**CBSE NOTES CLASS 11 PHYSICS CHAPTER 10**

**MECHANICAL PROPERTIES OF FLUIDS**

**Fluid Statics**

The study of the fluids at rest is called fluid statics.

**Density**

The density ρ for a fluid of is defined as the mass per unit volume.

For mass *m* occupying volume V, ρ = $\frac{\mathrm{m}}{\mathrm{V}}$

The SI unit of density is kg m^{–3}.

It is a positive scalar quantity.

The density of water at 4^{o}C (277 K) is 1.0 × 10^{3} kg m^{–3}.

**Relative density**

The relative density of a substance is the ratio of the density of a substance to the density of water. It is a ratio of similar quantities and has no unit.

$$\mathrm{Relative\; density\; =}\frac{\mathrm{Density\; of\; substance}}{\mathrm{Density\; of\; water}}$$

**Pressure**

When an object is submerged in a fluid at rest, the fluid exerts a force on the surface of the object, which is always normal to the surface of the object. The normal force per unit area is called pressure,

$${\mathrm{P}}_{\mathrm{a}\mathrm{v}\mathrm{}}=\frac{\mathrm{F}}{\mathrm{A}}$$

For irregular surfaces, pressure on a small area ΔA, due to normal force ΔF is given by,

$$\mathrm{P}=\mathrm{}\underset{\mathrm{\Delta}\mathrm{A\to 0}}{\mathrm{lim}}\mathrm{}\mathrm{}\frac{\mathrm{\Delta}\mathrm{F}}{\mathrm{\Delta}\mathrm{A}}$$

SI unit of pressure is is Nm^{-2} or Pascal (Pa).

Also,

1 atm = 1.013 × 10^{5} Pa

1 bar = 10^{5} Pa

Pressure is a scalar quantity. It is the component of the force normal to the area under consideration and not the (vector) force that appears in the numerator.

**Explain why the force by a fluid on a submerged object is always perpendicular to the surface of the object.**

The force exerted by the fluid on the object is always perpendicular to the surface of the object. This is because if there were a component of force parallel to the surface, the object will also exert a force on the fluid parallel to it; as a consequence of Newton’s third law. This force will cause the fluid to flow parallel to the surface. Since the fluid is at rest, this cannot happen.

Hence, the force exerted by the fluid at rest has to be perpendicular to the surface in contact with it.

**Pascal’s Law**

“Whenever external pressure is applied on any part of a fluid contained in a vessel, it is transmitted undiminished and equally in all directions. This is the Pascal’s law for transmission of fluid pressure.“

The pressure in a fluid at rest is the same at all points if they are at the same height.

This principle can be stated mathematically as follows,

ΔP = h ρ g

Where,

ΔP is the difference in pressure at two points within a fluid column, called hydrostatic pressure

ρ is the fluid density of the fluid

g is acceleration due to

h is the height of fluid above the point of measurement, or the difference in elevation between the two points within the fluid column.

**Proof of Pascal’s Law**

Let us consider a fluid element ABC-DEF in the form of a right-angled prism. In principle, this prismatic element is very small so that every part of it can be considered at the same depth from the liquid surface and therefore, the effect of the gravity is the same at all these points.

The forces are exerted by the rest of the fluid and they must be normal to the surfaces of the element as discussed above.

Thus, the fluid exerts pressures P_{a}, P_{b} and P_{c} on this element of area corresponding to the normal forces F_{a}, F_{b} and F_{c} on the faces BEFC, ADFC and ADEB denoted by A_{a}, A_{b} and A_{c} respectively. Then by equilibrium, we have,

F_{b} cos θ_{1} = F_{c},

F_{b} sin θ_{1} = F_{a}

And by geometry,

A_{b} cos θ_{1} = A_{c},

A_{b} sin θ_{1} = A_{a}

Thus, $\mathrm{}\mathrm{}\mathrm{}\frac{{\mathrm{F}}_{\mathrm{a}}}{{\mathrm{A}}_{\mathrm{a}}}=\frac{{\mathrm{F}}_{\mathrm{b}}}{{\mathrm{A}}_{\mathrm{b}}}=\frac{{\mathrm{F}}_{\mathrm{c}}}{{\mathrm{A}}_{\mathrm{c}}}$

Hence, pressure exerted is same in all directions in a fluid at rest.

**Variation of Pressure with Depth**

The pressures at points 1 and 2 are P_{1} and P_{2} respectively. Consider a cylindrical element of fluid having area of base A and height h. As the fluid is at rest the resultant horizontal forces should be zero and the resultant vertical forces should balance the weight of the element.

The forces acting in the vertical direction are due to the fluid pressure at the top (P_{1}A) acting downward, at the bottom (P_{2}A) acting upward.

If mg is weight of the fluid in the cylinder then

(P_{2} − P_{1}) A = mg

Now, if ρ is the mass density of the fluid, we have the mass of fluid

m = ρV= ρ h A

⇒ P_{2} − P_{1}= ρ g h

Pressure at depth P = P_{a} + ρ g h, where P_{a} is the atmospheric pressure.

The excess of pressure, P − P_{a}, at depth h is called a **gauge pressure **at that point.

**Hydrostatic paradox**

The height of the fluid column does not depend on the cross-sectional area of the container. Liquid pressure is the same at all points at the same horizontal level (same depth).

Consider three vessels A, B and C of different shapes. They are connected at the bottom by a horizontal pipe. On filling with water the level in the three vessels is the same though they hold different amounts of water. This is so, because water at the bottom has the same pressure below each section of the vessel.

