Del bar a Grindr. Lugares físicos y virtuales como espacios de la seducción

If we consider for a moment a simplified concept of the circulatory blood system in man, we can imagine that we have a pump delivering blood to a complicated network of pipes, which has innumerable connections. To develop an adequate mathematical model of this system and its behaviour is an almost impossible task. Thus, in order to make any progress, we attempt to model parts of the system separately. Here we concentrate on a small section of this circuit, say in the region of the aorta as shown in Figure 4.3.1. Indeed, we shall

consider the relatively straight section between A and B. One can imagine that blood flow in this section behaves in much the same way as water in a cylindrical tube. This, however, is a gross oversimplification of the situation. To see this, let us consider some salient facts regarding blood flow. First of all, unlike water, blood does not have constant viscosity and this varies with velocity. Thus blood may be claimed to be non- Newtonian; indeed the properties of blood change rapidly if removed from the system and so it is extremely difficult to perform experiments on it under laboratory conditions.

If we now consider the type of flow in an artery, it is apparent that because the heart delivers blood in short bursts during contraction into systole, the flow is pulsatile and not uniform. Furthermore. we do not know the velocity profile of the flow entering A in Figure 4.3.1 and consequently the velocity

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FIGURE 4.3.1: Schematic description of an aorta.

profile at B is also unknown. This observation is of fundamental importance in the mathematical description of blood flow. On the other hand, the hydrodynamic problem of considering the change of an initial velocity profile of a Newtonian fluid in a rigid pipe is fairly well understood and is based on the fundamental theory of Poiseuille (1846). One should remark here that Poiseuille, whose contributions to hydrodynamics are well known to engineers and mathematicians, was in fact a physician and his interest was precisely the problem we are considering here, namely, the study of blood flow.

Let us now focus on the arteries themselves. We know them to be elastic and a typical cross section may change significantly with time due to the pulsating nature of the flow of blood. Thus once again it may be unreasonable to treat the arteries as rigid tubes. Nevertheless we find it necessary to assume this as a first appoximation.

Referring to Figure 4.3.1 consider the flow of blood delivered into an aorta. The blood is pumped in an asymmetrical fashion and there are large cross-channel components of velocity in the arch region and consequently large cross-channel components in the pressure gradient. This is well known from thoracic surgery on animals. However, away from the arch itself, say in section A−B, the cross-channel components of velocity are considerably reduced and the flow is almost entirely longitudinal but, of course, still pulsatile. In the arch region it is found in thoracic surgery that the arch is very pliant and yields easily to the cross- channel pressure gradients. Thus it is reasonable to assume that the recurrent upward yielding of the arch

region in response to pressure changes and the “general give” radially of all cross sections of the aorta cause changes in pressure to be dampened, especially the radial components. We shall assume then that, as the blood proceeds down the trunk of the aorta, radial velocity components may be neglected. This assumption is known to physiologists as the Windkessel effect assumption, an idea introduced by the German

physiologist Otto Frank.

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FIGURE 4.3.2: Section of a rigid tube.

To begin our development of a mathematical model, consider pulsatile flow in a rigid tube of constant cross section A and boundary ∂A. A typical section is shown in Figure 4.3.2.

At time t assume that the velocity of flow along the tube is V(x, y, z, t). In accordance with the Windkessel assumption above we assume that there are no velocity components across the tube and that the pressure P depends only on x and t. That is, P does not vary radially with position, but only longitudinally. To derive a mathematical model we proceed, as Poiseuille did, to balance inertial forces

where ρ is the density of the blood, with the drag FD due to viscous shear and the pressure force FP, which can be written in terms of P(x, t) as

Note that FP is a surface force and body forces such as gravity are neglected. Equating these forces and dividing the resulting expression by ρ gives

(4.3.1) Now

using the chain rule for differentiation of a function of several variables. Since we assume that the velocity V is always in the x direction, the velocity

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components and are zero and . Thus (4.3.1) now takes the form

(4.3.2)

We now assume that the drag FD is proportional to the sum of the second partial derivatives of the velocity, that is

(4.3.3) where ν is the coefficient of viscosity. Using (4.3.3) in (4.3.2) gives the final form

(4.3.4)

and Marsden (1979).

If the pressure gradient is known and ν and ρ are also known then (4.3.4) defines a partial differential equation satisfied by the velocity V(x, y, z, t). In fact (4.3.4) is a second order partial differential equation because it involves second order partial derivatives. It is also non-linear since it contains the nonlinear term The term in parentheses on the left-hand side of (4.3.4) is called the “Laplacian” of V after the French

mathematician Laplace, and we often use the symbol (written Δ by some authors) to denote the Laplacian operator, i.e.,

Under the assumption that the walls of the tube are rigid and the pressure is the only driving force directed along the tube, the velocity does not change with position x along the tube, only with position across the tube, i.e., V depends only on y, z and t. In this case (4.3.4) simplifies and reduces to the linear equation

(4.3.5) where depends only on t.

The boundary condition to be applied here is that

(4.3.6)

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on ∂A. Notice that we cannot, as discussed above, provide an “initial” velocity profile, i.e., (4.3.7)

is not given. If (4.3.7) was known then it can be proved that (4.3.5) together with the boundary condition (4.3.6) leads to a unique solution V(y, z, t). That is, the problem (4.3.5)−(4.3.7) is said to be well posed. In the case of blood flow we seek a different type of uniqueness which says essentially that if u(y, z, t) and w(y, z, t) are two solutions of (4.3.5) with each satisfying (4.3.6), then, as t becomes large, the two flows become indistinguishable from one another. This condition may be conveniently called the Windkessel condition. To give some idea of the types of solution to be expected from (4.3.5) and (4.3.6) let us suppose the

coefficient of viscosity ν and the density ρ are constant. Suppose also that the rigid tube is a circular cylinder of radius a. Introducing polar co-ordinates we have, from Figure 4.3.2,

(4.3.8)

The lateral surface of the cylinder is then described by r=a. Using the transformation (4.3.8) and writing we can write (4.3.5) as

(4.3.9) and furthermore

(4.3.10)

If we assume the flow is axially symmetric then the velocity V is independent of θ and (4.3.9) reduces to That is

(4.3.11) Integrating (4.3.11) with respect to r gives

i.e., under the assumption that exists and is continuous at r=0 we have

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Integrating once more we arrive at

which on using the boundary condition (4.3.10) gives

(4.3.12) In order to make further progress towards a solution of (4.3.12) let us assume as a first approximation that

(4.3.13) This approximation is precisely the parabolic velocity profile well known in Poiseuille flow. As a better approximation, we substitute V0 under the integral sign in (4.3.12) to arrive at the next approximation Repeating this idea we arrive at the iterative method:

(4.3.14) If this scheme converges as n→∞, then we obtain a solution to the given problem. Of course this is not the only possible solution; indeed the method defined by (4.3.14) will in general only converge if is sufficiently small. If this is not the case then we must seek solutions by other means. Thus, for example, it may be possible to use the method of separation of variables as discussed in Chapter 11.

Let us now return to the problem of pulsating blood flow in a section of the trunk of the aorta. The model governing the velocity V is again (4.3.4) but now we cannot neglect the non-linear term . In addition, the situation is further complicated by the fact that both ρ and ν depend both on position and time t; also due to the elasticity of the aorta and the pulsating nature of the flow, the boundary ∂A of the tube is time dependent and may also vary with position. Thus in summary, we are asked to solve a non-linear partial differential equation subject to a moving boundary constraint, a problem well beyond the scope of this book. Finally we should not forget the Windkessel uniqueness requirement.

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