An intravenous (IV) system is supplying saline solution to a patient at the rate of 0.06 cm3/s through a needle of radius 0.2 mm and length 6.32 cm.

What gauge pressure (in Pa) is needed at the entrance of the needle to cause this flow?
Assume that the viscosity of the saline solution to be the same as that of water, η = 1.0*10-3 Pa-s, and that the gauge pressure of the blood in the vein is 1500 Pa.

Enter an integer

The gauge pressure (in Pa) needed at the entrance of the needle to cause this flow is; 7492 Pa

We are given;

Rate of flow; Q' = 0.06 cm³/s = 0.06 × 10⁻⁶ m³/s

Radius; r = 0.2 mm = 0.0002 m

Length; L = 6.32 cm = 0.0632 m

viscosity of the saline solution; η = 1 × 10⁻³ Pa.s

Gauge Pressure; P = 1500 Pa

According to Hagen–Poiseuille's law, the equation for Flow rate of liquid through a cylindrical pipe is;

Q' = ΔP*πr⁴/(8ηL)

ΔP is the pressure difference between the two ends

Plugging in the relevant values gives;

0.06 × 10⁻⁶ = ΔP(π * 0.0002⁴)/(8 * 1 × 10⁻³ * 0.0632)

ΔP = [0.06 × 10⁻⁶ × (8 * 1 × 10⁻³ * 0.0632)]/(π * 0.0002⁴)

ΔP = 0.030336 × 10⁻⁹/(5.0625 × 10⁻¹⁵)

ΔP = 5992 Pa

Thus;

gauge pressure needed at the entrance of the needle to cause this flow is;

5992 Pa + 1500Pa = 7492 Pa

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Pressure applied to the needle is 7528 Pa

Explanation:

As we know by poiseuille's law of flow of liquid through a cylindrical pipe

the rate of flow through the pipe is given as

now we know that

radius = 0.2 mm

Length = 6.32 cm

now we have

now we have