Poiseuille's Law: Definition, Statement, Laminar Flow, Viscosity, Derivation and Formula

Poiseuille's Law Overview

You may wonder how fluid flows through a pipe or tube. The complete explanation lies in Poiseuille's law. As an engineer or student of fluid dynamics, understanding this fundamental law of fluid flow is essential. Poiseuille's law describes the pressure-driven flow of an incompressible fluid through a pipe with constant circular cross-section. It relates the flow rate, viscosity, and pressure difference between two points. With Poiseuille's law, you can calculate the flow rate and pressure loss in a pipe for any fluid. This article provides a comprehensive overview of Poiseuille's law, including the equation, variables, and applications. You will learn how to apply this important law in calculations and understand the factors affecting fluid flow in pipes.

What Is Poiseuille's Law?

Poiseuille's Law describes the relationship between the factors that affect the flow of fluid through a pipe. According to this law, the flow rate of an incompressible fluid through a pipe is directly proportional to the pressure difference across the pipe and the fourth power of the pipe's radius. It is inversely proportional to the length of the pipe and the viscosity of the fluid.

Mathematically, Poiseuille's Law can be expressed as:

Q = (πr4ΔP)/(8ηL)

Where Q is the volumetric flow rate, r is the radius of the pipe, ΔP is the pressure difference across the pipe, η is the dynamic viscosity of the fluid, and L is the length of the pipe.

This law shows that flow rate can be increased by increasing the pressure difference, increasing the radius of the pipe, decreasing the length of the pipe, or decreasing the viscosity of the fluid. It highlights how even small changes in these parameters can have a significant impact on flow rate due to the radius being raised to the fourth power.

Poiseuille's Law applies to laminar flow of incompressible fluids in long pipes of constant circular cross-section. It provides an important theoretical basis for understanding fluid flow and has many applications in mechanical and biomedical engineering.

How Poiseuille's Law Works: The Equation

Poiseuille's Law describes the relationship between the pressure drop (ΔP) and flow rate (Q) of an incompressible fluid through a pipe. The equation is:

ΔP = 8μLQ/πr^4

Where,

• μ is the dynamic viscosity of the fluid 
• L is the length of the pipe
• r is the radius of the pipe

This equation demonstrates that flow rate is directly proportional to the pressure drop along a pipe and inversely proportional to both fluid viscosity and pipe length. Flow rate increases with higher pressure drop, lower viscosity, and shorter pipe length.

Conversely, as pipe radius increases, flow rate also increases due to a decrease in resistance. Resistance is inversely proportional to the fourth power of the pipe radius. So, doubling the radius results in a 16-fold decrease in resistance.

What is a laminar flow?

A laminar flow refers to the smooth, steady flow of a fluid in parallel layers with no disruption between the layers. In a laminar flow, the fluid particles move in smooth paths parallel to each other. The flow velocity remains constant with no mixing of fluid across the layers. Laminar flow occurs at lower velocities, where the viscous forces in the fluid dominate the inertial forces. The Reynolds number, which is the ratio of inertial forces to viscous forces, is less than 2000 for laminar flow.

Some key characteristics of laminar flow are:

• Smooth and constant fluid motion
• Parallel streamlines
• No mixing of fluid across layers
• Constant flow velocity across the cross-section
• Governed by viscous forces.
• Low Reynolds number

Laminar flow is highly ordered and predictable. It allows for simplified analysis using Poiseuille's law. In contrast, turbulent flow is chaotic, irregular, and difficult to predict or model mathematically. Understanding the difference between laminar and turbulent flow is crucial in the study of fluid dynamics and applications like the Poiseuille's law.

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What do you mean by viscosity?

Viscosity refers to a fluid's resistance to flow. In other words, it is a measure of a fluid's thickness or internal friction. The more viscous a fluid is, the more slowly it flows. Viscosity depends on the strength of intermolecular forces in a fluid which determine how easily its molecules slide past each other.

Some factors that affect a fluid's viscosity include:

• Temperature: Viscosity decreases with increasing temperature as the molecules move faster, reducing intermolecular forces.
• Molecular weight: Fluids with higher molecular weights typically have higher viscosities as their molecules are larger and interact more.
• Branching: Fluids with more branched molecules often have higher viscosities due to increased intermolecular interactions.
• Polarity: Polar fluids typically have higher viscosities than nonpolar fluids due to stronger intermolecular forces like hydrogen bonds.

The viscosity of a fluid is measured using a viscometer and is reported in units of poise (P) or centipoise (cP) where 1 P = 0.1 Pa∙s. Water has a viscosity of 1 cP at room temperature while honey can have a viscosity of 10,000 cP or higher. Understanding how viscosity varies with different factors and materials is important for applications like lubrication, transport, and coating flows.

