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Mechanics

Flow of Liquid in Capillary Tube

The flow of liquid through a narrow cylindrical tube, or capillary tube, is governed by viscous forces under a pressure gradient. It is fundamentally described by Poiseuille's equation for streamlined, laminar flow.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Qualitative understanding of capillary rise, Jurin's law, factors affecting surface tension, and basic dimensional analysis of viscous flow.

PGT extension

Rigorous derivation of Poiseuille's formula, series and parallel combination of capillary tubes, end corrections, and kinetic energy corrections.

Prerequisites

Newton's law of viscosity
Bernoulli's theorem and equation of continuity
Surface tension and contact angle

Nature of Flow in Capillary Tubes

The flow of an incompressible Newtonian liquid through a long narrow tube of uniform circular cross-section is streamline or laminar.
Fluid velocity is maximum at the axis of the tube and drops to zero at the inner walls due to the no-slip condition.
The velocity profile across the cross-section of the capillary tube is parabolic in shape.

Poiseuille's Law of Liquid Flow

The volume of liquid flowing per second (rate of flow, Q) through a capillary tube depends on the pressure gradient per unit length, radius of the tube, and coefficient of viscosity of the liquid.
Q is directly proportional to the fourth power of the radius (r⁴) and the pressure difference (P), and inversely proportional to the length of the tube (l) and viscosity (η).
It applies strictly to steady, laminar flow without turbulence or turbulent kinetic energy losses.

Capillary Combinations

When capillary tubes are joined in series, the rate of flow through each tube is the same, and the total pressure difference is the sum of pressure drops across individual tubes.
When capillary tubes are joined in parallel, the pressure difference across each tube is the same, and the total rate of flow is the sum of rates through individual tubes.
Equivalent viscous resistance for series combination is R_s = R₁ + R₂, analogous to electrical resistance.

Correction Factors in Capillary Flow

Hagen-Poiseuille formula assumes zero velocity at the entry and exit, which is idealized.
Traverse kinetic energy requires a correction term known as Hagenbach's correction, modifying the effective pressure head.
End corrections account for the constriction of streamlines just before entering the capillary tube.

Derivations

Derivation of Poiseuille's Equation

Conditions: Steady, laminar, streamline flow of a Newtonian fluid through a cylindrical tube of radius R and length l under pressure difference P.

Consider a coaxial cylindrical fluid shell of inner radius x and thickness dx moving with velocity v. The viscous drag force acting on the outer surface of this shell is given by $F = - \eta (2\pix l) (dv/dx)$.
This viscous force is balanced by the net pressure force acting on the cross-section of the shell, which is F_p = (P / l) (πx²). Equating the two forces gives -η(2πxl)(dv/dx) = (P/l)(πx²).
Rearranging and integrating for velocity v(x) with the boundary condition $v = 0 at x = R yields v(x) = (P / 4\etal) (R^{2} - x^{2})$.
To find the total volume rate of flow Q, integrate the discharge through elementary annular rings of area 2πx dx from x = 0 to $x = R: Q = \int v(x) (2\pix dx)$.

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

Capillary Tubes in Series and Parallel

Conditions: Two tubes of lengths l_1, l_2 and radii r_1, r_2 connected end-to-end (series) or side-by-side (parallel).

For series combination, rate of flow Q is identical through both tubes: $Q = \frac{\pi P_1 r_1^4}{8 \eta l_1} = \frac{\pi P_2 r_2^4}{8 \eta l_2}$.
Total pressure drop $P = P_1 + P_2 = Q \left(\frac{8 \eta l_1}{\pi r_1^4} + \frac{8 \eta l_2}{\pi r_2^4}\right)$.
Define hydrodynamic resistance $R_v = \frac{8 \eta l}{\pi r^4}$. Thus, $P = Q (R_1 + R_2)$.
For parallel combination, total flow $Q = Q_1 + Q_2 = P \left(\frac{\pi r_1^4}{8 \eta l_1} + \frac{\pi r_2^4}{8 \eta l_2}\right)$.

