# week2 - 1.7 The Lagrangian derivative(a.k.a the convective...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1.7 The Lagrangian derivative (a.k.a. the convective derivative, or the material derivative). We know how to measure the time derivative of a physical quantity associated with the fluid, for example that of the temperature T ( x ,t ), at a fixed point in space (the Eulerian derivative). It’s just ∂T ∂t . This quantity will describe the change of temperature at a fixed location, for example air temperature in Bristol. However, this quantity will not be a measure of how a mass of air heats up or becomes colder. The reason is that air is swept away by the prevailing flow field u ( x ,t ). In other words, to describe the change of temperature of a piece of air, we need to consider the rate of change of T ( x ,t ), following a fluid particle . This is called the Lagrangian derivative. δ x+ x t+ δ t x=u δ δ t O x t Suppose a particle at time t is positioned at x , then at time t + δt it has moved to x + u δt . The change in T in the instant δt is T ( x + u δt,t + δt ) − T ( x ,t ) (6) Taylor expanding about x , t we get T ( x ,t ) + uδtT x ( x ,t ) + vδtT y ( x ,t ) + wδtT z ( x ,t ) + δtT t ( x ,t ) + O (( δt ) 2 ) − T ( x ,t ) . The Lagrangian derivative is (6) divided by δt as δt → 0, so that we get ∂T ∂t + ( u ·∇ ) T ≡ DT Dt . (7) A particularly important example is the change in velocity (the acceleration) of a fluid particle, which we need to apply Newton’s equations to fluid motion. Of course, the velocity is a vector quantity, which means we have to apply (7) to each component: D u Dt = parenleftbigg Du 1 Dt , Du 2 Dt , Du 3 Dt parenrightbigg . The the acceleration of a fluid particle becomes D u Dt = ∂ u ∂t + ( u ·∇ ) u . (8) Note: The Lagrangian derivative (i.e. following fluid particles) is given in terms of Eulerian (i.e. fixed point) measurements. It is vital to understand exactly how to com- pute expressions like ( u ·∇ ) u for a given velocity. Example: Consider an accelerating fluid flow, such as the log flowing through a narrowing channel. Supposenarrowing channel....
View Full Document

## This note was uploaded on 01/20/2012 for the course MATH 33200 taught by Professor Eggers during the Fall '11 term at University of Bristol.

### Page1 / 6

week2 - 1.7 The Lagrangian derivative(a.k.a the convective...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online