Definitions

Eulerian flow descriptions entail spatio-temporal descriptions for any parameter, and correspond to control volume analysis. $a=a(x,y,z,t)$

Lagrangian flow descriptions entail temporal descriptions for each parameter, specific to each particle, and correspond to a system analysis. $a_i=a_i(t)$

Control Volumes are spaces defined by geometric boundaries which we analyze while fluid flows through them.

Systems are collections of masses which we analyze through movement and deformation.

Flow Description

Flow Field

A Eulerian description of flow is given by a flow field.

$$ \bold u=\begin{bmatrix} u(x,y,z,t)&v(x,y,z,t)&w(x,y,z,t) \end{bmatrix} $$

Attributed of Flow

A particular flow field may have attributes that allow us to make simplifications in analysis.

• Steady Flow means that the velocity does not change with time at any locale, $\frac{\partial \bold u}{\partial t}=0$

Flow Lines

In addition to a flow field, a flow may be described by the following lines. For steady flow, all of these lines are identical.

• Streamlines, which are tangent to velocity at every point.
• Streaklines, which include every particle which has passed through a certain locale.
• Pathlines, which include every locale though which a certain particle has passed.

Streamlines may be found from a given two dimensional flow by solving the following ODE.

$$ \bold u\Rarr\frac{dy}{dx}=\frac vu\Rarr s(x,y,C)=0 $$

Material Derivative