**Atmospheric Pressure and Gauge Pressure**

The pressure of the atmosphere at any point is equal to the weight of a column of air of unit cross sectional area extending from that point to the top of the atmosphere.

At sea level it is 1.013 × 10^{5} Pa (1 atm).

**Mercury Barometer**

The device used to measure atmospheric pressure is known as mercury barometer.

It consists of a long glass tube closed at one end, filled with mercury and inverted into a tub of mercury as shown.

The space above the mercury column in the tube contains only mercury vapour whose pressure P* *is so small that it may be neglected. The pressure inside the column at point A must equal the pressure at point B, which is at the same level.

Pressure at B = atmospheric pressure = P_{a}

P_{a} = ρ g h

where ρ is the density of mercury and h* *is the height of the mercury column in the tube.

The mercury column in the barometer has a height of about 76 cm at sea level equivalent to 1 atm.

A pressure equivalent of 1 mm of mercury is called a torr (after Torricelli).

1 torr = 133 Pa.

**Open Tube Manometer**

It is a U shaped tube filled with a low density liquid (such as oil) for measuring small pressure differences and a high density liquid (such as mercury) for large pressure differences.

One end of the tube is open to the atmosphere and other end is connected to the system whose pressure we want to measure.

The pressure P at A is equal to pressure at point B. What we normally measure is the gauge pressure, which is P − P_{a}, proportional to manometer height h.

**Mechanical Advantage**

**Hydraulic lift **and **hydraulic brakes **are based on the Pascal’s law. In these devices fluids are used for transmitting pressure.

Two pistons are separated by the space filled with a liquid. A piston of small cross section A1 is used to exert a force F1 directly on the liquid.

The pressure P = $\frac{{\mathrm{F}}_{1}}{{\mathrm{A}}_{1}}$ is transmitted throughout the liquid to the larger cylinder attached with a larger piston of area A_{2}, which results in an upward force of P × A_{2}.

Therefore, the piston is capable of supporting a large force (large weight of, say a car, or a truck, placed on the platform)

$${\mathrm{F}}_{2}\mathrm{}=\mathrm{}\mathrm{P}\times {\mathrm{A}}_{1}\mathrm{}=\frac{{\mathrm{F}}_{1}}{{\mathrm{A}}_{1}}\times {\mathrm{A}}_{2}$$

By changing the force at A_{1}, the platform can be moved up or down. Thus, the applied force has been increased by a factor of $\frac{{\mathrm{A}}_{2}}{{\mathrm{A}}_{1}}$. This factor is called the **mechanical advantage** of the device.

#### Buoyancy

When an object is immersed in a fluid it experiences an upward force called buoyant force. This property is called buoyancy or upthrust.

- Example, an empty bottle held under the water, bounces upwards when released.

**Why objects float or sink when placed on the surface of water?**

When we place a piece of cork and an iron nail, of the same mass, on the surface of water the cork floats and the nail sinks.

If the density of an object is less than the density of a liquid, it will float on the liquid and if the density of an object is more than the density of a liquid, it will sink in the liquid.

**Archeimedes’ Principle**

Archimedes’ principle states that, when a body is partially or fully immersed in a fluid it experiences an upward force that is equal to the weight of the fluid displaced by it.

F* = m*_{f}* g = ρ*_{f}* g *V

Where, F = Buoyant force on a given body,

V = Volume of the displaced fluid

*g* = Acceleration due to gravity

*ρ*_{f}* * = Density of the fluid

*m*_{f}* = *mass of the fluid displaced

**Apparent weight of the body (inside the fluid)**

= Weight in air or vaccum – Upthrust

⇒ W_{a} = (*ρ* - *ρ*_{f}) *g* V

Where *ρ* is the density of body.

Archimedes principle is helpful in finding the buoyant force, volume of displaced body, density of body or density of fluid if some of these quantities are known.

**Fluid Dynamics**

The study of the fluids in motion is known as fluid dynamics.

**Steady Flow**

The flow of the fluid is said to be **steady **if at any given point, the velocity of each passing fluid particle remains constant in time. The velocity of a particular particle may change as it moves from one point to another. Every particle which passes the point behaves exactly in the same manner

Each particle follows a smooth path, and the paths of the particles do not cross each other.

**Streamline Flow**

The path taken by a fluid particle under a steady flow is a **streamline**. It is defined as a curve whose tangent at any point is in the direction of the fluid velocity at that point.

No two streamlines can cross, for if they do, an oncoming fluid particle can go either one way or the other and the flow would not be steady. Hence, in steady flow, the map of flow is stationary in time.

Spacing between the streamlines shows the relative velocity of the fluid. Higher the velocity, closer is the streamlines.

**Equation of Continuity**

When an incompressible fluid is in motion, it must move in such a way that mass is conserved.

Volume flux or flow rate remains constant throughout the pipe of flow.

Thus, at narrower portions where the streamlines are closely spaced, velocity increases and vice versa.

Consider planes perpendicular to the direction of fluid flow e.g., at three points P, R and Q.

The plane pieces are so chosen that their boundaries be determined by the same set of streamlines.

This means that number of fluid particles crossing the surfaces as indicated at P, R and Q is the same.

