FLUID MECHANICS (Part-2)

What is Viscosity?

Viscosity is the property of a fluid by which it opposes the relative motion between its adjacent layers.

Explanation of Viscosity

When a fluid flows, its different layers may move with different velocities. The faster-moving layer tends to drag the slower-moving layer along with it, while the slower-moving layer tends to retard the faster-moving layer.

As a result, an internal resisting force develops between adjacent layers of the fluid. This force acts opposite to their relative motion and is called viscous force or viscous drag.

A fluid having greater viscosity offers greater resistance to flow. For example, honey flows more slowly than water because honey has greater viscosity.

Thus, viscosity may be regarded as the internal friction of a fluid.

Physical Characteristics of Viscosity

Nature: Viscosity is a property of fluids and represents their internal resistance to relative motion between adjacent layers.

Cause: Viscosity arises from molecular interactions and momentum transfer between neighbouring layers of a fluid.

Effect: Greater viscosity means greater resistance to flow, whereas lower viscosity means that the fluid flows more easily.

Dependence on Temperature: The viscosity of liquids generally decreases as temperature increases, whereas the viscosity of gases generally increases with temperature.

Scalar Nature: The coefficient of viscosity is a scalar quantity for an ordinary isotropic fluid.

SI Unit and Dimensional Formula

Viscosity is quantified by the coefficient of viscosity, represented by \(\eta\). Its SI unit is pascal-second (Pa·s).

It can also be expressed as:

The dimensional formula of the coefficient of viscosity is:

In the CGS system, the unit of coefficient of viscosity is poise (P).

Causes of Viscosity

Viscosity is mainly caused by the intermolecular forces between the molecules of a fluid and the transfer of momentum between neighbouring layers.

When adjacent layers of a fluid move with different velocities, molecular interactions tend to oppose this relative motion.

In liquids, viscosity is mainly associated with cohesive intermolecular forces between neighbouring molecules.

In gases, viscosity is mainly due to the transfer of momentum by molecules moving between layers having different velocities.

Thus, viscosity acts as an internal resistance to the relative motion of adjacent fluid layers.

Coefficient of Viscosity

Definitions of Coefficient of Viscosity

Definition 1: The coefficient of viscosity of a fluid is defined as the ratio of shearing stress to the rate of shear strain.

Definition 2: The coefficient of viscosity is the tangential viscous force required to maintain a unit velocity gradient between two parallel layers of a fluid having unit area of contact.

Explanation

Consider a fluid enclosed between two large parallel surfaces. Let the lower surface be stationary and the upper surface move with a velocity \(v\).

The fluid layer in contact with the stationary surface has zero velocity, while the layer in contact with the moving surface has velocity \(v\). The intermediate layers have velocities that vary continuously between these two values.

Due to viscosity, a tangential force acts between adjacent layers and opposes their relative motion. This force is called viscous force.

If the area of contact is \(A\), the separation between the surfaces is \(y\), and the velocity difference between the layers is \(v\), the velocity gradient is \(\frac{dv}{dy}\).

Mathematical Derivation of the Coefficient of Viscosity

According to Newton’s law of viscosity, the viscous force \(F\) is directly proportional to the area \(A\) of contact between the fluid layers.

The viscous force is also directly proportional to the velocity gradient \(\frac{dv}{dy}\).

Combining these proportionalities:

Introducing the proportionality constant \(\eta\), called the coefficient of viscosity:

Therefore:

The viscous force per unit area is called shearing stress:

The rate of shear strain is equal to the velocity gradient:

Hence:

For a Uniform Velocity Gradient

If the velocity changes uniformly from \(0\) to \(v\) over a distance \(y\), then:

Substituting this into Newton’s law of viscosity:

Rearranging:

Physical Meaning of Coefficient of Viscosity

If the area of contact is \(1\,m^2\) and the velocity gradient is \(1\,s^{-1}\), then:

From:

We obtain:

Therefore, the coefficient of viscosity is numerically equal to the tangential viscous force required to maintain a unit velocity gradient between two parallel fluid layers having unit area of contact.

SI Unit of Coefficient of Viscosity

From:

Therefore:

Since \(1\,Pa=1\,N\,m^{-2}\):

CGS Unit of Coefficient of Viscosity

The CGS unit of coefficient of viscosity is called poise, represented by \(P\).

Conversion Between SI and CGS Units

The relation between the SI unit pascal-second and the CGS unit poise is:

Therefore:

Dimensional Formula of Coefficient of Viscosity

From:

The dimensional formula of force is:

Therefore:

Thus, the coefficient of viscosity quantitatively measures the internal resistance offered by a fluid to the relative motion of its adjacent layers.

Variation of Viscosity with Temperature

The viscosity of a fluid changes with temperature. However, the effect of temperature on viscosity is different for liquids and gases.

1. Variation of Viscosity of Liquids with Temperature

For liquids, the viscosity decreases as temperature increases. When a liquid is heated, its molecules acquire greater kinetic energy and can move more freely. The intermolecular attraction between neighbouring molecules becomes less effective, so the resistance to relative motion between adjacent layers decreases. Therefore, the coefficient of viscosity of a liquid decreases with increase in temperature.

