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Mechanics

Poiseuille's Formula for Viscous Flow

Poiseuille's formula describes the laminar, steady-state flow of an incompressible Newtonian fluid through a long cylindrical pipe of constant circular cross-section. It establishes that the volume flow rate is directly proportional to the pressure gradient and the fourth power of the radius of the pipe.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Direct application of Poiseuille's formula for volume flow rate, dependence on radius, length, viscosity, and pressure difference, along with units and dimensional verification.

PGT extension

Rigorous derivation using Navier-Stokes equation in cylindrical coordinates, hydrodynamic resistance analogy to Ohm's law, and series/parallel combination of capillary tubes.

Prerequisites

Newton's Law of Viscosity
Laminar versus turbulent flow and Reynolds number
Pressure gradient in fluid flow

Nature of Flow and Velocity Profile

The fluid flow must be steady, streamline (laminar), and incompressible with negligible entrance effects.
The velocity profile across the cross-section of a cylindrical pipe is parabolic, with maximum velocity at the axis and zero velocity at the inner wall due to the no-slip condition.
The velocity v(r) at a radial distance r from the axis is given by v(r) = &frac{Delta P}{4eta l} (R² - r²).

Dependence on Physical Parameters

Volume flow rate Q is directly proportional to the pressure difference (ΔP) and the fourth power of the radius (R⁴).
Volume flow rate Q is inversely proportional to the coefficient of viscosity (η) and the length of the pipe (l).
A small change in the radius of a blood vessel or pipe leads to a massive fourth-power change in the flow rate, highlighting the sensitivity of hydrodynamic resistance.

Analogy with Electrical Circuits

Poiseuille's law can be expressed in a form analogous to Ohm's Law (I = V/R), where volume flow rate Q acts as current and pressure difference ΔP acts as potential difference.
Hydrodynamic resistance R_h of a cylindrical pipe is defined as R_h = &frac{8eta l}{pi R⁴}.
For pipes connected in series, equivalent resistance is the sum of individual resistances; for parallel connections, reciprocals add up.

Limitations and Validity Conditions

The formula is valid only for streamline (laminar) flow where the Reynolds number remains well below the critical value (typically < 2000).
The fluid must be Newtonian, meaning its viscosity remains constant regardless of the shear rate applied.
The tube must be rigid, cylindrical, and sufficiently long so that end effects and velocity profile development length can be ignored.

Derivations

Derivation of Poiseuille's Equation using Force Balance

Conditions: Steady, laminar flow of a Newtonian fluid through a horizontal cylindrical pipe of radius R and length l under a pressure difference ΔP.

Consider a cylindrical fluid element of radius r and length dx coaxial with the pipe.
The driving force due to pressure difference acting on the cross-section is (ΔP) (πr^2).
The opposing viscous drag force acting on the curved surface of the fluid cylinder is -(η)(2πr dx)(dv/dr).
Equating the driving force and viscous drag for steady uniform flow: (ΔP)(π$r^2) = -2πr η dx (dv/dr)$, leading to dv/d$r = -(ΔP/(2η l)) r$.
Integrating with the no-slip boundary condition $v = 0 at r = R gives the velocity profile v(r) = (ΔP/(4η l))(R^2 - r^2)$.
Integrating the elementary volume flux dQ = v(r) (2πr dr) from r = 0 to r = R yields the total volume flow rate.

Result: $Q = \frac{\pi (\delta P) R^4}{8 \eta l}$

Derivation of Hydrodynamic Resistance

Conditions: Analogy between fluid flow and current electricity.

Start with Poiseuille's formula: $Q = \frac{\pi R^4 \delta P}{8 \eta l}$.
Rearrange the equation to isolate the pressure difference: Δ$P = Q \left(\frac{8 \eta l}{\pi R^4}\right)$.
Compare this equation with Ohm's Law $V = I R_elec$, where V corresponds to ΔP and I corresponds to Q.
Identify the term inside the parenthesis as the hydrodynamic resistance R_h.