If area of cross-sections at these points are A_{P}, A_{R} and A_{Q} and speeds of fluid particles are v_{P}, v_{R} and v_{Q}, then mass of fluid Δm_{P} crossing at A_{P} in a small interval of time Δt is,

Δm_{P} = ρ_{P}A_{P}v_{P}Δt

Similarly mass of fluid Δm_{R} flowing or crossing at A_{R} in a small interval of time Δt is,

Δm_{R} = ρ_{R}A_{R}v_{R}Δt

and mass of fluid Δm_{Q} crossing at A_{Q} is,

Δm_{Q} = ρ_{Q}A_{Q}v_{Q}Δt

For all the cases, the mass of liquid flowing out equals the mass flowing in

Therefore,

ρ_{P}A_{P}v_{P}Δt = ρ_{R}A_{R}v_{R}Δt = ρ_{Q}A_{Q}v_{Q}Δt

For flow of incompressible fluids

ρ_{P} = ρ_{R} = ρ_{Q}

⇒ A_{P}v_{P} = A_{R}v_{R} = A_{Q}v_{Q}

Or Av = Volume flux = constant.

**Turbulent Flow**

Flow of fluids, in which the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction, is called turbulent flow

Examples of Turbulent flow – When the stream encounters rocks, small foamy whirlpool-like regions called **‘white water rapids’** are formed. Circular movements of water causing a small whirlpool, are called **eddies**.

**Critical Speed**

Limiting value of speed beyond which, the flow loses steadiness and becomes **turbulent**, is called **critical speed**.

**Effects of Turbulence**

- Turbulence dissipates kinetic energy in the form of heat.
Aeroplanes are designed to minimize turbulence.

- Turbulence (like friction) is sometimes desirable.
Turbulence promotes mixing and increases the rates of transfer of mass, momentum and energy.

The blades of a kitchen mixer induce turbulent flow which helpful in making milk shake or beating the eggs.

**Bernoulli's Theorem**

When an **incompressible fluid **with** zero viscosity, **is flowing along a **streamline**, the total of the pressure, kinetic energy per unit volume and potential energy per unit volume remain constant.

$$\mathrm{P}+\frac{1}{2}\mathrm{\rho}{\mathrm{v}}^{2}+\mathrm{\rho}\mathrm{g}\mathrm{h}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

**Proof of Bernoulli's Theorem**

Consider a fluid moving in a pipe of varying cross-sectional area. Let the pipe be at varying heights.

Assume that an incompressible fluid is flowing through the pipe in a steady flow.

Its velocity must change as a consequence of equation of continuity.

A force is required to produce this acceleration, which is caused by the fluid surrounding it, i.e., the pressure must be different in different regions.

Consider the flow at two regions 1 (i.e. BC) and 2 (i.e. DE). Consider the fluid initially lying between B and D. In an infinitesimal time interval Δt, this fluid would have moved a distance v_{1}Δt to C (v_{1}Δt is small enough to assume constant cross-section along BC). In the same interval Δt the fluid initially at D moves to E, a distance equal to v_{2}Δt, where v_{1} and v_{2} are speeds at B and D respectively.

Pressures P_{1} and P_{2} act as shown on the plane faces of areas A_{1} and A_{2} binding the two regions.

The work done on the fluid at left end (BC) is

W_{1} = P_{1}A_{1}(v_{1}Δt) = P_{1}ΔV.

Since the same volume ΔV passes through both the regions (from the equation of continuity) the work done by the fluid at the other end (DE) is

W_{2} = P_{2}A_{2}(v_{2}Δt) = P_{2}ΔV

Or, the work done on the fluid is –P_{2}ΔV.

So the total work done on the fluid is

ΔW = W_{1} – W_{2 }= (P_{1}− P_{2}) ΔV

Part of this work goes into changing the kinetic energy of the fluid, and part goes into changing the gravitational potential energy.

If the density of the fluid is ρ and Δm = ρA_{1}v_{1}Δt = ρΔV is the mass passing through the pipe in time Δt, then change in gravitational potential energy is

ΔU = ρ g ΔV (h_{2} − h_{1})

The change in its kinetic energy is

$$\mathrm{\Delta}\mathrm{K}=\frac{1}{2}\mathrm{\rho}\mathrm{}\mathrm{\Delta}\mathrm{V}\mathrm{}\mathrm{}({{\mathrm{v}}_{2}}^{2}-{{\mathrm{v}}_{1}}^{2})\mathrm{}$$

As per work energy theorem,

Work done = Change in energy

i.e., ΔW = ΔK + ΔU

Putting the values we get,

$$\left({\mathrm{P}}_{1}-{\mathrm{P}}_{2}\right)\mathrm{\Delta}\mathrm{V}=\frac{1}{2}\mathrm{\rho}\mathrm{}\mathrm{\Delta}\mathrm{V}\left({{\mathrm{v}}_{2}}^{2}-{{\mathrm{v}}_{1}}^{2}\right)+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}\mathrm{\Delta}\mathrm{V}({\mathrm{h}}_{2}\mathrm{}-\mathrm{}{\mathrm{h}}_{1})\mathrm{}\mathrm{}$$

Dividing each term by ΔV we get,

$$\left({\mathrm{P}}_{1}-{\mathrm{P}}_{2}\right)=\frac{1}{2}\mathrm{\rho}\mathrm{}\left({{\mathrm{v}}_{2}}^{2}-{{\mathrm{v}}_{1}}^{2}\right)+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}({\mathrm{h}}_{2}\mathrm{}-\mathrm{}{\mathrm{h}}_{1})$$

Rearranging the terms,

$${\mathrm{P}}_{1}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{1}}^{2}+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}{\mathrm{h}}_{1}=\mathrm{}{\mathrm{P}}_{2}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{2}}^{2}+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}{\mathrm{h}}_{2}$$

$$\Rightarrow \mathrm{}\mathrm{P}+\frac{1}{2}\mathrm{\rho}{\mathrm{v}}^{2}+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}\mathrm{h}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

When a fluid is at rest i.e. its velocity is zero everywhere, Bernoulli’s equation becomes

P_{1} + ρgh_{1} = P_{2} + ρgh_{2 }

_{ }⇒_{ } (P_{1}− P_{2}) = ρg (h_{2} − h_{1})

This is Pascal’s law.

