Flow in a bended tube

Principle

A curvature increases flow resistance. This can be expressed as the curvature resistance coefficient or loss-factor ζ_c =. It is dependent on curvature angle β and the ratio of curvature radius (r_c) and tube diameter (D) r_c/D. Up to β is about 22.5 ^o there is no r_c/D dependency. At 45^o, normal values in vascular anatomy, a ratio of 10 yields a factor 0.07. With laminar flow, in the plane of the bend the flow obtains an asymmetric profile with the highest velocities in the outer bend (compare meanders of rivers). With turbulent flow the vortices are also asymmetrically distributed.

With a sharp bend over 90 with ratio 1.5, as occurs with the aorta bend, a curvature coefficient ζ of 0.17 is reached. In rest, with a cardiac output of 6 L/min and v is 0.15 m/s, Reynolds number of the aorta is about 1350, supposed the aorta is straight, there is no entrance effect (see Entrance effect and entrance Length) of the aortic valve and the flow is steady. Taking the pulsatile blood flow into account the maximal velocity is about 0.6 m/s. Taking the bending and the entrance effect (ζ at least 0.2) also into account, Re will reach a temporarily maximum of Re much higher than 1350. Although with pulsatile flow the critical Reynold’s number is higher than 2100, the flow in the aorta is supposed to be transitional, even at rest. With very high cardiac outputs (30 L/min), as occur during heavy endurance sport, the aortic flow is highly turbulent and this will proceed in the large arteries.

Application

In dynamics of blood flow through vessels and flow in the airways.

More info

In an 180⁰ strong bend, as in the aorta, and high flows, which occur during heavy exercise, there arise curling, spiralizing vortices (Dean vortices).