Result: $R_h = \frac{8 \eta l}{\pi R^4}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Poiseuille's Law (Volume Flow Rate)	$$Q = \frac{\pi \delta P R^4}{8 \eta l}$$	Q = Volume flow rate, ΔP = Pressure difference, R = Radius, η = Coefficient of viscosity, l = Length Q in m^3/s, ΔP in Pa, R and l in m, η in Pa·s	Laminar flow, Newtonian fluid, rigid cylindrical pipe
Velocity Profile in Pipe	$$v(r) = \frac{\delta P}{4 \eta l} (R^2 - r^2)$$	v(r) = Velocity at radius r, r = Radial distance from axis v in m/s, r in m	Steady laminar flow
Maximum Velocity	$$v_{max} = \frac{\delta P R^2}{4 \eta l}$$	$v_{max} = Velocity at the central axis (r = 0)$ m/s	At the axis of the pipe
Average Velocity	$$v_{avg} = \frac{Q}{\pi R^2} = \frac{\delta P R^2}{8 \eta l} = \frac{v_{max}}{2}$$	$v_{avg} = Cross-sectional average velocity$ m/s	Laminar flow in circular pipe
Hydrodynamic Resistance	$$R_h = \frac{8 \eta l}{\pi R^4}$$	$R_h = Resistance$ to fluid flow Pa·s/m^3	Constant viscosity and laminar flow
Mass Flow Rate	$$\frac{dm}{dt} = \rho Q = \frac{\pi \rho \delta P R^4}{8 \eta l}$$	ρ = Density of the fluid, dm/dt = Mass flow rate kg/s	Incompressible fluid
Series Combination of Capillary Tubes	$$R_{eq} = R_{1} + R_{2} = \frac{8eta l_1}{\pi R_1^4} + \frac{8eta l_2}{\pi R_2^4}$$	$R_{eq} = Equivalent hydrodynamic resistance$ Pa·s/m^3	Identical flow rate through connected tubes
Parallel Combination of Capillary Tubes	$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$$	$R_{eq} = Equivalent resistance$ for parallel tubes Pa·s/m^3	Same pressure difference across each tube

Graphs and experiments

Velocity Profile across Pipe Cross-section

Draw: A parabolic curve showing velocity on the horizontal axis and radial distance r (from -R to +R) on the vertical axis, with maximum velocity at r = 0 and zero at the walls.

Interpret: Illustrates that fluid layers move fastest at the center and slowest at the stationary boundaries due to viscous shearing.

Exam use: Frequently asked in conceptual questions identifying the shape of velocity distribution in laminar flow.

Volume Flow Rate Q versus Radius R

Draw: A steeply rising non-linear curve representing Q proportional to R^4.

Interpret: Demonstrates the extreme sensitivity of flow rate to the radius of the pipe.

Exam use: Numerical problem-solving where radius is doubled or halved and the resulting change in Q is calculated.

Capillary Flow Viscometer Experiment

Draw: A schematic setup showing a liquid reservoir, a horizontal capillary tube of known length and radius, a manometer or liquid column measuring pressure head, and a measuring cylinder.

Interpret: Used to experimentally determine the coefficient of viscosity (η) of a liquid by measuring Q and ΔP.

Exam use: Experimental physics viva and conceptual questions on measuring dynamic viscosity.

Solved numericals

TGT core

A liquid flows through a capillary tube of radius 1 mm and length 10 cm under a pressure head equivalent to 20 cm of water column. If the radius of the tube is doubled keeping all other parameters constant, find the ratio of the new volume flow rate to the original volume flow rate.

Poiseuille's formula states that volume flow rate Q is proportional to R^4 when ΔP, η, and l are constant.
Therefore, Q_2 / $Q_1 = (R_2 / R_1)^4$.
Given that $R_2 = 2 R_1$, substitute the ratio into the expression: Q_2 / $Q_1 = (2 R_1 / R_1)^4 = 2^4 = 16$.

Answer: 16 : 1

PGT extension

Two capillary tubes of radii r and 2r, and lengths l and 2l respectively, are connected in series. If the pressure difference across the combination is P, find the pressure drop across the first tube.

