Asymptotic uniqueness for elastic tube flows satisfying a windkessel condition

Asymptotic Uniqueness for Elastic Tube a Windkessel Condition* A. R. ELCRAT Mathematics Communicated

Flows Satisfying

H. M. LIEBERSTEIN**

AND

Deeartment,

Wichita

by Richard

State

University.

Wichita,

Kansas

Bellman

-1BSTKXCT

An improvement is presented of a theorem in [l] on mean square asymptotic uniqueness for viscous fiow of an incompressibie fiuid in an eiastic tube. As previousiy, radial velocity components are ignored and 20 mitial velocity profile is specified. Axial symmetry is dropped, however, and a “windkessel condition for uniqueness” is introduced to replace the former “normality relations.” The windkessel condition is

x2

.>

x1

(zG)~ d.4 <

=s

A(G.~)

(u”) t dA A(%,>4

where u is the difference of any two regular flows and A(x, t) is the cross section of the tube at position x. It has a heuristic relation to a condition that the capacity of the tube to absorb changes in square magnitude of the velocity d;es not increa% as the flow proceeds down the tube, and it replaces any necessity to impose artificial boundary conditions at the ends of a tube section. It is found that the mean square difference between any two regular solutions decays exponentially with timt, and a relation between a bound for the decay rate and the radius (and the kinematic The derivation of this decay rate utiiizes a wellviscosity coefficient) is studied. .411 conditions specified are intended known variational principle for eigeuvalues. to have direct relevance to blood flow in short, relatively

straight sections

of the

trunk of the aorta.

* Work sponsored by NIH grant HElO034. ** Work prepared while at Mathematics Department, Bloomington,

Ir,ciiana University,

Indiana. % athematica.l

Biosciences

1, 397-

411 (1967)

A.

I.

It. ELCRAT

AND

H. M. LIEBERSTEIN

IN’IXODUCTION

1n the first three decades of this century the windkesse’l theory was developed, by ,the noted cserman physiologist Gtto Frank and h:is associates, as an eisentially sratic modci for systemic circulation. In replacing the core of the body by a liquid-filled container with elastic sides, ilts originators sought to emphasize, rather than the usual fluid dynamic considerations, the significance o’f the fact that large pressure variations originating in the core of the body are damped as they pass to the periphery. The importance, however, of this absorption of pressure variations by the elastic reaction

of the walls of the arterial

tree was already

clear:ly recognized

as earlq. as the first half of the eighteenth century, and the name Windkessel was apparently chosen by Otto Frank ii. analogy to the, fire-fighting apparatus of an earlier period (see [2]). This paper contains an improvement of our theorem in [I] on mean square time asymptotic uniqueness for time-dependent regular flows in elastic tubes. The weakened relation that will define the set of functions in which uniqueness is sought will be called the windkessel condition for uniqueness. It has an intuitive connection to the condition that the capacity for the walls to dampen changes of magnitude of the flow velocity by elastic reaction to an increase of pressure does not increase as the fluid movts down the tube. It must be remembered that no initial velocity data are available on the kinds of flows that we wish to consider between reservoirs of time~~~1 some A.__ I_ _ -r~ieasurec~ ^^___--^1 -pressure --.-.-.-- gramem) -..- 3!- -I\ ana I__ uut: .VW-,.LCr V~LYIII~ .-.C?-._.-....._,. pc33~1~3 GUI LU a conditions other than those given in the differential equa.tion must be introduced. This justifies our introduction of the windkessel condition for uniqueness, since it is a weaker condition than prescription of another boundary value. Hence, although the wall elasticity allows for nonzero variations V, of velocity in the axial direction x, we prescribe no additional boundary condition to account for this. The windkessel condition for uniqueness suffices for our purpose of proving mean square asymptotic uniqueness. Moreover, this condition will be found to be t+ially satisfied whirh ,rm for rigid tubes, _A weakened Jnrm of . . -La---rcu ‘“au* VI the %.I*_ rnnrlitinll C”“UILI”II, VVIIILII YVL ,.~ll La,IA the L1,G ateakeized windkessel condition, will also be found to be trivially satisfied for tube sections with rigid ends and for tube sections where the velocity profile or the velocity flux profile is specified on the end:; for all time, Under this weaker condition a part of tht-! main theorem on mean square asymptotic uniqueness can still be prov,:n. .~iathemticaZ

