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A fluid with constant properties ( viscosity and density ) is kept between two concentric cylinders of height L , along which the pressure is

A fluid with constant properties (viscosity
and density ) is kept between two
concentric cylinders of height L, along
which the pressure is the same. The inner
cylinder has a radius of rl, and the outer one
has a radius of r2. The cylinders can rotate
around an axe in axial direction passing
through their center.
In Case 1, the outer cylinder rotates with
angular velocity while the inner cylinder
is kept fixed. In Case 2, the inner cylinder
rotates with angular velocity while the
outer cylinder is kept fixed. With the
rotational motion of the cylinders, the fluid
is set in motion in tangential direction with
Here, the tangential velocity sto only changes with the radial direction r. The velocity profile for both
cases are shown in the figure. Note that tangential velocity is the product of angular velocity and radius:
w=I.
a) Draw the forces acting on the control volume dr d d x dz given here.
Please do not make shell balance.
b) Using the Navier-Stokes equations given below, find the tangential velocity profile va(r) for Case 1
and Case 2.
r-direction:
(vrt+vrvrr+vrvr-v2r+vzvrz)=-Pr+gr+[r(1rr(rvr))+1r22vr2-2r2v+2vrz2]
-direction:
z-direction:
((hat(vz))t+vrvzr+vrvz+vzvzz)=-Pz+gz+[1rr(r(hat(del))vzr)+1r22vz2+2vzz2]
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