**Speed of Efflux: Torricelli’s Law**

**Efflux** means fluid outflow.

Torricelli's law states that the speed of efflux, v, of a fluid through a sharp-edged hole in a tank at a depth h is the same as the speed that a body would acquire in falling freely from a height h.

$\mathrm{v}=\mathrm{}\sqrt{2\mathrm{g}\mathrm{h}}$

**Proof of Torricelli’s Law**

Consider a tank containing a liquid of density ρ with a small hole in its side at a height y_{1} from the bottom. The air above the liquid, whose surface is at height y_{2}, is at pressure P.

From the equation of continuity

$${\mathrm{v}}_{1}\mathrm{}{\mathrm{A}}_{1}\mathrm{}=\mathrm{}{\mathrm{v}}_{2}\mathrm{}{\mathrm{A}}_{2}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\mathrm{v}}_{2}=\mathrm{}\mathrm{}{\mathrm{v}}_{1}\frac{{\mathrm{A}}_{1}}{{\mathrm{A}}_{2}}$$

If (A_{2} >>A_{1}), then we may take the fluid to be approximately at rest at the top, i.e. v_{2} = 0.

Applying the Bernoulli equation at points 1 and 2 and noting that at the hole P_{1} = P_{a},

$${\mathrm{P}}_{\mathrm{a}}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{1}}^{2}+\mathrm{\rho}\mathrm{g}{\mathrm{y}}_{1}=\mathrm{P}+\mathrm{\rho}\mathrm{g}{\mathrm{y}}_{2}\mathrm{}$$

$$\Rightarrow \mathrm{}\mathrm{}{\mathrm{v}}_{1}=\mathrm{}\sqrt{2\mathrm{g}\mathrm{h}+\frac{2(\mathrm{P}-{\mathrm{P}}_{\mathrm{a}})\mathrm{}}{\mathrm{\rho}}}\mathrm{}$$

Where h = y_{2} – y_{1}

When P >> P_{a}, 2gh may be ignored, the speed of efflux is determined by the container pressure. Such a situation occurs in rocket propulsion.

If the tank is open to the atmosphere, then P = P_{a} and_{ }we have, $\mathrm{v}=\mathrm{}\sqrt{2\mathrm{g}\mathrm{h}}$

**Venturi-meter**

The Venturi-meter is a device to measure the flow speed of incompressible fluid. It consists of a tube with a broad diameter and a small constriction at the middle. A manometer in the form of a U-tube is attached to it, with one arm at the broad neck point of the tube and the other at constriction.

The manometer contains a liquid of density ρ_{m}. The speed v_{1} of the liquid flowing through the tube at the broad neck area A is to be measured.

From equation of continuity the speed at the constriction becomes v_{2} = $\frac{\mathrm{A}}{\mathrm{a}}$ × v_{1}

Using Bernoulli’s equation, we get,

$${\mathrm{P}}_{1}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{1}}^{2}+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}{\mathrm{h}}_{1}=\mathrm{}{\mathrm{P}}_{2}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{2}}^{2}+\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}{\mathrm{h}}_{2}$$

Since centres of both th portions are at same height, h_{1} = h_{2}, the equation is reduced to,

$${\mathrm{P}}_{1}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{1}}^{2}=\mathrm{}{\mathrm{P}}_{2}+\frac{1}{2}\mathrm{\rho}{{\mathrm{v}}_{2}}^{2}$$

$$\Rightarrow \mathrm{}\left({\mathrm{P}}_{1}-{\mathrm{P}}_{2}\right)=\frac{1}{2}\mathrm{\rho}\mathrm{}\left({{\mathrm{v}}_{2}}^{2}-{{\mathrm{v}}_{1}}^{2}\right)$$

Putting the value of v_{2}, we get,

$$\Rightarrow \mathrm{}\left({\mathrm{P}}_{1}-{\mathrm{P}}_{2}\right)=\frac{1}{2}\mathrm{\rho}\mathrm{}{{\mathrm{v}}_{1}}^{2}\left[{\left(\frac{\mathrm{A}}{\mathrm{a}}\right)}^{2}-1\right]$$

This pressure difference causes the fluid in the U tube connected at the narrow neck to rise in comparison to the other arm. The difference in height (h) measures the pressure difference.

$$\left({\mathrm{P}}_{1}-{\mathrm{P}}_{2}\right)={\mathrm{\rho}}_{\mathrm{m}}\mathrm{}\mathrm{g}\mathrm{}\mathrm{h}=\frac{1}{2}\mathrm{\rho}\mathrm{}{{\mathrm{v}}_{1}}^{2}\left[{\left(\frac{\mathrm{A}}{\mathrm{a}}\right)}^{2}-1\right]$$

$$\Rightarrow \mathrm{}{\mathrm{v}}_{1}\mathrm{}=\sqrt{\left(\frac{2\mathrm{}{\mathrm{\rho}}_{\mathrm{m}}\mathrm{}\mathrm{g}\mathrm{}\mathrm{h}}{\mathrm{\rho}}\right)}{\left[{\left(\frac{\mathrm{A}}{\mathrm{a}}\right)}^{2}-1\right]}^{-\frac{1}{2}}\mathrm{}$$

The principle behind venture-meter has many applications.