Uiosciencfs

1, 397 - 411. ( 1967)

WINDKESSEL

CONDITION

399

As will be seen, what we have chosen to call mean square asymptotic uniqueness can be regarded as mean square asymptotic stability with respect to the initial velocity profile, and the rate at which the (timedependent) solution in “steady state” is acquired is of considerable importance for consideration of flows at finite times. This comment encompasses both considerations of stability with respect to initial data and the use of one solution (selected for its convenient form) in the processing of empirical data (see [l]). We find that there is an exponential decay of the mean square difference of any two regular flows and that the exponential decay rate depends linearly on the kinematic viscosity coefficient Y, whereas the Reynolds number OLIYdepends on it inversely, As in [l], we ignore velocities orthogonal to the tube axis while allowing for elastic variations of the tube walls. This is our only assumption that is similar to a linearization. The differential equation problem, however, is always nonlinear unless the wall is rigid. The assumption that radial velocities are negligible is also made in the classical onedimensional analysis of Delaval (1850) given for inviscid compressible flows through nozzles with small curvature. That the analysis of smallcurvature nozzle flows using this method has been very successful gives as good a justification of our assumption as one could hope to obtain in a physical situation where there is inadequate empirical information. It must be admitted, however, that in the case of inviscid small-curvature nozzle flows, because of the different kind of boundary conditions imposed, the axial component of velocity varies only slightly across the nozzle. This, together with the assumption of a small radial component of velocity, I gives a double reason to ignore terms that involve the procliuct of the radial velocity component and the radial derivative of the axial velocity i a product component. Our viscous tube flow problems do not involve suc,:h i of small quantities as is involved in the inviscid nozzle flows. $evertheless, to say that radial velocities are important, for example, for/ blood flows in the trunk of the aorta, would be to say that there are la,rge pressure gradients in the radial directions. and thus to say that pressure measurements cannot be made there by a probe that is allowed to float loosely in the stream, In turn, this is to say that we do not have and probably a mnmr,rmman+c r\f him-w-l flnrxrc in +hn t-rmnlz r\f Nolan,IIb”LA PQ~ bc&II hcatrn sx&vL rln~r airy +nl;c,hl LbUcx”IL; Ill\;c&JUI\;.IIILULJ v1 “l”“U II”*“J 111 LILti CLbatfin.“1 the aorta. The most reliable data on blood flow in the aorta are those taken from L pressure probe carried on a catheter introduced at the femoral artery and allowed to float freely in the stream. The arguments given in [3] against ignoring radial velocities for flows in elastic tubes are Mathematical

Biosciences

1, 3E7- 411 (1967)

A. R. ELr3RAT

400

AND

H. M. LIEBERSTEIN

clearly inadequate for some flows and, moreover, the authors of [3] find it impossible to treat the more general case that they propose. Of course, theories where radial components of velocity are ignored apply in the aorta ---- ~--onlv ----J in sections that are well removed from the arch region. We rely on a windkessel effect to dampen the wild flow conditions that must exist in the arch region before the blood reaches other portions of the aorta and thus to provide a flow that is essentially free of radial components of velocity. Of course, this leaves open the question of stability with respect to the occurrence of small nonzero components of the pressure gradient in radial direction, and we leave consideration of this question for a future study. For the uniqueness results given here we are able to eliminate the assumption of axial symmetry used in [l]. It is assumed that the segment of the tube studied is a simply connected Green’s region, a region where the divergence theorem is valid. It is further assumed that the cross sections perpendicular to the axis are simply connected planar Green’s regions bounded by Jordan curves, and that the lateral surface of the tube has a normal at every point. We assume the tube is “bounded in time”; that is, it is contained in the circular cylinder

((XTy, 2)Ix E

[a,bl,y2 +

z2szr2}

for all t. Finally, if d(t) is the distance of a point on the lateral surface to the axis, we assume d and d, are functions that are continuous on their domains. The elimination of axial symmetry is of some physical interest; since the aorta is large and lacks any structure that would give it rigidity, the cross sections may take odd shapes that even depend on orientation to gravity. 2. THE WINDKESSEL

