What is the pressure gradient (the drop of pressure per length unit) in a blood vessel with a volume flow rate of ÄV/Ät = 35 cm3/s and an inner diameter of 15 mm. Hint: the viscosity coefficient of blood in the specific case we study is çblood= 2.7 x 10-3 Ns/m2
From the volume flow rate (Q = 35 cm^3/s)and the vessel cross sectional area (A = 1.767 cm^2), get the average flow velocity:
V = Q/A = 19.8 cm/s
In cgs units, the viscosity is 0.027 g/cm*s
The Reynolds number is
Re = (rho*D*V/(viscosity)
= (1.00 g/cm^3*1.5 cm*19.8 cm/s)/0.027 g/cm*s)
= 1100 (dimensionless)
There will be laminar flow at this Reynolds number.
Pressure drop per unit length at this Reynolds number is
dP/dL = f*(1/2)*(rho)*V^2/D
where f = 64/Re = 0.058
Now you can go ahead and calculate the pressure drop per unit length
This is standard pipe flow engineering; you should have been exposed to it by now. Sometimes it is called Hagen-Poiseuille flow.
http://www.fas.harvard.edu/~scdiroff/lds/NewtonianMechanics/PoiseuillesLaw/PoiseuillesLaw.html flow.
It is important to calculate the Reynolds number first to make sure that the flow is laminar and not turbulent.
To find the pressure gradient in a blood vessel, you can use the Hagen-Poiseuille equation, which relates the pressure gradient to the flow rate and the properties of the blood vessel.
The Hagen-Poiseuille equation is given by:
ΔP = (8ηLQ) / (πr^4)
Where:
ΔP = pressure gradient (drop of pressure per length unit)
η = viscosity coefficient of blood (2.7 x 10^-3 Ns/m^2 in this case)
L = length of the blood vessel
Q = volume flow rate (ÄV/Ät = 35 cm^3/s)
r = inner radius of the blood vessel
But before using this equation, we need to convert the given values to the appropriate units.
1. Convert the flow rate from cm^3/s to m^3/s:
35 cm^3/s = 35 x 10^-6 m^3/s
2. Convert the inner diameter to radius:
Inner diameter = 15 mm
Radius = (15 mm) / 2 = 7.5 mm = 7.5 x 10^-3 m
Now, we can plug the values into the Hagen-Poiseuille equation:
ΔP = (8 × (2.7 x 10^-3 Ns/m^2) × L × (35 x 10^-6 m^3/s)) / (π × (7.5 × 10^-3 m)^4)
Simplifying further, we get:
ΔP = (8 × 2.7 x L x 35) / (π × (7.5)^4)
Note: We still need to know the length of the blood vessel (L) to calculate the pressure gradient accurately. Without that information, we cannot calculate the specific pressure gradient.