Question

A patient in the hospital is receiving a saline solution through a needle in their arm. The needle is 2.20 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 0.250 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 0.250 L/h. What is the average gauge pressure inside the vein where the needle is? Use g = 9.8 m/s^2.

Answer 1

Given:

Length of the needle, l = 2.20 cm = 0.022 m

Radius of the needle, r = 0.250 mm = 0.250 x -3 m

Density of fluid, d = 1025 kg/m3

Viscosity of fluid, ɳ = 0.0010 Pa s

Rate of flow of fluid, V = 0.250 L/h = 0.250 x 10-3 /3600 m3/s = 6.94 x 10-8 m3 /s

According to Poiseuille’s formula, the rate of flow of fluid through a needle is

            V = πP r4 /8ɳd

Difference in pressure, P = V 8ɳd/ π r4

            P = 6.94 x 10-8 x 8 x 0.0010 x 10.25/ 3.14 x (0.250 x -3 )4 = 463.9 x 105 Pa

Gauge pressure, p = P – Patm where, Patm is atmospheric pressure = 1.013 x 105 Pa

                            p = 463.9 x 105 – 1.013 x 105 = 462.8 x 105 Pa

Answer: the gauge pressure inside the vein where the needle is p = 462.8 x 105 Pa