Answer:
7270.7 Pa
Explanation:
g = Acceleration due to gravity = 9.8 m/s²
[tex]\rho[/tex] = Density of liquid = 1025 kg/m³
h = Height = 90 cm
[tex]P_a[/tex] = Pressure = [tex]\rho gh[/tex]
[tex]\eta[/tex] = Viscosity = 0.001 Pas
l = Length of needle = 2 cm
r = Radius of needle = 0.2 mm
P = Absolute pressure
Q = Flow rate = [tex]0.2\times \frac{10^{-3}}{3600}=5.56\times 10^{-8}\ m^3/s[/tex]
From Poiseuille's equation we have
[tex]Q=\frac{P\pi r^4}{8\eta l}\\\Rightarrow P=\frac{8Ql\eta}{\pi r^4}\\\Rightarrow P=\frac{8\times 5.56\times 10^{-8}\times 0.02\times 0.001}{\pi (0.2\times 10^{-3})^4}\\\Rightarrow P=1769.8\ Pa[/tex]
This pressure is the absolute pressure
[tex]P_{ab}=P_g+P_a\\\Rightarrow P_g=P_{ab}-P_{a}\\\Rightarrow P_g=1769.8-1025\times 9.81\times 0.9\\\Rightarrow P_g=-7270.7\ Pa[/tex]
The average gauge pressure inside the vein where the needle is is 7270.7 Pa (ignoring the negative sign)