Poiseuille’s Law statement

Poiseuille's Law states the flow rate of an incompressible fluid in a pipe is directly related to the pressure difference and the fourth power of the pipe's radius while being inversely related to pipe length and fluid viscosity.

Equation: Q = (πr^4ΔP) / 8μL

Where,

• Q: Volume flow rate
• r: Pipe radius
• ΔP: Pressure difference between the two ends of the pipe
• μ: Dynamic viscosity of the fluid
• L: Pipe length

Flow rate (Q) can be increased by:

• Increasing pressure difference (ΔP)

• Increasing pipe radius (r)
• Decreasing pipe length (L)
• Decreasing fluid viscosity (μ)

Flow rate sensitivity to changes in pipe radius:

• Flow rate is highly sensitive to changes in the pipe radius due to the fourth power relationship in the equation.
• Doubling the pipe radius leads to a 16-fold increase in the flow rate.

In summary, Poiseuille's Law quantitatively relates the physical properties of a fluid flowing through a pipe to its flow rate. It enables the calculation of laminar flow when relevant values are known and input into the formula.

In summary, Poiseuille's Law quantitatively relates the physical properties of a fluid flowing through a pipe to its flow rate. It enables the calculation of laminar flow when relevant values are known and input into the formula.In summary, Poiseuille's Law quantitatively relates the physical properties of a fluid flowing through a pipe to its flow rate. It enables the calculation of laminar flow when relevant values are known and input into the formula.In summary, Poiseuille's Law quantitatively relates the physical properties of a fluid flowing through a pipe to its flow rate. It enables the calculation of laminar flow when relevant values are known and input into the formula.In summary, Poiseuille's Law quantitatively relates the physical properties of a fluid flowing through a pipe to its flow rate. It enables the calculation of laminar flow when relevant values are known and input into the formula.

Poiseuille's Law Derivation

Poiseuille's law shows the relationship between flow rate (Q), pressure difference (ΔP), fluid viscosity (μ), and tube radius (r) and length (l). It can be expressed mathematically as:

Q = (πr4ΔP) / (8μl)

Consider a horizontal, cylindrical pipe through which an incompressible fluid is flowing steadily under a constant pressure difference between the two ends of the pipe. We want to find an expression for the flow rate of the fluid through the pipe.

1. Begin with the definition of flow rate: Flow rate (Q) = Volume of fluid passing a given point per unit time
2. Assume the fluid velocity varies with the radial distance from the center of the pipe (r). This variation is a result of the no-slip condition, which states that the fluid velocity is zero at the pipe's walls and maximum at the center of the pipe.
3. Now, consider a differential element of the fluid within the pipe at a radial distance (r) and with a thickness (dr). The volume of this differential element is given by: dV = A dr where A is the cross-sectional area of the pipe at radial distance (r).
4. The flow velocity at this radial distance (r) is denoted as v(r). Thus, the flow rate (Q) passing through the differential element is given by: dQ = v(r) dV
5. We know that flow velocity (v) can be expressed as the volumetric flow rate per unit cross-sectional area (A): v(r) = Q / A
6. Substituting the expression for v(r) in terms of Q / A into the equation (4), we get: dQ = (Q / A) * A dr dQ = Q dr
7. Integrate both sides of the equation (6) over the entire cross-sectional area (A) of the pipe, from the center (r = 0) to the pipe's radius (r = R): ∫dQ = Q ∫dr Q = ∫dr Q = ∫(r^0)^(r=R) dr Q = ∫(r^0)^(r=R) dr
8. Integrate ∫dr with respect to r from r = 0 to r = R: Q = [r^2/2]_(0 to R) Q = R^2/2 - 0^2/2 Q = R^2/2
9. Now, we need to consider the pressure drop (ΔP) across the pipe. According to Bernoulli's principle, the pressure difference is related to the fluid velocity: ΔP = (1/2)ρv^2
10. We know that flow velocity (v) at any radial distance (r) is given by: v(r) = Q / A where A is the cross-sectional area of the pipe at radial distance (r).
11. Substituting the expression for v(r) in terms of Q / A into the equation (10), we get: ΔP = (1/2)ρ(Q / A)^2 ΔP = (1/2)ρ(Q^2 / πr^2)^2
12. Rearrange the equation (11) to solve for Q: Q^2 = (2ΔP) / ρ * (πr^2)^2 Q^2 = (2ΔP * π^2 * r^4) / ρ Q = √((2ΔP * π^2 * r^4) / ρ)
13. Now, we have an expression for the flow rate (Q) in terms of the pressure difference (ΔP), fluid density (ρ), and pipe radius (r). To find the final form of Poiseuille's Law, we need to introduce viscosity (μ) into the equation.
14. The viscous force (F_v) acting on the fluid inside the pipe can be given by the Hagen-Poiseuille equation as follows: F_v = -πr^3 * ΔP / 8L
15. This force is balanced by the viscous drag (F_d) exerted on the fluid when it flows through the pipe: F_d = 6πrμv
16. Equate the two forces (F_v and F_d) to find the relationship between flow velocity (v) and the pressure drop (ΔP): -πr^3 * ΔP / 8L = 6πrμv
17. Solve for v: v = (-ΔP * r^2) / (8μL)
18. Now, substitute this expression for v back into the equation for Q from step 12: Q = √((2ΔP * π^2 * r^4) / ρ) Q = √((2ΔP * π^2 * r^4) / ρ) * (-ΔP * r^2) / (8μL)
19. Simplify and rearrange the equation: Q = (πr^4ΔP) / 8μL

