problem 3.1

One-dimensional unsteady flow

Write down the three conservation equations of mass, momentum and energy for one-dimensional unsteady flows (in the absence of external force) with x as the spatial coordinate and t the time when the density is ?( x, t), velocity is v = ( u( x, t) , 0 , 0), and so on.

Comment: Lagrange s form of equation of motion

Lagrangian representation of position of a fluid particle a = ( a, b, c) at time t is defined by X( a , t) in (1.11) with its velocity V _a( t) = ? _t X( a , t). Then, the x component of equation of motion (3.12) can be written as . We multiply this by ?x/ ?a. Similarly, multiplying y and z components of Eq. (3.12) and summing up the three expressions, we obtain

This is the Lagrange s form of equation of motion (1788), corresponding to the Eulerian version (3.13).

Defining the Lagrangian coordinates a = ( a _i) by the particle position at t = 0, the continuity equation for the particle density ? _a( t) is given by

where ?( x) / ?( a) = det( ? X _j / ?a _i) is the Jacobian determinant.

problem 3.1

From (3.7),...