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Mechanical Properties of Fluids

Ch. 2Std 12

Easy Overview

Fluids — liquids and gases — are everywhere around us, and they behave in ways that often defy our everyday intuition. Have you ever wondered how a massive ship made of steel floats on water, or how a tiny needle can rest on the surface of water without sinking? Why does honey pour so much more slowly than water, and why does a liquid jet break up into droplets? The mechanical properties of fluids give us the answers. This branch of physics deals with fluids at rest (fluid statics) and fluids in motion (fluid dynamics), covering everything from the pressure at the bottom of the ocean to the lift that keeps an aeroplane in the sky. Let us begin with pressure. Pressure is defined as force per unit area (P = F/A). Unlike solids, where a force can be applied in one direction, fluids exert pressure equally in all directions. A swimmer feels pressure from all sides — not just from above. This is Pascal's law: in an enclosed fluid, a pressure change at any point is transmitted undiminished throughout the fluid. This seemingly simple principle has enormous practical applications. Hydraulic lifts, hydraulic brakes in cars, and hydraulic presses all work on this principle. A small force applied to a small-area piston creates a pressure that is transmitted to a large-area piston, multiplying the force. Now consider what happens when you immerse an object in a fluid. The pressure increases with depth because the weight of the fluid above pushes down. The pressure at depth h in a fluid of density ρ is P = P₀ + ρgh, where P₀ is the pressure at the surface. This increasing pressure with depth causes an upward buoyant force — Archimedes' principle. An object feels lighter in water because the buoyant force partially cancels its weight. A ship floats because its shape displaces enough water that the buoyant force equals the ship's weight. A submarine adjusts its buoyancy by taking on or releasing water. And then there are fluids in motion. Bernoulli's principle is the superstar of fluid dynamics. It says that where the speed of a fluid is high, the pressure is low, and vice versa. This explains how an aeroplane wing generates lift, how a carburettor works, why a curved cricket ball swings, and why a roof flies off in a storm (wind above creates low pressure). But real fluids also have viscosity — internal friction. Honey is more viscous than water. Viscosity causes drag and energy loss. Surface tension makes water droplets spherical and allows insects to walk on water. Together, these properties determine how fluids behave in the natural and engineered world — from blood flow in your arteries to oil flow in pipelines, from raindrops to the design of ships and aircraft.

Pressure in Fluids

Pressure is defined as the normal force per unit area: P = F/A. In fluids, pressure acts equally in all directions at a given point. The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m². Atmospheric pressure at sea level is about 1.01 × 10⁵ Pa, also called 1 atm. Other common units include bar (1 bar = 10⁵ Pa), torr (1 torr = 133.3 Pa, approximately 1 mmHg), and psi (pounds per square inch). Pressure in a fluid at rest increases with depth: P = P₀ + ρgh. This is why dams are built thicker at the bottom — the pressure is much greater there. It also explains why a small hole near the bottom of a water tank shoots water much farther than a hole near the top.

Pascal's Law and Hydraulic Machines

Pascal's law states that pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. This is the operating principle of hydraulic systems. In a hydraulic lift, two cylinders of different cross-sectional areas are connected and filled with a fluid (usually oil). A force F₁ applied to the small piston of area A₁ creates pressure P = F₁/A₁. This pressure is transmitted through the fluid to the large piston of area A₂, producing a force F₂ = P × A₂ = (A₂/A₁)F₁. Since A₂ > A₁, the force is multiplied by the ratio of areas. Hydraulic brakes in cars work on the same principle — a small force on the brake pedal creates a much larger force on the brake pads. The work-energy principle holds: the small piston moves through a larger distance than the large piston, so input work equals output work.

Variation of Pressure with Depth

Consider a fluid in a container open to the atmosphere. Take a horizontal area element at depth h below the surface. The weight of the fluid column above this area is mg = ρVg = ρ(Ah)g. The force on the area from above is the weight of the column plus the force from atmospheric pressure: F = P₀A + ρghA. Therefore, P = P₀ + ρgh. This means pressure increases linearly with depth. At a depth of 10 m in water (density 1000 kg/m³), the pressure due to water alone is about 10⁵ Pa — roughly one additional atmosphere. At the deepest point in the ocean (the Mariana Trench, about 11 km deep), the pressure exceeds 1000 atm. The pressure is the same at all points at the same depth, regardless of the shape of the container (hydrostatic paradox).

