The mild-slope equation is a mathematical representation used in coastal engineering and fluid dynamics to describe the propagation of surface water waves over varying bathymetry (the underwater equivalent of topography). It is especially useful for analyzing wave behavior in coastal areas, where the depth of the water changes gradually.

The Kármán–Howarth equation is a fundamental relation in fluid dynamics, particularly in the study of turbulence. It describes the evolution of the second-order velocity correlation function in an incompressible flow. The equation provides insight into the relationships between different scales of motion in turbulent flows. In turbulent fluid mechanics, the velocity field can be characterized using correlation functions, which measure the statistical relationships between the velocities at different points in space.

The Kozeny-Carman equation is a mathematical model that describes the flow of fluids through porous media. It relates the permeability of a porous material to its porosity and specific surface area. It is widely used in fields such as hydrogeology, petroleum engineering, and soil science to analyze how fluids move through soils and rocks.

The Korteweg–de Vries (KdV) equation is a third-order nonlinear partial differential equation that describes the evolution of waves in shallow water. It is significant in various fields, including fluid dynamics, nonlinear wave theory, and mathematical physics.

Kelvin's circulation theorem is a fundamental principle in fluid dynamics, particularly in the study of inviscid (non-viscous) and irrotational flows. It states that the circulation around a closed contour moving with the fluid is constant in time, provided the flow is conservative and the fluid is incompressible and inviscid.

The Kadomtsev–Petviashvili (KP) equation is a fundamental nonlinear partial differential equation (PDE) that describes the propagation of waves in a quasi-one-dimensional medium. It arises in various fields such as fluid dynamics, plasma physics, and nonlinear optics. The equation serves as a higher-dimensional generalization of the Korteweg–de Vries (KdV) equation, which describes solitons in one dimension.

A Herschel–Bulkley fluid is a type of non-Newtonian fluid that exhibits both yield stress and shear-thinning (or shear-thickening) behavior. The defining characteristic of such fluids is that they do not begin to flow until a certain threshold stress, known as the yield stress, is exceeded. Once this yield stress is surpassed, the fluid flows according to a power-law relationship that describes its viscosity.

The Hazen–Williams equation is an empirical formula used to calculate the flow of water through pipes, specifically in civil engineering and hydraulics. It estimates the head loss (pressure loss due to friction) in a pipe based on the flow rate, pipe diameter, and the roughness of the pipe's interior surface. The equation is particularly applicable for water flow in pipes where the flow is turbulent. The general form of the Hazen–Williams equation is: \[ h_f = 0.

The Hasegawa–Mima equation is a nonlinear partial differential equation that arises in the study of plasma physics, particularly in the context of magnetically confined plasmas, such as those found in fusion reactors. It describes the evolution of certain wave phenomena in a magnetized plasma, specifically the dynamics of plasma turbulence and the behavior of density perturbations.

The Hadamard–Rybczynski equations describe the motion of a fluid in a gravitational field, particularly in the context of fluid dynamics. These equations are important in studying the behavior of inviscid and incompressible fluids, especially when analyzing potential flow around bodies. The Hadamard–Rybczynski equations relate the velocity potential or stream function to the shape of the body and the flow conditions around it.

Faxén's law describes the force experienced by a spherical particle suspended in a fluid when it is subjected to an external oscillating field, such as a pressure gradient or a fluid flow. It is particularly relevant in the study of colloidal suspensions and the behavior of particles in non-Newtonian fluids.

The Fanning friction factor is a dimensionless quantity used in fluid mechanics to characterize the frictional resistance to flow in a pipe or duct. It is defined as the ratio of the wall shear stress to the dynamic pressure of the fluid. The Fanning friction factor (\(f\)) is commonly used in the analysis of laminar and turbulent flow regimes and plays a crucial role in the calculation of pressure losses due to friction in piping systems.

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.