Shear-Thickening (Dilatancy)

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, \(\tau_{abs max} = (\sigma_1 - \sigma_3)/2\).

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.

In electrochemical cells like batteries and fuel cells, EMF is generated by chemical reactions. The separation of charge is driven by oxidation and reduction reactions occurring at two different electrodes. The maximum EMF of a cell is related to the change in Gibbs free energy (\(\Delta G\)) of the reaction by \(\mathcal{E} = -\frac{\Delta G}{nF}\), where \(n\) is moles of electrons and \(F\) is the Faraday constant.

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.

The Reynolds number (\(\text{Re}\)) is a dimensionless quantity in fluid mechanics used to predict flow patterns by representing the ratio of inertial forces to viscous forces. Low Reynolds numbers characterize smooth, orderly laminar flow, while high Reynolds numbers indicate chaotic, eddy-filled turbulent flow. It is crucial for determining the dynamic behavior of a fluid and for scaling experiments.

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 Nernst equation quantifies the reversible electromotive force (EMF) or open-circuit voltage of a fuel cell under non-standard conditions. It links the cell potential (\(E\)) to its standard potential (\(E^0\)), temperature, and the activities (approximated by partial pressures) of reactants and products. The equation is \(E = E^0 - \frac{RT}{nF} \ln Q\), where Q is the reaction quotient.

Motional EMF is generated when a conductor moves through a magnetic field. The magnetic component of the Lorentz force, \(\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\), acts on the charge carriers within the conductor, causing them to move and create a charge separation. This separation establishes an electric field and a potential difference. The resulting EMF is given by the line integral \(\mathcal{E} = \oint (\mathbf{v} \times \mathbf{B}) \cdot d\mathbf{l}\).