Final answer:
The local acceleration of the fluid is -9x/t^2 ft/s^2 while the convective acceleration is 81x/t^2 ft/s^2.
Step-by-step explanation:
The question involves finding the local acceleration and the convective acceleration of a fluid that has a velocity field given by V = (9x/t)i, where i is the unit vector in the x-direction. In fluid dynamics, the local acceleration is given by the time derivative of velocity, while the convective acceleration is associated with the spatial change in velocity due to a fluid element moving through a velocity field.
To determine the local acceleration (a), we differentiate the velocity function V with respect to time t, treating x as constant because local acceleration does not consider changes in position. Therefore, the local acceleration is dV/dt = d/dt (9x/t) = -9x/t2. The convective acceleration (b) involves taking the spatial derivative of V with respect to x, multiplied by V itself because it represents the rate of change of velocity with respect to space as the fluid element moves. Hence, the convective acceleration is (V · dV/dx), which equals (9x/t) · d/dx (9x/t) = (9x/t) · (9/t) = 81x/t2. The units for both accelerations are in feet per second squared (ft/s2).