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A study on the low Reynolds number flow in a valved vessel
A STUDY
ON THE LOW REYNOLDS NUMBER FLOW IN A VALVED VESSEL* H. S. LEW
Departments of Physiology. and Aerospace and Mechanical Engineering. The University of Arizona. Tucson. Ark 85721. U.S.A.
Y. C. FUNG Department
of AMES
(Bioengineeringj. University of California. San Diego. La Jolla, Calif. 92037. U.S.A.
Abstract-The understanding of the duid flow in valved vessels is of importance pertaining to its application to the problem of fluid transport in valved vessels of the living body, such as veins and lymphatic ducts. In order to have a quantitative understanding of fluid flow through valved vessels, the Newtonian fluid flow through a rigid circular cylindrical tube involving a series of uniformly spaced plate orifices of axisymmetric geometry is studied at low Reynolds number. The solution of the above problem is posed as a series solution of the Stokes equation and the equation of continuity. satisfying the appropriate boundary conditions. The velocity distributions and the average rate of the mean pressure change along the tube axis are computed for a number of different combinations of the orifice-tube radius ratio and the ratio of the inter-orifice distance to the tube radius. The results of the analysis are discussed for possible application to certain physiological systems.
INTRODWTION
vessels constituting the passage of body fluids in the living body have valves. which play the role of check valves. It is well known that many veins of medium caliber are provided with valves that allow blood flow toward the heart only (Bloom and Fawcett. 1968). The lymphatic ducts connecting the lymphatic capillaries to larger blood veins also have valves, which allow the lymphatic fluid flow only from tissue to blood vessel (Davson. 1965). The motion of blood in the valved veins, as well as that of lymphatic fluid in ducts, is very difficult to analyze theoretically. Such vessels not only lack regularity and finesse in geometry, but their geometry also undergo a continuous change under the influence of fluid pressure, and stress of the vessel wall and surrounding tissues. In the regime of very small Reynolds number in SOME
*Received 22 December 1969.
HM
Vol
4.No.?-A
which the inertial force of fluid is much smaller than pressure and viscous force. the governing equations become linear. Hence. the solution of a problem involving complicated boundary conditions can be split into several solutions satisfying less complicated boundary conditions. This implies that the physics of fluid flow in a vessel, created by a number of different mechanisms. can be investigated by studying the isolated flow system involving a single individual mechanism. Of course, such an approach is not applicable to blood flow in large veins where the Reynolds number is high. In this study a theoretical solution is addressed to the problem of a low Reynolds number flow of the Newtonian fluid, through a rigid circular. cylindrical tube involving a series of plate orifices uniformly distributed along the axis of the tube. In order to avoid
a6
H. S. LEW and Y. C. FUNG
algebraic complications in finding the solution. we assume that the geometry of the plate orifices is axisymmetric and infinitely sharp. The purpose of our analysis is to study the effect of fluid flow through tubes with valves by investigating the simplest mathematical model of the problem. Because the employed mathematical model does not realistically represent the physiological vessels geometrically and functionally, the result of our analysis does not provide data which can be directly applied to the fluid flow through some physiological valved vessels. The information provided by this analysis is helpful for a better understanding of the transport of body fluids through valved vessels in a qualitative fashion. Some capillary blood vessel wails have irregularities such as outbulging of the nucleus of the endothelial cells and imperfect joining of these cells. The effect of such wall irregularities on the plasma flow and on the exchange of gases across the blood vessel wall is also a problem of interest. If the orifice wall is regarded as an analogy of wall irregularities, our analysis can provide some qualitative information regarding such effects. Altshuler used a valved vessel similar to that investigated in this study to discuss the finite value of the effective velocity of the low Reynolds number air flow at the periphery of the alveolar ducts open to the alveoli, and the deviation of the velocity profile from the Poiseuille flow ( 1969). Our results can be used to quantify such a concept. The problem of fluid flow through a plate orifice has an engineering application in designing a device for measurement of flow in tubes. Mills studied the flow through a single plate orifice in a pipe by finding a numerical solution of the Navier-Stokes equation for low Reynolds number range and discussed his results in connection with the problem of the flow-rate measurement (1968). Our solution for large inter-orifice distance may serve as a standard by which the numerical solution at zero Reynolds number may be compared.
GOVERNING
EQUATIONS
AND BOUNDARY
CONDITIONS
The Navier-Stokes equations and ‘the equation of continuity are the governing equations. Since we are limiting our investigation to the low Reynolds number flow, the inertia force term in the Navier-Stokes equation can be neglected; thus we have the Stokes equation. -vp+j.Lv2v
= 0,
(1)
where p is the pressure, p is the shear viscosity, v is the velocity of the fluid and V is the gradient operator. The valves in the circular cylindrical tube under investigation have axisymmetric geometry and are uniformly spaced along the tube axis. We assume that the flow is axisymmetric based on the axisymmetry of the tube geometry. If we assume that the fluid is incompressible, the equation of continuity can be written in the form
(2) where r, 8, x are a set of stationary cylindrical polar coordinates with the x-axis coinciding with the axis of the tube and the origin of the coordinate system located on the middle plane between two adjacent orifices. The components of the velocity u and 2: are in the direction of x and r axis. respectively. We assume that the flow is symmetric with respect to the middle plane between two adjacent orifices, i.e.
v(x, f) = - c(- x, f).
