# Poiseuilles Law Formula

The Poiseuille’s law states that the flow of liquid depends on following factors like the pressure gradient (**∆P)**, the length of the narrow tube (L) of radius (r) and the viscosity of the fluid **(η) **along with relationship among them.

The entire relation or the Poiseuille’s Law formula is given by,

Q = ΔPπr4 / 8ηl

Wherein,

**The Pressure Gradient ****(∆P)** : Shows the difference in the pressure between the two ends of the tube, determined by the fact any fluid will always flow from high pressure (p1) to low pressure region(p2) and the flow rate is determined by the pressure gradient (P1 – P2)

**Radius of tube**: The liquid flow varies directly with the radius to the power 4.

**Viscosity ****(η)**: The flow of the fluid varies inversely with the viscosity of the fluid and as the viscosity of the fluid increases, the flow decreases vice versa.

**Length of the Tube (L):** The liquid flow is inversely proportional to the length of the tube, therefore longer the tube, greater is the resistance to the flow.

Resistance(R): The resistance is described by 8Ln / πr4 and therefore the Poiseuille’s law becomes

Q= (ΔP) R

**Example 1:**

**The blood flow through a large artery of radius 2.5 mm is found to be 20 cm long. The pressure across the artery ends is 380 Pa, calculate the blood’s average speed.**

**Solution:**

The blood viscosity η = 0.0027 N .s/m2

Radius = 2.5 mm

Difference of pressure = 380 Pa ( P1 – P2)

The average speed is given by, =

380 x (0.0025)2 / 8(0.0027)(0.20m)

The average speed becomes 1.6031 m / s

**The water flow through a large tube of radius 3 mm is found to be 10 cm long. The pressure across the two ends of the tube is 300 Pa, calculate the water’s average speed.**

**Solution:**

The viscosity η = 0.0027 N .s/m2

Radius = 3 mm

Difference of pressure = 300 Pa ( P1 – P2)

The average speed is given by, =

300 x (0.0003)2 / 8(0.0027)(0.10m)

The average speed becomes 0.911 m / s