reynolds numbers with the method of boundary layer approximation (ignoring radial equation of momentum, linearizing axial equation of momentum). The influence of unsteadiness depends, of course, on the ‘reduced frequency’ wa/U, ,where w is the frequency of oscillation, a is the radius of the tube, (a characteristic length) and W is the mean speed of flow. When the parameter oa/U is small, the entry flow is quasi-steady. Then the inlet length can be calculated at each instant as if the flow were steady instantaneously. Therefore, our result should be valid as a quasi-steady approximation at small and moderate Reynolds numbers when the parameter oa/V is small. In this sense our result complements those of Atabek and Chang. CONCLUSlON

A method of solving the problem of entry flow into a cylindrical tube is presented. Its effectiveness for numerical calculations is demonstrated. The results on the uniform entry flow match our previous analysis at

ENTRY

FLOW INTO BLOOD

zero Reynolds number on the one hand, and Targ’s result at high Reynolds numbers on the other. The entry length is shown to vary smoothly from L= 1-3~ (a = radius of the tube) at very small Reynolds numbers (say, R, < I) to L = O-16 R,a for Reynolds numbers > 50. The radial velocity is not negligible at small Reynolds numbers (say, R, between 0 and 50), and the boundary layer theory should not be used. The method proposed here can be used for the entry flow which has a given distribution of the radial component of velocity at the entry section as well as the axial component. The pressure drop in the inlet region should depend on the type of distribution of the velocity in the entry section, but the dependence of the inlet length on the entry condition should be more or less insignificant, because the inlet length depends more on the exponential factors eehT in equation (19) et seq., and the factor X is essentially independent of the entry condition. On the other hand the pressure drop depends on the constants A,,, B, which are solely responsible for satisfying the entry condition. Indeed, the entry region is where the significant perturbation of the flow due to entry section exists. In case of a more general type of entry condition in which an arbitrary steady or unsteady distribution of axial and radial components of the velocity is given at the entry section, the entry length should be defined in terms of the relative decay of the perturbation of the flow caused by the imposed condition at the entry section instead of by a direct comparison of the flow with the Poiseuillean flow. It should be mentioned that the series derived in the present study is complete, and all types of flow can be represented by this series. Therefore, the method of splitting the solution into two parts (one corresponding to the developed flow and the other corresponding to the transition flow), as was employed in the present study, is merely a matter of convenience.

37

VESSELS

A cknowledgemenrs - The authors wish to express their appreciation lo Mrs. Claudia Lowenstein for her help in numerical calculations. This work is supported by the United States Air Force Ollice of Scie&fic Research under Grant No. 1186-67 and the National Science Foundation under Grant No. GK-1415. Dr. Lew also holds a fellowship from the San Diego County Heart Association granted in the name of the late Marjorie Siner.

REFERENCES M. and Stegun, I. A. (Editors) (1964) HandFunctions. National Bureau of Standards, U.S. Dept. of Commerce. Atabek, H. B. (1962) Develooment of flow in the inlet length of a circuI& tube stahng from rest. Z. anger. Abramowitz,

book of Mathematical

Math. Mech. 13 417-430.

Atabek,

H. B. (1962) On an eigenvalue

problem.

Q.

appl. Math. 20 188-191.

H. B. (1964). End effects. In Pulsatile Blood Flow. (Edited by E. Attinger), Chap. 10. McGraw-Hill,

Atabek.

New York. Atabek, H. B. and Chang, C. C. (1961) Oscillatory flow near the entry of a circular tube. Z. anger. Math. Mech. 12.185-201.

Atabek. H. B.. Chang. C. C. and Fingerson. L. M. (1964) Measurement of laminar oscillatory flow in the inlet length of a circular tube. Physics Med. Biol. 9.2 19-227. Bodoia, J. R. and Osterle, J. F. (1961) Finite difference analysis of plane Poiseuille and Couette flow developments. Appl. scient. Rex AlO. 265-276. Boussingeq,J. (1891) Compf. rend. 113,9,49. Campbell, W. D. and Slattery, J. C. (1963). Flow in the entrance of a tube. J. bas. Engng 85.4 I-46. Chang, C. C. and Atabek, H. B. (1961) The inlet length for oscillatory flow and its effect on the determination of the rate of flow in arteries. Physics Med. Biol. 6. 303-3 17.

Christiansen, E. B. and Lemmon, H. E. (1965) Entrance region flow. A. I. Ch. E. JI 11.995-999. Collins. M. and Schowalter, W. R. (1962) Laminar flow in the inlet region of a straight channel. Physics Fluids 5.1122-1124. Goldstein, S. (Editor) (1965) Modern Deoelopmenrs in Fluid Dynamics, Vol. 1, pp. 304-308. Dover, New York. Han, L. S. (1960) Hydrodynamic entrance lengths for incompressible laminar flow in rectangular ducts. J. appl. Mech. 27.403-409. Hombeck, R. W. (1963) Laminar flow in the entrance region of a pipe. Appl. scient. Res. A13,224-232. Langhaar, H. L. (1942) Steady flow in the transition length of a straight tube. J. appl. Mech. 9,55-X Lew, H. S. and Fung, Y. C. (I 968) On the entry flow in a circuiar cylindrical tube at arbitrary Reynolds numbers. AMES Dept.. Univ. Carif. San Diego. AFOSR Rep. No. 69-0061.

Tech.

Lew, H. S. and Fung, Y. C. (1968) The motion of the plasma between the red blood cells in the bolus flow Biorheology Journal. In Press. More complete numerical results are presented in the AMES Dept., Unia. Calif. San Diego. AFOSR

Tech. Rep. No. 68-2359.

38

H. S. LEW and Y. C. FUNG

Lew, H. S. and Fung, Y. C. (1969) On the low-Reynoldsnumber entry flow into a circular cylindrical tube. J. Biomechanics 2,105-l 19. Lundgren, T. S., Sparrow, E. M. and Starr, J. B. (1964) Pressure drop due to the entrance region in ducts of arbitrary cross section. 1. bas. Engng 8&620-626. Mills. R. D. (1%8)J. mech. Enana Sci. 10.133-140. Oseen, C. W. (1910) “Ueberdie Stokes’ sche Formel, und iiber eine venvandte Aufgabe in der Hydrodynamik”. Arch Math. 6. See also his book Hydrodynamik. Leipzig (1927). Pfenninger, W. (l%l) Transition experiments in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control. (Edited by G. V. Lachmann), Vol. 2, pp. 970-980. Pergamon Press, Oxford. Prandtl, L. and Tietjens, 0. G. (1957) Applied Hydroandderomechanics. Dover, New York.

Rosenhead, L. (Editor) (1963) Laminar Boundary Layers. Oxford University Press, Oxford. Schlichting, H. ( 1934) Laminare Kanaleinlaugstrijmung. 2. angew. Math. Mech. 14,368-373. Schiller, L. (1922) Die Entwicklung der Laminaren Geschwindigkeitsverteihmg und ihre Bedeutung Rlr Aiihigkeitsmessungen. Z. angew. Math. Mech. 2, %-106. Slezkin, N. A. (1955) Dynamics of Viscous Incompressible Fluid (In Russian). Gostenhizdat, Moscow. Sparrow, E. M., Lin, S. H. and Lundgren, T. S. (1964) Flow development in the hydrodynamic entrance region of tubes and ducts. Physics Fluids 7,338-347. Targ, S. M. (1951) Basic Problems ofthe Theory oflam_ inar Flows (In Russian). Moskva. Wang, Y. C. and Longwell, P. A. ( 1964) A. I. Ch. E. ./I 10,323-330.