(4)
This assumption, based on the smallness of the Reynolds number. has been demonstrated in many well-known examples of the low Reynolds number flow such as the creeping flow around sphere, cylinders, etc. Further, the flow has to be periodic because
LOW REYNOLDS
NUMBER FLOW IN A VALVED
the orifices are uniformly spaced. The symmetry and periodicity relationships involved in this problem have been well demonstrated in our previous study of related problems such as the bolus flow and the flow in tubes involving a series of suspended discs (Lew and Fung. 1969 and 1970). Therefore. it is sufficient to formulate the problem in the right half of the region between any two adjacent orifices. The condition of adherence of the fluid to the tube wall and the orifice plate requires that
VESSEL
boundary value problem at hand. As the following section will show, some of the above boundary conditions are not independent of each other and, consequently. our boundary value problem will be much more simple than that presently formulated. METHOD OF ANALYSIS
The equation of continuity (2) is satisfied if the velocity vector v is generated from a scalar functionf(x-, r) as follows: [ij’(x.r)]
v(x.r)=Vxxcx
II(X..r)IrZn = 0
for
0 c x c L.
(5)
V(X. T)/+=~~ = 0
for
0 6 x G L.
(6)
u(x, r)lfPL = 0
for
b C r C a.
(7)
C(X. r)lFL = 0
for
b C r c a,
(8)
where a is the radius of the tube. 2L is the distance between two adjacent orifices and b is the radius of the orifice opening. On the circular plane of the orifice opening, all quantities involved in the governing equations have to be continuous. The reason for this is because the circular plane of the orifice opening is not a natural boundary, but is an artificial one created in the formulation of the mathematical model of the problem. v(x. r) IpL- = v(X,r)lr+,T
for 0 G r C b, (9) Vv(x. r)ls_L_ = Vv(x, r)ls++ for 0 c r < b, (10) Vv(x. r)lr+ = ‘7%(x. r)j.,.,=,_+for 0 G r < b. p(x, r)ls+ vpcx. r)ls+
=p(x,
r)jFL+
= Vp(x. r)ls-,,+
= _i_1 a;af+ ;3axar-' r ar
where x = L- and x = L+ designate the planes defined by the left- and right-hand side surface of the orifice, respectively. The equations (1) through (13) define the
(14)
ar
where i and F-are unit base vectors of the x and r axes. respectively. Substitution of (14) into ( I 1yields -
( pv+; )
v p-
0.
-ipV4f=
(15)
Equation ( 15) is satisfied if p = pv&{.
(16)
V’tf = 0.
(17)
An x-symmetric solution of (17). which has bounded values in the region [-L c x s L, 0 s r s a] and satisfies the zero-velocity condition on the wall of the tube, can be found as follows (see Lew and Fung, 1970): f(x.r)
=dU
2
,4n{[-
(l+hk,coth
(A&))
“=I
(II) for 0 G r < b.
(12) for 0 L r < 6, (13)
87
x cash (k, (x/a) 1 cash (hk, 1 +k xsinh (k,(xla)) “u cash (Ak,) + i
B,,
fPl=*
(k, 1 1kn2J, J,(k,(r/a).l
~0s [ (mr/A 1 Ix/a ) 1 (rn~/A)~
88
H. S. LEW and Y. C. FUNG
Substitution
of (18)
into
(16)
yields the
following equation for pressure:
-2&,(mlr/h)
+ (m~/A)l,(m?r/A)
P --=2i
’
/.d(rr/a)
A.[k. W-1
sinh [k&/a)] cash (Ak,)
(18) where J,,(K,) = 0, B Rrn =
n = 1, 2, 3, . . . .
mdh) cos (mr -;k2tanh6%) (fk.” +(ma,A)‘]!*
The axial component of the velocity of ( 19) satisfies the no-slip condition (5). In equation (18), the values of B,, were selected so that the radial component of the velocity vanishes on the wall of the tube. Hence, u of (20) satisfies the condition (6). The radial component of the velocity u of (20) is antisymmetric with respect to the plane x = 0 and vanishes identically at x = L. Thus, the condition (8) and the r-component of equation (9) are satisfied. The x-component of (9) is also satisfied because of the symmetricity of U. Equation ( 10) is satisfied by u and v given respectively by (19) and (20) because au/ax = 0 and av/ar = 0 at x = + L and -L. and au/ar and &/ax are symmetric. The Stokes equation
m = 1, 2, 3, . . . A,‘s are arbitrary constants, Ji and I( are the i-th order Bessel function and modified Bessel function of the first kind, respectively (Hildebrand, 1962). A designates the ratio of one half of the inter-orifice distance to the radius of the tube. The characteristic velocity U is introduced to make the series of the RHS of ( 18) dimensionless. When (18) is substituted into (14), the following equations for the x and r components of the velocity result:
- Cl+ Aktco~ (AL))
cos5,
ZO[ (mr/A)
-ii
m=I A,,
“a
III=*
(Ak,)
(da)
1 _
lo(m~/A)
cash (Ak,)
2ld(m7dAW/a)l+ (mtr/A)(r/u)f,E(m~/A)(rla)l 2Z,(m7r/A)+ (mrr/A)l, (mrr/A) (19)
9 II
that (11) is the same condition as (13). Therefore, only three conditions are
left to be satisfied. i.e. (7), ( 12) and ( 13). As seen in (2 I), p ‘is anti-symmetric and, consequently aplax is symmetric. Hence.
1
J,[k&/d~ J, k)
the x-component of equation (13) is satisfied. If (12) is satisfied, the r-component of equation (13) will be satisfied. Therefore, only the boundary conditions (7) and (12) need be
IJ(m4A)(Ml f,(mr/A)
_ (m4A) (r/a)LC(m4A) (r/a)] ]] 210(m7r/A) + (mrr/A)fl(mv/A) *
1
xsinh Ck,(x/a)l Jdk,(r/a)] cash (Ak,) J, (kz)
‘u
( 1) indicates
Ak,coth ( Ak,,)Sinh [kn(x’u)l cash (Ak,)
_k xcosh CM-G)1
-2
cash t(~/a)l+~
(20)
considered.
