I am given, in each of three cases, an angular velocity [tex]\Omega(r)[/tex] and am told to assume no axial (z) velocity i.e., [tex]u_z = 0[/tex]. I am asked to(adsbygoogle = window.adsbygoogle || []).push({});

(1) find the velocity field in cartesian coordinates

(2) find the vorticity distribution in threee cases.

(1) As setup, the problem asks me to "Show the velocity components are given by [tex](u_x,u_y,u_z) = (-\Omega(r), \Omega(r), 0)[/tex]".

This doesn't seem possible to me....how can both x and y components of the velocity field be the same? I keep coming up with the like of, for a given r:

[tex]u_x = - r * \Omega(r) * \sin(\theta)[/tex][tex]u_x = r * \Omega(r) * \cos(\theta)[/tex]where [tex]\theta = \Omega(r) * t;

[/tex]

Nevertheless, the supplied answer to (1) is [tex](u_x, u_y, u_z) = (-\Omega(r), \Omega(r), 0)[/tex] and it seems to contradict my later assertion that (in cyl coords) [tex]u_r = 0[/tex].

For (2), I am asked to find the vorticity in three cases:

a) [tex]\Omega = q/r[/tex] (typ. flow around strong concentrated vortex)

b) [tex]\Omega r^2 = constant = k [/tex](fluid parcels slowly spiraling towards origin while conserving angular momentum)

c) [tex]\Omega^2 r = G*M/r^2[/tex] (velocity distribution inside accretion disk in black hole or neutron star)

So, I chose to ignore the bogus part (1) and solve in cyl coord where [tex]v = (u_r, u_{\theta}, u_z)[/tex]. For all three cases I assumed that [tex]u_z = 0[/tex] (given) and [tex]u_r = 0[/tex] ("flow is the form a circular 'swirl' about the origin in the x-y plane"). (If [tex]u_r[/tex] is not zero, I haven't a clue as to how to come up with a u_r.....)

Then, blithely proceeding:

* In cyl. coor, [tex]u_{\theta} = r * \Omega(r)[/tex], where [tex]\Omega(r) =[/tex] angular velocity.

* Then, vorticity [tex]w = \nabla \times v[/tex] is [tex]w_z[/tex] only and reduces to

[tex]w_z = \frac {1} {r} \frac {d} {dr} (r * u_{\theta}) = \frac {1} {r} \frac {d} {dr}(r * r * \Omega(r))[/tex]

Using this approach I get,

a) [tex]\Omega(r) = q/r[/tex] gives [tex]w = \frac {1} {r} \frac {d} {dr}(r * r * q/r) = \frac {1} {r} \frac {d} {dr}(r*q) = q/r[/tex]

b) [tex]\Omega r^2 = k \Rightarrow \Omega(r) = k/ r^2 \Rightarrow

w = \frac {1} {r} \frac {d} {dr}(r * r * k / r^2) = \frac {1} {r} \frac {d} {dr}(k) = 0[/tex]

c) [tex]\Omega^2 r = G*M/r^2 \Rightarrow \Omega(r) = \frac {k_2} {r^{3/2}} \Rightarrow w = \frac {1} {r} \frac {d} {dr}(r * r * \frac {k_2} {r^{3/2}}) = \frac {k_2} {r} \frac {d} {dr}(r^{1/2}) = \frac {k_2} {2} r^{-3/2}[/tex]

Am I even close here?

Confused in Seattle,

/catboy

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# Vorticity where angular velocity is function of r

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