The Euler–Tricomi equation is a second-order partial differential equation (PDE) that arises in various fields, including fluid dynamics and mathematical physics. It is named after the mathematicians Leonhard Euler and Francesco Tricomi.

The drag equation is a mathematical formula used to calculate the drag force experienced by an object moving through a fluid, such as air or water. The drag force (F_d) is the resistance experienced by the object due to the fluid surrounding it.

The Navier-Stokes equations describe the motion of fluid substances and are fundamental in fluid mechanics. They are derived from the principles of conservation of mass, momentum, and energy. Here, I'll summarize how these equations are derived step-by-step. ### 1. Conservation of Mass (Continuity Equation) The principle of mass conservation states that the mass of a fluid in a control volume must remain constant over time if no mass enters or leaves the volume.

The Davey-Stewartson equation is a nonlinear partial differential equation that arises in the study of wave phenomena, particularly in the context of two-dimensional surface water waves. It is a generalization of the nonlinear Schrödinger equation and describes the evolution of complex wave packets in a two-dimensional setting.

The Darcy–Weisbach equation is used in fluid mechanics to calculate the pressure loss (or head loss) due to friction in a pipeline or duct. It is an essential equation for engineers and designers working with fluid flow systems to assess the efficiency and performance of piping and ductwork.

The Darcy friction factor, often denoted as \( f \), is a key component in the Darcy-Weisbach equation, which is used to calculate pressure loss (or head loss) due to friction in a pipe or duct.

The continuity equation is a fundamental principle in fluid dynamics and other fields that describes the transport of some quantity (such as mass, energy, or charge) in a system. It expresses the idea that, in a closed system, the rate at which a quantity enters a volume must equal the rate at which it leaves that volume, plus any accumulation of that quantity within the volume.

The Camassa-Holm equation is a nonlinear partial differential equation that describes the dynamics of shallow water waves. It was first introduced by Roberta Camassa and Darryl Holm in their 1993 paper. The equation models unidirectional wave propagation and is noteworthy for its ability to describe solitary waves, which can maintain their shape while traveling at constant speeds.

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.