The Cahn–Hilliard equation is a partial differential equation that describes the phase separation and motion of interfaces in a binary mixture or alloy. It is particularly important in materials science, as it models the process by which two phases of a material (such as solid and liquid) separate from each other, leading to the formation of distinct microstructures over time. The equation was introduced by John W. Cahn and John E.

Burgers' equation is a fundamental partial differential equation in fluid mechanics and mathematics. It is named after the Dutch physicist Johannes Burgers, who introduced it in his study of turbulence and other fluid dynamics phenomena. The equation can be seen as a simplification of the Navier-Stokes equations, which govern fluid motion.

The Buckley–Leverett equation is a fundamental equation in petroleum engineering and reservoir engineering that describes the movement of two-phase fluids (typically oil and water) in porous media. It models the flow behavior of immiscible fluids in a reservoir when one fluid displaces another, commonly used to analyze waterflooding operations during oil recovery. The equation is derived from the conservation of mass principle and reflects the dynamics of the interfaces between the two fluids.

The Boussinesq approximation is a mathematical simplification used in fluid dynamics, particularly in the study of weakly non-linear and dispersive wave phenomena, such as water waves. Named after the French physicist Joseph Boussinesq, this approximation is particularly useful for analyzing the behavior of surface waves in fluids where the amplitude of the waves is small compared to the wavelength.

The Bosanquet equation is a mathematical expression used in the field of fluid dynamics and rheology to model the steady-state flow of non-Newtonian fluids. It is particularly relevant for describing the flow behavior of viscoelastic fluids, which exhibit both viscous and elastic characteristics.

The Borda-Carnot equation describes the relationship between the temperature, pressure, and specific properties of a fluid in a thermodynamic context, particularly for a fluid undergoing adiabatic (no heat transfer) expansion or compression. It is commonly associated with the performance of turbines and compressors. The equation itself typically relates how the enthalpy, pressure, and temperature of the fluid change during these processes.

The term "black oil equations" refers to a set of mathematical relations used in reservoir engineering and petroleum production to model the behavior of black oil, a type of crude oil characterized by its relatively high viscosity and the presence of dissolved gases and lighter hydrocarbon components. Black oil models help in understanding and predicting the behavior of oil reservoirs during production.

Bernoulli's principle is a fundamental concept in fluid dynamics that describes the behavior of a fluid moving along a streamline. Formulated by the Swiss mathematician Daniel Bernoulli in the 18th century, the principle states that in a steady flow of an incompressible, non-viscous fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy in that flow.

The Benjamin–Bona–Mahony (BBM) equation is a mathematical model that describes wave propagation in shallow water. It is a simplified equation derived from the full water wave equations, particularly focusing on long waves with small amplitude. The BBM equation incorporates both nonlinearity and dispersion, making it a significant model in the study of wave phenomena.

The Benedict–Webb–Rubin (BWR) equation is a thermodynamic model used to describe the behavior of gases, particularly mixtures and non-ideal gas mixtures. It is a more complex equation of state compared to the ideal gas law, allowing for the incorporation of molecular interactions and the effects of pressure and temperature on gas behavior.

The Batchelor–Chandrasekhar equation is a fundamental equation in the field of fluid dynamics, specifically in the study of turbulence and the behavior of suspensions of small particles in a fluid. It describes the way that particles, such as bubbles or solid particles, interact with the surrounding fluid flow, particularly under conditions of sedimentation or dispersion.

The Basset–Boussinesq–Oseen (BBO) equation is a mathematical model that describes the motion of small particles suspended in a viscous fluid. This equation accounts for the effects of inertial and viscous forces acting on the particles, along with the interaction between the particles and the surrounding fluid. It is particularly important in the fields of fluid mechanics and particle dynamics, especially in scenarios where the Reynolds number is low.

The Allen-Cahn equation is a partial differential equation that describes the evolution of phase interfaces in materials science and represents the dynamics of gas-liquid phase transitions typically in the context of, but not limited to, crystallization processes. It is an example of a conserved order parameter system and is derived from the principles of thermodynamics and variational calculus.

The Vis-viva equation is an important equation in orbital mechanics that relates the speed of an object in orbit to its distance from the center of the body it is orbiting and the gravitational parameter of that body. It provides a way to calculate the orbital velocity of an object at any point in its orbit, given its distance from the center of mass of the central body.

The Virbhadra–Ellis lens equation describes the behavior of light in the gravitational field of a massive object, such as a star or galaxy, and is used in the context of gravitational lensing in general relativity. This lens equation accounts for the effects of both the classical lensing mass and any relativistic effects that might arise due to the curvature of spacetime.

Velocity dispersion is a measure of the range of velocities within a group of objects, such as stars in a galaxy or galaxies in a cluster. It quantifies how much the velocities of the objects deviate from the average velocity of the group. In a more technical sense, it is defined as the standard deviation of the velocities of the objects in the sample. In astrophysics, velocity dispersion is an important metric because it provides insights into the dynamics and mass distribution of celestial bodies.

The Van Cittert–Zernike theorem is a fundamental result in the field of imaging and optics, particularly relevant to the theory of image formation in astronomy and other fields where diffraction-limited imaging is important. The theorem provides a mathematical framework for understanding how the intensity distribution of a diffraction-limited image can be reconstructed from the visibility of spatial frequencies in an observed object.

Universal Variable Formulation (UVF) is a mathematical approach used in astrodynamics, particularly in the analysis of orbital mechanics and trajectory optimization. The formulation provides a way to describe the motion of a spacecraft or an object in space by using a set of universal variables that can simplify the computations involved in trajectory analysis. UVF is particularly beneficial for three-body problems, such as spacecraft flybys or transfers between celestial bodies, because it allows for the integration of equations of motion under varying gravitational influences.

The Sérsic profile is a mathematical function used to describe the brightness distribution of astronomical objects, particularly galaxies and bulges of galaxies. It was introduced by the Argentine astronomer José Sérsic in 1963. This profile is an extension of the simpler exponential (for disk-like structures) and de Vaucouleurs (for elliptical structures) profiles, allowing for a more flexible representation of the surface brightness of an object.