Archimedes' Principle and Buoyancy

Archimedes' principle: any object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. The buoyant force arises because the pressure at the bottom of the object is greater than the pressure at the top (since pressure increases with depth). This pressure difference creates a net upward force. Mathematically, F_b = ρ_fluid × V_displaced × g. An object floats if its density is less than the fluid's density (ρ_object < ρ_fluid), sinks if ρ_object > ρ_fluid, and hangs suspended at any depth if ρ_object = ρ_fluid. This is why ice (density 917 kg/m³) floats on water (density 1000 kg/m³) — about 92% of an iceberg is submerged. The apparent loss of weight of an object in a fluid equals the weight of the fluid displaced.

Fluid Dynamics and Streamline Flow

Fluid dynamics distinguishes between two types of flow: streamline (or laminar) and turbulent. In streamline flow, every particle of fluid follows the same smooth path (streamline), and different streamlines do not cross. The velocity at each point is constant in time. In turbulent flow, the paths become irregular, chaotic, and change with time — eddies and vortices form. Whether flow is laminar or turbulent is determined by the Reynolds number: Re = ρvD/η, where D is a characteristic length (like pipe diameter). Low Re (below about 2000) gives laminar flow. High Re (above about 3000) gives turbulent flow. Between them is a transition region. The equation of continuity for incompressible fluids states A₁v₁ = A₂v₂ — the product of cross-sectional area and velocity is constant along a pipe. Where the pipe is narrower, the fluid must flow faster.

Bernoulli's Equation

Bernoulli's equation for an ideal fluid (incompressible, non-viscous, steady, irrotational flow) is P + ½ρv² + ρgh = constant along a streamline. It is essentially a statement of conservation of energy per unit volume. The three terms represent: pressure energy (P), kinetic energy per unit volume (½ρv²), and potential energy per unit volume (ρgh). If the pipe is horizontal (h constant), the equation simplifies to P + ½ρv² = constant — where velocity is high, pressure is low, and vice versa. This explains the Venturi effect, the working of a carburettor, the lift on an aeroplane wing, and the curved trajectory of a spinning ball (the Magnus effect). Bernoulli's equation has limitations — it does not apply to turbulent flow, compressible flow at high speeds, or when there is significant viscous dissipation.

Viscosity and Newton's Law of Viscous Flow

Viscosity is a measure of a fluid's resistance to shear or flow. Consider a fluid between two parallel plates separated by distance dy. The lower plate is stationary, the upper plate moves with velocity v. The fluid in contact with each plate has the same velocity as the plate (no-slip condition). The velocity gradient (or shear rate) is dv/dy. The shear stress τ (force per unit area required to maintain the flow) is proportional to the shear rate: τ = η(dv/dy), where η is the coefficient of dynamic viscosity. Its SI unit is pascal-seconds (Pa·s) or poise (P) in the CGS system (1 P = 0.1 Pa·s). Water at 20°C has η ≈ 1.0 × 10⁻³ Pa·s; honey has η ≈ 2-10 Pa·s; air at 20°C has η ≈ 1.8 × 10⁻⁵ Pa·s. Fluids that obey this linear relationship are called Newtonian fluids (water, air, thin oils). Non-Newtonian fluids (ketchup, toothpaste, blood) have more complex behaviour.

Stokes' Law and Terminal Velocity

When a sphere moves through a viscous fluid at low speeds, the drag force opposing its motion is given by Stokes' law: F = 6πηrv, where η is the viscosity of the fluid, r is the radius of the sphere, and v is its velocity relative to the fluid. This drag force is proportional to velocity, not velocity squared (which is the case for high-speed turbulent drag). For a sphere falling under gravity through a fluid, three forces act: weight (mg = ρ_sVg) downward, buoyancy (ρ_fVg) upward, and viscous drag (6πηrv) upward. Initially, the sphere accelerates. As speed increases, drag increases until the net force becomes zero. At terminal velocity: mg = ρ_fVg + 6πηrv_t. Solving gives v_t = (2/9)(r²/η)(ρ_s - ρ_f)g. Terminal velocity is proportional to r² — a larger sphere falls much faster.