CONDITION

Consider a segment of a tube (Fig= ij that at time t is represented by a simply connected Green’s region R(t) C E3 between planes x = a and x = b. Let the points of the boundary of R(t) that are not contained in the planes x = cc,b be denoted by S(t) and the cross section at x by A (x,8). For flow of an incompressible fluid in a rigid tube (time dependent or otherwise) there is no variation rl, of velocity in the axial direction x because the pressure gradient does not vary down the length of the tube. If the pressure is raised on one end of the tube, there is no axial relief Mufhett2tctid

Biosciences

1, 397-411

(1967)

WINDKESSEL

401

CONDITION

of pressure because the fluid is not compressible and no lateral relief beca.use the walls are rigid. Thus, the velocity in each cross section is simply altered (by the same amount) either up or down, depending on whether the rise in pressure on one end raises or lowers the (overall) pressure gradient in the tube. For an elastic wall, the provision for a

s(t)

FIG. 1

lateral relief of pressure tends to dampen locailyr the change of the magnitude of the velocity and this dampening introduces nonzero values of V, as well as a pressure gradient fi, that depends on axial distance x. Our windkessel condition for velocities would simply state that the capacity of the fluid to provide for such a change mqst no’c increase as the flow proceeds down the tube. It is the change lvix of the magnitude of II that is involved in this phenomenon, not II,. However, since it is hard to utilize II& in a mathematical formulation, we look simply at (v2),. Since at any given time and position, to its maximum somewhere in that the integral of (v2)%over We write, then, that for

x2 >

(v2), varies from zero at the tube boundary the interior, we prefer to write a condition the cross section does not increase with x. every t E [to, co) and every x1, x2 E [a, b]

(v2)xdA<

Xl *

(v2LdA

(2.1)

A(%,, t)

A(xpr t)

where v(x, y, z, t) is the velocity of the fluid. Here the equality holds triv!ally for rigid tubes. This is ow windkessel condition for velocities. Recognizing that the flow velocity is zero on S(t) and thus on the boundaries of A (x, t) for every x, we see that for v a regular solction (see Section 3) of the equation of motion

( 1 v2~+jtj(v2). cu. A b-at)

(2.2)

--G

Maihematicwl Biosciewes

1, 397 - 411 (1967)

A. R. ELCRAT

402

AND H. M. LlEBERSTEfN

Therefore, for reguIar solutions of the requisite boundary-value problem the windkessel condition for velocities (2.1) is equivalent to the condition (24

Actuafiy we utilize a somewhat different condition; where ZJand w are regular flow solutions (see Section 3) of the boundary-value problem

y(v*, +

vt + vv, =

for every x, y, z E R(t) and

vyy

c vzz)+ 7f(% 4

P.4)

t E [to, OCJ), and

v(x, y, 2, t) = 0

(2.5)

for every x, y, z E S(t) and t E [&,,CQ),and at = v - w,

we require that for x1, x2 E [a, b]

(u2)xdA<

x1 *

x2 >

A(% f)

(u2)x dA

(2.6)

A(% 0

or, equivalentIy, \. u2dA’

(5 A(r,t) T&s is OUYwindkessel

) xn


for

XE [a, b].

conditionp for ulzigueness.

(2.7)

Note that eqataiity

is

triviaE for rigid tubes. 3. MEAN SQUARE

ASYMP1’OTIC UNIQUENESS

Let 9 be the set in space-time defined by

92- ((x, y, z, t) /(x y, 2)E R(t)*t E: p()*+. A function rg is said to be a regular solution of (2.4) and (2.5) in W if 0) 9, (W

(iii)

qt, plxz, yyyl 4722E C(@,

vxI py, yz E C(@,

and

IJ = 9(x, y, z, t) satisfies (2.4) and (2.5).

Mat?rematical Biosciences

1, 397-

411

(1967)

WINDKESSEL

CONDITION

403

A regular solution q~is said to be a regular flow solution if u = ~(x, y, x, t) represents fluid velocities in Z(i) so that dX/dt = D where x = X(t) is the time trajectory of a point of R(t). On such a path dv/dt = vt + vv,. Szlppose (i) ‘o alad w are regzclar flow so&ions of (2.4) and (2.5); (ii) the region R(t) and its cross sections in planes perpendicular to the &be axis satisfy the conditions mentioned in Section 1, and (iii) for u=vw, condition. (2.7) is satisfied. Thert if we define THEOREM.

