What is a fluid flux?

Question

I need a concise definition of a fluid flux and an accompanying example. I've never taken a single physics course before, but I'm required to understand this concept so I can do the calculations for a Complex Analysis class.

Answer 1

The flux through a surface is the amount of fluid that crosses the surface in a flow per unit time at any one instant. If the velocity field is $v(x)$, and the surface is S, it is the integral over the surface

$$\int_S v \cdot n $$

where n is the normal to the surface. This is the general definition of flux of a vector field, applied to the special case of the velocity field.

Examples

Suppose the velocity field is

$$ v_x = \omega y $$ $$ v_y = - \omega x $$ $$ v_z = 0 $$

This is the fluid rigidly rotating with angular frequency $\omega$. Suppose you want the flux through the surface defined by $0<x<L$, $0<z<L$ at $y=0$ at the instant the fluid has this velocity profile. This has a normal in the y-direction, so the integral is of the y-component of the velocity over the surface:

$$ \int_0^L \int_0^L -x dx dz = L\int_0^L x dx = - L^2 {\omega L\over 2}$$