Question

In: Physics

Couette Flow: flow between 2 long concentric cylinders of radii Ro and Ri, created by the...

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.

Solutions

Expert Solution

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)


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