- The carburetor of automobiles has a Venturi channel (nozzle) through which air flows with a large speed. The pressure is then lowered at the narrow neck and the petrol (gasoline) is sucked up in the chamber to provide the correct mixture of air and fuel necessary for combustion.
- Filter pumps or aspirators, Bunsen burner, atomisers and sprayers used for perfumes or to spray insecticides work on the same principle.

**Blood Flow and Heart Attack**

Bernoulli’s principle helps in explaining blood flow in artery.

The artery may get constricted due to accumulation of plaque on its inner walls. In order to drive the blood through this constriction a greater demand is placed on the activity of the heart.

The speed of the flow of the blood in this region is raised which lowers the pressure inside and the artery may collapse due to the external pressure.

The heart exerts further pressure to open this artery and force the blood through.

As the blood rushes through the opening, the internal pressure once again drops due to same reasons leading to a repeat collapse.

This may result in heart attack.

**Dynamic Lift**

Dynamic lift is the force that acts on a body, such as airplane wing, a hydrofoil or a spinning ball, by virtue of its motion through a fluid.

**Ball moving without spin**

The streamlines around a non-spinning ball moving relative to a fluid are symmetrical and the velocity of fluid (air) above and below the ball at corresponding points is the same resulting in zero pressure difference. The air therefore, exerts no upward or downward force on the ball and the ball moves in straight line.

**Ball moving with spin**

A ball which is spinning drags air along with it. If the surface is rough more air will be dragged.

The ball is moving forward and relative to it the air is moving backwards. Therefore, the velocity of air above the ball relative to it is larger and below it is smaller.

The streamlines of air for a ball which is moving and spinning at the same time get crowded above and rarified below.

This difference in the velocities of air results in the pressure difference between the lower and upper faces and there is a net upward force on the ball. This dynamic lift due to spining is called **Magnus effect**.

**Aerofoil or lift on aircraft wing**

An aerofoil is a solid piece shaped to provide an upward dynamic lift when it moves horizontally through air.

The cross-sections of the wings of an aeroplane look like the aerofoil with streamlines around it.

When the aerofoil moves against the wind, the orientation of the wing relative to flow direction causes the streamlines to crowd together above the wing more than those below it.

The flow speed on top is higher than that below it and there is an upward force resulting in a dynamic lift of the wings which balances the weight of the plane.

**Viscosity**

The resistance to fluid motion due to internal friction is called viscosity. It is the measure of being thick, sticky, and semi-fluid in consistency

Consider a fluid like oil enclosed between two glass plates. The bottom plate is fixed while the top plate is moved with a constant velocity **v **relative to the fixed plate.

The fluid in contact with a surface has the same velocity as that of the surfaces. Hence, the layer of the fluid in contact with top surface moves with a velocity **v **and the layer of the liquid in contact with the fixed surface is stationary. The velocities of layers increase uniformly from bottom (zero velocity) to the top layer (velocity **v**).

For any layer of liquid, its upper layer pulls it forward while lower layer pulls it backward. This results in force between the layers. This type of flow is known as laminar or stream line flow.

Let us assume that a portion of liquid, having shape ABCD, takes the shape of AEFD after short interval of time (Δt).

During this time interval the liquid has undergone a shear strain of Δx/l. Since, the strain in a flowing fluid increases with time continuously, the stress is found experimentally to depend on ‘rate of change of strain’ or ‘strain rate’ i.e. Δx/(lΔt) or v/*l* instead of strain itself.

The coefficient of viscosity (pronounced ‘eta’) for a fluid is defined as the ratio of shearing stress to the strain rate.

$$\mathrm{\eta}=\frac{\mathrm{F}/\mathrm{A}}{\mathrm{v}/\mathrm{l}}\mathrm{}=\frac{\mathrm{F}\mathrm{l}}{\mathrm{v}\mathrm{A}}$$

The SI unit of viscosity is poiseiulle (Pl). Its other units are Nsm^{-2} or Pas.

- When a fluid is flowing in a pipe or a tube, then velocity of the liquid layer along the axis of the tube is the maximum and decreases gradually as we move towards the walls where it becomes zero. The velocity on a cylindrical surface in a tube is constant.

- Thin liquids like water, alcohol etc. are less viscous than thick liquids like coal tar, blood, glycerin etc.
- Blood is ‘thicker’ (more viscous) than water.
- Relative viscosity $\left(\frac{{\mathrm{\eta}}_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{d}}}{{\mathrm{\eta}}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}\right)$ of blood remains constant between 0
^{o}C and 37^{o}C. - The viscosity of liquids decreases with temperature while it increases in the case of gases.

**Poiseiulle’s Formula **

For a system with a pipe of radius r, fluid viscosity η, distance between the two points along the pipe Δ*x* = *x*_{2} – *x*_{1}, and the volumetric flow rate φ, of the fluid, the pressure difference between the two points along the pipe Δp is given by

$$\mathrm{\Delta}\mathrm{p}=\frac{8\mathrm{}\mathrm{}\mathrm{\eta}\mathrm{}\mathrm{}\mathrm{\Delta}\mathrm{x}}{\mathrm{\pi}{\mathrm{r}}^{4}}$$

**Stokes Law**

The force of viscosity on a small sphere moving through a viscous fluid is given by:

$${\mathrm{F}}_{\mathrm{d}}=\mathrm{}6\mathrm{\pi}\mathrm{}\mathrm{\eta}\mathrm{}\mathrm{r}\mathrm{}\mathrm{v}$$

where:

- F
_{d}is the frictional force – known as**Stokes' drag**– acting on the interface between the fluid and the particle - η is the coefficient of viscosity
- r is the radius of the spherical object
- v is the flow velocity relative to the object.

