Fluid Mechanics & Surface Tension — Pressure, Buoyancy, Viscosity & Bernoulli's Principle

Overview

Fluid mechanics is the branch of physics that studies the behaviour of fluids (liquids and gases) at rest (fluid statics) and in motion (fluid dynamics). It underpins technologies ranging from hydraulic brakes and water supply systems to airplane flight and weather prediction. For UPSC Prelims, questions frequently test Archimedes' principle, Pascal's law, Bernoulli's theorem, and everyday applications of fluid properties such as surface tension and viscosity.

Exam Strategy: Focus on understanding the principles (Pascal's law, Archimedes' principle, Bernoulli's theorem) and their real-world applications. Prelims questions are typically application-based — e.g., "Why does a ship float?", "How does a hydraulic brake work?", "Why does a spinning cricket ball swing?" Tables of key concepts and formulae serve as excellent last-minute revision tools.

Properties of Fluids

Density and Specific Gravity

• Water has maximum density at 4 °C (approximately 1000 kg/m³ or 1 g/cm³)
• Objects with specific gravity less than 1 float in water; greater than 1 sink
• Hydrometer — instrument used to measure the specific gravity of liquids (e.g., lactometer for milk purity)

Pressure in Fluids

Key Fact: Atmospheric pressure was first measured by Evangelista Torricelli in 1643 using a mercury barometer. Standard atmospheric pressure supports a mercury column of 760 mm (29.92 inches) at sea level and 0 °C.

• Pressure at a point in a fluid depends only on depth, density, and gravitational acceleration — not on the shape of the container (this is called the hydrostatic paradox)
• Pressure acts equally in all directions at a given depth in a fluid (a consequence of Pascal's law)

Pascal's Law

Statement: A change in pressure applied to an enclosed, incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.

Formulated by French mathematician Blaise Pascal in 1653 (published 1663).

Applications of Pascal's Law

Prelims Tip: The hydraulic press does NOT violate the law of conservation of energy. The force is amplified, but the distance moved by the larger piston is proportionally smaller. Work input equals work output (in an ideal system): F₁ × d₁ = F₂ × d₂.

Archimedes' Principle and Buoyancy

Archimedes' Principle

Statement: When a body is wholly or partially immersed in a fluid, it experiences an upward force (buoyant force) equal to the weight of the fluid displaced by the body.

Discovered by the Greek mathematician Archimedes of Syracuse (c. 287–212 BCE).

Buoyant Force: F_b = ρ_fluid × V_displaced × g

Conditions for Floating and Sinking

Law of Flotation

A floating body displaces a weight of fluid equal to its own weight. This is why:

• A steel ship floats — the hull encloses a large volume of air, making the average density of the ship less than water
• An iron nail sinks — its density (~7,800 kg/m³) is much greater than water
• Icebergs float with about 90% submerged — because ice density (~917 kg/m³) is about 90% of seawater density (~1025 kg/m³)

The Plimsoll Line

The Plimsoll line (or load line) is a marking on a ship's hull indicating the maximum depth to which the vessel may be safely immersed when loaded with cargo. Named after Samuel Plimsoll (1824–1898), a British MP who campaigned against overloading of cargo ships. The Merchant Shipping Act 1876 made the load line mark compulsory. The International Convention on Load Lines (1966) standardised these markings globally.

Different markings account for:

Why different marks? Seawater is denser than fresh water, and cold water is denser than warm water. A ship sits higher in denser water, so it can carry more cargo safely.

Viscosity

Definition and Concept

Viscosity is the internal friction of a fluid — its resistance to flow. A fluid with high viscosity (e.g., honey) flows slowly; a fluid with low viscosity (e.g., water) flows easily.

Newton's Law of Viscosity

The shear stress between adjacent fluid layers is proportional to the velocity gradient between the layers:

τ = η (dv/dx)

where τ = shear stress, η = dynamic viscosity (coefficient of viscosity), dv/dx = velocity gradient.

• Fluids that obey this law are called Newtonian fluids (e.g., water, air, glycerine)
• Fluids that do not obey this law are Non-Newtonian fluids (e.g., ketchup, blood, quicksand)

SI Unit of Viscosity

Stokes' Law and Terminal Velocity

Stokes' Law (George Gabriel Stokes, 1851): The drag force on a small sphere moving through a viscous fluid is:

F = 6πrηv

where r = radius of sphere, η = viscosity, v = velocity.

Terminal Velocity: When a sphere falls through a viscous fluid, it accelerates until the drag force plus buoyant force equals its weight. At this point, it moves at constant speed called terminal velocity:

v_T = 2r²(ρ_s − ρ_f)g / 9η

where ρ_s = density of sphere, ρ_f = density of fluid.

Applications of Viscosity and Stokes' Law

Surface Tension

Definition

Surface tension is the property of a liquid surface that causes it to behave like a stretched elastic membrane. It arises because molecules at the surface experience a net inward cohesive force (they are pulled inward by neighbouring molecules but have no molecules above to balance).

SI Unit: N/m (Newton per metre)

Phenomena Due to Surface Tension

Capillarity

Capillarity (capillary action) is the rise or fall of a liquid in a narrow tube (capillary) due to the combined effect of surface tension, cohesion, and adhesion.

Applications of Capillarity:

• Absorption of water by a towel or sponge
• Rising of oil in a lamp wick
• Movement of water from roots to leaves in plants (along with transpiration pull)
• Ink rising in blotting paper
• Groundwater movement through soil pores

Effect of Detergents and Temperature

• Detergents (surfactants) reduce the surface tension of water, allowing it to spread and penetrate fabrics — this is why soapy water cleans better than plain water
• Hot water has lower surface tension than cold water — another reason hot water cleans better
• Adding impurities generally decreases the surface tension of water

Exam Tip: The cleaning action of soap/detergent is a frequently tested concept. Soap molecules have a hydrophilic (water-attracting) head and a hydrophobic (water-repelling) tail. The tail attaches to grease/oil, while the head remains in water, forming micelles that lift dirt away from surfaces.

Bernoulli's Theorem

Statement

For an ideal, incompressible, non-viscous fluid in steady (streamline) flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline:

P + ½ρv² + ρgh = constant

where P = pressure, ρ = fluid density, v = velocity, g = acceleration due to gravity, h = height.

This is essentially the law of conservation of energy applied to fluid flow.

Key Implication

Where fluid velocity increases, pressure decreases — and vice versa (for horizontal flow where h is constant).

Applications of Bernoulli's Theorem

Common Misconception: Bernoulli's principle alone does not fully explain airplane lift. The Coanda effect (air following curved surfaces) and Newton's third law (downward deflection of air) also contribute. However, for UPSC purposes, the Bernoulli explanation is the standard expected answer.

Reynolds Number — Laminar vs Turbulent Flow

Types of Fluid Flow

Reynolds Number (Re)

A dimensionless number that predicts whether fluid flow will be laminar or turbulent:

Re = ρvD / η

where ρ = fluid density, v = flow velocity, D = characteristic length (e.g., pipe diameter), η = dynamic viscosity.

Prelims Tip: Reynolds number is the ratio of inertial forces to viscous forces in a fluid. A high Reynolds number means inertial forces dominate (turbulent flow); a low number means viscous forces dominate (laminar flow). It is a dimensionless quantity — it has no units.

Fluid Mechanics in Daily Life and Technology

Frequently Asked Questions (Prelims Pattern)

Key Terms for Quick Revision

Sources: Physics LibreTexts (OpenStax University Physics); Britannica; Wikipedia; NOAA (oceanservice.noaa.gov); NASA Glenn Research Center; TeachEngineering (University of Houston).