An infinite cylinder of radius a rotates in a viscous, incompressible fluid with angular

velocity $.$ Show that the vortex of a following form:

u $=($a$^2/$r$)$e$\_$theta r $>=$ a $(1)$

is in this case an exact solution of the Navier$-$Stokes equation satisfying the boundary conditions. Show that there is a nonzero torque exterted on such a cylinder by a fluid and find the value of this torque. Next, find a mistake in the following reasoning:

$$Navier$-$Stokes eqution for viscous, incompressible fluid

$\backslash$rho $*$Du$/$Dt$=$p $+\backslash$rho g $+\backslash$eta $^2$ u

can be transformed, using a formula

$^2$ u $=$ grad divu $$ rot rotu,

into the form

$\backslash$rho $*$Du$/$Dt$=$p $+\backslash$rho g $\backslash$eta rot rotu.

However the flow $(1)$ is irrotational $($check$!),$ thus the viscous term in the Navier$-$Stokes equations vanishes; therefore the viscous forces are zero; and so the torque on the cylinder is zero.$$