In SI units, F_{d} is given in Newtons, η in Pa s, r in meters, and v in m/s.

Stokes' law makes the following assumptions for the behavior of a particle in a fluid:

- Laminar Flow
- Spherical particles
- Homogeneous (uniform in composition) material
- Smooth surfaces
- Particles do not interfere with each other.

For molecules Stokes' law is used to define their Stokes radius.

**Terminal velocity of raindrop**

Consider a raindrop falling through air. It accelerates initially due to gravity. As the velocity increases, the retarding force also increases. Finally when **viscous force plus buoyant force** becomes equal to force due to gravity, the net force becomes zero and so does the acceleration. The sphere (raindrop) then descends with a constant velocity.

In equilibrium, this terminal velocity is given by

$$6\mathrm{\pi}\mathrm{}\mathrm{\eta}\mathrm{}\mathrm{r}\mathrm{}{\mathrm{v}}_{\mathbf{t}}\mathrm{}=\mathrm{}\mathrm{}\mathrm{gravitational\; force\; -\; buoyant\; force}$$

$$\Rightarrow 6\mathrm{\pi}\mathrm{}\mathrm{\eta}\mathrm{}\mathrm{r}\mathrm{}{\mathrm{v}}_{\mathbf{t}}=\frac{4\mathrm{\pi}\mathrm{}{\mathrm{r}}^{3}\left(\mathrm{\rho}-\mathrm{\sigma}\right)\mathrm{g}}{3}\mathrm{}\mathrm{}\mathrm{}$$

$$\Rightarrow \mathrm{}\mathrm{}\mathrm{}{\mathrm{v}}_{\mathrm{t}}=\frac{2}{9}\frac{{\mathrm{r}}^{2}\left(\mathrm{\rho}-\mathrm{\sigma}\right)\mathrm{g}}{\mathrm{\eta}}$$

where ρ and σ are mass densities of sphere and the fluid respectively.

The terminal velocity depends on the square of the radius of the sphere and inversely on the viscosity of the medium.

**Reynolds Number** is defined as

$${\mathrm{R}}_{\mathrm{e}}\mathrm{}=\frac{\mathrm{\rho}\mathrm{v}\mathrm{d}}{\mathrm{\eta}}$$

where ρ is the density of the fluid flowing with a speed *v*, *d *stands for the dimension of the pipe, and η is the viscosity of the fluid.

R_{e }is a dimensionless number and therefore, it remains same in any system of units.

R_{e} < 1000 - flow is streamline or laminar

1000 < Re < 2000 – Flow is unsteady

R_{e} > 2000 - flow is turbulent

The critical value of *R*e (known as critical Reynolds number), at which turbulence sets, is found to be the same for the geometrically similar flows. For example when oil and water with their different densities and viscosities, flow in pipes of same shapes and sizes, turbulence sets in at almost the same value of *R*e.

R_{e} can also be written as,

$${\mathrm{R}}_{\mathrm{e}}=\frac{\mathrm{\rho}{\mathrm{v}}^{2}}{\mathrm{\eta}\mathrm{v}/\mathrm{d}}=\frac{\mathrm{\rho}\mathrm{A}{\mathrm{v}}^{2}}{\mathrm{\eta}\mathrm{A}\mathrm{v}/\mathrm{d}}$$

$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}=\frac{\mathrm{inertial\; force}}{\mathrm{fforce\; due\; to\; viscocity}}.$$

[We know that,

If a fluid element of mass ‘m’ is moving with a velocity ‘v’, then

moment p = mv.

Inertial force,

F = mass flow rate × velocity

= ρ × A × v × v = ρ A v^{2}]

**Surface Tension**

Surface tension is the property of any liquid by virtue of which it tries to minimize its free surface area and behaves like a taught membrane.

Surface tension of a liquid is measured as the force acting per unit length on an imaginary line drawn tangentially on the free surface the liquid.

Surface tension S = $\frac{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}}{\mathrm{L}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}$

$$=\frac{\mathrm{F}}{\mathrm{l}}$$

$$=\frac{\mathrm{F}.\mathrm{d}}{\mathrm{l}.\mathrm{d}}$$

$$=\mathrm{}\frac{\mathrm{Work\; Done}}{\mathrm{Change\; in\; Area}}$$

Its SI unit is Nm^{-1} or Jm^{-2} and its dimensional formula is [MT^{-2}].

It is a scalar quantity.

Surface tension of a liquid depends only on the nature of liquid and independent of the surface area of film or length of the line.

**Examples:**

Small liquid drops are spherical due to the property of surface tension.

Hair of a wet brush stick together.

- Surface tension is a molecular phenomenon which is due to intermolecular force.

**Intermolecular Forces**

There are two types of intermolecular forces.

**Adhesive Force**

The force of attraction between the molecules of different substances is called adhesive force, e.g., the force of attraction acting between the molecules of paper and ink, water and glass, etc.

**Cohesive Force**

The force of attraction between the molecules of same substance is called cohesive force, e.g., the force of attraction between molecules of water.

Cohesive and adhesive forces are van der Waals’ forces.

**Molecular Range**

The maximum distance up to which a molecule can exert a force of attraction on other molecules is called molecular range.

Molecular range is different for different substances. In solids and liquids it is of the order of 10^{-9} m.

If the distance between the molecules is greater than 10^{-9} m, the force of attraction between them is negligible.

