Voltaic Pile Principle

These are two ways to describe motion in continuum mechanics:

• the Lagrangian specification follows individual material particles, tracking their properties over time, like watching a specific car in traffic
• the Eulerian specification focuses on fixed points in space, observing the properties (velocity, density) of whatever particles pass through those points, like a traffic camera observing a fixed intersection.

Solid mechanics is a branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, or other external loads. It is fundamental to engineering for the design and analysis of structures. Key areas include elasticity (recoverable deformation), plasticity (permanent deformation), and fracture mechanics (crack initiation and propagation).

The Voltaic pile established the principle of adding voltages by connecting electrochemical cells in series. Each zinc-copper pair, or cell, generates a characteristic electromotive force (EMF) of approximately 0.76 volts. By physically stacking these cells, Volta demonstrated that the total voltage across the pile is the sum of the individual EMFs of each cell, as described by [latex]V_{total} = n \times V_{cell}[/latex].

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 [latex]\rho (\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}[/latex].

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.

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.

In continuum mechanics, the principle of mass conservation states that the mass of a closed system must remain constant over time. For a fluid, this is expressed by the continuity equation. In its Eulerian differential form, it is written as [latex]\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0[/latex], where [latex]\rho[/latex] is the density and [latex]\mathbf{u}[/latex] is the velocity field.

A reformulation of classical mechanics based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy ([latex]L = T - V[/latex]). The equations of motion are derived from the Euler-Lagrange equation, [latex]\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0[/latex], using generalized coordinates, which simplifies analysis of complex systems with constraints.

Galvanic corrosion is an electrochemical process where one metal corrodes preferentially when in electrical contact with another in the presence of an electrolyte. This occurs because the dissimilar metals create a bimetallic couple, with the less noble (more active) metal acting as the anode and corroding, while the more noble metal acts as the cathode.

The electric current in a Voltaic pile is produced by a redox reaction. At the zinc anode, zinc metal is oxidized, releasing two electrons per atom ([latex]Zn \rightarrow Zn^{2+} + 2e^{-}[/latex]). These electrons travel through the external circuit to the copper cathode. There, hydrogen ions from the aqueous electrolyte are reduced, forming hydrogen gas ([latex]2H^{+} + 2e^{-} \rightarrow H_2[/latex]).

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 [latex]\boldsymbol{\sigma}[/latex], 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) [latex]\mathbf{T}[/latex] on any surface passing through that point to the surface's normal vector [latex]\mathbf{n}[/latex] via the linear relationship [latex]\mathbf{T} = \boldsymbol{\sigma} \cdot \mathbf{n}[/latex].

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

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 [latex]E = T + V[/latex] is conserved.