Thus,

\[ T \uparrow \quad \Rightarrow \quad \eta \downarrow \]

T ↑  →  Intermolecular attraction becomes less effective
     →  Molecules move more freely
     →  Internal friction decreases
     →  Viscosity decreases

Diagram: Viscosity of a Liquid vs Temperature

The graph shows that the coefficient of viscosity of a liquid decreases with increase in temperature.

2. Variation of Viscosity of Gases with Temperature

For gases, the viscosity increases as temperature increases. When a gas is heated, its molecules move with greater average speed. This increases the transfer of momentum between neighbouring layers of the gas having different velocities. As a result, the internal friction between the layers increases and hence the coefficient of viscosity increases.

Thus,

\[ T \uparrow \quad \Rightarrow \quad \eta \uparrow \]

T ↑  →  Molecular speed increases
     →  Momentum transfer between layers increases
     →  Internal friction increases
     →  Viscosity increases

Diagram: Viscosity of a Gas vs Temperature

The graph shows that the coefficient of viscosity of a gas increases with increase in temperature.

Why Does Viscosity Behave Differently in Liquids and Gases?

The main reason is the different mechanisms responsible for viscosity in liquids and gases. In liquids, viscosity is mainly associated with intermolecular cohesive forces. Increasing temperature weakens the effective influence of these forces and allows the molecules to move more freely. Hence, viscosity decreases.

In gases, viscosity is mainly due to the transfer of momentum between molecules moving between neighbouring layers. Increasing temperature increases the average molecular speed and therefore increases momentum transfer. Hence, viscosity increases.

Comparison of Liquids and Gases

Practical Applications and Uses of Viscosity

Viscosity has many important applications in everyday life, engineering, industry and medicine. Some important applications of viscosity are given below.

1. Rain Drops and Terminal Velocity

Rain drops are very small in size but fall from great heights. Due to gravity, their velocity tends to increase continuously. However, as the velocity of a rain drop increases, the viscous force offered by air also increases and opposes its motion.

After some time, the viscous force becomes equal to the effective gravitational force acting on the drop. The resultant force then becomes zero, and the rain drop falls with a constant velocity called terminal velocity.

2. Lubricants and Reduction of Friction

Viscosity is important in the selection of lubricants used between moving parts of machines. A lubricant of suitable viscosity forms a thin layer between the surfaces and reduces direct contact between them.

Thus, viscous lubricants help to reduce friction, wear and heating of moving parts. The required viscosity depends on the type of machine and its operating conditions.

3. Glycerine and Water Come to Rest

Suppose glycerine and water are stirred vigorously and then left undisturbed. Glycerine comes to rest earlier than water because the viscosity of glycerine is much greater than that of water.

The greater the viscosity of a liquid, the greater is its resistance to the relative motion of its layers. Hence, glycerine loses its motion more quickly.

4. Separation of Sand and Gold Particles

Viscosity can be used in the separation of particles having different densities. When water flows over a mixture of sand and gold particles, the lighter sand particles are more easily carried away by the flowing water, whereas the denser gold particles are less easily carried away.

Therefore, the difference in the motion of particles through the flowing water can be used to separate gold particles from sand.

5. Damping of Sea Waves

Sea water consists of different layers that undergo relative motion when waves travel through it. Viscous forces between these layers oppose their relative motion.

As a result, some of the energy of the waves is dissipated, and the waves gradually become smaller or are damped as they travel through the water.

6. Viscosity of Blood in Medical Diagnosis

Blood has viscosity, and its flow through blood vessels is affected by this viscosity. Therefore, measurements related to the viscosity and flow properties of blood are useful in medical studies.

The sedimentation behaviour of blood cells and changes in blood viscosity can provide useful information in the diagnosis and study of certain diseases.

7. Streamlined Shape of Ships, Automobiles and Aeroplanes

The viscous force offered by water or air opposes the motion of ships, automobiles and aeroplanes. Therefore, these bodies are designed with shapes that reduce the resistance offered by the surrounding fluid.

Such a shape is called a streamlined shape. Streamlined shapes reduce fluid resistance and help the body move more easily through air or water.

8. V-Shaped Canals

Canals are often constructed with a V-shaped cross-section. Water flowing through a canal moves in layers, and the area of contact between neighbouring layers is smaller near the bottom and larger towards the top.

Therefore, the viscous resistance associated with the flow is reduced near the narrower bottom region, allowing greater flow towards the wider upper part of the canal.

Quick Revision

1. Rain drops → Viscous force of air produces terminal velocity.

2. Lubricants → Reduce friction between moving machine parts.

3. Glycerine → Comes to rest earlier than water because its viscosity is higher.

4. Gold and sand → Can be separated using their different behaviour in flowing water.

5. Sea waves → Viscous forces cause damping of waves.

6. Blood → Viscosity and sedimentation behaviour are useful in medical studies.

7. Ships, automobiles and aeroplanes → Streamlined shapes reduce fluid resistance.

8. Canals → V-shaped construction helps in efficient flow of water.

Stokes’ Law and Terminal Velocity

Stokes’ Law

Definition: Stokes’ law states that the viscous force acting on a small spherical body moving slowly through a viscous fluid is directly proportional to the coefficient of viscosity of the fluid, the radius of the sphere, and its velocity relative to the fluid.