Result: Series: $R_{s} = R_1 + R_2 ; Parallel: \frac{1}{R_{p}} = \frac{1}{R_1} + \frac{1}{R_2}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Poiseuille's Equation	$$Q = \frac{\pi P r^4}{8 \eta l}$$	Q = Volume flow rate, P = Pressure difference, r = Radius, l = Length, \eta = Coefficient of viscosity m³/s (SI)	Laminar flow, Newtonian fluid, rigid cylindrical tube
Velocity Profile in Capillary Tube	$$v(r) = \frac{P}{4 \eta l} (R^2 - r^2)$$	v(r) = Velocity at radius r, R = Radius of tube m/s	Steady axisymmetric flow
Maximum Velocity	$$v_{max} = \frac{P R^2}{4 \eta l}$$	$v_{max} = Velocity at the axis (r = 0)$ m/s	At the central axis of the tube
Viscous Resistance of Capillary	$$R_v = \frac{8 \eta l}{\pi r^4}$$	$R_v = Hydraulic/viscous resistance$ kg m^{-4^{s^-1}}	Laminar flow regime
Jurin's Law (Capillary Rise Height)	$$h = \frac{2 T \cos\theta}{r \rho g}$$	T = Surface tension, \theta = Contact angle, \rho = Density, g = Acceleration due to gravity m	Narrow cylindrical tube, negligible meniscus weight
Rate of Flow in Series Combination	$$Q = \frac{P}{\frac{8\eta l_1}{\pi r_1^4} + \frac{8\eta l_2}{\pi r_2^4}}$$	l_1, $l_2 = lengths; r_1$, $r_2 = radii$ m³/s	End-to-end connection
Reynolds Number for Tube Flow	$$Re = \frac{\rho v_{avg} d}{\eta}$$	$v_{avg} = Average velocity$, d = Diameter of tube Dimensionless	Re < 2000 for laminar flow
Mass Flow Rate	$$M = \rho Q = \frac{\pi P \rho r^4}{8 \eta l}$$	M = Mass per second kg/s	Incompressible fluid

Graphs and experiments

Velocity Distribution Across Capillary Radius

Draw: A parabolic profile symmetric about the central axis, showing zero velocity at the walls (r = R) and maximum velocity at the center (r = 0).

Interpret: Demonstrates that velocity gradient (dv/dr) is zero at the center and maximum at the inner wall, confirming non-uniform shear stress.

Exam use: Frequently asked in PGT multiple-choice questions regarding the nature of velocity profile in viscous flow.

Rate of Flow (Q) versus Radius (r) Graph

Draw: A non-linear steep curve representing a power-law relationship, plotting Q on the y-axis and r on the x-axis.

Interpret: Confirms that volume flow rate scales steeply with the fourth power of the radius (Q proportional to r⁴).

Exam use: Tested via conceptual interpretation where radius is doubled and the resulting change in flow rate must be calculated.

Poiseuille's Constant Head Viscometer Experiment

Draw: Experimental setup showing a constant head tank, capillary tube horizontal mount, measuring flask, and a thermometer.

Interpret: Used to determine the coefficient of viscosity (η) of a liquid by measuring volume collected per unit time under a constant hydrostatic pressure head.

Exam use: Practical physics viva and conceptual numericals on laboratory determination of viscosity.

Solved numericals

TGT core

A capillary tube of radius 0.1 cm and length 10 cm is used to flow water under a constant pressure head of 20 cm of water. Calculate the volume of water flowing per second. (Take coefficient of viscosity of water = 0.001 Pa s, $g = 9.8 m/s^{2}$, density of wate$r = 1000 kg/m^{3})$

Identify given parameters: $r = 0.1 cm = 10^{-3} m$, l = 10 cm = 0.1 m, η = 10^-3 Pa s.
Calculate pressure difference $P = h\rhog = 0.2 m * 1000 kg/m^{3} * 9.8 m/s^{2} = 1960 N/m^{2}$.
Apply Poiseuille's formula: $Q = (\pi P r^{4}) / (8 \eta l)$.
Substitute values: $Q = (3.1416 * 1960 * (10^{-3})^{4}) / (8 * 10^{-3} * 0.1) = 7.697 * 10^{-7} m^{3}/s$.

