Question

Ideal fluid flows along a flat tube of constant cross-section, located in a horizontal plane and bent as shown in figure above (top view). The flow is steady. What is the shape of the streamlines?

• A. Density of streamlines is proportional to the velocity.
  
• B. Density of streamlines is not proportional to the velocity.
  
• C. Density of streamlines is constant.
  
• D. None of these.
  

Answer 1

The correct option is A Density of streamlines is proportional to the velocity.

In the figure, between 1 and 2 fluid-particle are nearly circular motion and have centripetal acceleration. The force for this acceleration, like for any other situation in an ideal fluid, can only come from the pressure variation along the line joining 1 and 2. This required that pressure at 1 should be less than the pressure at 2.

$P_1<P_2$.

So, the fluid particle can have required acceleration. If there is no turbulence, then the motion can be taken as irrotational.

$\int \overrightarrow{v}.\overrightarrow{dl}=0$ (for closed-loop)

The velocity at point 1 is greater than the velocity at point 2

$v_1>v_2$

The portion of the circle near 1 and 2 are streamlined while the other two arms are at the right angle triangle to the streamlines.

In an incompressible liquid, the Divergence of the vector is zero.

By electrostatic analogy, we find that the density of streamlines is proportional to the velocity at that point.

The correct option is A.