© by Ed Seykota, 1999
Consider the air moving between the plates as made up of thin sheets of paper sliding over one another. Then, dynamic viscosity [mu] is the interlamellar stickiness or friction between the sheets and also indicates how much a fluid particle is likely to be affected by aberrations in the movements of adjacent particles. Dynamic viscosity has the units of kg/m-s or nt-s/m2 or pa-s. The dynamic viscosity of air at 20 degrees C is about 1.8 e-5 kg/m-s and that of water at the same temperature is about 1 e-3 kg/m-s. These values are largely independent of density. Dynamic viscosity relates shear stress [t] to the local velocity gradient or shear rate [dU/dz], per:
T = mu * dU/dz
Consider one plate of area [S] moving over another with velocity [U] separated by a fluid of width [h] with dynamic viscosity [mu]. The force [F] necessary to sustain the motion is:
F = mu * S * U / h
From this, we can estimate the friction of fluid moving between two plates. We can visualize two half-width plate systems stacked vertically such that the two center plates are pulled through the outside plates. We can also visualize the two middle plates as functionally equivalent to a motive force equal to 2 * F pushing the fluid at the center twice as fast so as to achieve the desired average fluid velocity.
F = 2 * [mu * S * 2 * U / (h/2)]
F = 8 * mu * S * U / h
Now, the frictional area of the ring is just the circumference [2 * pi * r] times the length [dR], so
dF = 8 * mu * [2 * pi * r * dr] * U / h
dF = 16 * mu * pi * r * dr * U / h
As a ring expands, it gradually loses velocity while it gains area as the square of radius. Therefore, the overall resistive force would generally increase with radius. This result would help to explain the "negative resistance" effect of the levitator in action. Note: this equation applies only to laminar flow. At low Reynolds numbers where flow is laminar, the non-slip property insures there is no fluid motion at the surface of either disk, so the roughness of the disks is immaterial.
Kinematic viscosity [nu] is the ratio of viscosity to density [rho].
Nu = mu/rho
Density indicates the inertia of a particle or its tendency to continue in motion regardless of the motions of its neighbors. So kinematic viscosity is a stickiness-to-inertia ratio and indicates the ability of the fluid to eliminate non-uniform fluid velocity. Since water is about one thousand times denser than air, air is actually about fifteen times more kinematically viscous than water, a somewhat counterintuitive result.
The Reynolds number is the keystone of much of fluid mechanics. It indicates the relative importance of inertial drag to viscous drag. Inertial drag, per Newton, is F = rho * S * U2. The Reynolds number or ratio of inertial to viscous drag is:
Re = (rho * S * U2 )/( mu * S * U / z)
Re = rho * U * z / mu
Re = U * z / nu
When the Reynolds number is large, inertial phenomena predominate and air behaves like tiny hailstones. At low Reynolds numbers, air behaves more like sticky, viscous goo.
Engineers working on pipe flow problems represent the loss of energy per unit volume as a friction head [hf] in terms of the kinetic energy, where [hf] is in units of length and [g] is the acceleration of gravity:
hf = rho * f * L/D * U2/2g - in terms of height of water, rho = 1 implied
dP = f * L/D * rho * U2/2 - in terms of pressure
For laminar flow at low Reynolds numbers:
f = 64/Re
For example, at Re = 1000, f = .064.
The Moody diagram graphically represents the friction factor as a function of Reynolds number. At low Reynolds numbers, it shows the 64/Re relationship. For higher Reynolds numbers, the diagram also incorporates a factor for the relative roughness of the walls. Re = U z / nu. For a rectangular cross section, z = 4 * cross section / circumference. For the ring, the cross section is 2 * pi * r * h and the circumference is 4 * pi * r so z = 2 * h. Then,
Re = [2 * h] * U / nu
For low Re we have
f = 64 / (2 * h * U / nu)
f = 32 * nu / (U * h)
Substituting [f] and [z = 2h] into the equation for friction pressure [dP] , we have,
P = [32 * nu / (U * h)] * dr/[2 * h] * rho * U^2 / 2
P = 8 * mu * l * U / h^2
And, since F = P * A,
F = 8* mu * l * U * A / h^2
dF = 8 * mu * dR * U * (2 * pi * r * h) / h^2
dF = 16 * mu * pi * r * dR * U / h
And this confirms the result in the dynamic viscosity section. For low Reynolds numbers, friction is proportional to dynamic viscosity, area and velocity and inversely proportional to spacing.
For higher Reynolds numbers the formula for the friction factor is more complex. I was able to locate various approaches.
f = 1.325 /( ln( (e/3.7D) + (5.74/Re^0.9) ) ) Re>2100 Kaszeta by email
1/sqrt(f) = (-4 log(.27e/D + (7/Re)^.9) Re>4000 Churchill in Perry's
f = .25/[log(e/3.7D + 7/Re^.9)] rewriting Churchill
f = .25/[log(e/3.7D) + 5.47/Re^.9]^2 Swamee-Jain, pg 212 of Schaum's
f = 1.325/[ln(.27e/D) + 5.74/(Re^.9))]^2 , by Luiz at firstname.lastname@example.org
F = .079/Re^.25 for 4000<Re<100000 - Blasius
Estimates for absolute roughness, e for glass, range from .0015 mm in Perry's to < 0.0003 mm. in Granger.
Neither Kaszeta nor Churchill seemed to give results comparable with the Moody diagram. This may have been due to typographical errors in the formulas or to ambiguity in the units of measure. The Blasius equation defines only the lower boundary of the friction factor for turbulent flow through smooth pipes. All in all, a serviceable fit obtained by using Swamee-Jain.
The relative importance of friction in the Swamee-Jain equation develops for e/D at around .003. Based on a guess that acrylic plastic has a roughness of about 0.0015 mm to 0.0003 mm, the spacing, [D] would become important at about 0.5 to 0.1 mm. This is roughly in agreement with experimentally determined gap sizes.
In summary, I use the following in the model:
Re < 2500: f = 64/Re - laminar flow
2500 < Re < 4000: f = interpolation from laminar to turbulent
4000 < Re: f = .25/[log(e/3.7D) + 5.47/Re^.9]^2 - Swamee-Jain for turbulent flow.