Introduction

This chapter explores the mechanical properties of liquids and gases. Both liquids and gases have the ability to flow, and for this reason, they are collectively known as fluids. This property is what fundamentally distinguishes them from solids.

Fluids play a crucial role in our world and biological systems. The Earth is enveloped by air and its surface is largely covered by water. All life processes, in both plants and animals, are mediated by fluids.

Differences between Solids, Liquids, and Gases

Unlike solids, fluids do not have a definite shape of their own; they take the shape of their container.

• Solids and Liquids have a nearly fixed volume under standard atmospheric pressure. Their volume changes only slightly with large changes in external pressure, meaning they have low compressibility.
• Gases do not have a fixed volume and will expand to fill the entire volume of their container. They are highly compressible.
• A key property of fluids is that they offer very little resistance to shear stress. A very small shear stress can cause a fluid to change its shape, whereas a solid can resist significant shear stress.

Pressure

The impact of a force depends not only on its magnitude but also on the area over which it is applied. This concept is quantified by pressure.

When a fluid is at rest, it exerts a force on any surface in contact with it. This force is always directed normal (perpendicular) to the surface. If there were a tangential component of the force, the fluid would flow, which contradicts the condition that it is at rest.

Definition of Pressure

Pressure (P) is defined as the normal force (F) acting per unit area (A).

$P = \frac{F}{A}$

Pressure is a scalar quantity. It has magnitude but no direction. The force exerted by a fluid on a surface is always perpendicular to that surface, regardless of the surface's orientation.

• SI Unit: N/m², which is named the pascal (Pa).
• Other Units: atmosphere (atm), bar, torr. $1 \text{ atm} = 1.013 \times 10^5 \text{ Pa}$.
• Dimensional Formula: $[ML^{-1}T^{-2}]$.

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Density

Density ($\rho$) is another crucial property of fluids, defined as mass (m) per unit volume (V).

$\rho = \frac{m}{V}$

• SI Unit: kg/m³.
• Liquids are largely incompressible, so their density is nearly constant.
• Gases are highly compressible, and their density varies significantly with pressure.

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Pascal’s Law

This fundamental principle of fluid statics was discovered by Blaise Pascal.

Pascal's Law states that the pressure in a fluid at rest is the same at all points if they are at the same height.

Another form of the law states: Whenever external pressure is applied on any part of a fluid contained in a vessel, it is transmitted undiminished and equally in all directions.

Hydraulic Machines

Pascal's law is the principle behind hydraulic machines like the hydraulic lift and hydraulic brakes. These devices use an incompressible fluid to transmit force.

In a hydraulic lift with two pistons of areas $A_1$ and $A_2$ ($A_2 > A_1$), a small force $F_1$ applied to the smaller piston creates a pressure $P = F_1/A_1$. This pressure is transmitted to the larger piston, producing a large upward force $F_2 = P \times A_2$.

$F_2 = \left(\frac{F_1}{A_1}\right) A_2 = F_1 \left(\frac{A_2}{A_1}\right)$

The force is magnified by a factor of $A_2/A_1$, which is the mechanical advantage of the lift.

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Variation of Pressure with Depth

In a fluid under gravity, pressure increases with depth. Consider a fluid of constant density $\rho$ at rest. The pressure difference between two points separated by a vertical height $h$ is given by:

$P_2 - P_1 = \rho g h$

If the surface of the liquid (at height $h=0$) is open to the atmosphere, the pressure at the surface is atmospheric pressure, $P_a$. The absolute pressure $P$ at a depth $h$ below the surface is:

$P = P_a + \rho g h$

Gauge Pressure

The excess pressure above atmospheric pressure, $P - P_a$, is called the gauge pressure. So, Gauge Pressure $= \rho g h$.

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Streamline Flow

The study of fluids in motion is called fluid dynamics. We often analyze an idealized type of motion called steady or streamline flow.

The flow of a fluid is said to be steady if, at any given point, the velocity of each passing fluid particle remains constant over time. In steady flow, the path taken by a fluid particle is called a streamline. A streamline is a curve whose tangent at any point gives the direction of the fluid velocity at that point. In steady flow, streamlines do not cross each other.

