Flow between Two Concentric Rotating Cylinders

•  Another example which leads to an exact solution of Navier-Stokes equation is the flow between two concentric rotating cylinders.
• Consider flow in the annulus of two cylinders (Fig. 26.5), where r₁ and r₂ are the radii of inner and outer cylinders, respectively, and the cylinders move with different rotational speeds ω₁ and ω₂ respectively

.

FIG 26.5 - Flow between two concentric rotating cylinders

• From the physics of the problem we know, , .
• From the continuity Eq. and these two conditions, we obtain

which means is not a function of θ. Assume z dimension to be large enough so that end effects can be neglected and (any property) = 0.

• This implies . With these simplifications and assuming that " θ symmetry" holds good, Navier-Stokes equation reduces to

(26.17)

and

(26.18)

• Equation (26.17) signifies that the centrifugal force is supplied by the radial pressure, exerted by the wall of the enclosure on the fluid. In other words, it describes the radial pressure distribution.
  From Eq. (26.18), we get

 

or

(26.19)

• For the azimuthal component of velocity, , the boundary conditions are: at at .
• Application of these boundary conditions on Eq. (26.19) will produce

and

• Finally, the velocity distribution is given by

(26.20)