What is Poiseuille's Law? 

The law of Poiseuille or Poiseuille's equation states that the pressure drop of an incompressible fluid especially a liquid in a laminar flow that passes through a cylindrical tube of length L, radius r, pressure gradient ΔP, and mainly depends on the viscosity of the fluid is nothing but the pressure difference of the layers of fluids. ΔP=P₁-P₂ 

The Poiseuille law was stated by Jean Leonard Marie Poiseuille in 1838. But initially, without proven results, few assumptions were made like the fluid follows Newton's mechanic, it is incompressible and the diameter of the cylindrical tube is uniform throughout.  

Viscosity, Laminar Flow, and Turbulent Flow

To understand the difference between the viscosity of various liquids, let us take an example of water and blood, if the water is poured from a cup then notice the rate of its flow and when thick blood is poured from the cup, notice its rate of flow, you will notice that the blood takes more time to flow than water because blood has higher viscosity value or flow rate than the water. This is due to the fluid friction between the fluid and its surroundings. In rare cases, ideal fluids are considered to have no viscosity. The viscosity depends on the laminar flow and turbulent flow.

Laminar flow (or streamline flow) is the ease of flow of liquid in multiple layers that usually do not mix together and each layer moves smoothly past the adjacent layers while the turbulent flow is a fluid motion contrasting to the laminar flow where the fluid flows with less ease undergoing irregular fluctuation in parallel layers without any kind of disruption within the layers. Turbulent flow undergoes an instant variation of pressure and flow velocity in space and time. The flow is proportional to the pressure difference and inversely proportional to the resistance in the fluid.

An example of a laminar flow is the flow of air over an aircraft wing and a common example of turbulent flow or turbulence is smoke coming from a burnt cigarette where the initial flow is laminar but after few seconds it comes turbulent due to the pressure difference in the air. When turbulence occurs, that means the layers of the fluid are mixed and they flow with a velocity and direction, and the path they flow is known as streamlines. While the fluid flows, due to the obstruction and high-speed variation of the fluid, the turbulence is caused and the fluid swirls in eddies.   

Experimental Setup and Poiseuille's Equation 

To understand the viscosity of the fluid, let's set an experiment where it consists of two parallel plates which have specific fluid between them. Let the plates be placed one below the other. The bottom plate is fixed and the top plate is moved to the right, which drags the fluid along. The lamina of the fluid touches one of the plates. The uppermost layer moves with a velocity ‘v’, as the bottom most layer does not move. The top lamina exerts a force on the bottom, where there is a fluctuation of speed. Make sure the flow rate remains laminar. Since the fluid does not have any shear strength, the top and bottom layer undergoes shear deformation. So there is a force exerted in the top lamina which moves at a constant velocity hence this force is proportional to the velocity (v), cross-sectional area of the plate (A), and inversely proportional to the square of the distance between both the plates (L). The velocity is proportional to the force until its speed is maximum till the turbulence occurs. This force is also proportional to the coefficient of the viscosity (η).

Hence it is mathematically represented as : 

F=η vA/L.

This force is the viscosity of the liquid and its corresponding SI unit is (N/m²)s or Pa.s. 

The laminar flow which is confined to tubes indulges the pressure difference and tells about the proportionality of the flow and pressure. The flow equation for a viscous fluid can be represented as the flow rate, Q=P₂ - P₁/R, where ‘R’ is the resistance to airflow. So, the greater the pressure difference, the higher the flow rate of the fluid is.  

Turbulence increase the R, since R is high for a long-sized tube and when R is high, the viscosity of the fluid is also high. For a viscous liquid, the spread is high at the midstream due to the drag force at the boundaries of the cylindrical tube and the effect of viscosity changes as per the cross-section of the cylindrical tube especially its radius. So the laminar flow is denoted as R and fluid's viscosity is denoted as η, the length of the cylinder is L and the radius is r. So this laminar flow rate is inversely proportional to the fourth power of the radius. 

Thus we can write 

R=8ηl/πr⁴ 

The π and the radius's fourth power magnitude is the square of the area of the cross-section of the cylindrical tube. Hence the formula can also be R=8ηl/A² 

Hence this is the Poiseuille equation and explains the relation of the pressure difference and the viscosity of the fluid. 

Now from this, we can also deduce the laminar flow rate. Q= (ΔP)A²/8ηl or Q=(P₂ - P₁)πr⁴/8ηl 

Application of Poiseuille's Law 

1) The Poiseuille law is used vastly in the medical fields, especially it helps us understand the blood pressure of our body, used in predicting the vascular resistance and flow rate of the intravenous fluid. 

2) It is used in analyzing the viscosity of various fluids, especially liquids used for chemical analysis, DNA testing, and fuel for automobile engines 

3) It is used in analyzing and designing the internal combustion engine, gas turbines, and the external flow rate of the motors used in heavy vehicles or equipment. 

4) It is used in space vehicles, design of the spacecraft's configuration, and astronaut's types of equipment. 

Practice Problems 

1) The blood flow through a large artery of a radius of 2.5mm is 20 cm long. The blood pressure across the artery ends is 380 Pascals. Determine the blood's average speed. 

Answer: It is given that the change in the pressure is 380 Pascals, the radius is 2.5 mm and length is 20 cm. The average viscosity of the blood is 0.0027 Ns/m² .The formula used to calculate the average speed is

$Q=\frac{\Delta P\pi r^4}{8\eta L}$

Substitute 2.5 mm (0.0025 m) for radius, 20 cm (0.2 m) for length, 380 Pascals for pressure and 0.0027 Ns/m²

for viscosity to calculate the value of average speed.

2) Let's assume that the blood flow rate is reduced to half its normal value. By what factor has the radius of the artery reduced?. Neglect the turbulence and the answer can be in percentage also. 

Answer:  If all the factors are considered to be constant, the equation that is used is

$\frac{Q_1}{r_1^4}=\frac{Q_2}{r_2^4}$

If the flow rate is reduced by half, put the value of final flow rate to be equal to the half of the initial flow rate and develop the relation between the radii under different flows.  

Context and Applications    

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for  

• Bachelors in Science Physics 
• Masters in Science Physics 
• Bachelor of Engineering