Answer: 7.70 x 10^-7 m^3/s

PGT extension

Two capillary tubes of same length and radii r and 2r are connected in series. If the total pressure difference across the combination is P, find the pressure difference across the second tube (radius 2r).

Viscous resistance of first tube: R₁ = 8ηl / (π r⁴).
Viscous resistance of second tube: R₂ = 8ηl / (π (2r)⁴) = R₁ / 16.
In series, rate of flow Q is identical, so pressure drop is directly proportional to resistance: P₂ = Q * R₂ and $P = Q * (R_{1} + R_{2})$.
Ratio P₂ / $P = R_{2} / (R_{1} + R_{2}) = (R_{1} / 16) / (R_{1} + R_{1}/16) = (1/16) / (17/16) = 1/17$.
Therefore, P₂ = P / 17.

Answer: P / 17

PGT extension

If the radius of a capillary tube is increased by 10% while keeping the pressure difference and length constant, find the percentage increase in the rate of liquid flow.

Poiseuille's relation gives Q proportional to r⁴ when P, η, and l are constant.
Taking logarithmic differentiation: ΔQ / $Q = 4 * (\Deltar / r)$.
Given percentage change in radius Δr / $r = +10% = 0.1$.
Percentage change in $Q = 4 * 10% = 40%$.

Answer: 40%

Assumptions and limitations

The liquid is assumed to be Newtonian with a constant coefficient of viscosity unaffected by shear rate.
The flow must be strictly streamline and laminar (Reynolds number < 2000); turbulence invalidates Poiseuille's equation.
The inner walls of the tube are rigid, perfectly smooth, and cylindrical, with a zero slip condition at the boundary.
End effects, kinetic energy head loss at entry, and gravitational head variations across the cross-section are neglected in the elementary form.

Common misconceptions

Common error: Liquid velocity is uniform across the entire cross-section of the capillary tube.

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

Viscous drag creates a shear stress gradient from the stationary walls to the central axis.

Common error: Doubling the radius of a capillary tube doubles the rate of liquid flow.

Correct: Doubling the radius increases the rate of flow by a factor of 2⁴ = 16.

Poiseuille's law establishes a fourth-power dependence of volume flow rate on the radius (Q &prop; r⁴).

Common error: Capillary flow rate depends on the total mass of liquid in the reservoir.

Correct: Flow rate depends only on the hydrostatic pressure difference ($P = h\rhog) between the ends of the tube$.

The absolute height of the liquid column above the entry does not drive flow; only the pressure drop across length l matters.

Quick revision

Poiseuille's formula states $Q = \piPr^{4} / 8\etal$ for laminar tube flow.
The volume flow rate is extremely sensitive to tube radius, scaling as r⁴.
Viscosity decreases with an increase in temperature for liquids, thereby increasing the rate of flow Q.
Hydrodynamic resistance in series adds algebraically: R_s = R₁ + R₂.
Velocity profile across the capillary diameter is parabolic with peak velocity at the center.

Must remember

The volume flow rate of a Newtonian fluid through a capillary tube is directly proportional to the fourth power of the radius ($Q \propto r^4$) and pressure gradient, and inversely proportional to the viscosity ($\eta$) and length ($l$).
The velocity profile across the cross-section of a capillary tube is parabolic, with maximum velocity at the axis ($v_{max} = \frac{P R^2}{4 \eta l}$) and zero velocity at the inner walls due to the no-slip condition.
Viscous resistance of a capillary tube is analogous to electrical resistance and is given by $R_v = \frac{8 \eta l}{\pi r^4}$.
For capillary tubes joined in series, the equivalent viscous resistance is the sum of individual resistances ($R_s = R_1 + R_2$), and the total pressure drop is the sum of pressure drops.
For capillary tubes joined in parallel, the total rate of flow is the sum of rates through individual tubes, and the pressure difference across each tube is identical.
Reynolds number ($Re = \frac{\rho v_{avg} d}{\eta}$) must remain below $2000$ to maintain steady laminar flow in the capillary tube.
Jurin's law defines the capillary rise height as $h = \frac{2 T \cos\theta}{r \rho g}$ in a narrow cylindrical tube.
Hagenbach's correction accounts for traverse kinetic energy losses at the entrance and exit of the capillary tube.