Equation of Continuity

For an incompressible fluid (density $\rho$ is constant) in steady flow, the mass of fluid entering a section of a pipe must equal the mass leaving it. This is a statement of the conservation of mass.

If the fluid flows with speed $v_1$ through an area of cross-section $A_1$ and speed $v_2$ through an area $A_2$, then:

Mass flow rate = $\rho A_1 v_1 = \rho A_2 v_2$

Since $\rho$ is constant, we get the equation of continuity:

$A_1 v_1 = A_2 v_2 \quad \text{or} \quad Av = \text{constant}$

The product $Av$ is the volume flow rate. This equation implies that where the pipe is narrower (smaller A), the fluid speed is greater, and where the pipe is wider (larger A), the speed is smaller.

Turbulent Flow

At low speeds, flow is typically steady (or laminar). Above a certain critical speed, the flow becomes unsteady and chaotic. This irregular flow is called turbulent flow and is characterized by the formation of eddies and whirlpools.

Bernoulli’s Principle

Bernoulli's principle is a fundamental principle in fluid dynamics that relates pressure, velocity, and height for a moving fluid. It is essentially a statement of the conservation of energy for an ideal fluid (incompressible and non-viscous) in steady flow.

Bernoulli's Equation

Consider an ideal fluid flowing through a pipe of varying cross-section and height. The work done on a volume of fluid by pressure differences goes into changing its kinetic and potential energy.

The relationship derived from the work-energy theorem is Bernoulli's equation:

$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$

This means that along a streamline, the sum of the pressure ($P$), the kinetic energy per unit volume ($\frac{1}{2}\rho v^2$), and the potential energy per unit volume ($\rho g h$) remains constant.

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$

A key consequence for horizontal flow ($h$=constant) is that where the speed is higher, the pressure is lower, and vice versa.

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Applications of Bernoulli's Principle

1. Speed of Efflux: Torricelli’s Law

Consider a tank filled with a liquid of density $\rho$, open to the atmosphere, with a small hole at a depth $h$ below the surface. Applying Bernoulli's equation between the surface of the liquid (point 2) and the hole (point 1), we find the speed of the liquid exiting the hole (efflux) is:

$v_1 = \sqrt{2gh}$

This is Torricelli's law. The speed of efflux is the same as the speed an object would acquire by freely falling through the same height $h$.

2. Venturi-meter

A Venturi-meter is a device used to measure the flow speed of a fluid. It consists of a tube with a narrow constriction (a "throat"). According to the equation of continuity, the fluid speeds up in the throat. According to Bernoulli's principle, the pressure in the throat is therefore lower than in the wider section. By measuring this pressure difference (often with a U-tube manometer), the flow speed can be calculated.

3. Dynamic Lift (Aerofoil and Magnus Effect)

The shape of an aeroplane wing (an aerofoil) is designed so that the air flows faster over its curved upper surface than its flatter lower surface. This difference in speed creates a pressure difference: the pressure above the wing is lower than the pressure below it. This pressure difference results in a net upward force called dynamic lift, which counteracts the weight of the aircraft.

A similar effect explains the curve of a spinning ball (Magnus effect). A spinning ball drags air with it, causing the air speed to be higher on one side and lower on the other, creating a pressure difference and a net force that makes the ball swerve.

Viscosity

Ideal fluids are non-viscous, but real fluids offer resistance to flow. This internal friction in a fluid is called viscosity. It arises from the frictional forces between adjacent layers of the fluid that are in relative motion.

Consider a fluid between two parallel plates, with the bottom plate fixed and the top plate moving at a constant velocity $v$. The layer of fluid in contact with the top plate moves with velocity $v$, while the layer in contact with the bottom plate is at rest. The velocity of intermediate layers varies linearly from 0 to $v$. This type of flow, where layers slide over one another, is called laminar flow.

The shearing stress ($F/A$) in the fluid is found to be proportional to the rate of change of strain (strain rate), which is given by $v/l$. The constant of proportionality is the coefficient of viscosity ($\eta$).

