Fluid properties

Statics dealt in rigid bodies and asked how forces balance on them. Nothing here is rigid, and nothing sits still under load without admitting it. Before any of that can be made precise, it is worth being exact about what separates a fluid from a solid, because the distinction is not about being runny or being a liquid rather than a gas — plenty of what this course treats as one fluid barely pours. It is about what happens under a shear force, and that idea, once fixed, hands you the rest of the subject’s vocabulary almost for free.

What makes a fluid

Push sideways on a solid block and it deforms by some amount and stops, holding a fixed strain against your force. Push sideways on a fluid and it deforms too, but it never stops — it keeps deforming for as long as the force is applied, however small that force is. There is no shear stress a fluid at rest can support, not even an infinitesimal one. That is the definition, and it is worth taking literally: a fluid is a substance that deforms continuously under any applied shear.

Two consequences follow immediately and both are used constantly. A fluid at rest has no shear stress anywhere in it, so the only stress it can carry is normal to a surface — pressure, which is the subject of the next article. And because the deformation never settles, what characterises a moving fluid is not how far it has sheared but how fast it is shearing — a rate, not an angle. Chasing that rate down at a point, across a thin layer of fluid dragged along by a moving surface, is exactly what the next section does.

This picture is built on a second assumption that is easy to skip past: the continuum assumption. Real fluids are made of molecules with empty space between them, colliding and moving individually, and none of that is what the equations of this course describe. Instead, a fluid is treated as a continuous distribution of matter, so that density, velocity and pressure are smooth functions defined at every point, with no gaps. That holds as long as the smallest length scale you care about is enormous compared with the mean free path between molecular collisions — true for almost every flow an engineer meets, and false for the rarefied gas around a spacecraft at very high altitude, where the individual molecules matter again and this whole framework stops applying.

Both liquids and gases satisfy the shear definition and count as fluids, though they differ in how they respond to a normal squeeze. A liquid’s molecules sit close enough together that compressing it takes an enormous pressure for a tiny change in volume, so for most purposes a liquid is treated as incompressible. A gas has no such restraint and will expand to fill whatever volume it is given; its density responds directly to pressure and temperature through an equation of state. That difference resurfaces repeatedly through the course — a liquid has a free surface and fills a container from the bottom, a gas does not — but it is a difference of degree, not of the defining property. Both flow because neither can resist a shear.

Density and specific weight

Once a substance qualifies as a fluid, the first thing worth knowing about a particular one is how much of it there is per unit volume:

Density is what makes water heavy and air light, and it is the single property from which most of the rest of this section is built by multiplication. Multiply by the local gravitational acceleration and you have specific weight, the weight of a unit volume rather than its mass:

The two are proportional, not identical, and the distinction matters the moment gravity is not exactly 9.81 m/s² — on the Moon the density of water is unchanged and its specific weight is a sixth of what it is here. Density is a property of the substance; specific weight also carries the local value of \(g\).

Different fluids are usually compared against water rather than against each other, which is where specific gravity comes in:

the reference is water at 4°C, where its density is at a maximum,\(\ 1000\ \text{kg/m}^3\). Specific gravity is a pure ratio with no units, which is exactly its use — a mercury manometer, a hydrometer reading, a table of relative densities all quote SG because it lets you carry a fluid’s density around as a single memorable number: oil sits around 0.85, seawater around 1.03, mercury at a startling 13.6. Multiply any of these by 1000 kg/m³ and you have the density back.

What trips students up is treating density as fixed. For a liquid it very nearly is, changing by a fraction of a per cent over ordinary temperature ranges and safely ignored in almost every calculation this course asks of you. For a gas it is not: density depends on both pressure and temperature through the ideal gas law, and doubling the absolute pressure at fixed temperature doubles the density outright. Carrying a gas’s density as a constant into a problem that changes its pressure is one of the most common errors made in the transition from statics, where nothing compresses, into this subject, where sometimes everything does.

Viscosity

Trap a thin film of fluid between two parallel plates, hold the bottom one still and drag the top one at a steady speed \(U\), and something definite happens inside the gap: the fluid touching each plate moves with that plate exactly — zero at the bottom, \(U\) at the top — a boundary condition called no-slip, and it holds at every solid wall a real fluid ever meets. Between the two, the velocity varies smoothly, and for this simple case, linearly. The rate at which one layer slides past its neighbour is the velocity gradient \(du/dy\), and it is this gradient, not the speed itself, that the fluid resists.

Newton’s law of viscosity says that resistance is proportional to the gradient:

\(\tau\) is the shear stress the moving plate must overcome, and the constant of proportionality, \(\mu\), is the dynamic viscosity — a property of the fluid, with units of pascal-seconds. Large \(\mu\) means a thick, syrupy fluid that fights being sheared hard; small \(\mu\) means something that gives way easily, like air. Divide by density and you get the kinematic viscosity \(\nu = \mu/\rho\), in m²/s, which turns up wherever viscous and inertial effects are compared directly, most importantly in the Reynolds number.

