Lecture 1: Introduction

Nothing on slides 1-24 is examinable as fluid mechanics, but four facts matter for planning.

Intuition. Water pouring from a pipe looks smooth (macroscopic). Zoom into a tiny cube of it and you can still talk about "the pressure on this face" (microscopic). Zoom further and it is a crowd of molecules moving randomly and colliding (molecular).

Physical meaning of each scale.

• Molecular: about Avogadro's number of molecules ($6.02 \times 10^{23}$ per mole) in constant random thermal motion. Collisions carry momentum, energy and mass from place to place. This is the origin of viscosity, heat conduction and diffusion.
• Microscopic (from molecules to continua): the properties you use in engineering are averages over many molecules - temperature is average molecular kinetic energy; densityDensity$\rho$Units: kg m^-3Mass per unit volume. Water is taken as 1000 kg/m^3 in this module's exams.Open in lectureslide 32Questions is mass per unit volume; viscosity is momentum transport between layers by molecules; diffusivity is the rate of molecular mixing (the link to the mass-transfer half of the module, Fick's law).
• Macroscopic: pipe flow, drag, lift, pumps and turbines; the systems you design.

The key idea for the whole module: we never track molecules; we use averaged properties defined at every point. The continuum section explains when that is allowed.

Slide 27 previews the three conservation laws the module is built on. You are not expected to use these yet; they are introduced properly in later weeks. What you should take from the slide is the structure: each physical phenomenon has a balance equation.

The second row of equations on the slide (with $D/Dt$ and $\nabla$) are the same laws written at a point, which is only possible because of the continuum hypothesisContinuum hypothesisTreat the fluid as a continuous medium so that density, velocity, pressure and temperature are defined at every point, even though it is really made of molecules. Valid when the Knudsen number is below about 0.001.Open in lectureslide 28. Those differential forms are what CFD solves (the coursework).

Intuition. Put a small box around point C in a glass of water and measure the mass inside divided by the box volume. Shrink the box. For a while the answer stays the same, but once the box is so small that it holds only a handful of molecules, the answer jumps around wildly depending on whether a molecule happens to be inside at that instant.

Reading the graph (right of the slide).

• Large $\delta V$ (right side): the density is a steady value. This is the continuum region.
• Very small $\delta V$ (left side): density fluctuates because of individual molecules. This is the non-continuum region.
• The dashed line marks the smallest volume, call it $\delta V_{min}$, that still contains enough molecules for a stable average.

The definition (slides 28 and 32).

$\rho = \lim_{\delta V \to \delta V_{min}} \frac{\delta m}{\delta V}$

The slides write the limit as $\delta V \to 0$. Read the "0" as "small compared with the flow, but still large compared with molecular spacing". Mathematically we then treat $\delta V_{min}$ as zero, which is what lets us write density, velocity, pressure and temperature as continuous functions $\rho(x,y,z,t)$, $v(x,y,z,t)$ and so on, and take derivatives of them.

What the continuum hypothesis lets you do (slide 30):

1. Each fluid element (a small control volume) contains a very large number of molecules.
2. Microscopic fluctuations average out.
3. Density, velocity, pressure and temperature become continuous functions of space and time.

Intuition. The model fails when the molecules are so far apart, relative to the size of the thing they flow through, that a molecule may cross the whole device without hitting another. Then "the velocity at a point" stops meaning anything. This happens in very low-pressure (rarefied) gases, such as high-altitude flight, vacuum systems and some micro- and nano-devices. It almost never happens in liquids.

The mean free pathMean free path$\lambda$Units: mAverage distance a molecule travels between collisions. About 68 nm for air at 1 atm and 20 deg C; inversely proportional to pressure at fixed temperature.Open in lectureslide 29 $\lambda$ is the average distance a molecule travels between collisions. Compare it with the size of the system $L$:

Molecular view (left of the slide). In a solid, molecules are tightly packed in fixed positions and strongly bonded. In a liquid they are close together but free to move past each other. In a gas they are far apart and move almost independently. This is why liquids and gases are both fluids: neither has fixed molecular positions.

Continuum view (right of the slide): the definition. Apply the same tangential (shear) force $F$ to the top of a block of height $h$:

• Steel (solid): the block deforms by a finite displacement $\delta x$ and then stops. The internal shear stressShear stress$\tau$Units: Pa (N m^-2)Tangential force per unit area. In a moving fluid it is set by the rate of deformation (viscosity); in a fluid at rest it is zero.Open in lectureslide 31 balances $F$. Remove $F$ and it springs back (elastic). A solid resists shear by deforming its shape by a fixed amount.
• Water (fluid): the top surface keeps moving, at a velocity $\delta u$. The deformation grows continuously for as long as the shear stress acts. A fluid cannot resist a shear stress at rest; it flows.

That is why the slide labels the solid with a displacement $\delta x$ but the fluid with a velocity $\delta u$. For a solid, shear stress is related to the amount of deformation (strain); for a fluid, shear stress is related to the rate of deformation (strain rate, roughly $\delta u / h$).

Two consequences you will use in Lecture 2.

1. A fluid at rest has no shear stress. If there were any, it would be flowing. So in a static fluid the only surface stress is pressure, which acts normal to any surface. This is the foundation of hydrostaticsHydrostaticsFluids at rest. Only pressure acts on any surface, and pressure increases with depth as p = p0 + rho g h.Open in lecture.
2. Shear stress needs motion. The fluid's resistance to the rate of deformation is viscosity. For a Newtonian fluid this is Newton's law of viscosity, $\tau = \mu\,\dfrac{dv}{dy}$, which is on the ENGR271 2026 formula sheet (rheology weeks). You do not need it yet; just see where it comes from.

Intuition. Density tells you how much mass is packed into a space. Specific weight tells you how heavy that space is, which is what matters in hydrostatics because pressure comes from the weight of fluid above. Relative density is just density compared with water, a quick number for "heavier or lighter than water".

The Week 1 exercise sheet mostly covers Lectures 2 and 3 (pressure, manometers, forces on surfaces, buoyancy); those exercises are solved in the L2 and L3 packs. Exercise 6 part 1 is a density calculation and fits here.

No real exam question tests the continuum hypothesis, the Knudsen number or the definition of a fluid, so the following are generated in the style of the ENGR271 2026 paper (short theory parts worth 5 marks, calculation parts worth 5-10 marks, data listed in the stem, $g = 9.81\ \mathrm{m/s^2}$, $\rho_w = 1000\ \mathrm{kg/m^3}$).