Glossary

Poiseuille's EquationAn expression giving the volume flow rate of a liquid through a cylindrical tube under laminar flow condition: $Q = \frac{\pi P r^4}{8 \eta l}$.

Viscous Resistance ($R_v$)The resistance offered by the capillary tube to the flow of fluid, defined as $R_v = \frac{8 \eta l}{\pi r^4}$.

Parabolic Velocity ProfileThe variation of fluid velocity across the radius of the tube where velocity follows a squared coordinate relation: $v(r) = \frac{P}{4 \eta l} (R^2 - r^2)$.

Jurin's LawThe law stating that the height of liquid rise in a capillary tube is inversely proportional to its radius: $h = \frac{2 T \cos\theta}{r \rho g}$.

Hagenbach's CorrectionA correction term applied to Poiseuille's formula to account for the kinetic energy of the fluid entering and leaving the capillary.

Reynolds Number ($Re$)A dimensionless quantity used to predict fluid flow patterns, determining whether flow is laminar or turbulent ($Re < 2000$ for laminar pipe flow).

Series vs Parallel Combination of Capillary Tubes

Parameter	Series Combination	Parallel Combination
Rate of Flow ($Q$)	Same through each tube ($Q = Q_1 = Q_2$)	Sum of individual rates ($Q = Q_1 + Q_2$)
Pressure Difference ($P$)	Sum of individual drops ($P = P_1 + P_2$)	Same across each tube ($P = P_1 = P_2$)
Equivalent Resistance ($R_v$)	$R_s = R_1 + R_2$	$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2}$

Practice Quiz

Choose an option to reveal the explanation.

1. What is the shape of the velocity profile across the cross-section of a capillary tube for a Newtonian fluid in laminar flow?

Explanation: The velocity profile across the cross-section of the capillary tube is parabolic in shape, with maximum velocity at the axis and zero at the walls due to the no-slip condition.

2. According to Poiseuille's law, the rate of flow of liquid $Q$ through a capillary tube depends on the radius $r$ as:

Explanation: Poiseuille's equation is $Q = \frac{\pi P r^4}{8 \eta l}$, showing that the volume flow rate is directly proportional to the fourth power of the radius ($r^4$).

3. Where does the maximum fluid velocity occur in a capillary tube during steady laminar flow?

Explanation: Fluid velocity is maximum at the axis of the tube ($v_{max} = \frac{P R^2}{4 \eta l}$) and drops to zero at the inner walls due to the no-slip condition.

4. If the radius of a capillary tube is doubled while keeping the pressure difference, length, and viscosity constant, the rate of liquid flow becomes:

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

5. What is the dimensional formula for the viscous resistance $R_v$ of a capillary tube?

Explanation: Viscous resistance is $R_v = \frac{8 \eta l}{\pi r^4}$. Since units are $\text{kg m}^{-4}\text{s}^{-1}$, the dimensional formula is $[M L^{-4} T^{-1}]$.

6. When two capillary tubes of resistances $R_1$ and $R_2$ are connected in series, the equivalent viscous resistance $R_s$ is:

Explanation: When capillary tubes are joined in series, the equivalent viscous resistance is $R_s = R_1 + R_2$, which is analogous to electrical resistance in series.

7. When two identical capillary tubes are joined in parallel, the effective rate of flow compared to a single tube under the same pressure difference is:

Explanation: In a parallel combination, total rate of flow is the sum of rates through individual tubes ($Q = Q_1 + Q_2$). For two identical tubes, $Q = 2Q_1$, doubling the flow rate.

8. What is the condition for the Reynolds number ($Re$) for fluid flow in a tube to remain laminar?

Explanation: For steady laminar flow in a cylindrical tube, the Reynolds number ($Re = \frac{\rho v_{avg} d}{\eta}$) must remain below $2000$.