A fluid that obeys Newton’s law with a constant \(\mu\), whatever the shear rate, is called Newtonian — water, air and most simple liquids and gases qualify, and almost everything in the rest of this course assumes it. Many everyday fluids do not. Paint and blood thin as they are sheared harder, their apparent viscosity falling with \(du/dy\); a cornstarch suspension does the opposite, thickening under a sudden shear until it behaves almost like a solid. Toothpaste and some muds will not flow at all below a threshold stress, then flow roughly linearly once past it — the Bingham plastic model, \(\tau = \tau_y + \mu_p\,du/dy\), adds that yield stress as an offset. None of these are exotic laboratory curiosities; they are the reason a shampoo bottle needs a squeeze to start pouring and then keeps going once it has.

Viscosity is also sharply temperature-dependent, and in opposite directions for liquids and gases, which is worth remembering before reaching for a table. In a liquid, molecules are held together by cohesive forces that weaken as temperature rises, so viscosity falls — honey pours more easily when warm. In a gas, viscosity comes from momentum exchange between molecules darting across layers, and that exchange intensifies as temperature rises, so gas viscosity increases with temperature. Reaching for the wrong intuition — assuming both behave like honey — is a specific, recurring mistake worth naming so you can catch yourself making it.

Surface tension and capillarity

A molecule deep inside a liquid is pulled equally in every direction by its neighbours and feels no net force. A molecule sitting at the liquid’s surface has neighbours only below and to the sides, not above, and the imbalance pulls it inward. The surface behaves, in aggregate, like a stretched elastic skin, under a tension measured as a force per unit length of any line drawn on it:

For water in air at room temperature \(\sigma\) is about 0.073 N/m — small, which is why surface tension only matters at small scales, where it competes on equal terms with gravity and viscosity rather than being swamped by them.

A flat surface under tension pulls sideways and produces no net force normal to itself, but curve the surface and that stops being true. On a spherical droplet of radius \(R\), the tension around every great circle has to be balanced by an excess pressure pushing outward from inside:

smaller droplets carry a larger internal excess pressure, which is why fine mist evaporates faster than a puddle and why soap bubbles — with two surfaces, inner and outer, each contributing — need \(4\sigma/R\) instead.

The same pull is behind capillary action: dip a narrow tube into a liquid that wets it — water in glass, not mercury in glass — and the liquid climbs the tube wall above the level of the surrounding reservoir. The curved meniscus at the top meets the wall at a contact angle \(\varphi\), small for a strongly wetting liquid, and the vertical pull of the surface tension around the tube’s circumference is what lifts the column, opposed by its own weight, until the two balance:

Halve the tube radius and the rise doubles — this is why capillary rise is negligible in a drinking glass and conspicuous in a narrow glass tube, and why a manometer reading a small pressure difference has to use a tube wide enough that the capillary contribution is smaller than the reading error you are willing to accept. A non-wetting liquid, mercury against glass being the standard example, has \(\varphi > 90^\circ\) and the formula gives a negative \(h\): the level is depressed inside the tube, not raised, for exactly the reason the same equation predicts the opposite for water.

Vapour pressure and cavitation

Every liquid has a vapour pressure \(P_v(T)\), the pressure at which, at a given temperature, liquid and its own vapour sit in equilibrium. Push the local pressure below \(P_v\) at whatever temperature the liquid happens to be, and it boils — not because it has been heated, but because the pressure holding it in liquid form has been removed. Water at room temperature boils at roughly 2.3 kPa, far below atmospheric, which is why nothing in your kitchen boils cold; it is also why water boils at a lower temperature on a mountain, where atmospheric pressure itself is lower and closer to \(P_v\) at an achievable temperature.

The mechanism matters here because a flowing liquid does not need heat to reach this condition — it only needs its pressure to drop, and pressure falls wherever a flow speeds up. Squeeze a liquid through a constriction, round the back of a fast spinning propeller blade, or into the low-pressure eye of a pump impeller, and the local pressure can fall far enough, fast enough, to reach \(P_v(T)\) even though the liquid is nowhere near its normal boiling point. Tiny vapour-filled bubbles form at that instant, entirely cold, carried along by the flow.

They do not last. Moments later the flow decelerates, or the bubble is swept somewhere the pressure has recovered above \(P_v\) again, and the vapour inside condenses back to liquid essentially instantly. The surrounding liquid rushes in to fill the space that has just vanished, and it does so from every direction at once, meeting at the centre in a microscopic, violent implosion. A single collapsing bubble is nothing. A pump impeller or a ship’s propeller enduring billions of them, over and over, in the same few square millimetres, pits and erodes the metal as surely as a slow sandblast — this is cavitation damage, and it is accompanied by a harsh crackling noise long before the metal itself shows anything.

The engineering response is entirely about staying clear of \(P_v\), not about the liquid’s temperature. Pumps are given a net positive suction head requirement precisely so their inlet pressure never falls that far; propellers and hydrofoils are shaped to keep the pressure minimum on their suction side above it. Once cavitation starts, an engine also loses performance quite apart from the damage, because the vapour pockets disrupt the smooth flow a pump or propeller depends on to do useful work — a machine cavitating is a machine already failing at its job, before it has failed as a piece of metal.