LOW REYNOLDS
NUMBER FLOW IN A VALVED
Since p is anti-symmetric, the condition (12) requires that p be a constant on the circular plane of the orifice opening, i.e. p(x. r)IrSL = const.
for
0 G r < 6.
(22)
Thus. conditions (7) and (22) provide mixed boundary conditions on the plane x = L, from which the only unknown set of constants A,, can be determined. NUMERICAL CALCULATIONS AND RESULTS
In computing for the velocity and the pressure from our series solution, the series summed over m is truncated after retaining the first 56 terms. Our numerical results indicate that the error introduced due to such a truncation in the Fourier series is negligible. We truncated the series summed over n after retaining the first 20 terms. Then. we have a solution with 20 unknown constants. i.e. 20 A n’~. involving a double infinite series in which the orthogonality relationship is lacking. The determination of the constants involved in such a series from the mixed boundary condition is very difficult. In this study. we determine A,, by the’collection method. When the mixed boundary conditions (7) and (22) are imposed at 20 uniformly distributed points of r/a in [0 s r/a s O-951.a set of 20 simultaneous linear algebraic equations result, which enable us to determine the value of 20 A,,‘s involved in this set of equations. Condition 17 I is imposed at uniformly distributed points in [‘I G r/a c 0.951 and condition (22) is applied to uniformly distributed points in [0 s r/a =Gv-0*05]. Here, the points are separated from each other by a distance of O*OSa and 7 is the ratio of the radius of the orifice opening to the tube radius: The magnitude of A,,‘s, which are determined by the above scheme. decreases quite rapidly in a more or less monotonic fashion as II increases. For example. the ratio of the magnitude of A, to A*,, is of the order of IO-” or smaller for all cases of different combina!ions of the value of A (inter-orifice distance) and 77 (radius of the orifice opening). In this
VESSEL
89
computation, we used twelve different combinations of four values of A: 0.25, 0.50, 140 and 1.50; and three values of 7: O-25,0-50 and O-75. We normalized the right-hand side of (22) and set it equal to p(Ula) so that the non-dimensionalized pressure p/M U/a)1 becomes unity at the orifice opening. The velocity computed from the truncated form of (19) and (20) after substituting the value of A, is presented in Figs. 2(a-c). where the flows of Figs. 2(a) and (b) are driven by the same amount of pressure gradient, while the pressure gradient of the flow of Fig. 2(c) is 10 times that of the flows of Figs. 2(a) and (b). It is interesting to note: (1) contrary to the flow at finite Reynolds number, no separated vortex pattern at the comer of the orifice is observable; and, (2) the existance of a nearstationary fluid layer near the wall of the tube when the inter-orifice distance is sufficiently small (e.g. less than 0.5a). The velocity profile becomes Poiseuillean if the distance from either orifice to the point of observation is equal to or greater than 1a3 times the tube radius. The pressure can be computed from the truncated form of (21). Since the pressure has anti-symmetric distribution with respect to the middle plane between two adjacent orifices. the pressure in the next region to the right of the region under consideration is given by [p + 2p(x, r) jrcL, ,.+J. where p is the same pressure as (21). The pressure of (21) vanishes at x = 0. Hence, the mean pressure gradient is simply equal to [p(x. r) lmL, ,,/L]. The non-dimensionalized mean pressure gradient is shown in Fig. 3 as a function of the dimensionless inter-orifice distance (2h) and the dimensionless radius of the orifice opening (71). This figure shows the resistance of the tube with a series of plate orifices in comparison with that of the smooth tube. The resistance of the tube to the flow strongly depends on the size of the orifice opening in the range of parameters selected in the present computation. The U used in the expression of the dimensionless pressure
90
H. S. LEW and Y. C. FUNG
t
-_--irleill__ 0
Fig. 1.Geometry of the flow system.
IO
u u
--
0
s
2fP -
0.51
/
I
_,
05: _
yc.75
X050
IO-x/o 0 0.1 0.2 0.3 0.4 0.5
x,0 0
0.1 0.2 0.3 0.4 0.5 0.6
0.7 08
09
I-0
r/a I.O_I\I\r\R\P\r\r\nr\l\hh,,,
A
0
01
0.2 0.3 04
0.5 0.6 0.7 0.0 o-9
I.0 1.1 I.2 I.3 1.4 1.5
Fig. 2(a). Velocity distribution of the fluid in the RHS of the region between two adjacent orifices for r) = O-75. Figure shows in each cross section the axial component of velocity II above the center line. and the radial component of velocity u below the center line. The parameter n is the ratio of the diameter of the orifice opening to the tube diameter. The pressure drop over the length A (not the pressure gradient) is the same for all figures.
LOW REYNOLDS
x/o
0
01
C2925
0
0 I
0.2
._ x/c1
NUMBER
FLOW IN A VALVED
AGo 0
0.3
0.4
0.5
0.6
0.7
0.0
0.1
0.9
02
I.0
0.3
I I
0.4
I.2
VESSEL
0 5
, 3
I.4
l-5
Fig. 2(b). Velocity distribution for TJ= 0.50. The pressure drop over the length A is the same as that of Fig. 2(a).
gradient is the mean velocity averaged over the cross’ section of the tube (not the value averaged over the orifice opening). When A approaches zero. the solution should approach the flow through a smooth tube of radius equal to qa. The points on Fig. 3 corresponding to A = 0 are calculated in this fashion, and the large value of the pressure gradient for the small value of v at A = 0 is due to the fact that they are non-dimensionalized, based on the nominal tube of radius a instead of the effective tube of radius r)a. CONCLUSION
The low Reynolds number flow of the Newtonian fluid through a rigid circular
cylindrical tube involving a series of plate orifices of axisymmetric geometry. which are uniformly distributed along the tube axis. is studied, based on the Stokes approximation. The results of this analysis show that: ( 1f the resistance of the vessel with valves to the very slow viscous flow through it becomes very large when the inter-valve distance is small or the valve opening is narrow; (2) in contrast to the flow at finite Reynolds number, the zero-Reynolds number flow does not create a separated vortex flow at the corner of the orifice: and. (3) the fluid near the wall of the vessel is almost stagnant when the inter-orifice distance is sufficiently small (e.g. less than 0.5~). We conclude from the above statements
92
H. S. LEW end Y. C. FUNG
1'025 A=050 i,o 0
01
Xm
.?I 0.2 0.3 7.4 0.5 96
0
Ol20.25
3.7 0.6 P-9
I.0 I.1 I2
I.3 1.4 I.5
Fig. Z(c). Velocity distribution for I) = 0.25. The pressure drop over rhe length ,\ is IO times that of Figs. I(n) or 2(b).
