Starting from the results of Couette flow we see that for a viscous fluid with laminated flow along a straight line we should have a shear force between planes given by:
[1]
(With μ = coefficient of dynamic viscosity,
= velocity of the fluid orthogonal to the r direction and hence actually in the plane, A = Area of the plane, r = direction orthogonal to the plane, directional derivative shown with grad operator if you prefer it).
For motion mainly in circular rings we just consider infinitesimal sections as planes (which is the comment you made, "bend your model into a circle!"). The direction r orthogonal to this plane is precisely the radial direction r in the original co-ordinates used to specify the problem. (Diagram should appear below)
Circular flow.jpg (53.72 kB . 907x421 - viewed 1276 times) Then we have:
=
[2]
which we can take as the force, δF, on the infinitesimal plane due to the plane just below it (at radial position R>r in the diagram).
We can simplify the above expression by assuming either that we are going to work in plane polar co-ordinates and the z-axis is redundant or that the entire vortex is contained within some vessel of height ΔZ along the z-axis. If we do the latter then, the factor ΔZ just appears as a multiplicative constant in our formulas.
We are also only interested in the stable or equilibrium flow which the vortex will settle down to. By symmetry arguments we can see that the velocity
v(
x) of this final state of the fluid flow will only depend on r (not on angle θ or position on the z-axis, where we can assume we're only interested in the flow in the x-y plane through the middle of the vessel in the z-axis - or that the vortex extends to infinity in the z-axis if you prefer, this is just a simplified and idealized model - the real flow would not be a perfect free vortex near the ends of a finite length container because the walls will interfere and slow down the fluid there).
Additionally we can see that (in the final steady flow state) the velocity at any position
x cannot have any component of velocity radially or along the z-axis. Since the flow is symmetric any inward flow at say position
x0 is an inward flow at all positions with r = r co-ordinate of
x0. We cannot sustain such an inward flow, even the most compressible fluid cannot support a region r< r
0 that becomes infintely dense. Similar arguments can apply to a sustained outward flow or flow in the z-axis direction. Hence, the final steady flow solution must be such that the velocity field
v =
.
So we have, velocity is perpendicular to the radius
= v = v(r) and ΔZ is a constant (or redundant) and we can make these amendements to equation [2].
[3]
(where η = a constant = μ or μΔZ if you keep the z-axis).
Equation [2] describes the force (in the positive
direction) that is due to shear from the fluid at a greater radius. (We haven't considered shear from fluid at a lower radius yet - because the original "Couette flow" formulation of shear forces describes only the shear force on the top plate due to fluid below it, there is no fluid above the top plate). This force was acting on a small piece of our fluid at radius r from the z-axis. So there is a torque about the z-axis. We will be interested in the torque applied to a complete ring of the fluid. From equation [3] we have a small element of torque
on a section of the ring at radius r and with angle between θ and θ+dθ. Hence,
[4]
since v=v(r) so nothing in that integrand has any θ dependance.
Equation [4] describes the torque on a ring of fluid at radius r due only to the fluid at a greater radius. The critical point is that this has to be independant of r, despite the appearance of an r
2 term on the RHS of equation [4], which obviously tells us exactly what the derivative dv/dr must be. However, we'll need to proceed slowly and explain why that expression can't have r dependance. Unlike the situation with Couette flow where the top plate didn't have any fluid above it, we do have some fluid with a radius less than r. So that fluid at smaller radius gets some torque from the fluid at a greater radius than itself. Specifically, that fluid which is interior to our ring at radius r is being pulled by our ring, and in accordance with Newton, our ring at radius r is experiencing the equal but opposite force. So there are two torques acting on our ring at radius r.
Let's explain it this way: Consider a thin ring of the fluid around the z-axis with radius between r and r+δr. It gets a torque (anti-clockwise around the z-axis) from the fluid at a greater radius given by eqn [4]:
The fluid interior to our ring is also getting some anti-clockwise torque from the fluid at a greater radius than itself, however that interior fluid is not in contact with any fluid at a radius bigger than our ring (because our ring was a width δr wide). We can make arguments about viscous forces being well approximated as contact (or microscpically close contact) forces - which is what we'll do because this post is already too long. It's sufficient for an idealisation of some viscous shear forces. (LATE EDITING: surplus discussion removed).
By Newton's third law (with equal and opposite forces), our ring experiences a clockwise torque from the fluid interior to it.
Hence the net torque on our ring of width δr is given by
to a first order approximation (and becomes exact for a fluid where viscous shear forces are treated as pure contact forces and δr → dr ). If you're happy working with a ring of width δr >0 that is small but
not a differential, then skip straight to the paragraph labelled [Jump here]. If you're keen to take δr down to a differential dr then you will notice that the net torque just described was itself just a differential and it will vanish as δr → 0. However, so does the mass of that ring and hence it's inertia. We have I = moment of Inertia = ρ(r) .2πr.δr.r
2.ΔZ with ρ(r) = density of the fluid. (Drop the ΔZ and replace ρ(r) with mass per unit area, if you're working in plane polar co-ords and have no z-axis). Thus (Net Torque) = I α => angular acceleration, α, of the ring can be expressed as a ratio of two differentials.
α(r) =
. That denominator is finite for all finite values of r. So we establish that α(r) = 0 <=>
= 0. Hence, we can find a streamline at radius r (a ring of infinitesimal width if you like) where the flow velocity changes with time while the simplified discussion just continues to talk about rings of a small width and a small net torque on that ring.
[Jump here]
For the final steady flow solution there can't be any torque on the fluid in this ring. If there was, then the tangential velocity would be changing here. Hence, for the steady flow solution, the gradient
is 0 (just to be clear it's everywhere 0, there's no radius r where it could be non-zero or else we can find a ring with net torque on it).
Finally,
= 0 =>
= constant and from equation [4] we must then have
for some constant c => v = (k/r) + c for some constants c,k and v = k/r is the simplest solution.
I think that does show that
is the only velocity field that could describe a steady flow free vortex when the fluid has some coefficient of viscosity μ > 0. Thanks again for your time.