Conservation of Mass

A Newtonian fluid's shear stress is directly proportional to the rate of shear strain. This linear relationship is defined by Newton's law of viscosity: \(\tau = \mu \frac{du}{dy}\), where \(\tau\) is the shear stress, \(\mu\) is the dynamic viscosity (a constant of proportionality), and \(\frac{du}{dy}\) is the shear rate or velocity gradient.

Bernoulli's principle states that for an inviscid flow, an increase in a fluid's speed occurs simultaneously with a decrease in pressure or a decrease in its potential energy. It is a statement of the conservation of energy for a moving fluid, commonly expressed as \(p + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}\) along a streamline.

Fluid mechanics is the branch of applied mechanics concerned with the statics (fluids at rest) and dynamics (fluids in motion) of liquids and gases. It applies fundamental principles of mass, momentum, and energy conservation to analyze and predict fluid behavior. Its applications are vast, ranging from aerodynamics and hydraulics to meteorology and oceanography.

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).

Electromotive force (\(\mathcal{E}\)) is the work done per unit of electric charge by a non-electrical source, such as a battery or generator. Despite its name, it is not a mechanical force but an electric potential, measured in volts. It represents the energy converted from another form (chemical, mechanical) into electrical energy as a charge traverses the source.

This law states that every particle attracts every other particle in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is \(F = G \frac{m_1 m_2}{r^2}\), where \(G\) is the gravitational constant. It unified terrestrial and celestial mechanics.

In an isolated system, the total momentum remains constant. An isolated system is one not subject to external forces. This principle follows from Newton's laws of motion. For a system of particles, the total momentum \(\vec{p}_{\text{total}} = \sum_{i} m_i \vec{v}_i\) is conserved if the net external force is zero. This is fundamental to analyzing collisions.

In a reference frame rotating with an angular velocity \(\boldsymbol{\omega}\), the centrifugal force \(\mathbf{F}_{\mathrm{cf}}\) acting on an object of mass \(m\) at a position vector \(\mathbf{r}\) from the origin is given by the vector formula: \(\mathbf{F}_{\mathrm{cf}} = -m \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r})\). This formula shows the force is directed perpendicular to the axis of rotation and outwards.

Coulomb's Law quantifies the electrostatic force between two stationary, electrically charged particles. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is \(F = k_e \frac{q_1 q_2}{r^2}\). This force can be either attractive or repulsive.

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 (\(L = T - V\)). The equations of motion are derived from the Euler-Lagrange equation, \(\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0\), 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 first electric battery, it produces a direct current by stacking pairs of dissimilar metal discs (e.g., zinc and copper) separated by brine-soaked cloth. Each pair forms a galvanic cell, and stacking them in series increases the overall voltage. This arrangement demonstrated the first continuous conversion of chemical energy into electrical energy, paving the way for modern electrochemistry.