Calculate the hydrodynamic resistance of the first tube: $R_1 = 8ηl / (π r^4)$.
Calculate the hydrodynamic resistance of the second tube: $R_2 = 8η(2l) / (π (2r)^4) = 16ηl / (16π r^4) = ηl / (π r^4) = R_1 / 8$.
Equivalent resistance for series combination: $R_eq = R_1 + R_2 = R_1 + (R_1 / 8) = (9/8) R_1$.
The pressure drop across the first tube (ΔP_1) is given by Q * R_1, and since $Q = P / R_eq$, we have Δ$P_1 = P * (R_1 / R_eq) = P * (R_1 / ((9/8) R_1)) = (8/9) P$.

Answer: (8/9) P

PGT extension

Water at 20°C (viscosity η = 1.0 × 10^-3 Pa·s) flows through a pipe of radius 2.0 cm and length 2.0 m at a volume flow rate of 3.14 × 10^-4 m^3/s. Calculate the pressure difference maintaining this flow.

Write Poiseuille's formula for pressure difference: Δ$P = \frac{8 \eta l Q}{\pi R^4}$.
Substitute the given values: η = 1.0 × 10^-3 Pa·s, l = 2.0 m, $Q = 3.14 × 10^-4 m^3/s$, R = 0.02 m.
Calculate R^4: (0.02)^4 = 1.6 × 10^-7 m^4$.
Substitute into the expression: Δ$P = \frac{8 \times (1.0 \times 10^{-3}) \times 2.0 \times (3.14 \times 10^{-4})}{3.14 \times (1.6 \times 10^{-7})}$.
Simplify the numerator: 8 \times 2.0 \times 3.14 \times 10^{-7} = 50.24 \times 10^{-7}.
Simplify the denominator: 3.14 \times 1.6 \times 10^{-7} = 5.024 \times 10^{-7}.
Divide: Δ$P = 50.24 \times 10^{-7} / (5.024 \times 10^{-7}) = 10 Pa$.

Answer: 10 Pa

Assumptions and limitations

Assumption: The fluid is strictly Newtonian and incompressible. Limitation: Fails for non-Newtonian fluids like blood under certain shear rates or polymer solutions.
Assumption: Flow is completely laminar. Limitation: If velocity exceeds the critical Reynolds number, turbulence sets in and Poiseuille's law becomes invalid.
Assumption: The pipe is rigid and has a uniform circular cross-section. Limitation: Fails in elastic vessels (like arteries) that expand with pressure or pipes with varying diameters.

Common misconceptions

Common error: Volume flow rate is inversely proportional to the square of the radius.

Correct: Volume flow rate is directly proportional to the fourth power of the radius (R^4).

Integrating the parabolic velocity profile introduces an extra r^2 factor, transforming the area dependence into R^4.

Common error: Fluid velocity is uniform across the entire cross-section of the pipe.

Correct: Fluid velocity varies parabolically, being maximum at the center and zero at the walls.

The no-slip condition at the solid boundary combined with internal viscous shear creates a parabolic velocity gradient.

Common error: Poiseuille's formula can be applied directly to turbulent flow in rough pipes.

Correct: Poiseuille's formula holds exclusively for laminar flow in smooth, cylindrical pipes.

Turbulent flow involves eddies and momentum mixing, drastically altering the pressure drop versus flow rate relationship.

Quick revision

Poiseuille's formula: $Q = (π ΔP R^4) / (8 \eta l)$.
Hydrodynamic resistance: $R_h = (8 \eta l) / (π R^4)$.
Velocity profile is parabolic with $v_max = 2 v_avg$.
Flow rate depends extremely sensitively on radius as R^4.

