Coaxial Cylinders Flow: Examining Velocity & Stress in a Rotating Fluid

[ettapandora]

A real incompressible fluid is contained in the region between two coaxial cylinders of radii R1 and R2. The outsider cylinder rotates with angular velocity w, in stationary regime, and the flow is purely circular. Neglect the action of gravity.

a) Show that vφ (azimuthal velocity component) obeys the differential equation:

d²vφ/dr² + (1/r) . (dvφ/dr) - vφ/r² = (d/dr). [(1/r). (d/dr)(rvφ)] = 0

and determine vφ.b) Calculate the stress tensor in the fluid;

c) Obtain the moment about the axis of rotation of the frictional forces in each of the cylinders;

Thank you so much for your attention. I'm studying biomedical engineering and I have a biomechanics exam tomorrow and I'm really in trouble. I'd be very thankfull if someone could help me.

 

[Chestermiller]

Start out with the theta component of the Navier stokes equations in cylindrical coordinates. The inertial terms, the pressure term, and the gravitational term are going to be zero for your problem. This leaves only the viscous term. The viscous term leads to the equation you gave.

Chet