Reynolds Number

This foundational formula connects the macroscopic thermodynamic quantity of entropy (S) with the number of possible microscopic arrangements, or microstates (W), corresponding to the system's macroscopic state. The equation, \(S = k_B \ln W\), reveals that entropy is a measure of statistical disorder or randomness. The constant \(k_B\) is the Boltzmann constant, linking energy at the particle level with temperature.

Sublimation, the direct phase transition from solid to gas, occurs at temperatures and pressures below a substance's triple point. The triple point is the unique thermodynamic state where the solid, liquid, and gas phases can coexist in equilibrium. On a phase diagram, the sublimation curve separates the solid and gas phases, originating at the origin and ending at the triple point.

Mohr's circle is a two-dimensional graphical representation of the Cauchy stress tensor. It visualizes the transformation of normal stress (\(\sigma_n\)) and shear stress (\(\tau_n\)) on an arbitrarily oriented plane at a point. The abscissa of each point on the circle is the normal stress, and the ordinate is the shear stress, allowing for easy determination of principal stresses.

Electromagnetic fields can carry momentum. The momentum density of the electromagnetic field is given by the Poynting vector \(\vec{S}\) divided by the speed of light squared, \(\vec{g} = \vec{S}/c^2 = (\vec{E} \times \vec{B})/(\mu_0 c^2)\). For momentum to be conserved in a system of charged particles and fields, the momentum of the fields must be included alongside the mechanical momentum of the particles.

The Mach number (M) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound: \(M = v/a\), where v is the flow velocity and a is the speed of sound. It is the primary indicator of compressibility effects. As Mach number approaches and exceeds 1, the density of the air changes significantly, altering aerodynamic forces.

The AC induction motor operates on the principle of a rotating magnetic field generated by the stator. This field is created by supplying polyphase AC currents to spatially distributed stator windings. The rotating field induces currents in the rotor conductors (e.g., a squirrel cage), which create an opposing magnetic field. The interaction between these fields produces the rotational torque.

The ideal Otto cycle is a thermodynamic model for spark-ignition engines. It consists of four internally reversible processes: isentropic compression (1-2), constant volume (isochoric) heat addition (2-3), isentropic expansion (3-4), and constant volume (isochoric) heat rejection (4-1). This cycle forms the theoretical basis for analyzing the performance of gasoline engines, assuming an ideal gas as the working fluid.

This gas liquefaction process, pioneered by Raoul Pictet, liquefies a gas by using a series of other gases with progressively lower boiling points. The first gas is liquefied by pressure at ambient temperature. Its evaporation is then used to cool and liquefy a second, more volatile gas, which in turn is used to cool and liquefy the target gas, creating a 'cascade' of cooling stages.

The Velocity of Detonation (VoD) is the speed at which the shock wave front travels through a detonating explosive. It is a primary measure of an explosive's performance and power, differentiating high explosives from low explosives. VoD is a characteristic property influenced by density, grain size, and confinement, with typical values reaching thousands of meters per second for high explosives.

For a general three-dimensional state of stress, the analysis is represented by three Mohr's circles. These circles are drawn in the \(\sigma_n - \tau_n\) plane using the three principal stresses (\(\sigma_1, \sigma_2, \sigma_3\)) as diameters. The largest circle, defined by \(\sigma_1\) and \(\sigma_3\), encloses the other two and determines the absolute maximum shear stress, \(\tau_{abs max} = (\sigma_1 - \sigma_3)/2\).

Shear-thickening, or dilatancy, is a non-Newtonian behavior where viscosity increases with the rate of shear stress. A classic example is a suspension of cornstarch in water (oobleck), which feels liquid when stirred slowly but becomes almost solid when struck or stirred rapidly. This phenomenon is caused by the jamming of suspended particles under high shear.

The Nernst equation relates the reduction potential of a half-cell (or the total voltage of an electrochemical cell) to the standard electrode potential, temperature, and the activities (often approximated by concentrations) of the chemical species undergoing redox. The equation is \(E = E^{\circ} - \frac{RT}{nF} \ln Q\), where Q is the reaction quotient.