Question

A patient in the hospital is receiving a saline solution through a needle in their arm. The needle is 2.00 cm long, horizontal, with one end in a vein in the arm, and the other end attached to a wide tube that extends down from the bag of solution, which hangs from a pole so that the fluid level is 90.0 cm above the needle. The inner radius of the needle is 0.200 mm. The top of the fluid is exposed to the atmosphere, and the flow rate of the fluid (which has a density of 1025 kg/m^3 and a viscosity of 0.0010 Pa s) through the needle is 0.200 L/h. What is the average gauge pressure inside the vein where the needle is? Use g = 9.8 m/s^2. _______ Pa.

Answer 1

Answer:

7270.7 Pa

Explanation:

g = Acceleration due to gravity = 9.8 m/s²

$\rho$ = Density of liquid = 1025 kg/m³

h = Height = 90 cm

$P_a$ = Pressure = $\rho gh$

$\eta$ = Viscosity = 0.001 Pas

l = Length of needle = 2 cm

r = Radius of needle = 0.2 mm

P = Absolute pressure

Q = Flow rate = $0.2\times \frac{10^{-3}}{3600}=5.56\times 10^{-8}\ m^3/s$

From Poiseuille's equation we have

$Q=\frac{P\pi r^4}{8\eta l}\\\Rightarrow P=\frac{8Ql\eta}{\pi r^4}\\\Rightarrow P=\frac{8\times 5.56\times 10^{-8}\times 0.02\times 0.001}{\pi (0.2\times 10^{-3})^4}\\\Rightarrow P=1769.8\ Pa$

This pressure is the absolute pressure

$P_{ab}=P_g+P_a\\\Rightarrow P_g=P_{ab}-P_{a}\\\Rightarrow P_g=1769.8-1025\times 9.81\times 0.9\\\Rightarrow P_g=-7270.7\ Pa$

The average gauge pressure inside the vein where the needle is is 7270.7 Pa (ignoring the negative sign)