Blood takes about 1.25 s to pass through a 2.00 mm long capillary. If the diameter of the capillary is 5.00 x 10^-6 m and the pressure drop is 2.45 kPa, calculate the viscosity n of blood. Assume laminar flow.
We can use the Hagen-Poiseuille equation to find the viscosity of blood:
ΔP = (8 * η * L * Q) / (π * r^4)
Where:
ΔP = pressure drop = 2.45 kPa = 2.45 * 10^3 Pa
η = viscosity of blood (what we want to find)
L = length of the capillary = 2.00 mm = 2.00 * 10^-3 m
Q = flow rate = volume / time = A * v = (π * r^2) * v
r = radius of the capillary = diameter / 2 = 5.00 * 10^-6 m / 2 = 2.50 * 10^-6 m
Let's first find the flow rate Q:
Q = (π * (2.50 * 10^-6 m)^2) * v
1.25 s = 2.00 * 10^-3 m / v
v = 1.6 * 10^-3 m/s
Q = (π * (2.50 * 10^-6 m)^2) * 1.6 * 10^-3 m/s = 1.963 * 10^-12 m^3/s
Now substitute the values into the equation:
2.45 * 10^3 = (8 * η * 2.00 * 10^-3 m * 1.963 * 10^-12 m^3/s) / (π * (2.50 * 10^-6 m)^4)
2.45 * 10^3 = (8 * η * 3.926 * 10^-15) / π * (39.06 * 10^-24)
2.45 * 10^3 = (31.41 * 10^-15 * η) / 0.01227 * 10^-24
η = (2.45 * 10^3 * 0.01227 * 10^-24) / 31.41 * 10^-15
η = 0.95 * 10^-3 Pa s
η = 0.95 mPa s
Therefore, the viscosity of blood is 0.95 mPa s.