Fluids

Fluids have molecules that are able to move about relative to each other, giving them the ability to flow.

The forces between the molecules are not strong enough to counter this motion; it is only resisted.

The inertia of a small blob of fluid causes it to continue to move and rotate, but it feels a force and torque due to differences in pressure and friction (viscosity) which resists fluid slip.

The pressure depends on the density, temperature, and fluid material.

The flow carries along the particle density, momentum, angular momentum and energy. Friction can change molecular translational energy into internal energy (as vibrations and rotations), and dissipates momentum and vorticity.

(Click on the fluid to start/pause flow.)

Turbulence

Blobs of fluid carry vorticity with them; in places where the fluid converges, the vorticity increases, while the viscosity dissipates it.

Near to fixed surfaces, the flow changes in character depending on the ratio of the flow velocity to the viscosity (Reynolds number = \(vL/\nu\)): from smooth laminar flow to highly non-smooth turbulent flow with eddies.

Objects feel a drag due to the pressure difference in front \(\rho v^2\) and behind, and skin friction, as well as a "lift" and torque. The drag is proportional to the speed \(v\) during laminar flow, but to \(v^2\) in turbulent flow.

Reynolds number =

Click on the image to change the type of flow. Click "Run" or drag inside the fluid to make it flow.

Reynolds number = 200

The viscosity depends on temperature and pressure.

Diffusion

Concentrations of particles diffuse away, helped by the fluid flow (mixing). Depends on the diffusivity which is inversely proportional to the viscosity, so the higher the viscosity, the less the diffusion away of particles (and the more the diffusion of local momentum).

Heat Conduction

Heat flow is the diffusion of internal energy density, again helped by mixing; it is measured by the heat conductivity of the material.

Sound

Fluids allow the transmission of particle density and momentum as pressure waves, called sound, where the frequency is perceived as pitch and the amplitude as loudness. Like all waves, sound reflects, refracts, diffracts, is partially absorbed at boundaries, and can form standing waves and shock waves. It is produced by vibrating solids (reed, drum/speaker, bell, string) or rapid flow variations (siren, explosion).

The speed of sound is roughly equal \(\sqrt{\kappa/\rho}\) (ratio of compressibility to density), e.g., 1450 m/s for water and 340 m/s for air at 290K; which are comparable to the speed of the particles. But the fluid flow distorts the waves, as seen in this demo:

Sound of a high enough frequency cannot be transmitted, e.g., 100 MHz in water, 10 MHz in air: molecules do not have enough time to pass on the momentum to the next molecule, and the high energies involved are absorbed more readily by molecules.

Pressure under Gravity

In an external force potential (such as gravity), a fluid settles so that the pressure equals the potential energy density.

Pressure in a fluid under gravity is proportional to the depth as \(p = \rho gh\). Increasing the pressure at one place transmits it elsewhere.

Convection

Any region inside the fluid (whether solid or not) experiences a total force equal to what the fluid itself would experience, e.g., \(\rho g\times \mathrm{volume}\). So if the region has a density lower or higher than \(\rho\) then the net force on the region causes it to move towards a region with the same density (generally, upwards/floats or downwards/sinks).

If the region is a fluid itself with a different density (because, say, warmer or colder), then if the difference in the density is large enough (\(\frac{\Delta\rho}{\rho} \times \frac{gL^3}{D\nu} > 1000\), \(D=\textrm{diffusion constant}\), \(\nu = \textrm{viscosity}\)) to overcome diffusion, the momentum of the fluid in the direction towards lesser density, increases. This flow transfers the internal energy of the molecules with them: convection.

Liquids

In a liquid, the molecules' energy is not enough to overcome their attraction to each other, so most are unable to break free from each other. Molecules move, rotate, and vibrate, separated by about 3-4 10^-10m, while colliding with each other at about a trillion times a second.

Simple atoms can only move about.
Molecules with 2 atoms can also vibrate and rotate (10¹¹Hz high microwaves).
Molecules with many atoms have most of their energy as internal molecular vibrations (10¹³Hz mid-infrared) if the temperature is high enough.

Electromagnetic Properties

Magnetic field: Because of their random motions, liquids are usually diamagnetic, even if the molecules are magnetic.

Electric field: Similarly, neutral molecules do not usually react to electric fields. However, polar molecules rotate with the field at frequencies < 10¹⁰ — 10¹¹ Hz, making the liquid a dielectric.

Ions move along or opposite to the field, depending on their charge, creating a current proportional to e^xE/T. This current is quite low because the motion is Brownian, especially if the ions have a high mass compared to the other molecules. Electric fields increase the reaction rates for ions.

Surface

The close proximity of molecules in a liquid allows them to attract each other, especially if they are polar. Molecules thus have a potential energy within the liquid. At the surface these forces appear as a net inward force countered by the liquid pressure. The liquid thus contracts as much as possible, creating surface tension. Higher temperature increases the molecules' speeds, and so lowers the surface tension. The surface has an energy per unit area, and produces a pressure difference between the inside and outside proportional to the mean curvature. So small bubbles and droplets are difficult to form because of the large difference in pressure needed.