Mathematically, the viscous force is

\[ F_v = 6\pi\eta r v \]

where \(F_v\) is the viscous force, \(\eta\) is the coefficient of viscosity of the fluid, \(r\) is the radius of the sphere, and \(v\) is the velocity of the sphere relative to the fluid.

Conditions for Stokes’ Law

Stokes’ law is applicable when a small spherical body moves slowly through a viscous fluid, so that the flow around the sphere is smooth or laminar and the Reynolds number is very small.

Terminal Velocity

When a small sphere falls through a viscous fluid, its velocity initially increases due to gravity. As its velocity increases, the upward viscous force also increases.

After some time, the downward effective gravitational force becomes equal to the upward viscous force. The resultant force on the sphere then becomes zero, and the sphere falls with a constant velocity called terminal velocity.

Mathematical Derivation of Terminal Velocity

Consider a small spherical body of radius \(r\) and density \(\rho_s\) falling through a liquid of density \(\rho_l\).

Let the terminal velocity of the sphere be \(v_t\).

Step 1: Weight of the Sphere

The volume of the sphere is

\[ V=\frac{4}{3}\pi r^3 \]

Therefore, the mass of the sphere is

\[ m=\rho_s V \]

Hence, the weight of the sphere is

\[ W=mg \]

\[ W=\frac{4}{3}\pi r^3\rho_s g \]

Step 2: Buoyant Force

The buoyant force acting upward on the sphere is equal to the weight of the liquid displaced by the sphere.

\[ F_B=\rho_l Vg \]

Therefore,

\[ F_B=\frac{4}{3}\pi r^3\rho_l g \]

Step 3: Viscous Force

According to Stokes’ law, the viscous force opposing the motion of the sphere is

\[ F_v=6\pi\eta r v \]

At terminal velocity \(v=v_t\), therefore

\[ F_v=6\pi\eta r v_t \]

Step 4: Condition at Terminal Velocity

At terminal velocity, the net force acting on the sphere is zero. Hence, the effective downward gravitational force is equal to the upward viscous force.

\[ W-F_B=F_v \]

Substituting the above expressions,

\[ \frac{4}{3}\pi r^3\rho_s g – \frac{4}{3}\pi r^3\rho_l g = 6\pi\eta r v_t \]

Step 5: Simplification

Taking the common terms on the left side,

\[ \frac{4}{3}\pi r^3g(\rho_s-\rho_l) = 6\pi\eta r v_t \]

Dividing both sides by \(6\pi\eta r\),

\[ v_t= \frac{2r^2g(\rho_s-\rho_l)} {9\eta} \]

Final Expression for Terminal Velocity

Therefore, the terminal velocity of a small spherical body falling through a viscous fluid is

\[ \boxed{ v_t= \frac{2r^2g(\rho_s-\rho_l)} {9\eta} } \]

where \(r\) is the radius of the sphere, \(g\) is the acceleration due to gravity, \(\rho_s\) is the density of the sphere, \(\rho_l\) is the density of the liquid, and \(\eta\) is the coefficient of viscosity of the liquid.

Important Conclusions

From the expression for terminal velocity,

\[ v_t\propto r^2 \]

Thus, the terminal velocity is proportional to the square of the radius of the sphere.

Also,

\[ v_t\propto(\rho_s-\rho_l) \]

Therefore, the terminal velocity increases with the difference between the density of the sphere and the density of the liquid.

Further,

\[ v_t\propto\frac{1}{\eta} \]

Hence, the terminal velocity decreases when the viscosity of the fluid increases.

Reynolds Number

Definition of Reynolds Number

Reynolds number is a dimensionless quantity that determines the nature of fluid flow and indicates whether the flow is predominantly laminar or turbulent.

It is defined as the ratio of inertial force to viscous force acting in a flowing fluid.

Mathematically,

\[ R_e=\frac{\text{Inertial force}}{\text{Viscous force}} \]

Explanation of Reynolds Number

When a fluid flows through a pipe, two types of effects are important. The inertial effect tends to maintain the motion of the fluid, while the viscous effect opposes the relative motion between neighbouring layers of the fluid.

At low velocities, viscous effects are dominant and the fluid flows smoothly in parallel layers. Such flow is called laminar flow.

When the velocity of the fluid increases sufficiently, inertial effects become more important. The smooth layered motion becomes unstable and the flow may become irregular. Such flow is called turbulent flow.

Reynolds number gives a measure of the relative importance of these two effects.

Critical Velocity

Critical velocity is the maximum velocity up to which the flow of a fluid through a pipe remains laminar. When the velocity exceeds this value, the flow tends to change from laminar flow to turbulent flow.

The critical velocity is represented by \(v_c\).

Mathematical Derivation of Reynolds Number

Consider a fluid of density \(\rho\), coefficient of viscosity \(\eta\), and velocity \(v\) flowing through a pipe of diameter \(d\).

