Introduction

This chapter explores the physical properties of liquids and gases, collectively known as fluids due to their ability to flow.

Fluids differ from solids in a fundamental way: they do not have a definite shape of their own, instead taking the shape of their container. Liquids maintain a fixed volume (under atmospheric pressure), while gases expand to fill the entire volume available.

Another key distinction is their response to shear stress. Unlike solids which resist shear stress to maintain their shape, fluids offer very little resistance to shearing; they continuously deform under even small shear stresses.

While the volume of solids can be changed by stress, the change in volume (compressibility) for solids and liquids is generally much smaller compared to gases.

Understanding fluid behavior is essential as fluids are ubiquitous and involved in many natural processes and technological applications.

Pressure

The concept of pressure is used to describe the impact of a force distributed over an area, particularly relevant for fluids.

When a fluid is at rest, it exerts a force on any surface it is in contact with (like the walls of a container or a submerged object). This force is always normal (perpendicular) to the surface. If there were a tangential force component, the fluid would flow, but a fluid at rest cannot sustain a tangential force.

Average Pressure ($P_{av}$): Defined as the normal force ($F$) acting per unit area ($A$).

Pressure (P) at a point: Defined as the limit of the average pressure as the area approaches zero.

Pressure is a scalar quantity. It acts equally in all directions at a point within the fluid. Its direction is specified by the orientation of the surface it acts upon (always normal to the surface).

The SI unit of pressure is N/m$^2$, called the pascal (Pa).

Common units of pressure include atmosphere (atm), bar, millibar, and torr (mm of Hg).

• 1 atm $= 1.013 \times 10^5$ Pa
• 1 bar $= 10^5$ Pa
• 1 torr $= 1$ mm Hg $\approx 133$ Pa

Density ($\rho$): A scalar quantity defining the mass per unit volume of a fluid.

SI unit is kg/m$^3$. Dimensions are $[ML^{-3}]$.

Liquids are nearly incompressible, so their density is relatively constant with pressure. Gases are highly compressible, and their density varies significantly with pressure and temperature.

Relative density of a substance is its density divided by the density of water at $4^\circ$C ($1.0 \times 10^3$ kg/m$^3$). It is a dimensionless ratio.

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Pascal’s Law

Pascal's law describes how pressure behaves in a fluid at rest.

Form 1: Pressure in a fluid at rest is the same at all points at the same height.

This can be shown by considering a small fluid element in the shape of a prism. For the element to be in equilibrium, the forces on its faces must balance. Since the forces are normal to the faces and related to pressure ($F = PA$), balancing forces shows that the pressure must be equal on faces at the same depth, regardless of their orientation. This confirms pressure is a scalar.

Form 2: A change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

If pressure is increased at one point in an enclosed fluid (e.g., by pushing a piston), this pressure increase is transmitted equally and instantaneously to all other points in the fluid and to the walls of the container. This is demonstrated by interconnected vessels where pushing a piston causes the liquid level to rise equally in all tubes.

Variation Of Pressure With Depth

In a fluid at rest under the influence of gravity, pressure increases with depth.

Consider a cylindrical column of fluid of height $h$ and cross-sectional area $A$. The pressure at the bottom of the column ($P_2$) must support the weight of the fluid column plus the pressure at the top ($P_1$).

Forces in vertical equilibrium: $(P_2 - P_1)A = \text{Weight of fluid column} = mg$.

The mass of the fluid column is $m = \rho V = \rho (Ah)$.

So, $(P_2 - P_1)A = \rho (Ah)g$.

This shows that the pressure difference between two points in a fluid at rest depends on the vertical distance between them, the fluid density, and the acceleration due to gravity.

If point 1 is at the surface of the fluid exposed to the atmosphere ($P_1 = P_a$, the atmospheric pressure), and point 2 is at a depth $h$ ($P_2 = P$), then the absolute pressure $P$ at depth $h$ is:

The term $\rho g h$ is the additional pressure due to the weight of the fluid column above the point. This additional pressure (above atmospheric pressure) is called the gauge pressure ($P_g = P - P_a = \rho g h$).

Key implications of pressure variation with depth:

• Pressure is the same at all points at the same horizontal level within a continuous fluid at rest.
• The pressure at a given depth depends only on the height of the fluid column above it, not on the shape or total volume of the container (hydrostatic paradox).

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Atmospheric Pressure And Gauge Pressure

The atmospheric pressure ($P_a$) at any point is the pressure exerted by the column of air above that point extending to the top of the atmosphere. It varies with altitude and weather conditions.