This is the final form of Poiseuille's Law, which describes the relationship between the flow rate (Q) of an incompressible fluid through a pipe and the pressure difference (ΔP), pipe radius (r), pipe length (L), and dynamic viscosity of the fluid (μ).

Poiseuille's Law Formula & Equation

Poiseuille's law formula relates the flow rate (F) of an incompressible fluid through a pipe to factors like pressure difference (ΔP), pipe radius (r), pipe length (l), and fluid viscosity (η). The equation is:

F = (πr4ΔP) / (8ηl)

Where π is the mathematical constant (approximately 3.14), r is the radius of the pipe, ΔP is the pressure difference between the two ends of the pipe, η is the dynamic viscosity of the fluid, and l is the length of the pipe.

This law states that the flow rate is directly proportional to the pressure difference and the fourth power of the radius, but inversely proportional to the length of the pipe and the viscosity of the fluid. In other words, increasing the pressure difference, the radius of the pipe, or decreasing the length of the pipe or the viscosity of the fluid will increase the flow rate.

Some key points about Poiseuille’s law:

• It applies to steady, laminar flow of incompressible fluids in a pipe.
• The flow must be fully developed - the velocity profile across any cross-section is parabolic.
• It does not apply to turbulent flow.
• The law can be used to determine the dimensions of pipes required to achieve a desired flow rate.
• It helps calculate the resistance to flow in a pipe.
• The law is very useful in fields like fluid engineering, chemical engineering, and biomedical engineering.

Poiseuille’s law is a fundamental principle in fluid dynamics with many practical applications. Understanding this concept is important for optimizing flow systems and calculating pressure drops in pipes or tubes.

Factors That Affect Fluid Flow According to Poiseuille's Law

According to Poiseuille's law, several factors affect the flow of fluid through a pipe:

1. The viscosity of the fluid is inversely proportional to its flow rate. More viscous fluids, like honey, flow more slowly than less viscous fluids, such as water. The greater the viscosity, the slower the fluid will flow through the pipe.
2. The length of the pipe also impacts the flow rate. The longer the pipe, the slower the fluid will flow through it due to greater friction between the fluid and the pipe walls.
3. The radius of the pipe is directly proportional to the flow rate. The wider the pipe, the faster the fluid can flow through it. Doubling the radius of a pipe increases its cross-sectional area by a factor of four, allowing much more fluid to pass through.
4. The pressure difference between the two ends of the pipe drives the fluid flow. The greater the pressure difference, the faster the fluid will flow to equalize the pressure. Increased pressure provides more energy to overcome friction and viscosity, enabling higher flow rates.
5. These primary factors—viscosity, length, radius, and pressure—determine the flow of fluid through a pipe according to Poiseuille's law. By understanding how each component influences flow rate, we can optimize fluid flow systems and analyze how fluids will travel through different pipe configurations.

Real-World Applications of Poiseuille's Law

Practical Applications of Poiseuille's Law:

Engineering Applications:

1. Determine Liquid Flow: Poiseuille's Law is used in engineering to calculate the flow of liquids through pipes.
2. Size Piping Systems: Engineers rely on the law to properly size piping systems by considering the impact of pressure gradient, viscosity, and pipe diameter on flow rate.
3. Flow Rate Calculation: By knowing the relevant parameters, engineers can calculate flow rates and ensure efficient piping for various applications.

Medical Applications:

Understanding Blood Flow: Poiseuille's Law is essential in the medical field to comprehend blood flow through the circulatory system.

1. Impact of Stenosis: The law reveals how narrowing of blood vessels (stenosis) reduces flow rate and affects oxygen delivery in the body.
2. Diagnosing Conditions: Doctors use the law to diagnose conditions like atherosclerosis, where plaque buildup narrows arteries and alters blood flow.
3. Design of Medical Devices: Poiseuille's Law guides the design of medical devices like IV tubes, catheters, and cannulas, ensuring controlled flow of fluids and medications into patients.