Couette Flow - 3 | Fluid Mechanics for Mechanical Engineering PDF Download

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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

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

which means v_θ 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 v_{θ =} v_{θ (y)} . With these simplifications and assuming that " θ symmetry" holds good, Navier-Stokes equation reduces to

                                                                     (26.17)                                                      

                                       (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

                                ( 26.19)

• For the azimuthal component of velocity, v_θ, 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)

Calculation of Stress and Torque Transmitted

Now,  is the general stress-strain relation.

• In our case,

                                                        (26.21)

• Equations (26.20) and (26.21) yields

                                      (26.22)         

• Now,

and,

• For the case, when the inner cylinder is at rest and the outer cylinder rotates, the torque transmitted by the outer cylinder to the fluid is

                                      (26.23)

where l is the length of the cylinder.

• The moment T₁, with which the fluid acts on the inner cylinder has the same magnitude. If the angular velocity of the external cylinder and the moment acting on the inner cylinder are measured, the coefficient of viscosity can be evaluated by making use of the Eq. (26.23).