that, if flow in the tube is created by an upstream pressure head only, the tubes without valves constitute a much more efficient passage for the fluid than valved tubes. In general, it is true that the arrangement of the physiological system in the living body is highly efficient for performing appropriate functions. Since the valves in the vessel increase the resistance to fluid flow driven by the pressure gradient supplied by an upstream pressure head. the existence of valves in some physiological vessels implies that the driving forces of the fluid in such vessels include some energy source in addition to the consistent pressure gradient along the vessel axis. This additional driving force in the vein is the localized compressions and
expansions by the tissues surrounding the veins. It is believed that the peristaltic contractions of the wall of the lymphatic duct constitute the main driving force of the lymph flow (Hewson. 1774 and Hall et trl. 1965). Our results do not explain the role of valves in physiological valved vessels. In such vessels, valves play two contradicting roles; a part of the pumping device, and that of blocking fluid flow. We investigated the latter role, while the physiological valves predominantly play the former one. As shown by the existence of the highly distensible and collapsible wall of valved veins, the role of valves in physiological valved vessels is closely coupled with the deformation of the vessel wall. Consequently, the effects of the
LOW REYNOLDS
NUMBER
2x Fig. 3. The average mean pressure gradient as a function ofhandn.
valves and the deformation of the vessel wall have to be taken into consideration simultaneously. even for the simplest mathematical model of physiological valved vessels. For example. an investigation of fluid flow through a tube. similar to that considered in this study. created by a peristalsis of the tube wall should useful information for a better provide understanding of lymph flow in lymphatic ducts. It is interesting to note the result that near the wall of an intensely valved tube the fluid is nearly stagnant. This qualitatively suggests that a layer of very slow moving plasma should exist when the wall of the capillary blood vessel is unsmooth. Such a near-stagnant fluid layer near the blood vessel wall would hamper the gas exchange across the capillary blood vessel wall. In this respect. the smoothness of the capillary blood vessel can be vitally important, and irregularity can be a source of trouble. The idea of modeling the alveolar ducts by a valved vessel is an
FLOW IN A VALVED
VESSEL
93
interesting concept which deserves some consideration (Aitshuler, 1969). If the inter-valve distance is much smaller than the dimensions of the valve opening. flow through such a valved vessel can be effectively represented by a constant velocity profile (e.g. a space-averaged value of the velocity). If the inter-valve distance is comparable to the dimensions of the valve opening. the representation of flow by a constant macroscopic velocity profile becomes less useful. Our numerical results suggest that the slip velocity at the periphery of the hypothetical tube. whose cross section is defined by the valve opening, should be negligible if the intervalve distance is much smaller than the radius of the valve opening. However, if the intervalve distance is comparable to the radius of the valve opening. the average value of fluid velocity at the periphery of the hypothetical tube is quite different from zero and the velocity profile differs from the parabola significantly. The dimension of the alveoli opening is comparable to the radius of an alveolar duct and. consequently. the deviation of flow characteristics in the alveolar ducts from that in the conventional tube has to be considered. Of course. the question of how valid Altshuler’s model is for fluid flow in alveolar ducts is another matter. When one employs such a model. one must consider the change of tube diameter timewise because pulsation of the valved tube is a dominating parameter pertaining to the physics of air flow in alveolar ducts. For example. the nearly stagnant fluid layer near the wall of the valved tube of a constant diameter would not exist in the pulsating valved tubes. Ac~nolc,ledpemfnts-The authors wish to express their appreciation to Mrs. Claudia Lowenstein for her help in numerical calculations. This work is supported partially by the United States Public Health Service. National Institute of Health under Grant No. USPHS HE 12494-01. partially by the San Diego County Heart Association under a Fellowship granted in the name of the late Marjorie Sitter and partially by the General Research Grant Support from the College of Medicine. the University of Arizona.
H. S. LEW and Y. C. FUNG
94
REFERENCES Altshuler. B. ( 1969) Behavior of airborne particles in the respiratory tract. Circulatory and Respiratory Mass Transport, p. 223. Ciba Foundation. Bloom, W. and Fawcett, D. W. ( 1968) Blood vascular system. A Textbook of Histology. 9th Edn. p. 376. Sanders. Philadelphia. Davson. H. (1965) Interstitial fluid and lymph. A Textbook of General Physiology, 3rd Edn, p. 439. LittleBrown, Boston. Hail, J. C., Morris, B. and Woolley. G. (1965) Intrinsic rhythmic propulsion of lymph in the unanaesthetized sheep.J. Physiol. 180.336-349.
Hewson, W. (1774) E.\perimenfaf inquiries. Part 2. p. 126. A description of lymphatic system. Hildebrand. F. B. (I 962) Advanced Calculus forApplications. Prentice-Hall. New Jersey. Lew. H. S. and Fung Y. C. (1969) The motion of the plasma between the red cells in the bolus flow. J. Biorheol.
6. 109-
119.
Lew, H. S. and Fung Y. C. (1970) Plug effect of erythrocytes in capillary blood vessels. Biophvs. J. 10. I. Mills, R. D. (1968) Numerical solutions of vkcous flow through a pipe orifice at low Reynolds numbers. kfech. Engng. Sci. 10. 133- 140. Rosenhead. L. (Ed.) ( 1963) Laminar Boundury Layers. Oxford Universities Press, Oxford.