Step 1: Inertial Force

Inertial force is related to the mass of the fluid and its acceleration.

\[ F_i=ma \]

The mass of fluid is proportional to

\[ m\propto\rho d^3 \]

The characteristic acceleration is proportional to

\[ a\propto\frac{v^2}{d} \]

Therefore,

\[ F_i\propto\rho d^3\frac{v^2}{d} \]

Hence,

\[ F_i\propto\rho d^2v^2 \]

Step 2: Viscous Force

The viscous force is given by

\[ F_v=\eta A\frac{dv}{dy} \]

For a characteristic length \(d\),

\[ A\propto d^2 \]

and

\[ \frac{dv}{dy}\propto\frac{v}{d} \]

Therefore,

\[ F_v\propto\eta d^2\frac{v}{d} \]

Hence,

\[ F_v\propto\eta dv \]

Step 3: Ratio of Inertial Force to Viscous Force

Reynolds number is defined as the ratio of inertial force to viscous force.

\[ R_e=\frac{F_i}{F_v} \]

Substituting the proportional expressions,

\[ R_e\propto \frac{\rho d^2v^2}{\eta dv} \]

Therefore,

\[ R_e\propto\frac{\rho vd}{\eta} \]

The proportionality constant is taken as unity in the definition of Reynolds number. Hence,

Formula for Reynolds Number

\[ \boxed{R_e=\frac{\rho vd}{\eta}} \]

where

\(\rho\) = density of the fluid

\(v\) = velocity of flow

\(d\) = diameter of the pipe

\(\eta\) = coefficient of viscosity of the fluid

Reynolds Number at Critical Velocity

At critical velocity \(v_c\), the Reynolds number is called the critical Reynolds number.

Therefore,

\[ R_{ec}=\frac{\rho v_c d}{\eta} \]

Hence, the critical velocity is

\[ \boxed{ v_c=\frac{R_{ec}\eta}{\rho d} } \]

Nature of Flow

For flow through a pipe, Reynolds number is used to identify the nature of flow.

\[ R_e \lesssim 2000 \quad\Rightarrow\quad \text{Laminar flow} \]

\[ R_e \gtrsim 3000 \quad\Rightarrow\quad \text{Turbulent flow} \]

For values between these limits, the flow is generally regarded as a transition region.

Important Point

Reynolds number has no unit and no dimensions because it is a ratio of two forces.

Poiseuille’s Formula

Definition of Poiseuille’s Formula

Poiseuille’s formula gives the volume of a liquid flowing per second through a narrow cylindrical tube under a pressure difference, when the flow is steady and laminar.

The volume flow rate is given by

\[ \boxed{ Q=\frac{\pi P r^4}{8\eta l} } \]

where \(Q\) is the volume of liquid flowing per second, \(P\) is the pressure difference between the two ends of the tube, \(r\) is the radius of the tube, \(\eta\) is the coefficient of viscosity of the liquid, and \(l\) is the length of the tube.

Explanation

Consider a liquid flowing through a narrow cylindrical tube of radius \(r\) and length \(l\). Let the pressure difference between the two ends of the tube be \(P\).

Due to the pressure difference, the liquid flows through the tube. The layers of liquid near the wall move more slowly because of viscous resistance, while the layer along the central axis moves with maximum velocity.

Thus, the velocity of the liquid is not the same at all points across the tube. It is maximum at the centre and zero at the wall under the no-slip condition.

Conditions for Poiseuille’s Formula

Poiseuille’s formula is applicable when:

1. The tube is narrow and cylindrical.

2. The liquid is incompressible.

3. The flow is steady and laminar.

4. The liquid is Newtonian and its coefficient of viscosity remains constant.

5. The tube is sufficiently long so that the flow is fully developed.

Mathematical Derivation of Poiseuille’s Formula

Consider a liquid flowing through a cylindrical tube of radius \(R\) and length \(l\).

Let the pressure difference between the two ends of the tube be \(P\).

Consider a cylindrical layer of liquid of radius \(x\) and length \(l\), moving with velocity \(v\).

Step 1: Pressure Force

The pressure force acting on the cylindrical layer is

\[ F_P=P\pi x^2 \]

Step 2: Viscous Force

The surface area of the cylindrical layer in contact with the surrounding liquid is

\[ A=2\pi xl \]

The viscous force is

\[ F_v=\eta A\frac{dv}{dx} \]

Therefore,

\[ F_v=2\pi\eta xl\frac{dv}{dx} \]

Step 3: Equilibrium of the Liquid Layer

For steady flow, the pressure force is balanced by the viscous force.

\[ P\pi x^2 = -2\pi\eta xl\frac{dv}{dx} \]

The negative sign indicates that the viscous force opposes the direction of increase of velocity.

Cancelling \(\pi x\),

\[ Px=-2\eta l\frac{dv}{dx} \]

Therefore,

\[ dv=-\frac{P}{2\eta l}x\,dx \]

Step 4: Integrating

At the wall of the tube, \(x=R\) and \(v=0\). At a distance \(x\) from the axis, the velocity is \(v\).

Integrating,

\[ \int_v^0 dv = -\frac{P}{2\eta l} \int_x^R x\,dx \]

Therefore,

\[ -v = -\frac{P}{2\eta l} \left[ \frac{R^2-x^2}{2} \right] \]

Hence,

\[ v= \frac{P}{4\eta l}(R^2-x^2) \]

Step 5: Volume Flow Rate

Consider a thin cylindrical ring of radius \(x\) and thickness \(dx\).