$\frac{F}{A} = \eta \frac{v}{l} \implies F = \eta A \frac{v}{l}$

• SI unit: poiseiulle (Pl) or Pa·s.
• Viscosity of liquids generally decreases with increasing temperature, while that of gases increases.

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Stokes' Law and Terminal Velocity

When a solid object moves through a viscous fluid, it experiences a drag force that opposes its motion. For a small sphere of radius $a$ moving at a low speed $v$ through a fluid of viscosity $\eta$, the viscous drag force $F_v$ is given by Stokes' Law:

$F_v = 6\pi \eta a v$

When an object falls through a fluid, it initially accelerates due to gravity. As its speed increases, the viscous drag force also increases. Eventually, the upward drag force plus the buoyant force will balance the downward gravitational force. At this point, the net force is zero, the acceleration ceases, and the object falls with a constant maximum velocity called the terminal velocity ($v_t$).

Surface Tension

The free surface of a liquid behaves like a stretched elastic membrane. This phenomenon is called surface tension. It is responsible for many effects, such as the spherical shape of small liquid drops, the ability of insects to walk on water, and capillary action.

Microscopic Origin and Surface Energy

Molecules inside a liquid are attracted equally in all directions by their neighbors. However, molecules at the surface experience a net inward pull from the molecules below them. This net inward force pulls the surface molecules closer together and causes the surface to contract to the minimum possible area.

To bring a molecule from the interior to the surface, work must be done against these cohesive forces. This work is stored as potential energy in the surface. Thus, the molecules at the surface have a higher potential energy than those in the bulk. This extra energy per unit area of the surface is called surface energy.

Definition of Surface Tension

Surface tension (S) is defined as the force per unit length acting in the plane of the liquid surface, perpendicular to a line drawn on the surface. It is also numerically equal to the surface energy per unit area.

$S = \frac{\text{Force}}{\text{Length}} = \frac{\text{Surface Energy}}{\text{Area}}$

• SI Unit: N/m or J/m².
• Surface tension generally decreases as temperature increases.

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Angle of Contact

When a liquid is in contact with a solid surface, the liquid surface is generally curved near the point of contact. The angle of contact ($\theta$) is the angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid.

• If $\theta < 90^\circ$ (acute), the liquid is said to wet the solid (e.g., water on clean glass). The adhesive forces (liquid-solid) are stronger than cohesive forces (liquid-liquid).
• If $\theta > 90^\circ$ (obtuse), the liquid does not wet the solid (e.g., mercury on glass, water on a waxy surface). Cohesive forces are stronger than adhesive forces.

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Pressure Difference across a Curved Surface (Laplace Pressure)

Due to surface tension, there is a pressure difference across any curved liquid-air interface. The pressure is greater on the concave side of the surface.

• For a spherical liquid drop of radius $r$: The excess pressure inside is $\Delta P = P_{in} - P_{out} = \frac{2S}{r}$.
• For a soap bubble (which has two surfaces, inner and outer): The excess pressure inside is $\Delta P = \frac{4S}{r}$.

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Capillary Action

The rise or fall of a liquid in a narrow tube (a capillary) is called capillary action or capillarity. This is a result of surface tension and the angle of contact.

• If the liquid wets the tube (acute angle of contact, like water in glass), the surface inside the capillary is concave, and the liquid rises. The upward force from surface tension balances the weight of the lifted liquid column.
• If the liquid does not wet the tube (obtuse angle of contact, like mercury in glass), the surface is convex, and the liquid level is depressed.

The height $h$ to which a liquid of density $\rho$ and surface tension $S$ rises in a capillary of radius $a$ is given by:

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

The rise is greater for narrower tubes.

What is Blood Pressure?

The circulatory system of upright animals, including humans, has evolved to handle the demands of moving blood against gravity. The venous system, which returns blood from the lower parts of the body to the heart, is particularly adapted for this task. Without these adaptations, blood would pool in the lower extremities, as seen in some animals like snakes or rats, which cannot survive being held in an upright position for long.