9. What does Hagenbach's correction account for in capillary flow measurements?

Explanation: Traverse kinetic energy requires a correction term known as Hagenbach's correction, modifying the effective pressure head because velocity is non-zero upon entry and exit.

10. According to Jurin's Law, the height $h$ of liquid rise in a capillary tube is inversely proportional to:

Explanation: Jurin's Law states that $h = \frac{2 T \cos\theta}{r \rho g}$, showing that capillary rise height is inversely proportional to the radius $r$ of the tube.

11. If the length of a capillary tube is doubled while keeping all other parameters constant, the rate of liquid flow becomes:

Explanation: Poiseuille's law states $Q \propto \frac{1}{l}$. Doubling the length $l$ halves the rate of liquid flow.

12. What is the mass flow rate $M$ of an incompressible fluid through a capillary tube in terms of volume flow rate $Q$ and density $\rho$?

Explanation: Mass flow rate is the product of density and volume flow rate: $M = \rho Q = \frac{\pi P \rho r^4}{8 \eta l}$.

13. In a capillary tube experiment, what is the boundary condition for fluid velocity at the inner wall of the tube?

Explanation: Due to the no-slip condition, fluid velocity drops to zero at the inner solid walls of the capillary tube.

14. Which of the following is assumed in the derivation of Poiseuille's Law?

Explanation: Poiseuille's Law applies strictly to steady, laminar flow of an incompressible Newtonian fluid without turbulence in a long narrow tube of uniform circular cross-section.

15. If two capillary tubes of radii $r_1$ and $r_2$ and same length are joined in series, what is the equivalent flow rate expression under pressure difference $P$?

Explanation: For series combination, $Q = \frac{P}{R_{v1} + R_{v2}} = \frac{P}{\frac{8\eta l}{\pi r_1^4} + \frac{8\eta l}{\pi r_2^4}}$. (Assuming equal lengths $l$).

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 capillary tube of radius $r$ and length $l$ under a pressure difference $P$ is $Q$. If the radius and pressure are both doubled while length and viscosity are halved, the new rate of flow $Q'$ will be:

Explanation: From Poiseuille's equation, $Q \propto \frac{P r^4}{\eta l}$. Therefore, $Q' = Q \times \left(\frac{P'}{P}\right) \times \left(\frac{r'}{r}\right)^4 \times \left(\frac{\eta}{\eta'}\right) \times \left(\frac{l}{l'}\right) = Q \times (2) \times (2^4) \times (2) \times (2) = Q \times 2 \times 16 \times 2 \times 2 = 64 Q$.

2. Two capillary tubes of radius $r$ and $2r$ and lengths $l$ and $2l$ respectively are connected in series across a pressure difference $P$. The ratio of pressure drops across the first tube to the second tube is:

Explanation: In series combination, flow rate $Q$ is the same. Pressure drop $P = Q R_v$. Therefore, $\frac{P_1}{P_2} = \frac{R_{v1}}{R_{v2}} = \frac{8\eta l_1 / \pi r_1^4}{8\eta l_2 / \pi r_2^4} = \left(\frac{l_1}{l_2}\right) \left(\frac{r_2}{r_1}\right)^4 = \left(\frac{l}{2l}\right) \left(\frac{2r}{r}\right)^4 = \left(\frac{1}{2}\right) \times 16 = 8:1$.

3. Water flows through a capillary tube of radius $r$ at a rate $Q$. If the tube is replaced by another tube of radius $r/2$ and length half of the original, keeping the pressure head constant, the new rate of flow is:

Explanation: Using $Q \propto \frac{r^4}{l}$, we have $Q' = Q \times \left(\frac{r/2}{r}\right)^4 \times \left(\frac{l}{l/2}\right) = Q \times \left(\frac{1}{16}\right) \times 2 = \frac{Q}{8}$.

4. The maximum velocity of a Newtonian liquid flowing through a capillary tube of radius $R$ under pressure difference $P$ is $v_{max}$. The average velocity $v_{avg}$ of the liquid flow is equal to:

Explanation: The velocity profile is parabolic: $v(r) = v_{max}(1 - \frac{r^2}{R^2})$. The average velocity is obtained by integrating over the cross-section, yielding $v_{avg} = \frac{v_{max}}{2}$.