J(x, t) = (219-l

9 dA, A(%,4

and I(t) = (219-l c u2dV, RiJtI we have

(1) J(K t) =GJ(K O)e-“5

XE

[a,b],

and

(2) I(t) < I(0)e-T wheve k is a certain positive constant. Preliminaries

We give several classical results that will be needed in the proof of the theorem. Consider a plane Jordan region 52. We will call functions f such that f E Cz(Q) fl Cl(o) and f = 0 on the boundary of D admissible. Suppose we have a nonnegative number ;1 and an admissible function ~1 solving the problem in

0,

(3.la)

c-0

in

Q,

(3.lb)

fp=o

in

SG=!Q--Q.

(3.lc)

Mathematical

Biosciences

1, 397- 411 (1967)

A. R. ELCRAT

AND H. M. LIEBERSTEIN

Then,

(3.2a)

where the greatest lower bound (glb) is taken over the class of admissible functions f (see [4], page 89). The number A is positive (i.e., not zero) since (3.lb) and (3.1~) are incompatible for harmonic functions. It can be shown that 1%is the smahest eigenvalue of the problem (3.1s) and (3.lc), and that the value of Afor the Same problem on a (proper nonempty) subdomain of 52 is larger than that for Sz; that is

(see [63, Chapter VI). If 9 is a disk of radius Y and if we then denote 1 by il,, we have

a

I=----#

6% Y

(33

where ,?.+,is the first positive zero of the Bessel function JO(~) (see [5], page 213). We wiil take for granted rhe existence of Aand p for an arbitrary bounded Jordan region Q. Proof of tlze T/zeorem. Let x E [a, b]. Then, since v and zv satisfy (2.4) and, by (2.5), time variations do not contribute to derivatives of the integral of ~2, we have

4~

c

+ @,y+ ‘uzz)dA.

‘4k 4

MathematicaE Biosciences

1, 397- 411 (1567)

(3.4)

WINDKESSEL

406

CONDITION

Also, from (2.7) and (2.5),

Then from (3.4) and (3.5), we obtain

-+,t)

A(.%4 and, by the divergence theorem, Jt(x,

t) <

J

(N; + u:) dA.

(ti; + iv:) dA = -

ugdS--

aA@,t)

(3.6)

A(,4

A1%4

From the minimum property (3.2a) and the iaoperimetric inequality (3.2b) we have, since A(x, t) is contained in a disk of radius Y, (g”,+ ti:) dA > a2(x, t)

I 0.t)

5

A(%,4

&dA

>A,”

Jti2 dA,

(3.7)

A@,t)

where J(x, t) is the greatest lower bound given in (3.2a) for &?= A(%, t) and 5 is given by (3.3). Now with (3.6) and (3.7) we have J(U)

<

-

:=Jg&?&j 2&(x,

2

t),

Ab,t)

which yields the conclusion

(1) with k = 2&.

Turning to the second case, we find that b

b

J(x, t) dx I= ; Jt(x, t) dx

I’(t) = -g J

a

a b P

<-

24

J(x, t) dx = - 2&1(t), J

a

and the theorem is proven. Mathematzcal

Biosciences

1, 397 - 411 (1967)

.4.

406

R. ELCRAT AND H. M. LIEBERSTEIN

REMARK. In the physical problem treated in Cl], blood flow in the trunk of the aorta, if we assume a segment of an aorta fairly near the heart to be contained in a right circular cylinder of rad.ius 1 cm and take a representative value of v to be 0.035 in cgs units (see [Z], page 41), we obtain

and, for this segment, I(t) decreases to less than 20% of its initial value during the first second. On the other hand, if farther down the aorta, a segment of the trunk is contained in a right circular cylinder of diameter 4 cm, then k = 6.48,

ilI = 9.620,

and during the first secand r(t) decreases to less than 0.2% of its initial value. For arterial segments contained in right circular cylinders of smaller cross section, the exponential decay rate will be faster in accordance with the inverse of the radius squared. This means that the percentage of I(0) that I(t) reaches in the first second is then lowered according to a power l/r2 since this percentage is given by

I 01 2

exp -. 2~ e 72

= [exp (- 2~&j)]~j~‘.

