Poiseuille's Law Formula: Derivation & Applications

Poiseuille’s law formula states that the flow of a fluid depends on different variables such as the radius (r) and length of the tube (L), pressure gradient (∆P), and the viscosity of the fluid (η) as per their relationship. 

• It is also known as Hagen–Poiseuille equation, Hagen–Poiseuille law, Poiseuille law or Poiseuille equation.
• The law determines the pressure drop in an incompressible fluid in laminar flow flowing through a long cylindrical pipe of the constant cross section.
• It was derived by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen.
• Poiseuille’s formula assumes that there is no acceleration of fluid in the pipe.
• It helps in designing heating, ventilation, and air conditioning systems.
• Poiseuille’s law Formula is represented as 

Q = ΔPπr⁴ / 8ηl. 

• where ∆P  is the pressure gradient
• η is the viscosity
• l is the length of the tube

Key Terms: Poiseuille's Law Formula, Pressure Gradient, Viscosity, Resistance, Speed, Narrow Tube, Cylindrical Pipe, Poiseuille Formula Derivation

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Poiseuille’s Law Formula

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Poiseuille Law Formula studied the streamlined flow of a liquid in capillary tubes. According to the equation the law concluded that the volume, V of the liquid flowing per second through a capillary tube is:

• Directly proportional to the difference of pressure, P between the two ends of the tube. i.e; V α P.
• Directly proportional to the fourth power of the radius, r of the capillary tube. i.e; V α r⁴. 
• Inversely proportional to the coefficient of viscosity, η of the liquid. i.e; V α 1/η.
• Inversely proportional to the length, l of the capillary tube. i.e; V α 1/l.

Combining these factors, Poiseuille’s Formula can be given as 

V α Pr⁴/ηl

Or

\( V = {KP r^4 \over η l}\)

• Where k = π/8, a constant of proportionality.

So, we get \( V = {\pi P r^4 \over 8η l}\)

• Where, p is the pressure difference across the two ends of the tube.
• r is the radius of the tube
• η is the coefficient of viscosity,
• l is the length of the tube

This equation is called Poiseuille’s equation, which consists of the following components.

Pressure Gradient (P) 

Here the pressure differential between the two ends of the tube is defined by the fact that every fluid will always flow from the High-Pressure Area (P1) to the Low-Pressure Area (P2). The flow rate is calculated by P = P₁ - P₂.

Radius of the Narrow Tube (r)

The flow of liquid directly changed with the radius to the power four.

Viscosity (η)

The rate of flow of the fluid is inversely proportional to the viscosity of the fluid.

Length of the Narrow Tube (L)

The rate of flow of the fluid is inversely proportional to the length of the narrow tube.

Resistance (R)

The resistance is calculated by 8Ln/πr⁴ and hence the Poiseuille’s law is Q= ΔP*R

The video below explains this:

Poiseuille's Formula Detailed Video Explanation:

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Derivation of Poiseuille’s Law Formula

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Poiseuille’s found that the volume of a liquid flowing through a capillary tube per second depends upon

• The pressure gradient (P/l) (i.e. the rate of change of pressure with length)
• The radius of the capillary tube, r
• Coefficient of the viscosity of the liquid, η

Poiseuille’s Law Formula

Therefore, V = K (P/l)^a r^b η^c …… (i)

Where K is proportionality constant.

• The dimension of V per second = [L³ T^-1]
• The dimension of P/l = [M L^-1 T^-2] / [L] = [M L^-2 T^-2]
• The dimension of r = [L]
• The dimension of η = [M L^-1 T^-1]

Putting the dimensions of the quantities in Equation (i)

we get,

[L³ T^-1] = [M L^-2 T^-2]^a [L]^b [M L^-1 T^-1]^c

or

[M⁰ L³ T^-1] = [M^a+c L^-2a+b-c T^2a-c]

• Applying the principle of homogeneity of dimensions
• we get,
• Powers of M, a+c = 0 ….......... (ii)
• Powers of L, -2a+b-c = 3 ….... (iii)
• Powers of T, -2a-c = -1 …....... (iv)
• Solving these three equations, we get a = 1, b = 4 and c = -1 and substituting the values of a, b and c in Equation (i)
• we get,
• V = K (P/l)¹ (r)⁴η^-1
• = K P r4/ η l
• The value of K is found experimentally to be π/8 and the above equation changes to

 \( V = {\pi P r^4 \over 8η l}\)

• This equation is called Poiseuille’s equation or Poiseuille’s formula. 

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Application of Poiseuille's Law Formula

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The applications of Poiseuille's formula can be given as:

• The Poiseuille law formula has been used vastly in the medical fields, especially since it helps us to understand the blood pressure of our body.
• It is used to predict the vascular resistance and flow rate of the intravenous fluid. 
• The law is used to analyze the viscosity of various fluids, especially liquids used for chemical analysis, DNA testing, and fuel for automobile engines. 
• It is used to analyze and design the internal combustion engine and gas turbines.
• The law helps determine the external flow rate of the motors used in heavy vehicles or equipment. 
• The poiseuille law formula is used in designing space vehicles, spacecraft configurations, and astronaut equipment.

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Limitations of Poiseuille Law Formula

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Poiseuille law has certain limitations when applied to circulating blood.

• Blood vessels are not rigid tubes and are quite distensible, so their size depends on the blood pressure within them as well as the contraction of smooth muscles in the vessel walls.
• Blood is a non-Newtonian fluid, and fluid viscosity is not constant.
• The flow is not steady but pulsatile in most parts of the vascular bed.

The assumptions of Poiseuille’s equation are

• A long rigid cylinder is used with a length much greater than the radius
• Fluid has constant viscosity and is incompressible
• Steady Laminar flow is not pulsatile and turbulent
• The fluid velocity at the edges of the tube is zero

Read Also:

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Things to Remember

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• Poiseuille's Law Formula is an easy way to take out the correct viscosity of the liquid that goes through a narrow tube.
• The equation uses the velocity of the fluid to calculate the pressure with which the liquid flows through the tube.
• The fourth power of the tube's radius is also a varying factor.
• One must keep in mind that there is an inverse proportionality between the velocity of the liquid and the length of the tube.
• It is also inversely proportional to the coefficient of viscosity.

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