Publisher's Synopsis
In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier-Stokes equations with zero viscosity and zero thermal conductivity. In fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier-Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow - assuming the flow velocity is a solenoidal field, or using another appropriate energy equation respectively (the simplest form for Euler equations being the conservation of the specific entropy). Historically, only the incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set - including the energy equation - of the more general compressible equations together as "the Euler equations". To complete the description of the flow we need to specify the boundary conditions on the velocity field at a solid surface. In our dealings with nonviscous flow we imposed the condition that the normal component of the fluid velocity at a solid surface must be zero -- this just ensures that no fluid flows through the surface. Once we deal with real (viscous) fluids, we must also require that the tangential component of the velocity of the fluid be zero on the surface. This condition is often called the no-slip boundary condition, and is the result of experimental observations. The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances such as liquids and gases. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They may be used to model weather, ocean currents, water flow in a pipe, flow around an airfoil (wing), and motion of stars inside a galaxy. As such, these equations in both full and simplified forms, are used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc. Coupled with Maxwell's equations they can be used to model and study magnetohydro dynamics.