At sea level, average atmospheric pressure is about $1.013 \times 10^5$ Pa.

A mercury barometer is used to measure atmospheric pressure. It was invented by Torricelli. A column of mercury in a tube is supported by the atmospheric pressure acting on the surface of the mercury in a trough. The height of the mercury column is proportional to the atmospheric pressure ($P_a = \rho g h$, where $\rho$ is the density of mercury). Standard atmospheric pressure supports a column of about 76 cm (760 mm) of mercury.

1 mm of Hg is also called a torr (1 torr $\approx 133$ Pa).

An open tube manometer measures gauge pressure (pressure difference). It consists of a U-tube filled with a liquid. One end is open to the atmosphere ($P_a$), and the other is connected to the system whose pressure ($P$) is to be measured. The difference in the liquid levels ($h$) in the two arms indicates the gauge pressure: $P - P_a = \rho_m g h$, where $\rho_m$ is the density of the manometer fluid.

Gauge Pressure ($P_g$): The difference between the absolute (actual) pressure ($P$) and the atmospheric pressure ($P_a$). $P_g = P - P_a$. Gauge pressure is commonly measured by devices like tire pressure gauges.

Absolute Pressure (P): The total pressure at a point, including atmospheric pressure. $P = P_a + P_g$.

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Hydraulic Machines

Hydraulic machines like lifts and brakes use the principle of Pascal's law to multiply force. They work by transmitting pressure through an enclosed fluid.

In a hydraulic lift, a small force ($F_1$) is applied to a small piston (area $A_1$). This creates a pressure $P = F_1/A_1$ in the fluid. According to Pascal's law, this pressure $P$ is transmitted undiminished to a larger piston (area $A_2$). The force exerted by the fluid on the larger piston is $F_2 = P \times A_2 = (F_1/A_1) \times A_2$.

Since $A_2 > A_1$, the force $F_2$ on the larger piston is greater than the applied force $F_1$. The factor $A_2/A_1$ is the mechanical advantage.

Archimedes' Principle:

• When an object is submerged in a fluid, the pressure on its lower surfaces is greater than on its upper surfaces due to the increase in pressure with depth.
• This pressure difference results in a net upward force exerted by the fluid on the object, called the buoyant force.
• Archimedes' principle states that the buoyant force is equal to the weight of the fluid displaced by the submerged portion of the object.
• $F_{buoyant} = \rho_f V_{submerged} g$, where $\rho_f$ is the fluid density and $V_{submerged}$ is the volume of the object submerged in the fluid.
• If the object's weight is greater than the buoyant force, it sinks. If less, it floats (partially submerged until buoyant force equals weight). If equal, it is in equilibrium within the fluid.

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Streamline Flow

The study of fluids in motion is called fluid dynamics. Fluid flow can exhibit different patterns depending on the speed and properties of the fluid.

Steady Flow (Streamline Flow or Laminar Flow):

• A type of fluid flow where the velocity of each fluid particle passing through a particular point in space remains constant in time.
• While the velocity may differ at different points, any particle arriving at a specific point will have the same velocity as the particle that just left that point.
• Each particle follows a smooth path, and these paths do not cross each other.

A streamline is a curve such that the tangent at any point gives the direction of the fluid velocity at that point. In steady flow, streamlines represent the actual paths of the fluid particles.

No two streamlines can cross, otherwise a particle arriving at the intersection would have ambiguous velocity.

Equation of Continuity:

For the steady flow of an incompressible fluid through a pipe of varying cross-sectional area, the mass of fluid passing through any cross-section per unit time is constant. This is a statement of conservation of mass.

If $A$ is the area of cross-section and $v$ is the fluid speed at that cross-section, the volume of fluid passing per unit time is $Av$. The mass of fluid passing per unit time (mass flow rate) is $\rho Av$, where $\rho$ is the fluid density.

For incompressible flow ($\rho$ is constant), if $A_1$ and $v_1$ are values at point 1 and $A_2$ and $v_2$ at point 2:

This means the product $Av$ (volume flow rate or volume flux) is constant throughout the pipe:

This implies that where streamlines are crowded (smaller area), the fluid speed is higher, and where they are spread out (larger area), the speed is lower.

Turbulent Flow:

• When the fluid speed exceeds a certain critical speed, steady flow breaks down and becomes turbulent.
• Turbulent flow is characterized by irregular, chaotic, and fluctuating velocities and pressures. Streamlines become complicated and may form eddies or vortices.