Must remember

Volume flow rate $Q$ in Poiseuille's formula is directly proportional to the fourth power of the radius ($R^4$).
The velocity profile of a viscous fluid in a cylindrical pipe is parabolic, with maximum velocity at the axis ($v_{max} = \frac{\Delta P R^2}{4 \eta l}$) and zero at the walls.
Hydrodynamic resistance $R_h$ is defined as $R_h = \frac{8 \eta l}{\pi R^4}$, analogous to electrical resistance in Ohm's Law.
Poiseuille's formula is valid exclusively for steady, laminar, streamline flow of Newtonian fluids in rigid cylindrical pipes where the Reynolds number is less than 2000.
For capillaries connected in series, the equivalent hydrodynamic resistance is the sum of individual resistances ($R_{eq} = R_1 + R_2$).
For parallel connection of capillaries, reciprocals of hydrodynamic resistance add up ($\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$).

Glossary

Coefficient of Viscosity ($\eta$)A measure of a fluid's resistance to gradual deformation by shear or tensile stress, measured in $\text{Pa}\cdot\text{s}$.

Volume Flow Rate ($Q$)The volume of fluid passing through a cross-section per unit time, measured in $\text{m}^3/\text{s}$.

Hydrodynamic Resistance ($R_h$)The opposition to fluid flow through a pipe, expressed as $R_h = \frac{8 \eta l}{\pi R^4}$.

Parabolic Velocity ProfileThe distribution of velocities across a pipe cross-section given by $v(r) = \frac{\Delta P}{4 \eta l} (R^2 - r^2)$.

Newtonian FluidA fluid whose viscosity remains constant regardless of the shear rate applied.

Poiseuille's Law vs. Ohm's Law

Physical Property	Poiseuille's Law (Fluid Dynamics)	Ohm's Law (Electricity)
Flow / Current Driver	Pressure Difference ($\Delta P$)	Potential Difference ($V$)
Flow Rate / Current	Volume Flow Rate ($Q$)	Electric Current ($I$)
Opposition Term	Hydrodynamic Resistance ($R_h = \frac{8\eta l}{\pi R^4}$)	Electrical Resistance ($R = \rho \frac{l}{A}$)

Practice Quiz

Choose an option to reveal the explanation.

1. What is the dependence of volume flow rate $Q$ on the radius $R$ of a capillary tube in Poiseuille's formula?

Explanation: According to Poiseuille's law, $Q = \frac{\pi \Delta P R^4}{8 \eta l}$, meaning the volume flow rate is directly proportional to the fourth power of the radius ($R^4$).

2. What is the nature of the velocity profile of a viscous fluid flowing through a cylindrical pipe under laminar conditions?

Explanation: The velocity profile is given by $v(r) = \frac{\Delta P}{4 \eta l} (R^2 - r^2)$, which describes a parabola with maximum velocity at the axis ($r = 0$) and zero velocity at the wall ($r = R$).

3. If the radius of a capillary tube is doubled while keeping the pressure difference, length, and viscosity constant, by what factor does the volume flow rate change?

Explanation: Since $Q \propto R^4$, doubling the radius ($R \to 2R$) increases the flow rate by a factor of $2^4 = 16$.

4. What is the formula for hydrodynamic resistance $R_h$ of a cylindrical pipe of radius $R$, length $l$, and fluid viscosity $\eta$?

Explanation: By comparing Poiseuille's equation $Q = \frac{\Delta P}{R_h}$ with $Q = \frac{\pi \Delta P R^4}{8 \eta l}$, we get the hydrodynamic resistance $R_h = \frac{8 \eta l}{\pi R^4}$.

5. At what radial distance $r$ from the axis does the fluid velocity equal the average velocity $v_{avg}$ in a laminar pipe flow?

Explanation: Equating $v(r) = \frac{\Delta P}{4 \eta l}(R^2 - r^2)$ to $v_{avg} = \frac{\Delta P R^2}{8 \eta l}$ yields $R^2 - r^2 = \frac{R^2}{2}$, which gives $r = \frac{R}{\sqrt{2}}$.

6. Which of the following conditions is NOT required for Poiseuille's formula to hold true?

Explanation: Poiseuille's formula is valid exclusively for laminar flow where the Reynolds number is well below the critical value (typically $< 2000$). It fails for turbulent flow.

