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catboy
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I am given, in each of three cases, an angular velocity \Omega(r) and am told to assume no axial (z) velocity i.e., u_z = 0. I am asked to
(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 (u_x,u_y,u_z) = (-\Omega(r), \Omega(r), 0)".
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:
Nevertheless, the supplied answer to (1) is (u_x, u_y, u_z) = (-\Omega(r), \Omega(r), 0) and it seems to contradict my later assertion that (in cyl coords) u_r = 0.
For (2), I am asked to find the vorticity in three cases:
a) \Omega = q/r (typ. flow around strong concentrated vortex)
b) \Omega r^2 = constant = k(fluid parcels slowly spiraling towards origin while conserving angular momentum)
c) \Omega^2 r = G*M/r^2 (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 v = (u_r, u_{\theta}, u_z). For all three cases I assumed that u_z = 0 (given) and u_r = 0 ("flow is the form a circular 'swirl' about the origin in the x-y plane"). (If u_r is not zero, I haven't a clue as to how to come up with a u_r...)
Then, blithely proceeding:
* In cyl. coor, u_{\theta} = r * \Omega(r), where \Omega(r) = angular velocity.
* Then, vorticity w = \nabla \times v is w_z only and reduces to
w_z = \frac {1} {r} \frac {d} {dr} (r * u_{\theta}) = \frac {1} {r} \frac {d} {dr}(r * r * \Omega(r))
Using this approach I get,
a) \Omega(r) = q/r gives w = \frac {1} {r} \frac {d} {dr}(r * r * q/r) = \frac {1} {r} \frac {d} {dr}(r*q) = q/r
b) \Omega r^2 = k \Rightarrow \Omega(r) = k/ r^2 \Rightarrow <br /> w = \frac {1} {r} \frac {d} {dr}(r * r * k / r^2) = \frac {1} {r} \frac {d} {dr}(k) = 0
c) \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}
Am I even close here?
Confused in Seattle,
/catboy
(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 (u_x,u_y,u_z) = (-\Omega(r), \Omega(r), 0)".
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:
u_x = - r * \Omega(r) * \sin(\theta)
u_x = r * \Omega(r) * \cos(\theta)
where \theta = \Omega(r) * t;<br />Nevertheless, the supplied answer to (1) is (u_x, u_y, u_z) = (-\Omega(r), \Omega(r), 0) and it seems to contradict my later assertion that (in cyl coords) u_r = 0.
For (2), I am asked to find the vorticity in three cases:
a) \Omega = q/r (typ. flow around strong concentrated vortex)
b) \Omega r^2 = constant = k(fluid parcels slowly spiraling towards origin while conserving angular momentum)
c) \Omega^2 r = G*M/r^2 (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 v = (u_r, u_{\theta}, u_z). For all three cases I assumed that u_z = 0 (given) and u_r = 0 ("flow is the form a circular 'swirl' about the origin in the x-y plane"). (If u_r is not zero, I haven't a clue as to how to come up with a u_r...)
Then, blithely proceeding:
* In cyl. coor, u_{\theta} = r * \Omega(r), where \Omega(r) = angular velocity.
* Then, vorticity w = \nabla \times v is w_z only and reduces to
w_z = \frac {1} {r} \frac {d} {dr} (r * u_{\theta}) = \frac {1} {r} \frac {d} {dr}(r * r * \Omega(r))
Using this approach I get,
a) \Omega(r) = q/r gives w = \frac {1} {r} \frac {d} {dr}(r * r * q/r) = \frac {1} {r} \frac {d} {dr}(r*q) = q/r
b) \Omega r^2 = k \Rightarrow \Omega(r) = k/ r^2 \Rightarrow <br /> w = \frac {1} {r} \frac {d} {dr}(r * r * k / r^2) = \frac {1} {r} \frac {d} {dr}(k) = 0
c) \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}
Am I even close here?
Confused in Seattle,
/catboy
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