Steady flow typically occurs at low speeds, while turbulent flow occurs at high speeds or around obstacles.

Bernoulli’s Principle

Bernoulli's principle relates the pressure, speed, and height of a fluid in steady, incompressible, non-viscous flow. It is a consequence of the conservation of energy applied to fluid dynamics.

Consider an ideal fluid (incompressible and non-viscous) flowing steadily through a pipe of varying cross-section and height. As the fluid moves from a wider section at lower height to a narrower section at higher height, its speed changes (due to continuity), height changes, and pressure must also change to conserve energy.

By applying the Work-Energy theorem to a volume of fluid moving along a streamline, considering the work done by pressure forces and changes in kinetic and potential energy, we arrive at Bernoulli's equation.

Comparing points 1 and 2 along a streamline:

where $P$ is pressure, $\rho$ is fluid density, $v$ is speed, $g$ is acceleration due to gravity, and $h$ is height above a reference level.

Bernoulli's Principle (General Statement):

As we move along a streamline in the steady flow of an ideal fluid, 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.

Restrictions and Applications:

• Applies to ideal fluids (incompressible, non-viscous). Viscosity causes energy loss, which is not accounted for in this ideal form.
• Applies to steady (streamline) flow, not turbulent flow.
• Despite restrictions, it's very useful for low-viscosity, incompressible fluids in steady flow.
• If the fluid is at rest ($v_1 = v_2 = 0$), Bernoulli's equation reduces to the hydrostatic pressure variation with depth: $P_1 + \rho g h_1 = P_2 + \rho g h_2 \implies P_2 - P_1 = \rho g (h_1 - h_2)$, or $P_2 - P_1 = \rho g \Delta h$.

Speed Of Efflux: Torricelli’s Law

Efflux is the outflow of fluid from an opening. Torricelli's law describes the speed of fluid leaving a tank through a small hole.

Consider a tank filled with liquid to a height $h$ above a small hole. Let point 1 be at the hole (speed $v_1$, height $y_1$) and point 2 be on the liquid surface (speed $v_2$, height $y_2$). Applying Bernoulli's equation between points 1 and 2:

If the hole is open to the atmosphere, $P_1 = P_a$. If the liquid surface is also open to the atmosphere, $P_2 = P_a$. If the area of the surface is much larger than the area of the hole ($A_2 \gg A_1$), then $v_2 \approx 0$ by the equation of continuity ($A_1v_1 = A_2v_2$). Let $h = y_2 - y_1$ be the height of the liquid surface above the hole.

This is Torricelli's Law. The speed of efflux from a small hole at a depth $h$ below the free surface of a liquid is the same as the speed a body would acquire by falling freely through a height $h$.

If the pressure $P_2$ above the liquid surface is significantly higher than atmospheric pressure $P_a$, the speed of efflux is increased: $v_1 = \sqrt{2gh + 2(P_2-P_a)/\rho}$.

Venturi-Meter

The Venturi-meter is a device used to measure the flow speed of an incompressible fluid in a pipe based on Bernoulli's principle and the continuity equation.

It consists of a tube with a wider section (area $A$) and a narrower constriction (area $a$). A manometer measures the pressure difference ($P_1 - P_2$) between the wide and narrow sections.

By continuity, $A v_1 = a v_2$, where $v_1$ is speed in the wide section and $v_2$ in the narrow section. $v_2 = v_1 (A/a)$. Since $A > a$, $v_2 > v_1$.

Applying Bernoulli's equation horizontally ($h_1=h_2$): $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$.

The pressure difference $(P_1 - P_2)$ is measured by the manometer height difference $h$, so $P_1 - P_2 = \rho_m g h$, where $\rho_m$ is the manometer liquid density.

This equation allows calculating $v_1$ if the areas and manometer reading are known.

The Venturi principle (speed increases where area decreases, leading to pressure decrease) is applied in carburetors, spray guns, aspirators, etc.

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Blood Flow And Heart Attack

Bernoulli's principle has medical applications, helping to explain phenomena in the circulatory system.

A healthy artery can become constricted due to plaque buildup (atherosclerosis). According to Bernoulli's principle, the speed of blood flow increases through this narrower section (continuity equation $Av = \text{constant}$), leading to a decrease in pressure ($P + \frac{1}{2}\rho v^2 = \text{constant}$).