Its cross-sectional area is

\[ dA=2\pi x\,dx \]

Therefore, the volume of liquid flowing per second through this ring is

\[ dQ=v\,dA \]

Substituting the values of \(v\) and \(dA\),

\[ dQ= \frac{P}{4\eta l}(R^2-x^2) (2\pi x\,dx) \]

Therefore,

\[ dQ= \frac{\pi P}{2\eta l} (R^2-x^2)x\,dx \]

Step 6: Total Volume Flow Rate

Integrating from the centre of the tube \((x=0)\) to the wall \((x=R)\),

\[ Q= \frac{\pi P}{2\eta l} \int_0^R (R^2-x^2)x\,dx \]

Expanding the integral,

\[ Q= \frac{\pi P}{2\eta l} \int_0^R (R^2x-x^3)\,dx \]

Therefore,

\[ Q= \frac{\pi P}{2\eta l} \left[ \frac{R^2x^2}{2} – \frac{x^4}{4} \right]_0^R \]

Substituting the limits,

\[ Q= \frac{\pi P}{2\eta l} \left( \frac{R^4}{2} – \frac{R^4}{4} \right) \]

Hence,

\[ Q= \frac{\pi P}{2\eta l} \frac{R^4}{4} \]

Poiseuille’s Formula

Therefore, the volume of liquid flowing per second through the tube is

\[ \boxed{ Q=\frac{\pi P R^4}{8\eta l} } \]

If the pressure difference is written as \(\Delta P\), the formula becomes

\[ \boxed{ Q=\frac{\pi \Delta P R^4}{8\eta l} } \]

Important Results from Poiseuille’s Formula

From

\[ Q=\frac{\pi \Delta P R^4}{8\eta l} \]

we obtain

\[ Q\propto\Delta P \]

Thus, the volume flow rate is directly proportional to the pressure difference.

Also,

\[ Q\propto R^4 \]

Therefore, the flow rate is extremely sensitive to the radius of the tube.

Further,

\[ Q\propto\frac{1}{\eta} \]

Hence, the flow rate decreases when the viscosity of the liquid increases.

Finally,

\[ Q\propto\frac{1}{l} \]

Therefore, the flow rate decreases as the length of the tube increases.

Arrangement of Capillary Tubes in Series and in Parallel

Capillary tubes can be connected in series or in parallel. The flow of liquid through such arrangements can be studied using Poiseuille’s formula.

For a capillary tube of length \(l\), radius \(r\), and coefficient of viscosity \(\eta\), Poiseuille’s formula gives

\[ Q=\frac{\pi r^4\Delta P}{8\eta l} \]

Therefore,

\[ \Delta P = Q\frac{8\eta l}{\pi r^4} \]

The quantity

\[ R=\frac{\Delta P}{Q} \]

is called the hydraulic resistance of the capillary tube.

Hence,

\[ \boxed{ R=\frac{8\eta l}{\pi r^4} } \]

1. Capillary Tubes in Series

When two or more capillary tubes are connected end-to-end so that the same liquid flows successively through each tube, they are said to be connected in series.

Diagram: Capillary Tubes in Series

In a series arrangement, the same volume flow rate \(Q\) passes through every capillary tube.

Let the pressure differences across the three capillary tubes be \(\Delta P_1\), \(\Delta P_2\), and \(\Delta P_3\).

Derivation for Series Arrangement

The total pressure difference is

\[ \Delta P=\Delta P_1+\Delta P_2+\Delta P_3 \]

Since

\[ R=\frac{\Delta P}{Q} \]

for the first tube,

\[ \Delta P_1=QR_1 \]

for the second tube,

\[ \Delta P_2=QR_2 \]

and for the third tube,

\[ \Delta P_3=QR_3 \]

Therefore,

\[ \Delta P=Q(R_1+R_2+R_3) \]

If \(R_s\) is the equivalent hydraulic resistance of the series arrangement,

\[ \Delta P=QR_s \]

Comparing the two expressions,

\[ QR_s=Q(R_1+R_2+R_3) \]

Hence,

\[ \boxed{ R_s=R_1+R_2+R_3 } \]

Equivalent Resistance in Terms of Radius and Length

For each capillary tube,

\[ R_1=\frac{8\eta l_1}{\pi r_1^4} \]

\[ R_2=\frac{8\eta l_2}{\pi r_2^4} \]

\[ R_3=\frac{8\eta l_3}{\pi r_3^4} \]

Therefore,

\[ \boxed{ R_s= \frac{8\eta}{\pi} \left( \frac{l_1}{r_1^4} + \frac{l_2}{r_2^4} + \frac{l_3}{r_3^4} \right) } \]

2. Capillary Tubes in Parallel

When the ends of two or more capillary tubes are connected to the same two points, the tubes are said to be connected in parallel.

Diagram: Capillary Tubes in Parallel

In a parallel arrangement, the pressure difference across every capillary tube is the same.

Let the volume flow rates through the three capillary tubes be \(Q_1\), \(Q_2\), and \(Q_3\).

