In: Physics
Couette Flow: flow between 2 long concentric cylinders of radii Ro and Ri, created by the rotation of one of them. Solve the flow problem for a Newtonian, incompressible fluid for the following two cases:
1. The outer cylinder of radius Ro is rotating with an angular velocity Ω (s^-1 for units). Also calculate the torque for this.
2. The inner cylinder of radius Ri is rotating with an angular velocity -Ω (s^-1 for units). Also calculate the torque for this.
let there be two concentric cylinders, radii r1 < r2
a. now, let vr be the radial velocity, vz = velocity along the axis, vt = tangential flow speed
from physics
vr = vz = 0 m/s
from equation of continuity and these two euations we obtain
d(vt)/d(theta) = 0
now
assuming d(any property)/dz = 0 ( very long pipes)
hence
vt = f(r)
from symmetry and equation for newtonian compressible flow we get
rho*vt^2/r = dP/dr
and
d^2vt/dr^2 + (1/r)dvt/dr - vt/r^2 = 0
hence
d/dr ((1/r)d/dr(rvt)) = 0
vt = Ar/2 + B/r
now
for outer cylinder rotating at w
vt = wr2 for r = r2
vt = 0 for r = r1
hence
w(r2) = A*r2/2 + B/r2
0 = Ar1/2 + B/r1
A = -B/r1^2
A = wr2^2/(r2^2 - r1^2)
B = w*r1^2*r2^2/(r2^2 - r1)^2
torque is given by ( per ujnit length)
T = -4*pi*mu(w)/(1/r1^2 - 1/r2^2)
b. similiarly
vt = Ar/2 + B/r
A = -wr1^2/(r2^2- r1^2)
B = -wr1^2r2^2/(r2^2-r1^2)
T = -4*pi*mu(w)/(1/r1^2 - 1/r2^2)