Ohm’s Law

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.

Lenz's law provides the direction of the induced electromotive force (EMF) and current resulting from electromagnetic induction. It states that the induced current will flow in a direction that creates a magnetic field opposing the change in magnetic flux that produced it. This principle is a consequence of the conservation of energy and is represented by the negative sign in Faraday's law.

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.

Engineering stress (\(\sigma\)) is the applied load divided by the original cross-sectional area (\(A_0\)), while engineering strain (\(\epsilon\)) is the change in length (\(\Delta L\)) divided by the original length (\(L_0\)). These definitions, \(\sigma = \frac{F}{A_0}\) and \(\epsilon = \frac{\Delta L}{L_0}\), are fundamental for plotting stress-strain curves but assume the specimen's dimensions remain constant during the test.

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 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.