# Euler's approach

The red point shows a possible standpoint for Euler's view of a rubber skin (gray) moving in a room (black grid)

The Euler approach represents a particular perspective in monitoring movement of a body , that represents a particular point of observation. In the Euler approach or Euler-image movement of the body is analyzed by a spatially fixed point, which is why this view also spatially or locally is called. For example, a buoy firmly anchored in the ground in a river would perceive the current from Euler's perspective. An example from solid mechanics is shown in the figure on the right. The question here is what conditions, e.g. B. what pressure or what temperature prevail at a certain place in the room. Euler's approach is mainly used in fluid mechanics , but is also used in solid mechanics, e.g. B. in forming processes.

Euler's approach was introduced by Jean-Baptiste le Rond d'Alembert in 1752.

## description

In the Euler picture, the observer of a movement stands at a fixed point in space. By marking all the particles that pass this point in space, a streak line is created , which is associated with Euler's approach. In Euler's approach, all physical quantities are represented with respect to the instantaneous configuration , which depicts the moving body for calculations at any point in time, which is consequently called Euler's representation . The equations established in the current configuration are then available in Euler's version . In continuum mechanics , the quantities related to the current configuration are mostly noted with lowercase letters. Euler's approach is mainly used in fluid mechanics.

In fluid dynamics, the velocity is the kinematic unknown and not the displacement that the particles cover during their movement. It is therefore not difficult to describe large deformations. In any case, an undeformed initial state is often neither known nor of interest, at least in fluid mechanics. The incompressibility of a material simplifies the finding of solutions to a boundary value problem , because in the Euler equations the pressure can then be eliminated by forming the rotation .

Because the physical laws of mechanics relate to matter and not to points in space, the equations of motion get convective , non-linear components through the substantial derivation . These proportions ensure a strong coupling of the equations. The formulation of objective time derivatives for stress and strain tensors is complex and closely interlinked with the velocity field. The formulation of boundary conditions on free areas causes great difficulties. Mass inflows and outflows must be accounted for precisely.

## Summary

The characteristics of Euler's perspective are summarized again in the table.

property Occupancy
Namesake Leonhard Euler
Originator Jean-Baptiste le Rond d'Alembert (1752)
Observer location Point in space
application Fluid mechanics, forming processes
Visualization Streak line
Cause of the kinematic non-linearity convection
Time derivative Substantial derivative , objective time derivative
Effort for balances of field sizes high
Associated configuration Current configuration
Designation of the variables in continuum mechanics lowercase letters

In fluid mechanics the following still applies:

property Occupancy
Kinematic unknowns speed
Capability for incompressibility high
Suitability for boundary conditions on open areas low