ON THE LOW REYNOLDS NUMBER FLOW IN A VALVED VESSEL* H. S. LEW
Departments of Physiology. and Aerospace and Mechanical Engineering. The University of Arizona. Tucson. Ark 85721. U.S.A.
Y. C. FUNG Department
of AMES
(Bioengineeringj. University of California. San Diego. La Jolla, Calif. 92037. U.S.A.
Abstract-The understanding of the duid flow in valved vessels is of importance pertaining to its application to the problem of fluid transport in valved vessels of the living body, such as veins and lymphatic ducts. In order to have a quantitative understanding of fluid flow through valved vessels, the Newtonian fluid flow through a rigid circular cylindrical tube involving a series of uniformly spaced plate orifices of axisymmetric geometry is studied at low Reynolds number. The solution of the above problem is posed as a series solution of the Stokes equation and the equation of continuity. satisfying the appropriate boundary conditions. The velocity distributions and the average rate of the mean pressure change along the tube axis are computed for a number of different combinations of the orifice-tube radius ratio and the ratio of the inter-orifice distance to the tube radius. The results of the analysis are discussed for possible application to certain physiological systems.
INTRODWTION
vessels constituting the passage of body fluids in the living body have valves. which play the role of check valves. It is well known that many veins of medium caliber are provided with valves that allow blood flow toward the heart only (Bloom and Fawcett. 1968). The lymphatic ducts connecting the lymphatic capillaries to larger blood veins also have valves, which allow the lymphatic fluid flow only from tissue to blood vessel (Davson. 1965). The motion of blood in the valved veins, as well as that of lymphatic fluid in ducts, is very difficult to analyze theoretically. Such vessels not only lack regularity and finesse in geometry, but their geometry also undergo a continuous change under the influence of fluid pressure, and stress of the vessel wall and surrounding tissues. In the regime of very small Reynolds number in SOME
*Received 22 December 1969.
HM
Vol
4.No.?-A
which the inertial force of fluid is much smaller than pressure and viscous force. the governing equations become linear. Hence. the solution of a problem involving complicated boundary conditions can be split into several solutions satisfying less complicated boundary conditions. This implies that the physics of fluid flow in a vessel, created by a number of different mechanisms. can be investigated by studying the isolated flow system involving a single individual mechanism. Of course, such an approach is not applicable to blood flow in large veins where the Reynolds number is high. In this study a theoretical solution is addressed to the problem of a low Reynolds number flow of the Newtonian fluid, through a rigid circular. cylindrical tube involving a series of plate orifices uniformly distributed along the axis of the tube. In order to avoid
a6
H. S. LEW and Y. C. FUNG
algebraic complications in finding the solution. we assume that the geometry of the plate orifices is axisymmetric and infinitely sharp. The purpose of our analysis is to study the effect of fluid flow through tubes with valves by investigating the simplest mathematical model of the problem. Because the employed mathematical model does not realistically represent the physiological vessels geometrically and functionally, the result of our analysis does not provide data which can be directly applied to the fluid flow through some physiological valved vessels. The information provided by this analysis is helpful for a better understanding of the transport of body fluids through valved vessels in a qualitative fashion. Some capillary blood vessel wails have irregularities such as outbulging of the nucleus of the endothelial cells and imperfect joining of these cells. The effect of such wall irregularities on the plasma flow and on the exchange of gases across the blood vessel wall is also a problem of interest. If the orifice wall is regarded as an analogy of wall irregularities, our analysis can provide some qualitative information regarding such effects. Altshuler used a valved vessel similar to that investigated in this study to discuss the finite value of the effective velocity of the low Reynolds number air flow at the periphery of the alveolar ducts open to the alveoli, and the deviation of the velocity profile from the Poiseuille flow ( 1969). Our results can be used to quantify such a concept. The problem of fluid flow through a plate orifice has an engineering application in designing a device for measurement of flow in tubes. Mills studied the flow through a single plate orifice in a pipe by finding a numerical solution of the Navier-Stokes equation for low Reynolds number range and discussed his results in connection with the problem of the flow-rate measurement (1968). Our solution for large inter-orifice distance may serve as a standard by which the numerical solution at zero Reynolds number may be compared.
GOVERNING
EQUATIONS
AND BOUNDARY
CONDITIONS
The Navier-Stokes equations and ‘the equation of continuity are the governing equations. Since we are limiting our investigation to the low Reynolds number flow, the inertia force term in the Navier-Stokes equation can be neglected; thus we have the Stokes equation. -vp+j.Lv2v
= 0,
(1)
where p is the pressure, p is the shear viscosity, v is the velocity of the fluid and V is the gradient operator. The valves in the circular cylindrical tube under investigation have axisymmetric geometry and are uniformly spaced along the tube axis. We assume that the flow is axisymmetric based on the axisymmetry of the tube geometry. If we assume that the fluid is incompressible, the equation of continuity can be written in the form
(2) where r, 8, x are a set of stationary cylindrical polar coordinates with the x-axis coinciding with the axis of the tube and the origin of the coordinate system located on the middle plane between two adjacent orifices. The components of the velocity u and 2: are in the direction of x and r axis. respectively. We assume that the flow is symmetric with respect to the middle plane between two adjacent orifices, i.e.
v(x, f) = - c(- x, f).
