Euler's Equation

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-Strokes equations with zero viscosity. and zero thermal conductivity. In fact, Euler equations can be obtained by linearization of some more precise continuity equation like Navier-Strokes 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".
This is equation of motion in which the forces due to gravity and pressure are taken into consideration. This is derived by considering the motion of a fluid element along a stream-line as: Consider a stream-line in which flow is taking place in s-direction. Consider a cylindrical element of cross-section dA and length ds. The forces acting on the cylindrical element are:
1. Pressure force pdA in the direction of flow.
2. Pressure force 

 opposite to the direction of flow.

Let θ is the angle between the direction of flow and the line of action of the weight of element.
The resultant force on the fluid element in the direction of s must be equal to the mass of fluid element × acceleration in the direction s.

Above given equation is known as Euler's equation of motion.

Credit:-https://en.m.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)

Book Author- Dr.R.K.Bansal