Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
3. Contents:
FLUID MOTION
METHODS OF DESCRIBING FLUID MOTION
LAGRANGE’S THEOREM
LAGRANGE’S METHOD FOR FLUID MECHANICS
COMPARISON BETWEEN LAGRANGE AND EULER’S THEOREM
LAGRANGE’S EQUATION
ADVANTAGES OVER NEWTONIAN MECHANICS
CONNECTION TO EULER-LAGRANGE EQUATION
PROPERTIES OF THE EULER–LAGRANGE EQUATION
EXAMPLES AND APPLICATIONS
4. FLUID MOTION
WHAT IS FLUID?
a substance, as a liquid or gas, that is capable of flowing and that changes its
shape at a steady rate when acted upon by a force tending to change its shape.
The motion of fluid depend upon the fluid and shape of the passage
through which the fluid particles moves.
• A fluid consist of an innumerable number of particles,
• Whenever a fluid is in motion, these particles move along certain lines.
5. MOTION OF FLUID PARTICLES
The molecule of liquid and gas have freedom to move.
The motion of fluid depend upon the fluid and shape of the passage through
which the fluid particles moves.
A fluid consist of an innumerable number of particles,
Whenever a fluid is in motion, these particles move along certain lines.
6. Velocity Field
Different fluid particles in fluid flow ,which move at different velocities and
may be subjected to different accelerations.
The velocity and acceleration of a fluid particle may change both w.r.t time
and space.
In the study of fluid flow it is necessary to observe the motion of the fluid
particles at various points in space and at a successive instant at a time.
7. METHODS OF DESCRIBING FLUID MOTION
There are two methods of motion of fluid paticles.
1) Lagrangian method:
This method deals with the individual particles.
Langrangian description of fluid flow tracks the position and velocity of
individual particles.
(E.g.: Track the location of migrating bird.
Motion is described based upon Newton's laws.
Named after Italian mathematician Joseph Louis
Lagrange (1736-1813).
8. 2) Eulerian method
Describes the flow field (velocity,acceleration, pressure, temperature, etc.)
as functions of position and time.
It deals with the flow pattern of all the particles.
Count the birds passing a particular location
If you were going to study water flowing in a pipeline
which approach would you use? Eulerian Description
now we are going to discuss the lagrange’s method.
9. LAGRANGE’S THEOREM
Lagrange theorem exists in many fields, respectively
Lagrange theorem in fluid mechanics
Lagrange theorem in calculus
Lagrange theorem in number theory
Lagrange's theorem in group theory
10. LAGRANGE’S METHOD FOR FLUID
MECHANICS
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the
Italian-French mathematician and astronomer Joseph-Louis Lagrangein 1788.
The Lagrange method is based on the study of single fluid particle movement
process as the basis, all of the particle motion, constitute the entire fluid
movement.
Coordinate position of each particle with a start time of (a, B, c), as a symbol of
the particle.
Any time any particles in the space position (x, y, z) can be seen as (a, B, C and T)
function,
The basic characteristics of Lagrange method: tracking the motion of the fluid
particles
Advantages: can directly use the particle dynamics in solid mechanics analysis
11. COMPARISON BETWEEN LAGRANGE AND
EULER’S THEOREM
LAGRANGETHEOREM
It is not possible to track each
"particle" in a complex flow field.
Thus, the Lagrangian description is
rarely used in fluid mechanics.
EULER’S THEOREM
The Eulerian Description is one in
which a control volume is defined,
within which fluid flow properties
of interest are expressed as fields.
14. Advantages Over Newtonian Mechanics
We are now going to use the ideas of the previous lecture to develop a new formalism for
mechanics, called Lagrangian mechanics, invented by Lagrange (1736-1813).
There are two important advantages of the Lagrange formalism over that of Newton.
First, as we have seen, Lagrange’s equations take the same form for any coordinate system, so
that the method of solution proceeds in the same way for any problem.
Second, the Lagrangian approach eliminates the forces of constraint. This makes the Lagrangian
formalism easier to solve in constrained problems.
This chapter is the heart of advanced classical mechanics, but it introduces some new methods
that will take getting used to. Once you master it, you will find it an extraordinarily powerful
way to solve mechanics problems.
16. Properties of the Euler–Lagrange equation
Non uniqueness
The Lagrangian of a given system is not unique. A Lagrangian L can be multiplied
by a nonzero constant a, an arbitrary constant b can be added, and the new
Lagrangian aL + b will describe exactly the same motion as L. Each Lagrangian
will obtain exactly the same equations of motion.
Invariance under point transformations
Given a set of generalized coordinates q, if we change these variables to a new
set of generalized coordinates s according to a point transformation q = q(s, t),
the new Lagrangian L′ is a function of the new coordinates. Similary…
Energy conservation
Mechanical similarity
Interacting particles
Cyclic coordinates and conserved momenta
17. EXAMPLES
Conservative force
Cartesian coordinates
Polar coordinates in 2d and 3d
Pendulum on a movable support
Two-body central force problem
Electromagnetism
Extensions to include non-conservative forces
18. ALTERNATIVE FORMULATIONS OF CLASSICAL
MECHANICS
Momentum space formulation
Higher derivatives of generalized coordinates
Optics
Relativistic formulation
Quantum mechanics
Classical field theory
Noether's theorem
19. APPLICATIONS…
LAGRANGE’S APPLICATIONS
Sprays particles
bubble dynamics
rarefied gasses etc.
EULUERIAN APPLICATION
Simulation of microscale airborne probes etc.