(4)
This assumption, based on the smallness of the Reynolds number. has been demonstrated in many well-known examples of the low Reynolds number flow such as the creeping flow around sphere, cylinders, etc. Further, the flow has to be periodic because
LOW REYNOLDS
NUMBER FLOW IN A VALVED
the orifices are uniformly spaced. The symmetry and periodicity relationships involved in this problem have been well demonstrated in our previous study of related problems such as the bolus flow and the flow in tubes involving a series of suspended discs (Lew and Fung. 1969 and 1970). Therefore. it is sufficient to formulate the problem in the right half of the region between any two adjacent orifices. The condition of adherence of the fluid to the tube wall and the orifice plate requires that
VESSEL
boundary value problem at hand. As the following section will show, some of the above boundary conditions are not independent of each other and, consequently. our boundary value problem will be much more simple than that presently formulated. METHOD OF ANALYSIS
The equation of continuity (2) is satisfied if the velocity vector v is generated from a scalar functionf(x-, r) as follows: [ij’(x.r)]
v(x.r)=Vxxcx
II(X..r)IrZn = 0
for
0 c x c L.
(5)
V(X. T)/+=~~ = 0
for
0 6 x G L.
(6)
u(x, r)lfPL = 0
for
b C r C a.
(7)
C(X. r)lFL = 0
for
b C r c a,
(8)
where a is the radius of the tube. 2L is the distance between two adjacent orifices and b is the radius of the orifice opening. On the circular plane of the orifice opening, all quantities involved in the governing equations have to be continuous. The reason for this is because the circular plane of the orifice opening is not a natural boundary, but is an artificial one created in the formulation of the mathematical model of the problem. v(x. r) IpL- = v(X,r)lr+,T
for 0 G r C b, (9) Vv(x. r)ls_L_ = Vv(x, r)ls++ for 0 c r < b, (10) Vv(x. r)lr+ = ‘7%(x. r)j.,.,=,_+for 0 G r < b. p(x, r)ls+ vpcx. r)ls+
=p(x,
r)jFL+
= Vp(x. r)ls-,,+
= _i_1 a;af+ ;3axar-' r ar
where x = L- and x = L+ designate the planes defined by the left- and right-hand side surface of the orifice, respectively. The equations (1) through (13) define the
(14)
ar
where i and F-are unit base vectors of the x and r axes. respectively. Substitution of (14) into ( I 1yields -
( pv+; )
v p-
0.
-ipV4f=
(15)
Equation ( 15) is satisfied if p = pv&{.
(16)
V’tf = 0.
(17)
An x-symmetric solution of (17). which has bounded values in the region [-L c x s L, 0 s r s a] and satisfies the zero-velocity condition on the wall of the tube, can be found as follows (see Lew and Fung, 1970): f(x.r)
=dU
2
,4n{[-
(l+hk,coth
(A&))
“=I
(II) for 0 G r < b.
(12) for 0 L r < 6, (13)
87
x cash (k, (x/a) 1 cash (hk, 1 +k xsinh (k,(xla)) “u cash (Ak,) + i
B,,
fPl=*
(k, 1 1kn2J, J,(k,(r/a).l
~0s [ (mr/A 1 Ix/a ) 1 (rn~/A)~
88
H. S. LEW and Y. C. FUNG
Substitution
of (18)
into
(16)
yields the
following equation for pressure:
-2&,(mlr/h)
+ (m~/A)l,(m?r/A)
P --=2i
’
/.d(rr/a)
A.[k. W-1
sinh [k&/a)] cash (Ak,)
(18) where J,,(K,) = 0, B Rrn =
n = 1, 2, 3, . . . .
mdh) cos (mr -;k2tanh6%) (fk.” +(ma,A)‘]!*
The axial component of the velocity of ( 19) satisfies the no-slip condition (5). In equation (18), the values of B,, were selected so that the radial component of the velocity vanishes on the wall of the tube. Hence, u of (20) satisfies the condition (6). The radial component of the velocity u of (20) is antisymmetric with respect to the plane x = 0 and vanishes identically at x = L. Thus, the condition (8) and the r-component of equation (9) are satisfied. The x-component of (9) is also satisfied because of the symmetricity of U. Equation ( 10) is satisfied by u and v given respectively by (19) and (20) because au/ax = 0 and av/ar = 0 at x = + L and -L. and au/ar and &/ax are symmetric. The Stokes equation
m = 1, 2, 3, . . . A,‘s are arbitrary constants, Ji and I( are the i-th order Bessel function and modified Bessel function of the first kind, respectively (Hildebrand, 1962). A designates the ratio of one half of the inter-orifice distance to the radius of the tube. The characteristic velocity U is introduced to make the series of the RHS of ( 18) dimensionless. When (18) is substituted into (14), the following equations for the x and r components of the velocity result:
- Cl+ Aktco~ (AL))
cos5,
ZO[ (mr/A)
-ii
m=I A,,
“a
III=*
(Ak,)
(da)
1 _
lo(m~/A)
cash (Ak,)
2ld(m7dAW/a)l+ (mtr/A)(r/u)f,E(m~/A)(rla)l 2Z,(m7r/A)+ (mrr/A)l, (mrr/A) (19)
9 II
that (11) is the same condition as (13). Therefore, only three conditions are
left to be satisfied. i.e. (7), ( 12) and ( 13). As seen in (2 I), p ‘is anti-symmetric and, consequently aplax is symmetric. Hence.
1
J,[k&/d~ J, k)
the x-component of equation (13) is satisfied. If (12) is satisfied, the r-component of equation (13) will be satisfied. Therefore, only the boundary conditions (7) and (12) need be
IJ(m4A)(Ml f,(mr/A)
_ (m4A) (r/a)LC(m4A) (r/a)] ]] 210(m7r/A) + (mrr/A)fl(mv/A) *
1
xsinh Ck,(x/a)l Jdk,(r/a)] cash (Ak,) J, (kz)
‘u
( 1) indicates
Ak,coth ( Ak,,)Sinh [kn(x’u)l cash (Ak,)
_k xcosh CM-G)1
-2
cash t(~/a)l+~
(20)
considered.