**Qualitative Explanation of Surface Tension **

A liquid stays together because of attraction between molecules.

Consider a molecule well inside a liquid. It is surrounded by liquid molecules on all sides. Due to attractive forces, the potential energy of molecules is negative.

Now let us consider a molecule near the surface. Only lower half side of it is surrounded by liquid molecules. The potential energy is still negative, but less than that of a molecule fully inside.

This is approximately half of the earlier case.

Molecules on a liquid surface have some extra energy in comparison to molecules in the interior. A liquid tends to have the least surface area to minimize the surface energy.

Since a liquid consists of molecules moving about, there cannot be a **perfectly sharp surface**.

The density of the liquid molecules drops rapidly to zero around *z *= 0 as we move along the direction indicated in a distance of the order of a few molecular sizes.

**Surface Energy**

If we increase the free surface area of a liquid then work has to be done against the force of surface tension. This work done is stored in liquid surface as potential energy,

The additional potential energy required to increase the area of free surface of liquid is called** surface energy.**

Hence, surface energy (E) = S × ΔA,

Where, S = surface tension and

ΔA = increase in surface area.

**Surface energy of a stretched thin film**

Consider a horizontal liquid film ending in bar free to slide over parallel guides

Let us apply a force F, due to which the slide moves a distance d.

The **work done** by the applied force is F.d = Fd.

From conservation of energy, this is stored as additional energy in the film.

If the surface energy of the film is S per unit area, the extra area is 2dl**. **A film has two sides and the liquid in between, so there are two surfaces.

So the extra energy is,

$$\mathrm{S}\mathrm{}\left(2\mathrm{d}\mathrm{l}\right)=\mathrm{}\mathrm{F}\mathrm{d}\mathrm{}\mathrm{}$$

$$\Rightarrow \mathrm{}\mathrm{S}=\frac{\mathrm{F}\mathrm{d}}{2\mathrm{d}\mathrm{l}}=\frac{\mathrm{F}}{2\mathrm{l}}$$

**Angle of Contact**

The angle between tangent to the liquid surface at the point of contact and solid surface inside the liquid is termed as angle of contact (θ).

It is different at interfaces of different pairs of liquids and solids.

The three interfacial tensions at all the three interfaces, liquid-air, solid-air and solid-liquid denoted by S_{la}, S_{sa} & S_{sl} respectively must be in equilibrium.

S_{la} cos θ + S_{sl} = S_{sa }

- θ is obtuse if S
_{sl}> S_{la}, the liquid will form droplets, e.g. water on lotus leaf. - θ is acute if S
_{sl}< S_{la}, the liquid will spread on the surface of a solid, e.g. water on clean plastic plate.

**Work Done in Blowing a Liquid Drop**

Consider a spherical drop of radius r in equilibrium and let the radius be increased from r to r + Δr*.*

The extra surface energy is

[4π(r + Δr)^{2}- 4πr^{2}] S_{la} = 8πr Δr S_{la}

In equilibrium this energy must be equal to the work done due to expansion under the pressure difference (P_{i} – P_{o}) between the inside of the drop and the outside.

The work done is

W = (P_{i} – P_{o}) 4πr^{2}Δr = 8πr Δr S_{la}

$$\Rightarrow \mathrm{}{\mathrm{P}}_{\mathrm{i}}\mathrm{}\u2013\mathrm{}{\mathrm{P}}_{\mathrm{o}}\mathrm{}=\mathrm{}\left(\frac{2{\mathrm{S}}_{\mathrm{l}\mathrm{a}}}{\mathrm{r}}\right)$$

For a liquid-gas interface, the convex side has a higher pressure than the concave side. For example, an air bubble in a liquid would have higher pressure inside it.

A drop and Cavity have only a single surface, a bubble has two interfaces.

So for a bubble,

$${\mathrm{P}}_{\mathrm{i}}\mathrm{}\u2013\mathrm{}{\mathrm{P}}_{\mathrm{o}}\mathrm{}=\mathrm{}\left(\frac{4{\mathrm{S}}_{\mathrm{l}\mathrm{a}}}{\mathrm{r}}\right)$$

This is why we need to blow hard, but not too hard, to form a soap bubble.

**Work Done in Splitting a Bigger Drop into n Smaller Droplets**

If a liquid drop of radius R is split up into n smaller droplets, all of same size, then radius of each droplet

$$\mathrm{r}=\mathrm{}\frac{\mathrm{R}}{\sqrt[3]{\mathrm{n}}}\mathrm{}$$

⇒ Work done W = $4\mathrm{\pi}\left(\mathrm{n}{\mathrm{r}}^{2}\mathrm{}\u2013\mathrm{}{\mathrm{R}}^{2}\right)\mathrm{}{\mathrm{S}}_{\mathrm{l}\mathrm{a}}$

$$=\mathrm{}4\mathrm{\pi}{\mathrm{S}}_{\mathrm{l}\mathrm{a}}{\mathrm{R}}^{2}(\sqrt[3]{\mathrm{n}}\mathrm{}\u2013\mathrm{}1)$$

**Coalescence of Drops**

If n small liquid drops of radius reach combine together so as to form a single bigger drop of radius R, then in the process energy is released. Release of energy is given by

$$\mathrm{R}=\mathrm{}\sqrt[3]{\mathrm{n}}\mathrm{}\mathrm{r}\mathrm{}\mathrm{}\mathrm{}$$

$$\Rightarrow \mathrm{\Delta}\mathrm{U}\mathrm{}=\mathrm{}{\mathrm{S}}_{\mathrm{l}\mathrm{a}}4\mathrm{\pi}\left(\mathrm{n}{\mathrm{r}}^{2}\mathrm{}\u2013\mathrm{}{\mathrm{R}}^{2}\right)$$

$$=\mathrm{}{\mathrm{S}}_{\mathrm{l}\mathrm{a}}4\mathrm{\pi}{\mathrm{r}}^{2}\mathrm{n}(1\mathrm{}\u2013\frac{1}{\sqrt[3]{\mathrm{n}}}\mathrm{})$$

**Capillarity**

Capillary is a tube which has very-very small internal diameter.

