Question

Consider the steady, incompressible blood flow through the following vascular network: 100 µm diameter which is 80 mm in length to the first branching point, which can be estimated as a standard tee (flow through run), the branch off the tee turns 45 degrees and undergoes a constriction to 65 µm diameter and the flow through run continues for another 80 mm. The velocity of the blood into the 100 µm diameter vessel is 100 mm/s and 20% of the mass flow rate exits via the branch.

a) What are the velocities of both the run and branch?

b) What is the change in pressure from the inlet to the end of the run?

c) What length would the branch need to be to achieve the same pressure drop?

d) What assumptions did you make in completing this problem.

Answer 1

a) We can consider the density of the blood very close to the density of water, then the mass flow rate will equal the volumetric flow rate.

The velocity of the run will be:

The velocity of the branch will be:

b) First we must determine the Re number:

This means that the flow is laminar. For laminar flow, the pressure drop is given by:

c) The pressure drop for the branch includes minor losses due to the constriction:

Where Kc is a coefficient related to the ratio of the areas after and before the constriction:

For this value te coefficient is close to 0.1. Replacing:

If we solve for L we get:

d) We assumed the blood has the same properties than water and that the veins are tubular pipes