Derivation for Parallel Arrangement

The total flow rate is the sum of the flow rates through the individual tubes.

\[ Q=Q_1+Q_2+Q_3 \]

Since the pressure difference across each tube is the same,

\[ Q_1=\frac{\Delta P}{R_1} \]

\[ Q_2=\frac{\Delta P}{R_2} \]

\[ Q_3=\frac{\Delta P}{R_3} \]

Therefore,

\[ Q= \frac{\Delta P}{R_1} + \frac{\Delta P}{R_2} + \frac{\Delta P}{R_3} \]

If \(R_p\) is the equivalent hydraulic resistance of the parallel arrangement,

\[ Q=\frac{\Delta P}{R_p} \]

Therefore,

\[ \frac{\Delta P}{R_p} = \frac{\Delta P}{R_1} + \frac{\Delta P}{R_2} + \frac{\Delta P}{R_3} \]

Cancelling \(\Delta P\),

\[ \boxed{ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} } \]

Equivalent Resistance in Terms of Radius and Length

Since

\[ R=\frac{8\eta l}{\pi r^4} \]

we have

\[ \frac{1}{R} = \frac{\pi r^4}{8\eta l} \]

Therefore, for three capillary tubes in parallel,

\[ \boxed{ \frac{1}{R_p} = \frac{\pi}{8\eta} \left( \frac{r_1^4}{l_1} + \frac{r_2^4}{l_2} + \frac{r_3^4}{l_3} \right) } \]

Special Case: Identical Capillary Tubes

If \(n\) identical capillary tubes, each having resistance \(R\), are connected in series, then

\[ \boxed{R_s=nR} \]

If \(n\) identical capillary tubes are connected in parallel, then

\[ \boxed{R_p=\frac{R}{n}} \]

Comparison of Series and Parallel Arrangements

Important Results

For capillary tubes in series, the same flow rate passes through every tube and the hydraulic resistances are added.

For capillary tubes in parallel, the pressure difference across every tube is the same and the total flow rate is the sum of the individual flow rates.

The hydraulic resistance of a capillary tube is

\[ \boxed{ R=\frac{8\eta l}{\pi r^4} } \]

Hence, a small change in the radius of a capillary tube produces a large change in its resistance because

\[ R\propto\frac{1}{r^4} \]

Surface Tension

Introduction

The surface of a liquid behaves differently from the interior of the liquid. A liquid surface tends to contract and occupy the minimum possible area. This property is responsible for several familiar phenomena, such as the formation of nearly spherical water drops, the floating of a small needle on the surface of water, and the movement of some insects on the water surface.

The behaviour of the liquid surface is explained in terms of the intermolecular forces acting between the molecules of the liquid. This property of the liquid surface is known as surface tension.

Definition of Surface Tension

Definition 1: Surface tension is the tangential force acting per unit length on either side of an imaginary line drawn on the free surface of a liquid, the force being perpendicular to the line and acting in the plane of the liquid surface.

Mathematically,

\[ \boxed{T=\frac{F}{l}} \]

where \(T\) is the surface tension, \(F\) is the force acting tangentially to the liquid surface, and \(l\) is the length of the line on which the force acts.

Definition 2: Surface tension is the property of a liquid surface by virtue of which it tends to contract to occupy the minimum possible surface area.

Explanation of Surface Tension

To understand surface tension, we need to consider the intermolecular forces acting on the molecules inside a liquid and on the molecules near its surface.

Molecular Range

The intermolecular forces between liquid molecules are effective only up to a certain small distance. This distance is called the molecular range.

In other words, molecular range is the maximum distance up to which a molecule can exert an appreciable attractive force on another molecule.

The molecular range is very small compared with the ordinary dimensions of a liquid.

Sphere of Influence

Consider a molecule \(A\) inside a liquid. Since intermolecular forces are effective only within the molecular range, the molecule \(A\) experiences appreciable forces only from molecules lying within a sphere of radius equal to the molecular range.

This imaginary sphere is called the sphere of influence of the molecule.

Surface Film

Consider a molecule deep inside a liquid. It is surrounded by other molecules in all directions within its sphere of influence. Therefore, the attractive forces acting on it are approximately balanced in all directions.

A molecule near the free surface, however, has fewer neighbouring molecules above it because there are no liquid molecules in the air above the surface. Hence, the attractive forces acting on the surface molecule are not balanced.

The thin region of liquid near the free surface in which the molecules experience this unbalanced intermolecular attraction is called the surface film.

Thus, molecules inside the liquid experience nearly balanced intermolecular forces, whereas molecules in the surface film experience a net inward attractive force.

Intermolecular Forces of Interaction

The molecules of a liquid interact with one another through intermolecular forces. Two important types of intermolecular forces are adhesive forces and cohesive forces.

1. Adhesive Forces

Adhesive forces are the attractive forces between molecules of different substances.

For example, the attractive force between water molecules and glass molecules is an adhesive force.

Adhesive forces are responsible for the tendency of a liquid to stick to the surface of another substance.

2. Cohesive Forces

Cohesive forces are the attractive forces between molecules of the same substance.

For example, the attractive forces between water molecules themselves are cohesive forces.

Cohesive forces are responsible for holding the molecules of a liquid together.