7. How is the maximum velocity ($v_{max}$) related to the average velocity ($v_{avg}$) for laminar flow in a circular pipe?

Explanation: The average velocity is given by $v_{avg} = \frac{v_{max}}{2}$, which means the maximum velocity at the axis is twice the cross-sectional average velocity.

8. If two capillary tubes of hydrodynamic resistances $R_1$ and $R_2$ are connected in series, what is the equivalent hydrodynamic resistance $R_{eq}$?

Explanation: Analogous to electrical resistance in series, hydrodynamic resistances in series add directly: $R_{eq} = R_1 + R_2$.

9. What is the SI unit of hydrodynamic resistance $R_h$?

Explanation: Since $R_h = \frac{\Delta P}{Q}$, its unit is pressure divided by volume flow rate, i.e., $\frac{\text{Pa}}{\text{m}^3/\text{s}} = \text{Pa}\cdot\text{s}/\text{m}^3$.

10. What is the mass flow rate $\frac{dm}{dt}$ of an incompressible fluid through a pipe obeying Poiseuille's law?

Explanation: Mass flow rate is given by $\frac{dm}{dt} = \rho Q = \rho \left(\frac{\pi \Delta P R^4}{8 \eta l}\right) = \frac{\pi \rho \Delta P R^4}{8 \eta l}$.

11. If two capillary tubes of resistances $R_1$ and $R_2$ are connected in parallel, how are their equivalent resistances combined?

Explanation: For parallel connections, reciprocals add up: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$, which gives $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$.

12. What happens to the volume flow rate $Q$ if the length of the capillary tube is doubled, keeping all other parameters constant?

Explanation: Since $Q$ is inversely proportional to length $l$ ($Q \propto \frac{1}{l}$), doubling $l$ halves the volume flow rate.

13. Why is fluid velocity zero at the inner wall of a cylindrical pipe?

Explanation: The no-slip condition states that a viscous fluid in contact with a solid boundary has zero relative velocity with respect to that boundary.

14. A Newtonian fluid is defined as one in which:

Explanation: A Newtonian fluid has a constant viscosity that does not change with shear rate or velocity gradient.

15. What is the velocity $v(r)$ at the central axis ($r = 0$) of a pipe according to the parabolic velocity profile?

Explanation: Substituting $r = 0$ into $v(r) = \frac{\Delta P}{4 \eta l}(R^2 - r^2)$ gives $v(0) = \frac{\Delta P R^2}{4 \eta l}$, which is $v_{max}$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The rate of flow of a liquid of viscosity $\eta$ through a pipe of radius $R$ and length $l$ under a pressure difference $\Delta P$ is proportional to:

Explanation: Poiseuille's formula states $Q = \frac{\pi \Delta P R^4}{8 \eta l}$, showing direct proportionality to $R^4$.

2. Two capillary tubes of same length and radii $r_1$ and $r_2$ are connected in series across a pressure difference. If the equivalent resistance is $R_{eq}$, how does it relate to individual radii?

Explanation: In series, $R_{eq} = R_1 + R_2 = \frac{8\eta l}{\pi} \left(\frac{1}{r_1^4} + \frac{1}{r_2^4}\right) = \frac{8\eta l}{\pi} \left(\frac{r_1^4 + r_2^4}{r_1^4 r_2^4}\right)$. Thus, $R_{eq}$ is inversely proportional to $\frac{r_1^4 r_2^4}{r_1^4 + r_2^4}$.

3. In a laminar flow of a viscous liquid through a horizontal pipe, the velocity of the liquid is:

Explanation: Due to the no-slip condition at the walls and internal viscous shear, the velocity is maximum at the axis ($r=0$) and zero at the walls ($r=R$).

4. If the pressure drop across a capillary tube is doubled, the volume flow rate of the liquid passing through it will:

Explanation: Volume flow rate $Q$ is directly proportional to pressure difference $\Delta P$. Doubling $\Delta P$ doubles $Q$.

