Lenz’s Law

The Navier–Stokes equations are a set of non-linear partial differential equations describing the motion of viscous fluid substances. They are a statement of Newton's second law, balancing momentum changes with pressure gradients, viscous forces, and external forces. For an incompressible fluid, the equation is \(\rho (\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}\).

Cathodic protection (CP) is a technique used to control the corrosion of a metal surface by making it the cathode of an electrochemical cell. This is achieved by supplying an external electrical current that suppresses the natural corrosion currents. The protected metal's potential is polarized to a more negative value, shifting it into a region of immunity or passivity.

This law states that the electromotive force (EMF, \(\mathcal{E}\)) induced in any closed circuit is equal to the negative of the time rate of change of the magnetic flux (\(\Phi_B\)) through the circuit. The mathematical expression is \(\mathcal{E} = -\frac{d\Phi_B}{dt}\). This principle is the basis for electric generators, transformers, and inductors, describing the macroscopic effect of induction.

A fundamental principle stating that the total energy of an isolated system remains constant over time. Energy can neither be created nor destroyed, only transformed from one form to another, such as from potential to kinetic energy. In classical mechanics, for systems with only conservative forces, the total mechanical energy \(E = T + V\) is conserved.

For a Newtonian fluid, viscosity is a function of temperature and pressure but not shear rate. In liquids, viscosity decreases significantly as temperature increases because higher thermal energy allows molecules to overcome cohesive intermolecular forces more easily. Conversely, in gases, viscosity increases with temperature as more frequent molecular collisions at higher speeds lead to greater momentum transfer.

This is the differential form of Faraday's law of induction, one of Maxwell's four equations. It states that a time-varying magnetic field (\(\mathbf{B}\)) always accompanies a spatially varying, non-conservative electric field (\(\mathbf{E}\)). The relationship is expressed as \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\). This equation governs how changing magnetic fields create electric fields at a specific point in space.

The continuum assumption treats fluids as continuous matter rather than as discrete molecules. This simplification is valid when the length scale of the problem is much larger than the intermolecular distance, allowing properties like density and velocity to be defined at infinitesimally small points. This enables the use of differential equations to describe the macroscopic behavior of fluid flow.

The Cauchy stress tensor, denoted \(\boldsymbol{\sigma}\), is a second-order tensor that completely defines the state of stress at a point inside a material. It relates the traction vector (force per unit area) \(\mathbf{T}\) on any surface passing through that point to the surface's normal vector \(\mathbf{n}\) via the linear relationship \(\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}\).

A primary source of electromotive force is electromagnetic induction, described by Faraday's law. It states that a time-varying magnetic flux \(\Phi_B\) through a circuit loop induces an EMF (\(\mathcal{E}\)). The magnitude of the EMF is proportional to the rate of change of the flux, given by the equation \(\mathcal{E} = - \frac{d\Phi_B}{dt}\). This principle is the foundation for electric generators, transformers, and inductors.

A reformulation of classical mechanics that uses generalized coordinates and their conjugate momenta. It is based on the Hamiltonian function, \(H(q, p, t)\), representing the system's total energy. The dynamics are described by Hamilton's equations: \(\dot{q}_i = \frac{\partial H}{\partial p_i}\) and \(\dot{p}_i = -\frac{\partial H}{\partial q_i}\). This framework is central to quantum mechanics and statistical mechanics.

An improvement on the Voltaic pile, the Daniell cell consists of a copper electrode in a copper(II) sulfate solution and a zinc electrode in a zinc sulfate solution, separated by a porous barrier. This two-fluid design prevents hydrogen gas buildup (polarization) on the copper electrode, resulting in a much more stable and reliable voltage source for a longer duration.

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials, like a rubber band, strain when stretched and quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both.

The first commercially successful rechargeable battery. It uses a lead (Pb) anode, a lead dioxide (PbO₂) cathode, and a sulfuric acid (H₂SO₄) electrolyte. During discharge, both electrodes are converted into lead sulfate (PbSO₄), and the sulfuric acid is consumed. This process is chemically reversible by applying an external current, making it a practical and robust energy storage system.

The Boltzmann distribution describes the probability that a system in thermal equilibrium at temperature T will be in a specific microstate with energy E. This probability is proportional to the Boltzmann factor, \(e^{-E / k_B T}\). It implies that states with lower energy are exponentially more likely to be occupied than states with higher energy, with temperature modulating this preference.