Surface Waves

The surface of a fluid allows waves to form, with the surface tension and gravity (say) trying to return it to equilibrium. In general, the speed of the waves depends on their wavelength and height, \[v=\sqrt{\tanh\left(\frac{2\pi h}{\lambda}\right)\left(\frac{g\lambda}{2\pi}+\frac{2\pi\sigma}{\rho\lambda}\right)}\] where \(\lambda\) is the wavelength, \(g\) gravity, \(h\) height of fluid, \(\sigma\) surface tension, \(\rho\) fluid density. So waves tend to disperse, with long wavelengths moving on faster than the rest.

Gravity waves — when the wavelength is relatively large, the only relevant force is gravity; the speed is then \(v\approx \sqrt{\frac{g\lambda}{2\pi}}\).

Properties of Liquids

The properties of liquids vary with the temperature and pressure. Liquids are mostly incompressible with a constant density, except near the critical point, when they resemble gases more closely, being more compressible.

At lower temperature, molecules have less kinetic energy, so they can be affected strongly by each other and move collectively.

Solvents

A molecule dissolves in a liquid when the attraction between the two types of molecules is larger or comparable to the attraction between the liquid molecules themselves.

When the molecule is not attracted enough to the liquid molecules, the surface tension of the liquid keeps the two sets of molecules apart. Non-soluble molecules accumulate at a surface.

Polar (especially ionic) solvents can dissolve polar/ionic molecules. A non-polar solute does not normally dissolve but "cages" of solvent molecules form about them (as bubbles).

Non-polar solvents dissolve some non-polar molecules.

Solid solutes take longer to dissolve because of the extra energy needed to overcome their potential energy; gaseous solutes dissolve less.

Osmosis

When a porous membrane is semi-permeable, i.e., it allows a solvent molecule through but not a solute molecule, then the solvent molecules move from the side of the membrane with a lower solute concentration to the side with a higher solute concentration, in an attempt to equalize the two concentrations.

Electro-osmosis: when an electric field exists across a porous membrane, the more mobile ions will drag with them some solvent molecules causing a pressure difference between the two sides. Conversely a flow through a membrane will cause mobile ions to cross at a faster rate than slow ones so that an electric field develops across the membrane \(\frac{n_A}{n_B} = e^{-eV/T}\).

Battery: When a conductor causes a solution (electrolyte) to dissociate into heavy and light ions, the light ones will diffuse out faster than the heavy ones, creating an electric field that can be maintained by an accepting conductor.

Special Liquids

Liquid crystals: Molecules can move but not rotate freely, e.g., long-chained or planar molecules. They align parallel to each other. At lower temperatures they may form layers that slide over each other. Long polar molecules have large polarization effects.

Ionic liquids: Most molecules are ions. They can dissolve more molecules than most other liquids and usually have a high melting point. A mixture of large and small ions may not settle as a solid but form a viscous liquid. They conduct electricity better than most liquids.

Gases

In a gas, molecules have more translational energy, as well as rotational (100 MHz) and vibrational (10 GHz), so they easily overcome the gas potential energy. Molecules are much farther apart, colliding with each other at 10¹⁰ — 10¹¹ Hz, with each collision taking about 10^-13s. The viscosity is much less than in liquids.

The energy of each molecule is proportional to the temperature, depending on the variety of vibrations and/or rotations:

Molecules in a gas are often assumed to be completely independent of each other (ideal gas). The pressure \(p\), temperature \(T\), and particle density \(n\) are theoretically related as \[p = nT\]

But in reality deviations can be seen at low pressures or high densities:

Other characteristics can be calculated: the speed of sound is proportional to \(\sqrt{T}\).

Usually gases are less dense than liquids, making them insulators, possibly slightly dielectric.

Plasmas

A plasma is a collection of ions, electrons, and neutral molecules in fluid form. At distances over \(\sqrt{T/n}\), the plasma is electrically neutral (where \(T\) is the temperature and \(n\) is the particle density). Plasmas occur either when the collisions between molecules are powerful enough to maintain them in an excited/ionized state, or when collisions are rare. At temperatures of at least \(1\,MK\), plasmas consist of nuclei and electrons.

Because plasmas consist of moving charged particles, they have associated magnetic fields. Electrons are continually emitting bremsstrahlung and synchrotron radiation as they collide or accelerate, as well as recombining with ions; so plasmas in general emit a glow. The whole motion is complex with fluid flow combined with magnetic field flow, as well as waves (sound, magneto-sound, Alfven waves, em waves). Plasmas often form cells with sharp discontinuities in temperature, density and ionization.

Electromagnetic Properties

An electric field causes the electrons (and to a much lesser extent the ions) to accelerate, maintaining the plasma by feeding ionisation from electrons that collide at \(\gt 10\,eV\). A high enough electric field causes a (Townsend) discharge of ions: As ions accelerate rapidly, they create more ions, causing an avalanche effect to form an arc or spark.

Electromagnetic waves of a characteristic plasma frequency \(\frac{e}{2\pi}\sqrt{\frac{n}{m_e}}\) cannot pass through a plasma, because they cause the electrons to resonate, oscillating about the positive ions.