A fluid motion has velocity potential given by
where is a constant. Find the corresponding velocity field . Determine .
The time-average of a quantity is defined as .
Show that the time-average of this velocity field at every point is zero.
Write down an expression for the fluid acceleration and find the time-average acceleration at .
Suppose now that . The material particle at at time is marked with dye. Write down equations for its subsequent motion and verify that its position at time is given (correct to terms of order ) as
Deduce the time-average velocity of the dyed particle correct to this order.