Kinematics of fluid flow: Eulerian and Lagrangian
approaches
Lagrangian approach
o Newton’s laws are used to describe the motion of objects, and we can
accurately predict where they go and how momentum and kinetic energy
are exchanged from one object to another.
o The kinematics of such experiments involves keeping track of the position
vector of each object, x → A, x → B, . . . , and the velocity vector of each
object, V → A, V → B, . . . , as functions of time.
o When this method is applied to a flowing fluid, we call it the Lagrangian
description of fluid motion after the Italian mathematician Joseph Louis
Lagrange (1736–1813).
o As you can imagine, this method of describing motion is much more
difficult for fluids.
o First of all we cannot easily define and identify particles of fluid as they
move around.
o Secondly, a fluid is a continuum (from a macroscopic point of view), so
interactions between parcels of fluid are not as easy to describe as are
interactions between distinct objects like billiard balls or air hockey pucks.
o Furthermore, the fluid parcels continually deform as they move in the flow.
From a microscopic point of view, a fluid is composed of billions of
molecules that are continuously banging into one another; but the task of
following even a subset of these molecules is quite difficult, even for our
fastest and largest computers.
o Nevertheless, there are many practical applications of the Lagrangian
description, such as the tracking of passive scalars in a flow, rarefied gas
dynamics calculations concerning re-entry of a spaceship into the earth’s
atmosphere, and the development of flow measurement systems based on
particle imaging.
approaches
Lagrangian approach
o Newton’s laws are used to describe the motion of objects, and we can
accurately predict where they go and how momentum and kinetic energy
are exchanged from one object to another.
o The kinematics of such experiments involves keeping track of the position
vector of each object, x → A, x → B, . . . , and the velocity vector of each
object, V → A, V → B, . . . , as functions of time.
o When this method is applied to a flowing fluid, we call it the Lagrangian
description of fluid motion after the Italian mathematician Joseph Louis
Lagrange (1736–1813).
o As you can imagine, this method of describing motion is much more
difficult for fluids.
o First of all we cannot easily define and identify particles of fluid as they
move around.
o Secondly, a fluid is a continuum (from a macroscopic point of view), so
interactions between parcels of fluid are not as easy to describe as are
interactions between distinct objects like billiard balls or air hockey pucks.
o Furthermore, the fluid parcels continually deform as they move in the flow.
From a microscopic point of view, a fluid is composed of billions of
molecules that are continuously banging into one another; but the task of
following even a subset of these molecules is quite difficult, even for our
fastest and largest computers.
o Nevertheless, there are many practical applications of the Lagrangian
description, such as the tracking of passive scalars in a flow, rarefied gas
dynamics calculations concerning re-entry of a spaceship into the earth’s
atmosphere, and the development of flow measurement systems based on
particle imaging.