Thus if r is replaced by r1 = (A)Y,the percentage of I(0) that .T(t) achieves in the first second is given by

which is a remarkable

advantage,

since it is always true that

exp -22y $ I 0

I<

1.

We see, then, that considerations of stability of blood flow with respect to initial velocity profiles are very dependent on channel size MatltematicaZ Biosciences

.I, 397 - 411 (1967)

WINDKESSEL

CONDITION

407

and this dependence may influence the occurrence of what has been called turbulence. It may be that it is no accident that only moderately large channek for blood have survived well in nature. It was noted in [l] (and [Z]) that steady-state pulsatile flows in rigid tubes could be expressed by the Poiseuille formula for flows with constant pressure gradient if the dimensionless parameter Q? = (ys/~)F (where P is frequency, see Section 5) were small. This fact is consistent with the foregoing in that both cases are very heavily damped. For pulsatile flow, small cross-sectional area and large viscosity have the same effect. 4. A

WEAKENED

WINDKESSEL

CONDITION

Suppose we have two regular flow solutions of (2.4), (23, v and r~. Suppose I(t) is defined as in the preceding section. We retain all our regularity assumptions on R(t). Since ‘uand w are zero on S(t) and since points of A(a, t), A (b, 1) do not move in the direction normal to the planes x = a, b, there is no boundary contribution to the derivative of I. We obtain

Using (2.4) and (2.5), we get * I’(t)

= +

= -

24(V#+

2121,-

[wsdA

+

I

Ah4

ze+ -

ww,)

dV =

at Azt dV

J tati.A (b, 1)

If, as in the theorem presented earlier, (2.7) were now assumed, then the sum of the first two terms on the left would be nonpositive and we would have

A. R. ELCRAT

408

AND H. M. LIEBEKSTEIN

a proposition that we wish to utilize. However, (2.7) is a stronger condition than we need. We use a weakened windkessel condition, which simply states that the sum referred to above is nonpositive; that is, that

This condition is the same as (2.6) (which, of course, is equivalent to (2.7)) except that it is assumed to be valid for the ends, X, = a and xs = b, of the tube only. The new inequality (4.2) is trivially satisfied for tube segments whose ends are rigid since there can be no radial pressure relief at each point at the ends of such tubes. If we prescribe either II or ~1,~ of the cross section of thr: tube ends for all times, the condition (4.2) is again trivially satisfied. Assuming (4.2), we have (4.1) and

so, utilizing

(3.7), b

I’(t) < -

A; J

a

dx

I

AND

ON EXISTENCE

ASYMPTOTIC

2&(t).

A(x,l)

Thus, with the weakened assumption and just as quickly. However, the retained. It is worth noting +hat if (4.1) thzn I(t) = 0 for t > t*, and in fact, 5. REMARKS

zt2da4 = -

(4.1), I(t) still decreases exponential13 conclusion concerning J(x, t) is not holds and if I’@*) = 0 for some t*, v z ‘wfor these values of t (see [l]).

AND UNIQUENESS

THEOREMS

IN FLUID DYNAMICS

STABILITY

Besides pointing out the relevance of asymptotic uniqueness to the blood flow considerations of [l], we would like to undertake certain remarks of a general nature. Our method of attack could almost be considered in counterthrust to the direction of modern movements in partial differential equations Malkewzaticai Biosciences

1, 397- 411 (1267)

WINDKESSEL

CONDITION

409

Although the classical uniqueness of regular solutions of boundary-value problems has been relatively easy to prove, difficulties with general theorics of existence of regular solutions have been so insurmountable as to cause most workers tir seek successively weaker senses of existence. In [l] and here, the concept of regular solutions is retained by weakening the sense of uniqueness. This is possible because the particular weakening undertaken of the sense of uniqueness has been such as to allow us to drop some boundary conditions, thus making it easier to undertake existence In the case of rigid tubes, it has even been possibls to considerations. produce an explicit series solution for velocity in a convenient form (see [l]). (It has been pointed out by Professor Michael Golomb of Purdue University that the series derived in Appendix B of 111for the voltlme flow rate, a(r, t) = l/(h)

