Fluid Statics

SYLLABUS

Fluid statics: Pressure in a fluid; Buoyancy, Surface tension: Theory of surface tension; Surface energy, Angle of contact, capillarity and its applications, Fluid Dynamics: Newton’s formula for viscosity in a liquid; Coefficient of viscosity, Poiseuille’s formula and its application, Stokes law and its applications, Equation of continuity and its applications, Bernoulli’s equation and its applications.

What is This Chapter About?

Why does a ship made of heavy steel float on water while a small pebble sinks? Why does water rise inside a thin glass tube? Why do deep-sea fish get crushed when brought to the surface? Why does blood flow smoothly through arteries but become turbulent under certain conditions?

All of these questions are answered by Fluid Statics — the study of fluids (liquids and gases) both at rest and in motion. Chapter 3 of NEB Grade 12 Physics is one of the broadest and most practically relevant chapters in the entire course, carrying 9 teaching hours. It bridges classical mechanics with real-world phenomena in engineering, medicine, and nature.

Key Concepts You Will Learn

Pressure in Fluids, Buoyancy, and Archimedes’ Principle

Pressure in a fluid at rest increases with depth: [latex]P = \rho g h[/latex]. Archimedes’ Principle states that any object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This principle explains floating and sinking — an object floats when its weight equals the buoyant force, and sinks when its weight exceeds it.

Pascal’s Law states that pressure applied to an enclosed fluid is transmitted equally in all directions. This is the working principle of hydraulic systems — brakes, lifts, and presses.

Surface Tension

At the surface of a liquid, molecules experience a net inward force because they have fewer neighbours above than below. This creates a thin elastic-like layer at the surface called surface tension. Surface tension explains why water forms droplets, why insects can walk on water, and why soap reduces the surface tension of water. The chapter establishes the relationship between surface energy and surface tension: [latex]T = \frac{W}{A}[/latex].

Angle of Contact and Capillarity

The angle of contact is the angle a liquid surface makes with a solid wall at their point of contact. For water in glass, it is less than 90° (wetting liquid), causing the liquid surface to curve upward — this leads to capillarity, the rise of liquid in narrow tubes. The height of capillary rise is given by [latex]h = \frac{2T\cos\theta}{r\rho g}[/latex]. Capillarity is responsible for water rising through plant roots and soil.

Viscosity and Newton’s Formula

Real fluids resist flow due to internal friction between layers — this property is called viscosity. Newton’s formula for viscous force is [latex]F = \eta A \frac{dv}{dx}[/latex], where [latex]\eta[/latex] is the coefficient of viscosity. Viscosity explains why honey pours slowly and water pours quickly.

Poiseuille’s Formula and Stokes’ Law

Poiseuille’s formula gives the rate of flow of a viscous fluid through a cylindrical pipe: [latex]Q = \frac{\pi r^4 P}{8\eta l}[/latex]. This has direct applications in blood flow through arteries and water through pipes.

Stokes’ Law states that the viscous drag force on a small sphere moving through a fluid is [latex]F = 6\pi\eta rv[/latex]. When this drag force balances gravity, the sphere reaches terminal velocity — the constant maximum speed of fall.

Streamlined Flow, Reynolds Number, and Turbulence

Fluid flow can be smooth (laminar) or chaotic (turbulent). The Reynolds number [latex]R_e = \frac{\rho v d}{\eta}[/latex] predicts which type of flow will occur — low values indicate laminar flow, high values indicate turbulence.

Equation of Continuity and Bernoulli’s Equation

The equation of continuity [latex]A_1v_1 = A_2v_2[/latex] states that for an incompressible fluid, the product of cross-sectional area and velocity is constant — fluid speeds up where a pipe narrows.

Bernoulli’s equation [latex]P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}[/latex] is one of the most powerful results in fluid mechanics. It states that in a streamlined flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant. This explains how aircraft wings generate lift, why a spinning ball curves in flight, and how carburettors and atomisers work.

Why This Chapter Matters for NEB Exams

Fluid Statics is one of the longest chapters in Grade 12 and is examined thoroughly. Derivations of Bernoulli’s equation, Stokes’ law terminal velocity, and capillary rise formula are frequent long-answer questions. Numerical problems on viscosity, surface tension, and Poiseuille’s formula appear regularly. The breadth of this chapter means that almost any subtopic can appear in a given year’s paper.

Important Topics to Focus On

• Archimedes’ principle and law of floatation with applications
• Surface tension, surface energy, and the relation between them
• Capillary rise formula derivation and applications
• Newton’s formula for viscosity and coefficient of viscosity
• Poiseuille’s formula and its applications
• Stokes’ law and terminal velocity derivation
• Reynolds number and types of fluid flow
• Equation of continuity
• Bernoulli’s equation — derivation and applications (venturimeter, aerofoil lift, atomiser)
• Numerical problems on all topics above

Connection to Other Chapters

Fluid Statics connects to Grade 11 Elasticity (both deal with material properties under stress) and to Gravitation (pressure depends on gravitational acceleration). Bernoulli’s equation is an application of energy conservation from Grade 11 Chapter 5. The concept of pressure and flow also appears in discussions of the human circulatory system, making this chapter a natural bridge between physics and biology.