Question

An incompressible fluid flows irrotationally through a tube of circular crosssection that has greater diameter at one end than the other. What conclusion can you reach about the speed of the fluid at the ends of the tube?A) The flow must be at a slower speed at the end with a greater diameter.
B) The speed of the flow must be the same at both ends.
C) The flow must be at a faster speed at the end with a greater diameter.
D) Pressures and tube heights must be known to relate the flow speeds at the two ends.
E) No conclusions can be reached based on this information.

Answer 1

In order to obtain information on the speed of the object with the variation of the diameter we will resort to the continuity equations. Mathematically, the continuity equation tells us that the input flow must be equal to the output flow, referring to the product between the area and the velocity. This is,

$A_1V_1 = A_2 V_2$

V = Velocity

A = Area

If we place in terms of speed at the beginning of the section the other terms we will have to

$V_1 = \frac{A_2}{A_1} V_2$

If diameter of $A_1$ is greater than $A_2$ we have that

$\frac{A_2}{A_1} <1$

$\therefore$ $V_1$ will be less than $V_2$

A) The flow must be at a slower speed as the end with greater diameter