Surface Tension and Surface Energy

Surface tension (γ) is the force per unit length acting along the surface of a liquid, perpendicular to any line drawn on the surface. Its SI unit is N/m. It arises because molecules at the surface experience a net inward pull from the molecules below, unlike molecules in the bulk which are surrounded on all sides. The surface is like a stretched membrane — it tends to minimise its area. This is why small liquid droplets are spherical (a sphere has the smallest surface area for a given volume). Surface tension also has an equivalent definition as surface energy: the work done per unit area to increase the surface area of the liquid. γ = W/ΔA. The units N/m and J/m² are equivalent. For water at 20°C, γ ≈ 0.073 N/m. For mercury, γ ≈ 0.465 N/m.

Capillary Rise and Depression

When a narrow tube (capillary) is dipped into a liquid, the liquid either rises or falls inside the tube relative to the outside level. This is capillary action. If the liquid wets the tube (adhesion > cohesion, e.g. water in glass), it rises. If it does not (cohesion > adhesion, e.g. mercury in glass), it is depressed. The height of rise or depression is given by h = 2γ cosθ / (ρgr), where γ is surface tension, θ is the contact angle, ρ is density, g is gravity, and r is the tube radius. For water in glass, θ ≈ 0° (cosθ = 1), so h = 2γ/(ρgr). Notice that h ∝ 1/r — narrower tubes give greater rise. This is how plants draw water from roots to leaves through tiny xylem vessels.

Excess Pressure in Drops and Bubbles

Due to surface tension, the pressure inside a liquid drop or a soap bubble is higher than the pressure outside. For a spherical liquid drop with one surface, the excess pressure is ΔP = 2γ/R. For a soap bubble, which has two surfaces (inner and outer), ΔP = 4γ/R. The excess pressure is inversely proportional to the radius — smaller drops have higher internal pressure. This is why two soap bubbles of different sizes joined by a tube behave interestingly: air flows from the smaller bubble (higher pressure) to the larger bubble (lower pressure), making the larger one grow and the smaller one shrink.

The Equation of Continuity

The equation of continuity expresses conservation of mass in fluid flow. For an incompressible fluid flowing through a pipe of varying cross-section, the mass flow rate must be constant: ρA₁v₁ = ρA₂v₂. Since ρ is constant for incompressible flow, this simplifies to A₁v₁ = A₂v₂, or Av = constant. This means the flow speed is inversely proportional to the cross-sectional area. Where the pipe is narrow, the fluid moves faster. Where it is wide, the fluid moves slower. This is why a river flows slowly where it is wide and fast where it narrows. It also explains why a gentle squeeze on a garden hose makes the water shoot out faster.

Torricelli's Law and Efflux Velocity

Torricelli's law gives the speed of efflux of a fluid through a small hole at the bottom of a tank. Applying Bernoulli's equation between the free surface of the liquid (where P = P_atm, v ≈ 0, h = H) and the hole (where P = P_atm, v = v, h = 0), we get v = √(2gH). Remarkably, this is exactly the speed an object would attain if it fell freely from a height H — the fluid's potential energy is completely converted to kinetic energy. The volume flow rate through the hole is Q = A_hole × v = A_hole × √(2gH). In reality, the actual flow rate is slightly less due to the vena contracta effect (the jet contracts slightly after leaving the hole), so a coefficient of discharge (C_d ≈ 0.6-0.7) is often included.