The reduced internal pressure in the constricted region can cause the artery to collapse further due to external pressure, leading to a cycle of collapse and forceful reopening by the heart. This increased strain on the heart and reduced blood flow can potentially lead to a heart attack.

Dynamic Lift

Dynamic lift is a force exerted by a fluid on a moving body due to its motion through the fluid. Bernoulli's principle can explain parts of this phenomenon.

Examples: Lift on aircraft wings, hydrofoils, deviation of spinning balls.

• Spinning Ball (Magnus Effect): When a ball spins while moving through air, it drags some air with it due to friction or viscosity. On one side of the ball, the air dragged by the spin adds to the forward motion of the air relative to the ball, increasing the relative air speed. On the other side, it opposes the relative air motion, decreasing the relative air speed. According to Bernoulli's principle, higher speed means lower pressure. The pressure on the side with lower relative air speed is higher than on the side with higher relative air speed, resulting in a net force (dynamic lift) perpendicular to the direction of motion, causing the ball to curve.
• Aerofoil (Aircraft Wing): An aerofoil is shaped such that when it moves through air, the air flowing over the top surface travels a longer distance than the air flowing under the bottom surface in the same amount of time. By the continuity equation, this means the air speed is higher on the top surface than on the bottom surface (assuming non-viscous flow over a shaped object). According to Bernoulli's principle, the pressure on the top surface is lower than the pressure on the bottom surface ($P_{top} + \frac{1}{2}\rho v_{top}^2 \approx P_{bottom} + \frac{1}{2}\rho v_{bottom}^2$). The pressure difference ($\Delta P = P_{bottom} - P_{top}$) results in a net upward force ($\Delta P \times A$), providing the dynamic lift that supports the aircraft.

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Viscosity

Viscosity is the property of fluids that causes them to resist relative motion between different layers of the fluid. It is analogous to friction in solids.

When a fluid flows, layers at different distances from a boundary or the center of flow move at different velocities. This velocity gradient exists perpendicular to the direction of flow.

Example: A fluid between a fixed plate and a moving plate. The fluid layer touching the moving plate moves with its velocity ($v$), while the layer touching the fixed plate is stationary ($v=0$). Layers in between have velocities that increase linearly from 0 to $v$ across the thickness of the fluid ($l$).

Due to relative motion, adjacent layers exert tangential forces on each other, opposing the relative sliding. These are viscous forces.

This type of flow, where layers slide over one another, is called laminar flow.

Experimentally, the viscous force ($F$) between layers is found to be proportional to the area of the layers ($A$) and the velocity gradient ($\Delta v / \Delta y$, where $\Delta v$ is velocity difference between layers separated by $\Delta y$). For the flow between plates, the velocity gradient is $v/l$.

The constant of proportionality is the coefficient of viscosity ($\eta$), a measure of the fluid's resistance to flow.

The shearing stress is $F/A$. The strain rate (rate of change of shear strain) is $(v \Delta t / l) / \Delta t = v/l$. The coefficient of viscosity is defined as the ratio of shearing stress to the strain rate:

The SI unit of viscosity is poiseiulle (Pl), which is equivalent to N s/m$^2$ or Pa s. Dimensions are $[ML^{-1}T^{-1}]$.

Liquids like honey and glycerine are more viscous (higher $\eta$) than water or alcohol (lower $\eta$).

Viscosity of liquids generally decreases with increasing temperature, while viscosity of gases generally increases with increasing temperature.

Example 10.9. A metal block of area 0.10 m$^2$ is connected to a 0.010 kg mass via a string that passes over an ideal pulley (considered massless and frictionless), as in Fig. 10.15. A liquid with a film thickness of 0.30 mm is placed between the block and the table. When released the block moves to the right with a constant speed of 0.085 m s$^{-1}$. Find the coefficient of viscosity of the liquid.

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Stokes’ Law

When a spherical object moves through a viscous fluid, it experiences a retarding viscous force. This force is described by Stokes' Law.

Stokes' Law states that the viscous drag force ($F$) on a sphere of radius $a$ moving with velocity $\mathbf{v}$ through a fluid of viscosity $\eta$ is:

The magnitude of the force is $F = 6\pi \eta a v$, and its direction is opposite to the velocity $\mathbf{v}$.

This law applies to laminar flow around a sphere and is valid for low speeds.

Terminal Velocity ($v_t$):

When an object falls through a viscous fluid under gravity, its speed increases initially. The viscous drag force, which depends on speed, also increases. Eventually, the downward force (gravity minus buoyant force) is balanced by the upward viscous drag force. At this point, the net force and acceleration become zero, and the object falls with a constant maximum velocity called the terminal velocity.

