ফ্লুইড ডায়নামিক্স এর Euler law কী?

Question

this is most important law in mathematics.

Answer 1

In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.[1]

Flow around a wing. This incompressible flow satisfies the Euler equations.

The Euler equations can be applied to incompressible or compressible flow. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations – including the energy equation – as "the compressible Euler equations".[2]

The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations.

The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid. The conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which is useful from a numerical point of view).

Answer 2

In fluid dynamics, the Euler Equations govern the motion of a compressible, inviscid fluid. They correspond to the Navier Stokes equations with zero viscosity, although they are usually written in the form shown here because this emphasizes the fact that they directly represent the conservation of mass, momentum, and energy.

Euler’s equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure, and density of a moving fluid. It is based on Newton’s Second Law of Motion. The integration of the equation gives Bernoulli’s equation in the form of energy per unit weight of the following fluid.

It is based on the following assumptions:

The fluid is non-viscous (i,e., the frictional losses are zero).
The fluid is homogeneous and incompressible (i.e., mass density of the fluid is constant).
The flow is continuous, steady, and along the streamline.
The velocity of the flow is uniform over the section.
No energy or force (except gravity and pressure forces) is involved in the flow.