The phenomenon of rise or fall of liquid column in a capillary tube as a result of surface tension is called **capillarity**.

The contact angle between water and glass is acute. Thus the surface of water in **the capillary is concave. **

The pressure difference between the two sides of the top surface is given by

$$\left({\mathrm{P}}_{\mathrm{a}}\u2013\mathrm{}{\mathrm{P}}_{\mathrm{o}}\right)=\mathrm{}\left(\frac{2\mathrm{S}}{\mathrm{r}}\right)$$

$$=\frac{2\mathrm{S}}{\mathrm{a}\mathrm{sec}\mathrm{\theta}}$$

$$=\left(\frac{2\mathrm{S}}{\mathrm{a}}\right)\mathrm{cos}\mathrm{\theta}$$

The pressure of the water inside the tube, just at the meniscus (air-water interface) is less than the atmospheric pressure.

Consider the two points A and B. They must be at the same pressure, namely

P_{0} + h ρ g = P_{i} = P_{A} = P_{a}

where ρ is the density of water and h is called the **capillary rise or ascent**.

h ρ g = (P_{i} – P_{0}) = $\frac{2\mathrm{S}\mathrm{cos}\mathrm{\theta}}{\mathrm{a}}$

For small θ, h = $\frac{2\mathrm{S}}{\mathrm{\rho}\mathrm{g}\mathrm{a}}$

Therefore, h = $\frac{2\mathrm{S}\mathrm{cos}\mathrm{\theta}}{\mathrm{\rho}\mathrm{}\mathrm{g}\mathrm{}\mathrm{a}}$

Here a = radius of capillary tube

- If θ < 90°, cos θ is positive, so h is positive, i.e., liquid rises in a capillary tube.
- If θ > 90°, cos θ is negative, so h is negative, i.e., liquid falls in a capillary tube.
- Rise of liquid in a capillary tube does not violate law of conservation of energy.

**Some Practical Examples of Capillarity**

- The kerosene oil in a lantern and the molten wax in a candle, rise in the capillaries formed in the cotton wick and burns.
- Coffee powder is easily soluble in water because water immediately wets the fine granules of coffee by the action of capillarity.
- The water given to the fields rises in the capillaries formed in the stems of plants and reaches the leaves.

**Zurin’s Law**

$$\mathrm{cos}\mathrm{\alpha}\mathrm{}\mathrm{}=\frac{\mathrm{h}}{\mathrm{l}}\mathrm{}$$

$$\Rightarrow \mathrm{}\mathrm{l}\mathrm{}=\frac{\mathrm{h}}{\mathrm{cos}\mathrm{\alpha}}$$

If a capillary tube of insufficient length is placed vertically in a liquid then never comes out from the tube its own, and radius of curvature increase, as

Rh = constant

⇒ R_{1}h_{1} = R_{2}h_{2}

where, R = radius of curvature of liquid meniscus and h = height of liquid column.

- When a tube is kept in inclined position in a liquid the vertical height of liquid column remains unchanged.

Liquid rises (water in glass capillary) or falls (mercury in capillary) due to property of surface tension

$$\mathrm{S}\mathrm{}=\frac{\mathrm{R}\mathrm{\rho}\mathrm{g}\mathrm{h}}{2\mathrm{cos}\mathrm{\theta}}$$

**Detergents and Surface Tension**

Washing with water does not remove grease stains. This is because water does not wet greasy dirt; i.e., there is very little area of contact between them. If water could wet grease, the flow of water could carry some grease away.

The molecules of detergents are hairpin shaped, with one end attracted to water and the other to molecules of grease, oil or wax, thus tending to form water-oil interfaces.

Addition of detergents, whose molecules attract water at one end and oil on the other, drastically reduces the surface tension *S *(water-oil). It becomes energetically favourable to form such interfaces.

Ultimately the globules (micelles) formed are washed away by water.

**Factors Affecting Surface Tension**

- Surface tension of a liquid decreases with increase in temperature and becomes zero at critical temperature.
- At boiling point, surface tension of a liquid becomes zero and becomes maximum at freezing point.
- Surface tension decreases when partially soluble impurities such as soap, detergent, dettol, phenol, etc are added in water.
- Surface tension increases when highly soluble impurities such as salt are added in water.
- When dust particles or oil spreads over the surface of water, its surface tension decreases.

- When charge is given to a soap bubble, its size increases and surface tension of the liquid decreases due to electrification.
- In weightlessness condition liquid does not rise in a capillary tube (why ?).

**Some Phenomena Based on Surface Tension**

- Medicines used for washing wounds, as dettol, have a surface tension lower than water.
- Hot soup is tastier than the cold one because the surface tension of the hot soup is less than that of the cold and so spreads over a larger area of the tongue.
- Insects and mosquitoes swim on the surface of water in ponds and lakes due to surface tension. If kerosene oil is sprayed on the water surface, the surface tension of water is lowered and the insects and mosquitoes sink in water and are dead.

- If we deform a liquid drop by pushing it slightly, then due to surface tension it again becomes spherical.