Fluids

The big picture

A fluid can support a pressure gradient while at rest, and a moving fluid can lose mechanical energy through viscosity. The first useful choice is therefore the regime: static, ideal flow, or a flow in which friction matters.

Keep the geometry visible. The height in a pressure difference is vertical; the radius in a pipe-flow law is the tube radius; the curvature in a surface-pressure jump belongs to the interface. Many difficult-looking questions are really tests of that translation.

What to read, and why

Start at the first stage if the subject is new. Otherwise, use the descriptions to find the missing step in your understanding. All chapter explanations are open; the labels identify chapters whose solutions and answer checks are also free.

1. STAGE 1

   Balance forces in matter

   Relate deformation to stress, then use pressure and buoyancy for fluids at rest.

   Elastic response and measuring Young’s modulus →Separate material stiffness from specimen geometry, then infer a modulus from a controlled extension experiment.6 worked examples · 8 checks · Open explanation · member practicePressure, floating and fluid support →Derive pressure from equilibrium, then use it to explain buoyancy and support readings.16 worked examples · 12 checks · Complete free chapter

2. STAGE 2

   Follow a moving fluid

   Begin with continuity and ideal energy accounting; then learn when viscosity changes the answer.

   Continuity and energy in flowing fluids →Track mass first, then connect pressure, speed and height along a valid flow model.9 worked examples · 8 checks · Open explanation · member practiceViscosity, terminal motion and flow regimes →Choose the drag model, balance forces and test the Reynolds number before trusting a speed.6 worked examples · 8 checks · Open explanation · member practiceViscous flow through pipes →Derive the velocity profile, track pressure losses and test what a change in radius really means.8 worked examples · 6 checks · Open explanation · member practice

3. STAGE 3

   Examine the interface

   Separate surface force, curved-surface pressure and capillary rise.

   Surface tension, curved interfaces and capillary rise →Count interfaces and connect boundary forces, surface work, pressure jumps and wetting.5 worked examples · 9 checks · Open explanation · member practiceCapillarity and curved interfaces →Read the meniscus, count the interfaces and connect pressure, height and surface energy.7 worked examples · 6 checks · Open explanation · member practice

Choose a method before an equation

Use the physical conditions to narrow the choices. A formula becomes useful after you know why its model applies.

Try a complete free problem

These problems include the hints, worked reasoning and answer feedback. Attempt one before opening the solution.