Question

Ideal fluid flows along a flat tube of constant cross-section, located in a horizontal plane and bent as shown in Fig. 1.80 (top view). The flow is steady. Are the pressures and velocities of the fluid equal at points 1 and 2? What is the shape of the streamlines?

Answer 1

Between 1 and 2 fluid particles are in nearly circular motion and therefore 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 requires that pressure at 1 should be greater than the pressure at 2 i.e.

 P₁>P₂ 

so that the fluid particles can have required acceleration. If there is no turbulence, the motion can be taken as irrotational. Then by considering 

along the circuit shown we infer that 

v₂>v₁

(The portion of the circuit near 1 and 2 are streamlines while the other two arms are at right angle to streamlines) 

In an incompressible liquid we also have div vector v=0.

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