Illustration of Adhesive and Cohesive Forces

Origin of Surface Tension

The molecules in the surface film experience a net inward cohesive force because they are attracted by molecules inside the liquid and have fewer neighbouring molecules on the outer side.

Therefore, the surface film tends to contract and reduce its area. This tendency of the liquid surface to occupy the minimum possible area gives rise to the phenomenon of surface tension.

Surface tension can therefore be understood as a manifestation of intermolecular cohesive forces acting on the molecules in the surface film.

Formula for Surface Tension

Surface tension is defined as the force acting per unit length on the liquid surface.

Therefore,

\[ \boxed{T=\frac{F}{l}} \]

Hence, the force acting on a length \(l\) of the liquid surface is

\[ \boxed{F=Tl} \]

SI Unit of Surface Tension

From

\[ T=\frac{F}{l} \]

The SI unit of force is newton (N) and the SI unit of length is metre (m).

Therefore, the SI unit of surface tension is

\[ \boxed{\text{N m}^{-1}} \]

It is also written as

\[ \boxed{\text{N/m}} \]

CGS Unit of Surface Tension

In the CGS system, force is measured in dyne and length in centimetre.

Therefore, the CGS unit of surface tension is

\[ \boxed{\text{dyne cm}^{-1}} \]

Conversion of Units

We know that

\[ 1\,\text{N}=10^5\,\text{dyne} \]

and

\[ 1\,\text{m}=100\,\text{cm} \]

Therefore,

\[ 1\,\text{N m}^{-1} = \frac{10^5\,\text{dyne}}{100\,\text{cm}} \]

Hence,

\[ \boxed{ 1\,\text{N m}^{-1} = 10^3\,\text{dyne cm}^{-1} } \]

Therefore,

\[ \boxed{ 1\,\text{dyne cm}^{-1} = 10^{-3}\,\text{N m}^{-1} } \]

Dimensional Formula of Surface Tension

Surface tension is given by

\[ T=\frac{F}{l} \]

The dimensional formula of force is

\[ [F]=[MLT^{-2}] \]

The dimensional formula of length is

\[ [l]=[L] \]

Therefore,

\[ [T] = \frac{[MLT^{-2}]}{[L]} \]

Hence,

\[ \boxed{ [T]=[MT^{-2}] } \]

Physical Characteristics of Surface Tension

1. Surface tension is a property of liquids. It arises because of intermolecular forces acting on molecules in the surface film.

2. Surface tension acts tangentially. The force associated with surface tension acts along the surface of the liquid.

3. Surface tension tends to minimise surface area. A liquid surface tends to contract and occupy the minimum possible area.

4. Surface tension is a scalar quantity. It is completely specified by its magnitude and does not have a fixed direction like a vector quantity.

5. Surface tension depends on the nature of the liquid. Different liquids have different values of surface tension because their intermolecular forces are different.

6. Surface tension generally decreases with increase in temperature. As temperature increases, molecular motion increases and the effective cohesive forces at the surface decrease.

7. Surface tension is affected by impurities. Dissolved substances can either increase or decrease the surface tension depending on the nature and concentration of the impurity.

8. Surface tension has the same dimensions as force per unit length. Its dimensional formula is

\[ \boxed{[MT^{-2}]} \]

and its SI unit is

\[ \boxed{\text{N m}^{-1}} \]

Important Formula and Units at a Glance

\[ \boxed{T=\frac{F}{l}} \]

SI unit: \(\text{N m}^{-1}\)

CGS unit: \(\text{dyne cm}^{-1}\)

Conversion: \[ \boxed{1\,\text{N m}^{-1}=10^3\,\text{dyne cm}^{-1}} \]

Dimensional formula: \[ \boxed{[MT^{-2}]} \]

Surface Energy

Definition of Surface Energy

Surface energy is the excess energy possessed by the molecules present at the surface of a liquid due to the unbalanced intermolecular forces acting on them.

Alternatively, surface energy can be defined as the work done in increasing the surface area of a liquid by one unit area, provided the temperature remains constant.

Formula for Surface Energy

If \(W\) is the work done in increasing the surface area by \(\Delta A\), then the surface energy per unit area is

\[ \boxed{E_s=\frac{W}{\Delta A}} \]

Therefore, the total surface energy associated with an increase in surface area \(\Delta A\) is

\[ \boxed{W=E_s\Delta A} \]

Since the work required to increase the surface area of a liquid is related to surface tension,

\[ \boxed{E_s=T} \]

Thus, surface energy per unit area is numerically equal to surface tension for a liquid surface.

Relation Between Surface Energy and Surface Tension

Consider a liquid surface whose area is increased by \(\Delta A\). If \(T\) is the surface tension, the work done in increasing the surface area is

\[ W=T\Delta A \]

Therefore,

\[ \frac{W}{\Delta A}=T \]

But surface energy per unit area is

\[ E_s=\frac{W}{\Delta A} \]

Hence,

\[ \boxed{E_s=T} \]

Explanation of Surface Energy

Molecules inside a liquid are surrounded by neighbouring molecules in all directions, so the intermolecular attractive forces acting on them are approximately balanced.

Molecules at the free surface, however, have fewer neighbouring molecules on the outer side. Therefore, they experience a net inward cohesive force and possess greater potential energy than molecules inside the liquid.