Applications of Bernoulli's Principle

Bernoulli's principle has numerous real-world applications. (1) Aeroplane wing (airfoil): the curved upper surface causes air to travel faster than along the flat lower surface, creating lower pressure above and generating lift. (2) Venturi meter: a constriction in a pipe causes a pressure drop proportional to the flow speed — used to measure flow rate. (3) Atomiser and spray gun: fast-moving air over a vertical tube reduces pressure, drawing liquid up and breaking it into a fine spray. (4) Bunsen burner: gas enters through a small jet, creating a low-pressure region that draws in air through side openings for combustion. (5) Pitot tube: measures flow speed by comparing stagnation pressure with static pressure. (6) Curving balls in sports: a spinning ball drags air on one side, creating a pressure difference that curves its trajectory (Magnus effect).

Reynolds Number and Flow Regimes

The Reynolds number (Re) is a dimensionless quantity that predicts whether fluid flow will be laminar or turbulent. It is defined as Re = ρvL/η, where ρ is fluid density, v is flow speed, L is a characteristic length (e.g., pipe diameter), and η is dynamic viscosity. Physically, Re = (inertial forces)/(viscous forces). When viscous forces dominate (low Re), the flow is smooth and laminar. When inertial forces dominate (high Re), the flow becomes turbulent with chaotic eddies. The critical Reynolds number for flow through a pipe is about 2000 — below this, flow is typically laminar; above 3000, it is turbulent. Between 2000 and 3000 is a transition region. The Reynolds number is crucial in engineering for scaling — if two flows have the same Re, they behave similarly even if their scales are vastly different (dynamic similarity).

Key Points

  • •Pressure P = F/A. In fluids, pressure acts equally in all directions.
  • •Pascal's law: pressure applied to enclosed fluid is transmitted undiminished throughout.
  • •Pressure variation with depth: P = P₀ + ρgh.
  • •Archimedes' principle: buoyant force = weight of fluid displaced.
  • •Equation of continuity: A₁v₁ = A₂v₂ (incompressible fluid).
  • •Bernoulli's equation: P + ½ρv² + ρgh = constant along a streamline.
  • •Viscosity η measures resistance to flow. Newton's law: τ = η(dv/dy).
  • •Stokes' law: drag force on a sphere F = 6πηrv.
  • •Terminal velocity: v_t = (2/9)(r²/η)(ρ_s - ρ_f)g.
  • •Surface tension γ = force per unit length along a surface.
  • •Capillary rise/depression: h = 2γ cosθ/(ρgr).
  • •Excess pressure in liquid drop: ΔP = 2γ/R. In soap bubble: ΔP = 4γ/R.
  • •Torricelli's law: v = √(2gH).
  • •Reynolds number Re = ρvL/η. Re < 2000 → laminar; Re > 3000 → turbulent.

Practice Questions

  • State Pascal's law. Explain the working of a hydraulic lift with a neat diagram. A hydraulic lift has a small piston of area 0.01 m² and a large piston of area 0.5 m². What force is needed on the small piston to lift a 1000 kg car?
  • Derive the expression for terminal velocity of a sphere falling through a viscous fluid. How does terminal velocity depend on the radius and density of the sphere?
  • State Bernoulli's principle. Derive Bernoulli's equation and explain any three applications from daily life.
  • What is surface tension? Explain why (a) a needle can float on water, (b) small drops of liquid are spherical, (c) soap bubbles have excess pressure inside.
  • Distinguish between laminar and turbulent flow. What is the physical significance of Reynolds number? Calculate the Reynolds number for water flowing at 1 m/s through a pipe of diameter 2 cm (ρ = 1000 kg/m³, η = 10⁻³ Pa·s).
  • Derive the expression for capillary rise in a tube. A capillary tube of radius 0.5 mm is dipped in water. Find the height of water rise in the tube. Surface tension of water = 0.072 N/m, contact angle = 0°, g = 9.8 m/s².
  • A large water tank has a small hole at a depth of 5 m below the water surface. Find the velocity of efflux of water from the hole. If the hole has an area of 1 cm², find the volume flow rate of water.
  • Explain Archimedes' principle. A solid sphere of density 2000 kg/m³ and radius 5 cm is dropped in a liquid of density 1000 kg/m³ and viscosity 1.5 Pa·s. Find the terminal velocity of the sphere.