LOW REYNOLDS
NUMBER FLOW IN A VALVED
Since p is anti-symmetric, the condition (12) requires that p be a constant on the circular plane of the orifice opening, i.e. p(x. r)IrSL = const.
for
0 G r < 6.
(22)
Thus. conditions (7) and (22) provide mixed boundary conditions on the plane x = L, from which the only unknown set of constants A,, can be determined. NUMERICAL CALCULATIONS AND RESULTS
In computing for the velocity and the pressure from our series solution, the series summed over m is truncated after retaining the first 56 terms. Our numerical results indicate that the error introduced due to such a truncation in the Fourier series is negligible. We truncated the series summed over n after retaining the first 20 terms. Then. we have a solution with 20 unknown constants. i.e. 20 A n’~. involving a double infinite series in which the orthogonality relationship is lacking. The determination of the constants involved in such a series from the mixed boundary condition is very difficult. In this study. we determine A,, by the’collection method. When the mixed boundary conditions (7) and (22) are imposed at 20 uniformly distributed points of r/a in [0 s r/a s O-951.a set of 20 simultaneous linear algebraic equations result, which enable us to determine the value of 20 A,,‘s involved in this set of equations. Condition 17 I is imposed at uniformly distributed points in [‘I G r/a c 0.951 and condition (22) is applied to uniformly distributed points in [0 s r/a =Gv-0*05]. Here, the points are separated from each other by a distance of O*OSa and 7 is the ratio of the radius of the orifice opening to the tube radius: The magnitude of A,,‘s, which are determined by the above scheme. decreases quite rapidly in a more or less monotonic fashion as II increases. For example. the ratio of the magnitude of A, to A*,, is of the order of IO-” or smaller for all cases of different combina!ions of the value of A (inter-orifice distance) and 77 (radius of the orifice opening). In this
VESSEL
89
computation, we used twelve different combinations of four values of A: 0.25, 0.50, 140 and 1.50; and three values of 7: O-25,0-50 and O-75. We normalized the right-hand side of (22) and set it equal to p(Ula) so that the non-dimensionalized pressure p/M U/a)1 becomes unity at the orifice opening. The velocity computed from the truncated form of (19) and (20) after substituting the value of A, is presented in Figs. 2(a-c). where the flows of Figs. 2(a) and (b) are driven by the same amount of pressure gradient, while the pressure gradient of the flow of Fig. 2(c) is 10 times that of the flows of Figs. 2(a) and (b). It is interesting to note: (1) contrary to the flow at finite Reynolds number, no separated vortex pattern at the comer of the orifice is observable; and, (2) the existance of a nearstationary fluid layer near the wall of the tube when the inter-orifice distance is sufficiently small (e.g. less than 0.5a). The velocity profile becomes Poiseuillean if the distance from either orifice to the point of observation is equal to or greater than 1a3 times the tube radius. The pressure can be computed from the truncated form of (21). Since the pressure has anti-symmetric distribution with respect to the middle plane between two adjacent orifices. the pressure in the next region to the right of the region under consideration is given by [p + 2p(x, r) jrcL, ,.+J. where p is the same pressure as (21). The pressure of (21) vanishes at x = 0. Hence, the mean pressure gradient is simply equal to [p(x. r) lmL, ,,/L]. The non-dimensionalized mean pressure gradient is shown in Fig. 3 as a function of the dimensionless inter-orifice distance (2h) and the dimensionless radius of the orifice opening (71). This figure shows the resistance of the tube with a series of plate orifices in comparison with that of the smooth tube. The resistance of the tube to the flow strongly depends on the size of the orifice opening in the range of parameters selected in the present computation. The U used in the expression of the dimensionless pressure
90
H. S. LEW and Y. C. FUNG
t
-_--irleill__ 0
Fig. 1.Geometry of the flow system.
IO
u u
--
0
s
2fP -
0.51
/
I
_,
05: _
yc.75
X050
IO-x/o 0 0.1 0.2 0.3 0.4 0.5
x,0 0
0.1 0.2 0.3 0.4 0.5 0.6
0.7 08
09
I-0
r/a I.O_I\I\r\R\P\r\r\nr\l\hh,,,
A
0
01
0.2 0.3 04
0.5 0.6 0.7 0.0 o-9
I.0 1.1 I.2 I.3 1.4 1.5
Fig. 2(a). Velocity distribution of the fluid in the RHS of the region between two adjacent orifices for r) = O-75. Figure shows in each cross section the axial component of velocity II above the center line. and the radial component of velocity u below the center line. The parameter n is the ratio of the diameter of the orifice opening to the tube diameter. The pressure drop over the length A (not the pressure gradient) is the same for all figures.
LOW REYNOLDS
x/o
0
01
C2925
0
0 I
0.2
._ x/c1
NUMBER
FLOW IN A VALVED
AGo 0
0.3
0.4
0.5
0.6
0.7
0.0
0.1
0.9
02
I.0
0.3
I I
0.4
I.2
VESSEL
0 5
, 3
I.4
l-5
Fig. 2(b). Velocity distribution for TJ= 0.50. The pressure drop over the length A is the same as that of Fig. 2(a).
gradient is the mean velocity averaged over the cross’ section of the tube (not the value averaged over the orifice opening). When A approaches zero. the solution should approach the flow through a smooth tube of radius equal to qa. The points on Fig. 3 corresponding to A = 0 are calculated in this fashion, and the large value of the pressure gradient for the small value of v at A = 0 is due to the fact that they are non-dimensionalized, based on the nominal tube of radius a instead of the effective tube of radius r)a. CONCLUSION
The low Reynolds number flow of the Newtonian fluid through a rigid circular
cylindrical tube involving a series of plate orifices of axisymmetric geometry. which are uniformly distributed along the tube axis. is studied, based on the Stokes approximation. The results of this analysis show that: ( 1f the resistance of the vessel with valves to the very slow viscous flow through it becomes very large when the inter-valve distance is small or the valve opening is narrow; (2) in contrast to the flow at finite Reynolds number, the zero-Reynolds number flow does not create a separated vortex flow at the corner of the orifice: and. (3) the fluid near the wall of the vessel is almost stagnant when the inter-orifice distance is sufficiently small (e.g. less than 0.5~). We conclude from the above statements
92
H. S. LEW end Y. C. FUNG
1'025 A=050 i,o 0
01
Xm
.?I 0.2 0.3 7.4 0.5 96
0
Ol20.25
3.7 0.6 P-9
I.0 I.1 I2
I.3 1.4 I.5
Fig. Z(c). Velocity distribution for I) = 0.25. The pressure drop over rhe length ,\ is IO times that of Figs. I(n) or 2(b).
