What is the pressure gradient (the drop of pressure per length unit) in a blood vessel with a volume flow rate of ΔV/Δt = 54 cm3/s and an inner diameter of 7 mm. Hint: the viscosity coefficient of blood in the specific case we study is ηblood= 2.7 x 10-3 Ns/m2
2476 Pa /m
To calculate the pressure gradient in a blood vessel, we need to use Poiseuille's Law, which relates the pressure gradient (∆P/∆L) to the flow rate (∆V/∆t), viscosity (η), and the dimensions of the blood vessel.
Poiseuille's Law is given by:
∆P/∆L = (8η∆V)/(πr^4)
Where:
∆P/∆L is the pressure gradient (drop of pressure per unit length)
η is the viscosity coefficient of blood
∆V/∆t is the volume flow rate
r is the radius of the blood vessel
Given:
∆V/∆t = 54 cm^3/s (volume flow rate)
d (diameter) = 7 mm = 0.7 cm (inner diameter)
ηblood = 2.7 x 10^-3 Ns/m^2 (viscosity coefficient of blood)
We need to calculate the pressure gradient (∆P/∆L).
First, we need to convert the volume flow rate from cm^3/s to m^3/s:
∆V/∆t = (54 cm^3/s) / (10^6 cm^3/m^3) = 5.4 x 10^-8 m^3/s
Next, we need to find the radius (r) of the blood vessel, which is half the diameter:
r = 0.7 cm / 2 = 0.35 cm = 0.0035 m
Now we can substitute these values into the equation to find the pressure gradient:
∆P/∆L = (8 * 2.7 x 10^-3 Ns/m^2 * 5.4 x 10^-8 m^3/s) / (π * (0.0035 m)^4)
Simplifying the equation, we get:
∆P/∆L = 8.08 x 10^5 N/m^4
Therefore, the pressure gradient in the given blood vessel is approximately 8.08 x 10^5 N/m^4.