Description can be an Imagination, but Action must be

Real ……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Material Derivative using Eulerian Description

Eulerian Description of Flow : The Concept

•Adopt the point of view that we will observe fluid properties at few selected points xi as a function of time.• The association with a given fluid particle must be broken.• Realize that as time flows different fluid particles will occupy the position xi.• Once properties are expressed as functions of (x; t) , it is said to be the Eulerian description of a fluid.

The Lagrangian Displacement of Particle in Eulerian Description

• It is of very high importance to compare these two descriptions of a fluid and validate Euler description.

• The most basic definition is the meaning of velocity: the definition is

ttxvt

x,,

That is to say, following the particle one need to calculate the rate of change of position with respect to time.

• Given the Eulerian velocity field, the calculation of Lagrangian displacement is therefore mathematically equivalent to solving the initial value problem .• A set of ordinary differential equations for the function x(t) , with the initial condition x(0) = . • This is of very high importance to compare these two descriptions of a fluid and validate Euler description.

Material Derivative with Imaginary Description

• Consider Eulerian quantity

t

txfttxxf

dt

txdf ppp

t

p

,,lim

,0

during any infinitesimal time t

• The time rate of change of attached to this fluid parcel is expressed as:

• Attribute this quantity to lgrangians’parcel, p.