Question

A fluid flows along the x axis with a velocity given by V = ( x / t ) ˆ i , where x is in feet and t in seconds. (a) Plot the speed for 0 ≤ x ≤ 10 ft and t = 3 s . (b) Plot the speed for x = 7 ft and 2 ≤ t ≤ 4 s . (c) Determine the local and convective acceleration. (d) Show that the acceleration of any fluid particle in the flow is zero. (e) Explain physically how the velocity of a particle in this unsteady flow remains constant throughout its motion.

Answer 1

Answer:

c)

 V_local = -x/t^2

 V_convec = x/t^2

d)

a =  V_local +  V_convec = 0

e) When a particle moves towards postive x direction its convective velocity increases, but at the same time the local velocity deacreases (at the same rate) when time increases

Explanation:

Hi!

You can see plots for a) and b) attached on this document

c)

The local acceleration is just teh aprtial derivative of the velocity with respect to t:

$\frac{dV}{dt} = \frac{d}{dt} \frac{x}{t}=- \frac{x}{t^2}$

And the convective acceleration is given by the product of the velocity times the gradient of the velocity, that is:

$\vec{v} \cdot \nabla \vec{v} = v ( \frac{dv}{dx} ) =\frac{x}{t} \frac{1}{t} = \frac{x}{t^2}$

d)

Since the acceleration of any fluid particle is the sum of the local and convective accelerations, we can easily see that it is equal to zero, since they are equal but with opposit sign

e)

This is because of teh particular form of the velocity. A particle will move towards areas of higher velocities (convectice acceleration), but as time increases,  the velocity is also decreasing (local acceleration), and the sum of these quantities adds up to zero