that, if flow in the tube is created by an upstream pressure head only, the tubes without valves constitute a much more efficient passage for the fluid than valved tubes. In general, it is true that the arrangement of the physiological system in the living body is highly efficient for performing appropriate functions. Since the valves in the vessel increase the resistance to fluid flow driven by the pressure gradient supplied by an upstream pressure head. the existence of valves in some physiological vessels implies that the driving forces of the fluid in such vessels include some energy source in addition to the consistent pressure gradient along the vessel axis. This additional driving force in the vein is the localized compressions and
expansions by the tissues surrounding the veins. It is believed that the peristaltic contractions of the wall of the lymphatic duct constitute the main driving force of the lymph flow (Hewson. 1774 and Hall et trl. 1965). Our results do not explain the role of valves in physiological valved vessels. In such vessels, valves play two contradicting roles; a part of the pumping device, and that of blocking fluid flow. We investigated the latter role, while the physiological valves predominantly play the former one. As shown by the existence of the highly distensible and collapsible wall of valved veins, the role of valves in physiological valved vessels is closely coupled with the deformation of the vessel wall. Consequently, the effects of the
LOW REYNOLDS
NUMBER
2x Fig. 3. The average mean pressure gradient as a function ofhandn.
valves and the deformation of the vessel wall have to be taken into consideration simultaneously. even for the simplest mathematical model of physiological valved vessels. For example. an investigation of fluid flow through a tube. similar to that considered in this study. created by a peristalsis of the tube wall should useful information for a better provide understanding of lymph flow in lymphatic ducts. It is interesting to note the result that near the wall of an intensely valved tube the fluid is nearly stagnant. This qualitatively suggests that a layer of very slow moving plasma should exist when the wall of the capillary blood vessel is unsmooth. Such a near-stagnant fluid layer near the blood vessel wall would hamper the gas exchange across the capillary blood vessel wall. In this respect. the smoothness of the capillary blood vessel can be vitally important, and irregularity can be a source of trouble. The idea of modeling the alveolar ducts by a valved vessel is an
FLOW IN A VALVED
VESSEL
93
interesting concept which deserves some consideration (Aitshuler, 1969). If the inter-valve distance is much smaller than the dimensions of the valve opening. flow through such a valved vessel can be effectively represented by a constant velocity profile (e.g. a space-averaged value of the velocity). If the inter-valve distance is comparable to the dimensions of the valve opening. the representation of flow by a constant macroscopic velocity profile becomes less useful. Our numerical results suggest that the slip velocity at the periphery of the hypothetical tube. whose cross section is defined by the valve opening, should be negligible if the intervalve distance is much smaller than the radius of the valve opening. However, if the intervalve distance is comparable to the radius of the valve opening. the average value of fluid velocity at the periphery of the hypothetical tube is quite different from zero and the velocity profile differs from the parabola significantly. The dimension of the alveoli opening is comparable to the radius of an alveolar duct and. consequently. the deviation of flow characteristics in the alveolar ducts from that in the conventional tube has to be considered. Of course. the question of how valid Altshuler’s model is for fluid flow in alveolar ducts is another matter. When one employs such a model. one must consider the change of tube diameter timewise because pulsation of the valved tube is a dominating parameter pertaining to the physics of air flow in alveolar ducts. For example. the nearly stagnant fluid layer near the wall of the valved tube of a constant diameter would not exist in the pulsating valved tubes. Ac~nolc,ledpemfnts-The authors wish to express their appreciation to Mrs. Claudia Lowenstein for her help in numerical calculations. This work is supported partially by the United States Public Health Service. National Institute of Health under Grant No. USPHS HE 12494-01. partially by the San Diego County Heart Association under a Fellowship granted in the name of the late Marjorie Sitter and partially by the General Research Grant Support from the College of Medicine. the University of Arizona.
H. S. LEW and Y. C. FUNG
94
REFERENCES Altshuler. B. ( 1969) Behavior of airborne particles in the respiratory tract. Circulatory and Respiratory Mass Transport, p. 223. Ciba Foundation. Bloom, W. and Fawcett, D. W. ( 1968) Blood vascular system. A Textbook of Histology. 9th Edn. p. 376. Sanders. Philadelphia. Davson. H. (1965) Interstitial fluid and lymph. A Textbook of General Physiology, 3rd Edn, p. 439. LittleBrown, Boston. Hail, J. C., Morris, B. and Woolley. G. (1965) Intrinsic rhythmic propulsion of lymph in the unanaesthetized sheep.J. Physiol. 180.336-349.
Hewson, W. (1774) E.\perimenfaf inquiries. Part 2. p. 126. A description of lymphatic system. Hildebrand. F. B. (I 962) Advanced Calculus forApplications. Prentice-Hall. New Jersey. Lew. H. S. and Fung Y. C. (1969) The motion of the plasma between the red cells in the bolus flow. J. Biorheol.
6. 109-
119.
Lew, H. S. and Fung Y. C. (1970) Plug effect of erythrocytes in capillary blood vessels. Biophvs. J. 10. I. Mills, R. D. (1968) Numerical solutions of vkcous flow through a pipe orifice at low Reynolds numbers. kfech. Engng. Sci. 10. 133- 140. Rosenhead. L. (Ed.) ( 1963) Laminar Boundury Layers. Oxford Universities Press, Oxford.