Consider a sphere of radius $a$ and density $\rho$ falling through a fluid of density $\sigma$ and viscosity $\eta$.

Forces acting on the sphere:

• Weight $W = mg = \rho (\frac{4}{3}\pi a^3) g$ (downwards).
• Buoyant force $F_B = \sigma (\frac{4}{3}\pi a^3) g$ (upwards, weight of displaced fluid).
• Viscous drag force $F_{viscous} = 6\pi \eta a v$ (upwards, opposing motion).

At terminal velocity $v_t$, the net force is zero:

Solve for $v_t$:

The terminal velocity depends on the densities of the object and fluid, the object's size ($a^2$), gravity, and the fluid's viscosity. Heavier or larger objects fall faster; more viscous or denser fluids lead to slower terminal velocities.

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Surface Tension

Liquids, having a definite volume but no definite shape, form a free surface when in a container. This free surface behaves as if it is under tension, a phenomenon called surface tension. Gases do not have free surfaces, so surface tension applies only to liquids and interfaces involving liquids.

Surface Energy

Molecules within a liquid are attracted by neighboring molecules. A molecule deep inside the liquid is surrounded by other molecules on all sides, resulting in a net attractive force towards the bulk of the liquid.

A molecule at the surface, however, is surrounded by liquid molecules only on the side towards the bulk and possibly some other substance (like air) on the other side. The net attractive forces on a surface molecule are directed inwards towards the bulk of the liquid.

Bringing a molecule from the interior to the surface requires doing work against these inward attractive forces. This work is stored as potential energy. Thus, molecules at the surface have higher potential energy compared to molecules in the interior of the liquid. This excess potential energy associated with the surface is called surface energy.

Liquids tend to minimize their surface area to minimize this surface energy.

Surface Energy And Surface Tension

Surface tension (S) is a property of the liquid surface that arises from surface energy. It can be viewed as a force per unit length acting along the interface, or as the surface energy per unit area.

Consider stretching a liquid film (like a soap film) on a frame with a movable bar. To increase the surface area by moving the bar, an external force $F$ must be applied to counteract an inward force due to surface tension.

If the bar of length $l$ is moved by a distance $d$, the surface area increases. Since the film has two surfaces, the increase in area is $2ld$. The work done by the applied force is $W = Fd$. This work increases the surface energy of the film. The increase in surface energy is $S \times (2ld)$.

By conservation of energy, $Fd = S(2ld)$.

Thus, surface tension is equal to the force per unit length exerted by the liquid surface on the boundary (in this case, the movable bar). The factor of 2 arises because the film has two surfaces (top and bottom) in contact with air.

Surface tension is also equal to the surface energy per unit area ($S = \frac{\text{Energy}}{\text{Area}}$). Its SI unit is N/m or J/m$^2$. Both units are equivalent.

Surface tension exists at the interface between two substances (e.g., liquid-air, liquid-solid, liquid-liquid) and depends on the attractive forces between molecules of both substances at the interface. Attractive forces across the interface reduce surface energy/tension.

Surface tension depends on temperature. Generally, surface tension of liquids decreases with increasing temperature, reaching zero at the critical temperature.

Surface tension can be measured by methods like measuring the force needed to detach a plate from the liquid surface.

Angle Of Contact

When a liquid surface meets a solid surface, the liquid surface is often curved near the line 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, measured *inside* the liquid.

The angle of contact depends on the nature of the liquid and the solid interface. It is determined by the balance of three interfacial tensions: $S_{la}$ (liquid-air), $S_{sa}$ (solid-air), and $S_{sl}$ (solid-liquid) along the line of contact:

The angle of contact indicates whether a liquid will wet a solid surface:

• If $\theta < 90^\circ$ (acute angle), $\cos \theta > 0$. $S_{sa} > S_{sl}$. Molecules of the liquid are more strongly attracted to the solid than to each other. The liquid tends to spread out and wet the solid surface (e.g., water on clean glass or plastic).
• If $\theta > 90^\circ$ (obtuse angle), $\cos \theta < 0$. $S_{sa} < S_{sl}$. Molecules of the liquid are more strongly attracted to each other than to the solid. The liquid tends to form drops and does not wet the solid surface (e.g., water on waxy surfaces, mercury on glass).

Wetting agents (like detergents) added to water decrease the angle of contact with dirt, allowing water to spread and clean effectively. Waterproofing agents increase the angle of contact, preventing water from wetting fibers.

