An artery branches into two smaller veins, identical arteries as shown in the diagram. The largest artery has a diameter of 10 mm. The two branches have the same diameter = 5 mm. The average speed of the blood in the main artery is given in the diagram. The blood has a density of 1004 kg/m^3.
Calculate v, the average blood flow speed in the two branches of smaller radius.
Main artery Vo = 0.50 m/s before it branches out into two veins.
To calculate the average blood flow speed in the two branches of smaller radius, we can use the principle of conservation of mass. According to this principle, the mass flow rate entering a section of a pipe must be equal to the mass flow rate leaving that section.
In this case, since the blood is incompressible, the mass flow rate is constant throughout the artery system.
The mass flow rate can be calculated using the equation:
mass flow rate = density × cross-sectional area × velocity
In the main artery, the cross-sectional area is given by:
A1 = πr1^2
where r1 is the radius of the main artery.
In the two branches, the cross-sectional area is given by:
A2 = πr2^2
where r2 is the radius of the two branches.
Since the mass flow rate is constant, we can equate the mass flow rate in the main artery to the sum of the mass flow rates in the two branches:
mass flow rate in main artery = mass flow rate in first branch + mass flow rate in second branch
or
density × A1 × V1 = density × A2 × V2 + density × A2 × V2
where V1 is the average blood flow speed in the main artery, and V2 is the average blood flow speed in the smaller branches.
Since the two branches have the same radius, we can simplify the equation to:
A1 × V1 = 2 × A2 × V2
Now, let's substitute the values given in the problem:
A1 = π(10 mm)^2 = 100π mm^2
A2 = π(5 mm)^2 = 25π mm^2
V1 = 0.50 m/s
Converting the cross-sectional areas to m^2:
A1 = 100π × 10^-6 m^2
A2 = 25π × 10^-6 m^2
Substituting these values into the equation, we get:
100π × 10^-6 × 0.50 = 2 × 25π × 10^-6 × V2
Simplifying the equation further:
0.05π = 0.05π × V2
Dividing both sides by 0.05π:
V2 = 1 m/s
Therefore, the average blood flow speed in the two branches of smaller radius is 1 m/s.