Flow in bifurcations

Nico A.M. Schellart, Dept. of Med. Physics, AMC

Principle

Fig. 1 Bifurcation

As Fig. 1 shows, a bifurcation changes the circular symmetric parabolic fluid flow (of a liquid or gas) in a circular mother tube to a skewed profile in the daughter branches. The radius ratio of mother and daughter branches, r_d/r_m, is energetically optimal for a ratio of 2^–1/3 = 0.79. This gives a daughters/mother area ratio of 1.25. For a symmetric bifurcation a semi-bifurcation angle a of 37.5⁰ (=arctan(2^–1/3)) is optimal. For this angle the resistance increase is only some 10%.

Fig. 2 Model of the flow profile of air in the trachea bifurcation

Application

Vascular and airways flow, and flow in the kidney. They play an important role in the study of the origination and grow of arteriosclerosis and aneurisms.

In the airways tree of the lungs some 26 generations (orders) of bifurcations, from trachea to the most remote alveoli can be distinguished. In mammals, the bifurcation ratio r_d/r_m, is on average 0.78 and the bifurcation angle 67 ^o, 8^o less than the optimum. Human midrange generations show higher angles: 79 ^o. The two semi-bifurcation angles are more similar for higher generations. From the 2^nd to the 10^th generation the human bifurcation ratio r_d/r_m is very close to 0.79, but for higher generations the ratio slowly increases and the highest generations hardly show a diameter decrease from mother to daughter. The first generation bifurcation shows daughter diameters which are much smaller than the optimum. However, above optimizations holds for Poisseuille flow (see Poisseuille’s Law), and this does not hold for the human trachea.

In the vascular system of mammals an averaged value of 0.68 is found for r_d/r_m. In the myocardium, r_d/r_m goes from 0.79 (the optimum) for the capillaries to about 0.72 for arteries of some mm diameter. Nevertheless, the optimization is poor since the variation is very large with many much too small or large daughters. Also the bifurcations are rather asymmetric, going from 0.8 (= r_{small daughter}/r_{large daughter}) of capillaries to about 0.2 of mm-sized arteries. For energetic reasons, asymmetry in diameter is accompanied by asymmetry in the semi-bifurcation angle. The human brain the mean bifurcation angle is 74^o with tens of degrees of variation. Branching angles in muscles can deviate substantial from their fluid dynamic minimum cost optimum. Classically, minimum cost according to Murray’s Law is obtained when:

(r_d1/r_m)³+ (r_d2/r_m)³ = 1.

With r_d1 = r_d1 the smallest minimum (with the ratio 0.79) is obtained. This solution is about the same as that on the basis of calculations with the Womersley number for laminar flow with minimal local wave reflection as criterion. For a given radius ratio, the optimal angles can be calculated.

The experimentally obtained exponents of Murray’s Law range from 2.5 to 2.8.

More info

Fig. 3 illustrates in more detail the flow patters in a bifurcation.

Fig. 3 a. The dashed profile in the upper daughter branch is the velocity profile perpendicular on the plane of the bifurcation. b. the dashed line is in the plane of the bifurcation (the symmetry plane). c. Stream lines with points of separation and reattachment indicated. d Streamline of flow impinging onto the apex.

When the direction of flow is inverted, in the mother tube four secondary loops can arise (Fig. 4).

Secondary flows are characterized by a swirling, helical component superimposed on the main streamwise velocity along the tube axis, see also Flow in a bended tube). A complicating factor is wall compliance, especially of importance with pulsatile flow, another factor which makes the pattern more complicated.

Fig. 4 Inversed flow in a bifurcation.

Literature

Canals M et. al. A simple geometrical pattern for the branching distribution of the bronchial tree, useful to estimate optimality departures. Acta Biotheor. 2004;52:1-16.

Frame MD, Sarelius IH.Energy optimization and bifurcation angles in the microcirculation. Microvasc Res. 1995 Nov;50(3):301-10.

Murray CD. The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc Nat Acad Sci 12: 207-214, 1926.

Pedley TJ et. al, Gas flow and mixing in the airways, In: West, J.B. (ed.) Bioengineering Aspects of the Lung. New York: Marcel Dekker, 1977.

VanBavel E, Spaan JA. Branching patterns in the porcine coronary arterial tree. Estimation of flow heterogeneity. Circ Res. 1992;71:1200-12.

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