Drops And Bubbles

Due to surface tension, free liquid drops and bubbles tend to be spherical in shape in the absence of external forces like gravity. A sphere has the minimum surface area for a given volume, thus minimizing surface energy.

Surface tension creates a pressure difference across a curved liquid-air interface:

• For a spherical liquid drop (one interface, liquid inside, gas outside) of radius $r$, the pressure inside ($P_i$) is higher than the pressure outside ($P_o$). $$ P_i - P_o = \frac{2S_{la}}{r} $$ The convex side (inside) has higher pressure.
• For a spherical air bubble inside a liquid (one interface, gas inside, liquid outside) of radius $r$, the pressure inside ($P_i$) is higher than the pressure outside ($P_o$). $$ P_i - P_o = \frac{2S_{la}}{r} $$ The convex side (inside) has higher pressure.
• For a spherical soap bubble in air (two liquid-air interfaces, liquid film between two gas regions) of radius $r$, there are two surfaces contributing to the inward tension. $$ P_i - P_o = \frac{4S_{la}}{r} $$ Higher pressure is needed inside a soap bubble compared to a liquid drop of the same radius.

Capillary Rise

One important consequence of surface tension and the pressure difference across a curved meniscus is capillary action, like water rising in a narrow tube dipped in a liquid.

If a capillary tube of radius $a$ is inserted into a liquid that wets the tube (acute angle of contact $\theta < 90^\circ$), the liquid surface inside the tube forms a concave meniscus. The pressure just under the concave meniscus ($P_i$) is less than the atmospheric pressure ($P_o$) above it. The pressure difference is $P_o - P_i = \frac{2S}{r_m}$, where $S$ is surface tension and $r_m$ is the radius of curvature of the meniscus. For a concave meniscus, $r_m = a/\cos\theta$.

So, $P_o - P_i = \frac{2S \cos\theta}{a}$. This means $P_i = P_o - \frac{2S \cos\theta}{a}$.

Consider a point A just under the meniscus inside the tube and a point B at the same horizontal level outside the tube surface. $P_B = P_o$. Due to the pressure difference, the liquid is pushed up the tube until the hydrostatic pressure of the liquid column of height $h$ balances this pressure difference.

The pressure at point A can also be expressed as $P_A = P_i + \rho g h$. Since A and B are at the same level in a continuous fluid at rest, $P_A = P_B = P_o$.

So, $P_o = P_i + \rho g h$. Substituting $P_i = P_o - \frac{2S \cos\theta}{a}$:

This formula shows that the capillary rise $h$ is directly proportional to the surface tension and the cosine of the contact angle, and inversely proportional to the liquid density, gravity, and the tube radius. Capillary rise is larger in narrower tubes.

If the liquid does not wet the tube (obtuse angle of contact $\theta > 90^\circ$, like mercury in glass), $\cos\theta$ is negative, and the liquid level inside the tube will be depressed below the level outside (capillary depression).

Detergents And Surface Tension

Detergents and soaps are examples of surfactants (surface-active agents) that significantly reduce the surface tension of water, particularly at interfaces with oil or grease.

Detergent molecules have a hydrophilic (water-attracting) end and a hydrophobic (water-repelling, oil-attracting) end. They work by surrounding oil/grease particles, forming interfaces between water and the dirt.

Adding detergents reduces the surface tension between water and greasy dirt. This lowers the angle of contact between water and the fabric/dirt, allowing water to wet the dirt more effectively. The water can then lift and wash away the dirt particles.

This process is an application of modifying surface tension to achieve desired wetting properties.

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Question 10.15. Figures 10.23(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?

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Question 10.18. Figure 10.24 (a) shows a thin liquid film supporting a small weight = $4.5 \times 10^{–2}$ N. What is the weight supported by a film of the same liquid at the same temperature in Fig. (b) and (c) ? Explain your answer physically.

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Question 10.22. A manometer reads the pressure of a gas in an enclosure as shown in Fig. 10.25 (a) When a pump removes some of the gas, the manometer reads as in Fig. 10.25 (b) The liquid used in the manometers is mercury and the atmospheric pressure is 76 cm of mercury.

(a) Give the absolute and gauge pressure of the gas in the enclosure for cases (a) and (b), in units of cm of mercury.

(b) How would the levels change in case (b) if 13.6 cm of water (immiscible with mercury) are poured into the right limb of the manometer ? (Ignore the small change in the volume of the gas).

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