V dA R

of a time-dependent

incompressible

flow in a rigid tube can be summed,

t - r/b _

a@, t) =

2Y2//4

1

(t-.$-S)/(S)di.

t

Here the pressure gradient is only required to be summable, not analytic, as required in [l], where the quoted series was acquired by iterating an unbounded operator. In order to obtain the velocity V that is needed in [l], the iter .Lrontechnique would still seem to be needed. An asymptotic formula for this volume flow rate t

lim t-+cQ

I

has also been given by Professor Golomb.) Also, it should be noted that it has so far turned out to be relatively easy to prove uniqueness in our weakened asymptotic sense. This is essentially mean square asymptotic stability with respect to an initial velocity profile, but does not require existence of a regular solution for a prescribed initial velocity profile. For most fluid dynamic problems where there is some nonzero viscosity that could or should be introduced, it would seem that mean square Mnt~!eaznticni

Biosciences

1, 397 - 411 (1967)

410

A. R. ELCRAT

AND

H. M. LIEBERSTEIN

asymptotic uniqueness would be both possible to provide and physically meaningful. Perhaps there is some clue here to an understanding of the natural phenomenon of turbulence. There are two basic tenets on which to found this hope. First, in our tube flow studies, the (possibly timedependent) steady-state flows are only taken on, so far as we know, in mean square. That is, we have been unable to prove that the steady or time-asymptotic state is uniformly approached, and we believe that it is not. However, it would seem to be of far more importance in looking at a solution at a finite time that (as we have already pointed out) the exponential rate at which the steady-state solution is taken on varies linearly with the kinematic viscosity, whereas the Reynolds number is inversely proportional to it. The Reynolds numbe: alone has been used flow; in the past to distinguish between “laminar” and “turbulent” however, the exponential decay rate is given by

where E’ = l/T and T is the period of a periodic pressure gradient (or of one component in a Fourier series expansion of a pressure gradient). Here

is a second ditasnsionless parameter, the first being the Reynolds number Re=A-v V

required to describe pulsatile flow in rigid tubes (see [l] and [2]). It is seen that the exponential decay rate is much more clearly understood in terms of as than in terms of Re, since k does not vary across the channel, whereas Re does. In the usual analysis of Poiseuille flow (flow in a rigid tube with constant pressure gradient) it is thought thxt turbulence develops-that is, the ~tlneoretical treatment fails-due to an increase of the Reynolds number, but all efforts to prescribe a critical Reynolds number have met with at best limited success. Zven though attempts are made in a laboratory to maintain a constant pressure gradient, this is not strictly possible, and thus -variations of a2 as well as of Re are involved. We further suggest Eriaiizemdi:r al Biosciences 1, 397- 411 (1967)

WINDKESSEL

411

CONDITION

that the empirical phenomenon

of turbulence

may be caused by a slow

rate of decav sf transients, and that such a slow rate of decay depends on there beng in the fluid a large value of a2, not Re. Both parameters are inversely confusion

proportional

to v, of course, which could have led to the

but a2 varies with area instead

juggested,

frequency of time perturbations

of radius and with

of the pressure gradient, not with velocity.

REFELENCES 1 H. M. Lieberstein.

Determination

the aorta from measurement ,4cln BiotWeoret. XVII,

of tile tension-stretch

relation

for a point in

in oivo of pressure at three equally spaced points,

II (1965). 50-94.

2 D. A. &IcDonald, Blood flow in a; teeries, Arnold, London, 1960. 3 W. P. Timlake, A. C. L. Barnard, W. A. Hunt, and E. Varley, A theory of fluid flow in compliant tubes, IBM Publ. 37.001 (Houston Scientific Center), May 18, 1966. 4 G. Polya and G. Szego, Isoperivnetric inequalities

in mathematical

plcysics, Princeton

Univ. Press, Princeton, New Jersey, 1951. 5 G. Tolstov, Fmviev series. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 6 R. Couranc. and D. Hilbert, Methods of mathematicaE physics, Vol. I, Wiley (Interscience),

Sew

York,

1953. Mathematical

Biosciences

i,

397-

411 (1967)