5. The hydrodynamic resistance of a pipe of radius $R$ and length $l$ is $R_h$. If the radius is halved and length is doubled, the new hydrodynamic resistance becomes:

Explanation: $R_h = \frac{8 \eta l}{\pi R^4}$. If $l' = 2l$ and $R' = R/2$, then $R_h' = \frac{8 \eta (2l)}{\pi (R/2)^4} = \frac{16 \eta l}{\pi R^4 / 16} = 32 \left(\frac{8 \eta l}{\pi R^4}\right) = 32 R_h$.

6. What is the ratio of maximum velocity to average velocity for streamline flow of a Newtonian fluid in a cylindrical pipe?

Explanation: Since $v_{max} = \frac{\Delta P R^2}{4 \eta l}$ and $v_{avg} = \frac{\Delta P R^2}{8 \eta l}$, the ratio $v_{max} : v_{avg}$ is $2 : 1$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which of the following expressions correctly represents the velocity $v(r)$ at a radial distance $r$ from the axis in Poiseuille's flow?

Explanation: The velocity profile derived from Navier-Stokes equations for steady laminar flow in a pipe is $v(r) = \frac{\Delta P}{4 \eta l} (R^2 - r^2)$.

2. Two capillary tubes of equal length and radii $R$ and $2R$ are connected in parallel. What is the ratio of volume flow rates through them for the same pressure difference?

Explanation: Since $Q \propto R^4$ for the same pressure difference, viscosity, and length, $\frac{Q_1}{Q_2} = \left(\frac{R}{2R}\right)^4 = \frac{1}{16}$.

3. What is the physical significance of the Reynolds number staying below 2000 in the context of Poiseuille's formula?

Explanation: A Reynolds number below 2000 is the primary criterion ensuring that inertial forces do not overcome viscous forces, thereby maintaining smooth laminar flow required for Poiseuille's law.

4. If the viscosity of a liquid flowing through a pipe increases by a factor of 3, how must the pressure difference across the pipe be adjusted to keep the volume flow rate constant?

Explanation: From $Q = \frac{\pi \Delta P R^4}{8 \eta l}$, to keep $Q$ constant when $\eta$ is multiplied by 3, $\Delta P$ must also be multiplied by 3.

5. In the electrical analogy of Poiseuille's law, what corresponds to electrical current $I$?

Explanation: In Ohm's law $I = \frac{V}{R}$, current $I$ is the flow of charge, which corresponds directly to volume flow rate $Q$ (flow of volume).

6. What is the average velocity $v_{avg}$ of a fluid flowing through a circular pipe of radius $R$ under a pressure gradient $\frac{\Delta P}{l}$?

Explanation: The average velocity is $v_{avg} = \frac{Q}{\pi R^2} = \frac{\Delta P R^2}{8 \eta l}$.

7. Why does Poiseuille's formula fail when entrance effects in a short pipe are significant?

Explanation: Poiseuille's formula assumes a fully developed parabolic velocity profile. In short pipes or entrance regions, the boundary layer is still developing, invalidating the standard formula.

8. What is the equivalent hydrodynamic resistance $R_{eq}$ for three identical capillary tubes of resistance $R_0$ connected in parallel?

Explanation: For three identical resistors in parallel: $\frac{1}{R_{eq}} = \frac{1}{R_0} + \frac{1}{R_0} + \frac{1}{R_0} = \frac{3}{R_0}$, so $R_{eq} = \frac{R_0}{3}$.

9. Which dimensional formula correctly represents the coefficient of viscosity $\eta$ derived from Poiseuille's law or general mechanics?

Explanation: Viscosity $\eta$ has units of $\text{Pa}\cdot\text{s} = \text{N}\cdot\text{s}/\text{m}^2$, giving dimensions $[M L T^{-2} T / L^2] = [M L^{-1} T^{-1}]$.

10. If a fluid is non-Newtonian, why cannot Poiseuille's equation be applied directly?

Explanation: Poiseuille's integration assumes a constant viscosity $\eta$. In non-Newtonian fluids, viscosity varies with shear rate, which changes radially across the pipe and alters the velocity profile.