English

3 Pages

Gain access to the library to view online

__
Learn more
__

Description

FINITE DIFFERENCES/FINITE VOLUMES & CONSERVATIONS LAWS Feb-April 09 Conservations laws : Derivation Let us consider a subset depending on time D(t) ? R3. Initially, for t = 0, any material particle in D(0) is identified by its coordinate ? . We define by x(?, t) the position at the time t of the particle that was initially at ? . The transformation (?, t) 7? x(?, t) is invertible and sufficiently regular. The material velocity u and jacobian of the transformation are : u(x, t) = ∂x ∂t and J(?, t) = ??x(?, t) = ( ∂xi ∂?j ) 1≤i≤3,1≤j≤3 For any function f(x, t) : R3?R+ 7? R continuously differentiable (that could represent a physical property), we define its particular derivative and sum over a moving volume : df dt = df(x(?, t), t) dt and If (t) = ∫ D(t) f(x, t)dx The aim here is to estimate the integral over the volume D(t) as a function of the initial position, and its variation in time in order to establish some conservation properties : 1.

- ∂ui ∂xk
- ∂? ∂t
- boundary integrals into
- ∂xk ∂?j
- finite volume
- cross product
- relation into
- conservation laws
- ∂f ∂xi
- ?x ·

Subjects

Informations

Published by | pefav |

Reads | 9 |

Language | English |

Report a problem