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Pressure Variation in the Human Body

The pressure in the arteries varies significantly at different points in the body due to gravity. We can understand this variation using a simplified form of Bernoulli's principle. For blood flow in major arteries, the velocity is relatively small and constant, so the kinetic energy term ($\frac{1}{2}\rho v^2$) can be ignored. Bernoulli's equation then reduces to the hydrostatic pressure relationship:

$P + \rho g y = \text{constant}$

This means the pressure difference between any two points depends on their vertical separation. Let's denote the gauge pressures at the brain, heart, and foot as $P_B$, $P_H$, and $P_F$ respectively. The relationship between them when a person is standing is:

$P_F = P_H + \rho g h_H = P_B + \rho g h_B$

where $\rho$ is the density of blood, $h_H$ is the height of the heart above the feet, and $h_B$ is the height of the brain above the feet.

Using typical values for an adult ($\rho \approx 1.06 \times 10^3 \text{ kg/m}^3$, $h_H \approx 1.3 \text{ m}$, and the height from heart to brain being 0.4 m, so $h_B \approx 1.7 \text{ m}$), and an average heart pressure of $P_H = 13.3 \text{ kPa} \approx 100 \text{ mm of Hg}$, we can estimate the pressures:

• Pressure at the foot ($P_F$): $P_F = P_H + \rho g h_H \approx 13.3 \text{ kPa} + (1.06 \times 10^3)(9.8)(1.3) \approx 13.3 + 13.5 = 26.8 \text{ kPa}$.
• Pressure at the brain ($P_B$): The brain is above the heart, so pressure is lower. $P_B = P_H - \rho g (\text{height difference}) \approx 13.3 \text{ kPa} - (1.06 \times 10^3)(9.8)(0.4) \approx 13.3 - 4.1 = 9.2 \text{ kPa}$.

When the person is lying down, all these points are at roughly the same horizontal level, so the pressures become nearly equal.

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Biological Adaptations for Blood Circulation

The human body has remarkable mechanisms to assist blood return to the heart against gravity:

• Veins in the lower limbs are equipped with valves that allow blood to flow towards the heart but prevent it from draining back down.
• The pumping action associated with breathing helps move blood through the veins.
• The contraction and relaxation of skeletal muscles during walking and other activities squeeze the veins, pushing blood upwards.

This is why a soldier standing at attention for a long time may faint; the lack of muscle movement reduces blood return to the heart, leading to insufficient blood supply to the brain. Lying the person down equalizes the pressure and restores consciousness.

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Measuring Blood Pressure

Blood pressure is typically measured using an instrument called a sphygmomanometer. The measurement is taken at the upper arm because it is at the same level as the heart, providing a pressure reading close to that at the heart.

Systolic and Diastolic Pressure

Blood pressure is not constant; it fluctuates with each heartbeat.

• Systolic Pressure: The maximum pressure in the arteries when the heart contracts and pumps blood out.
• Diastolic Pressure: The minimum pressure in the arteries when the heart is relaxed between beats.

The Measurement Process

The sphygmomanometer works by temporarily stopping and then releasing the blood flow in the brachial artery of the upper arm and listening to the resulting sounds with a stethoscope.

1. An inflatable cuff is wrapped around the upper arm and inflated to a pressure high enough to completely close the brachial artery, stopping blood flow.
2. The pressure in the cuff is then slowly released. A stethoscope is placed over the artery just below the cuff.
3. When the cuff pressure drops just below the systolic pressure, the artery opens briefly during the peak pressure phase of each heartbeat. The blood rushes through the constricted opening, creating turbulent flow, which is heard as a distinct tapping sound. The pressure reading at which this first sound is heard is recorded as the systolic pressure.
4. As the cuff pressure continues to decrease, the tapping sound continues.
5. When the cuff pressure drops to the diastolic pressure, the artery remains open throughout the entire heart cycle. The flow is still turbulent but the sound changes from a tapping to a continuous, dull roar or disappears altogether. The pressure reading at which the tapping sound ceases is recorded as the diastolic pressure.

The blood pressure reading is given as a ratio of these two values, for example, 120/80 mm of Hg for a healthy resting adult. Readings consistently above 140/90 mm of Hg are considered high blood pressure (hypertension) and require medical attention.

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Additional Exercises

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Calculator/Computer – Based Problem

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