5. Three capillary tubes of identical dimensions are connected in parallel. If the rate of flow through each tube is $Q_0$, the total viscous resistance of the parallel combination in terms of single tube resistance $R_v$ is:

Explanation: For three identical resistors in parallel, equivalent resistance $R_p = \frac{R_v}{3}$.

6. A liquid flows through a uniform capillary tube. The pressure drop across a length $l$ of the tube is $P$. If the tube is cut into three equal parts and connected in parallel across the same pressure source, the total rate of flow becomes how many times the original single-tube flow rate?

Explanation: For a single tube, $Q \propto \frac{1}{l}$. Each cut tube has length $l/3$ and resistance $R_v / 3$. When 3 such tubes are connected in parallel, equivalent resistance is $\frac{R_v/3}{3} = \frac{R_v}{9}$. Thus, flow rate becomes 9 times the original flow rate.

Timed Topic Test

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

1. What is the physical significance of the no-slip condition at the inner wall of a capillary tube?

Explanation: The no-slip condition states that viscous fluids have zero velocity relative to the boundary at solid interfaces.

2. Which of the following fluids does NOT strictly obey Poiseuille's Law?

Explanation: Poiseuille's law applies strictly to Newtonian fluids. Blood is non-Newtonian due to red blood cell aggregation and deformation.

3. If the pressure difference across a capillary tube is increased four times, the volume flow rate $Q$ increases by a factor of:

Explanation: Poiseuille's law gives $Q \propto P$. Thus, quadrupling $P$ increases $Q$ by a factor of 4.

4. What is the ratio of maximum velocity to average velocity for laminar flow in a circular capillary tube?

Explanation: Maximum velocity occurs at the center ($v_{max} = \frac{PR^2}{4eta l}$) and average velocity is half of that ($v_{avg} = \frac{PR^2}{8eta l}$), giving a ratio of $2:1$.

5. Two capillary tubes of radii $r$ and $2r$ having identical lengths are joined in series. What is the equivalent viscous resistance of the combination?

Explanation: Let resistance of first be $R_v = \frac{8eta l}{\pi r^4}$. Resistance of second is $\frac{8eta l}{\pi (2r)^4} = \frac{R_v}{16}$. Series equivalent is $R_s = R_v + \frac{R_v}{16} = \frac{17 R_v}{16}$.

6. In the study of liquid flow in capillary tubes, why are entrance and exit corrections (Hagenbach's correction) necessary?

Explanation: End and kinetic energy corrections account for the acceleration of fluid and constriction of streamlines upon entering and exiting the capillary.

7. If the coefficient of viscosity of a liquid is $\eta$, what are the SI units of viscous resistance $R_v$?

Explanation: Viscous resistance is $R_v = \frac{8\eta l}{\pi r^4}$, having units of $\frac{(\text{Pa s}) \times \text{m}}{\text{m}^4} = \text{Pa s m}^{-3} = \frac{\text{kg}}{\text{m s}^2} \frac{\text{s}}{\text{m}^3} = \text{kg m}^{-4}\text{s}^{-1}$.

8. What happens to the Reynolds number of flow if the tube radius is halved while keeping mass flow rate constant?

Explanation: Reynolds number $Re = \frac{\rho v_{avg} d}{\eta}$. Since $v_{avg} \propto \frac{Q}{r^2}$ and $d \propto r$, $Re \propto \frac{1}{r}$. Halving the radius doubles the Reynolds number.

9. Which parameter has the maximum percentage effect on the volume flow rate in a capillary tube if subjected to experimental error?

Explanation: Since $Q \propto r^4$, any small percentage error in radius $r$ gets magnified fourfold in the flow rate calculation, making radius the most sensitive parameter.

10. What is the primary assumption regarding the fluid state in the derivation of Poiseuille's equation?

Explanation: Poiseuille's law assumes the fluid is incompressible, meaning its density remains constant throughout the flow.