## Derivation of Hagen Poiseuille's Equation

The law of Poiseuille states that the flow of liquid through the narrow tube depends on the length of the tube (L), the radius of the tube (r), pressure gradient (∆P), and the viscosity of the fluid (η).

According to Poiseuille's equation derivation, the flow rate of the liquid through the narrow tube is directly proportional to the radius (r), pressure gradient (∆P) between the two points. And inversely proportional to the viscosity of the fluid (η) and length of the tube (l).

Derivation of Poiseuille's equation Q = ΔPπr4 / 8ηl

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The Pressure Gradient (∆P) denotes the pressure differential between the two points of the tube. P1 represents the high pressure in the tube and P2 represents the low-pressure area in the tube. So, the liquid flow rate can be calculated by the ∆P = P1-P2.

The radius of the narrow tube will change the pressure of the liquid flowing through it. The viscosity of the liquid will also change the liquid flow rate. The length of the narrow tube will increase or decrease the liquid flow rate. So, the derivation of poiseuille’s equation can be determined through all the above elements.

The Resistance (R) = 8Ln / πr4

The Poiseuille’s law flow rate Q = (ΔP) R

### SI Unit of Dynamic Viscosity

The SI unit of the dynamic viscosity η is Pascal-second (Pa-s). The viscosity of the liquid is dependent on the force (N) per unit area (M2) divided by the rate of shear (s-1).

### Application of Poiseuille's Law Formula

The Hagen-Poiseuille equation can be used to study the fluid feeding by insects that are sucking (haustellate) through mouthparts.

Can derive Poiseuille's equation to determine the blood flow through the veins in the body.

By using the poiseuille's equation it is possible to describe the mineral melt motion in mineral fibre production.

The Poiseuille's equation can apply to the flow of liquid through the drinking straw.

### Example Problem

Example: The blood flows through a large vein of radius 3.5 mm. And the length of the vein is found to be 20 cm long. The pressure across the ends of the vein is 480 Pa-s. Then calculate the average blood flow through the vein?

Solution:

The viscosity of the blood η = 0.0027 N .s/m2

The radius of the vein = 3.5 mm

Length of the vein (L)= 20 cm

The difference of pressure between two ends of the vein = 480 Pa (P_{1} – P_{2})

The average blood flow through the vein is

Q = ΔPπr4 / 8ηl

Q = (480 × 0.0035×0.0035×0.0035 × 3.14)/(8 × 0.0027 × 0.20)

The average blood flow through the vein is 1.4958611 m / s.

Q1. What Does Poiseuille's Law Explain?

Ans: Poiseuille's equation for viscosity used to calculate the viscosity of the liquid through the velocity of the fluid flow through a narrow tube, which varies directly with the pressure and the fourth power of the radius of the tube and inversely proportional to the length of the tube and coefficient of viscosity.

Poiseuille's law formula V = πpr^{4}/8lη

Q2. What are the Units of Poiseuille's Law?

Ans: Derivation of poiseuille's equation can determine the liquid’s viscosity (V) flows through the cylindrical tube. The unit of viscosity is (N/m^{2})s or Pascal Second (Pa⋅s). The flow of liquid is proportional to the difference between the pressure of two points and inversely proportional to resistance Q=(P_{2}−P_{1}) / R. For the laminar flow of liquid in a tube, Poiseuille’s equation resistance states that R=8ηlπr^{4}.