This excess energy associated with the surface molecules is called surface energy.

If the surface area of the liquid is increased, more molecules have to be brought to the surface. Work must therefore be done against the cohesive forces. This work is stored as additional surface energy.

Hence, a liquid surface naturally tends to reduce its area because a smaller surface area corresponds to lower surface energy.

Diagram: Increase in Surface Area and Surface Energy

The diagram represents an increase in the surface area of a liquid. Work must be done against surface tension, and this work is stored as additional surface energy.

SI Unit of Surface Energy

Surface energy is energy or work done per unit area. Therefore, its SI unit is

\[ \boxed{\text{J m}^{-2}} \]

Since

\[ 1\,\text{J}=1\,\text{N m} \]

we can write

\[ 1\,\text{J m}^{-2}=1\,\text{N m}^{-1} \]

Therefore, the SI unit of surface energy per unit area is also

\[ \boxed{\text{N m}^{-1}} \]

CGS Unit

In the CGS system, the unit of surface energy per unit area is

\[ \boxed{\text{erg cm}^{-2}} \]

It is equivalent to

\[ \boxed{\text{dyne cm}^{-1}} \]

Conversion of Units

Since

\[ 1\,\text{N m}^{-1} = 10^3\,\text{dyne cm}^{-1} \]

and

\[ 1\,\text{J m}^{-2} = 1\,\text{N m}^{-1} \]

therefore,

\[ \boxed{ 1\,\text{J m}^{-2} = 10^3\,\text{erg cm}^{-2} } \]

Dimensional Formula of Surface Energy per Unit Area

Surface energy per unit area is

\[ E_s=\frac{W}{A} \]

The dimensional formula of work is

\[ [W]=[ML^2T^{-2}] \]

The dimensional formula of area is

\[ [A]=[L^2] \]

Therefore,

\[ [E_s] = \frac{[ML^2T^{-2}]}{[L^2]} \]

Hence,

\[ \boxed{[E_s]=[MT^{-2}]} \]

Applications of Surface Energy

1. Formation of Spherical Drops: A liquid tends to minimise its surface area because a smaller surface area corresponds to lower surface energy. This is why small liquid drops tend to become spherical.

2. Formation of Bubbles: Surface energy is involved in creating the additional surface area of a soap bubble. Work must be done to increase the surface area of the bubble.

3. Spraying of Liquids: When a liquid is broken into very small droplets, its total surface area increases. Work is required to create this additional surface area.

4. Formation of Foam: Foam contains a large total surface area of liquid films. Energy is required to create and maintain these surfaces.

5. Wetting and Spreading: Surface energy plays an important role when a liquid spreads over a solid surface. The balance between adhesive and cohesive forces determines the extent of spreading.

6. Cleaning Action of Detergents: Detergents reduce the surface tension of water, making it easier for water to spread over surfaces and penetrate small spaces.

Important Point

Surface tension and surface energy per unit area are numerically equal and have the same dimensions and SI units:

\[ \boxed{ T=E_s } \]

Thus,

\[ \boxed{ \text{Surface tension}=\text{Surface energy per unit area} } \]

Shape of Drops

The shape of a liquid drop depends mainly on the competition between two forces: gravitational force and surface tension.

Effect of Gravitational Force and Surface Tension

The gravitational force acts vertically downward on a liquid drop due to its weight. It tends to pull the liquid downward and deform the drop.

Surface tension, on the other hand, acts along the free surface of the liquid and always tends to reduce the surface area. For a given volume of liquid, the sphere has the minimum possible surface area. Therefore, surface tension tends to make a free liquid drop spherical.

For a very small liquid drop, the effect of surface tension is much greater than the effect of gravity. Hence, a small free liquid drop tends to acquire a nearly spherical shape.

As the size of the drop increases, its weight and hence the gravitational effect become more important. Gravity then tends to deform the drop from its spherical shape.

Thus, the shape of a liquid drop is determined by the relative effects of surface tension and gravitational force. Surface tension tends to make the drop spherical, whereas gravity tends to deform it.

Liquid Drop on a Plane Solid Surface

Consider a small quantity of liquid placed on a plane solid surface. At the point of contact of the solid, liquid and air, three surface tensions act:

• \(T_{SA}\) = surface tension between solid and air
• \(T_{SL}\) = surface tension between solid and liquid
• \(T_{LA}\) = surface tension between liquid and air

Let \(\theta\) be the angle of contact between the liquid surface and the solid surface.

For equilibrium at the point of contact, the horizontal components of the surface tensions must balance. Therefore,

Hence,

The shape of the liquid and the angle of contact depend on the relative values of these surface tensions. The following three cases arise.

Case 1: \(T_{SA} > T_{SL}\)

When the surface tension between the solid and air is greater than the surface tension between the solid and liquid,

From

we have

Therefore, the angle of contact is acute:

This condition is fulfilled when the molecules of the liquid are strongly attracted towards the molecules of the solid surface. Therefore, the liquid tends to spread over the solid surface and wets it.

Examples: Water on glass and kerosene oil on glass.

Case 2: \(T_{SA} < T_{SL}\)

When the surface tension between the solid and air is less than the surface tension between the solid and liquid,