Viscosity - Meyer’s Oscillating Disc Method

Arun Naren GR

School of Physics IISER Thiruvananthapuram

Abstract Rotational viscometers use the fact that the torque required to turn an object in
a liquid depends on the viscosity of that fluid. They measure the torque required to rotate a disc
in a liquid at a calculated speed. Viscosity of a liquid is measured by Meyer’s Disc method which
measures the damping force produced by a liquid on a rotating disc and using the corresponding
second order differential equation to solve for it.
Keywords Viscosity : logarithmic decrement

1 Theory and this in turn causes a viscous force to act

and damp out the oscillations. Oscar Meyer
The viscosity of a liquid is a measure
suggested measuring the decay of these oscilla-
of its resistance to deformation at a given tions to find the viscosity of a liquid.
rate.Viscosity quantifies the internal frictional The equation to a harmonic oscillator un-
force between liquid layers that are in rela- dergoing torsional oscillations is
tive motion.In general, viscosity depends on a d2 θ dθ
fluid’s state, such as its temperature, pressure, I 2 + K + τθ = 0
dt dt
and rate of deformation.Zero viscosity (no re- Here I is the moment of inertia of the os-
sistance to shear stress) is observed only at very cillator, K is the damping coefficient, τ is the
low temperatures in superfluids; otherwise, the restoring torque per unit twist and θ is the
second law of thermodynamics requires all flu- oscillations angle. Solution of this differential
ids to have positive viscosity. equation is,
If a disc undergoes torsional oscillations −2λt 2πt
θ(t) = θ0 e T sin( + ϕ)
about its symmetry axis in a fluid medium, it T
does not push aside any additional fluid while KT 1 1
λ= , T = 2πI( 2 ) 2
executing this motion. The fluid in contact 4 τ − k4
with the disk then remains at rest with respect where, θ0 and ϕ are integration constants. θ(t)
to it, while the fluid far away is at rest with is shown in Fig 2.
respect to the enclosure/container. so a trans- The quantity λ, known as the logarithmic
verse velocity gradient is set up in the fluid, decrement, is the logarithm of the ratio of any
2 Meyer’s Oscillating Disc Method

Fig.1

Fig.2

two successive amplitudes on opposite sides of by measuring the amplitudes on either side of
the equilibrium position. Thus, the equilibrium position, we can find out the
B1C 1 B2C 2 B1C 1 + B2C 2 damping coefficient using Eq(1)
eλ = = =
B2C 2 B3C 3 B2C 2 + B3C 3 In the case of a disk oscillating inside a liq-
which by mathematical induction gives, uid, the damping is due to two causes: damp-
B 1 C 1 + B 2 C 2 + ... + B n C n ing due to the viscous forces of the liquid, and
eλ =
B 2 C 2 + B 3 C 3 + ... + B n+1 C n+1 damping due to the friction of the wire suspen-
Here Bj is the amplitude at the ith turn- sion at the support. Meyer suggested that the
ing point of the disk, as shown in Fig.1. Thus instrument be first used to find the logarithmic
Meyer’s Oscillating Disc Method 3

decrement λ0 in air, where the viscous damping here, m is the mass, a is the average radius
is negligible, followed by a measurement of the of the ring, i.e., a = (d1 +d2 )/4 where d1 and d2
logarithmic decrement λ in the liquid. As the are the inner and outer diameters of the ring,
frictional damping at the support is the same respectively. Using equations (4) and.(6) we
in both cases., this (unknown quantity)can be can find the viscosity of liquid
eliminated by taking the difference λ - λ0 . Us-
ing this, he was able to find a formula for the 2 Procedure
viscosity of the liquid as, 2.1 Finding the measurable values
16I 2 λ − λ0 λ − λ0 2 2
η= 4 3 2
( +( )) • The apparatus consists of a flat disk at-
πρT (r + 2r d) π π
tached to a short rod passing through its cen-
Here,
tre which is suspended (with the disk remain-
I - moment of inertia of the torsional pen-
ing horizontal) by employing a phosphor bronze
dulum about the suspension axis.
wire. The central rod has a perpendicular
T - time period for one complete oscillation.
screw with two movable masses on opposite
r - radius of the disk.
sides for leveling the disk. A small concave
d - thickness of the disk.
mirror with a radius of curvature of about one
ρ = density of liquid
meter is also mounted on this rod
λ = logarithmic decrement in liquid
A lamp and scale arrangement is adjusted
λ0 = logarithmic decrement in air
till a beam from the lamp after reflection from
The quantities mentioned above can all be
the concave mirror forms a well defined circular
measured directly, except the moment of iner-
patch of light on the scale. The image of the
tia of the disk which is a complex object. To
cross wires on the lamp should be visible on the
find the moment of inertia, the time period (T)
screen. The positions of the scale and the disk
of the disk in air is found and then a ring with
are adjusted till the equilibrium position of the
a known moment of inertia Ir is placed on the
spot of light is close to the center of the scale.
disk with its center on the suspension axis. The
• Taking care to avoid all transverse oscil-
time period of the disk and the ring together
lations (such as lateral swing or wobble), the
in air(T’ ) is again found, when the moment of
disk is rotated slightly to give a small torque
inertia of the ring-loaded disk is I + It . Then,
and left free to undergo torsional oscillations.
we have
By measuring the time of 25 oscillations, the
I r = ma2
period of the pendulum T is found. Repeat
T2
I = ma2 2 this step once more and take the mean value of
T’ − T2
r T.
I
T = • The given metallic ring is placed flat on
τ
I + I’ 1 the disk, so that it‘‘s center is as close as pos-
’
T =( )2 sible to the axis of suspension. The time pe-
τ
4 Meyer’s Oscillating Disc Method

riod of the pendulum T” is now found by the that it again lies at the center of the scale. The
procedure described above. The mass of the same procedure (as that to find the logarith-
ring, and the outer and inner diameters (d1 and mic decrement in air) is now repeated to find
d2 ) of the ring are measured. Make obser- va- the logarithmic decrement λ in water. Since the
tion tables for these measurements. (The ring oscillations in this case are very much damped,
may not be exactly circular: therefore measure the experiment has to be performed for smaller
the diameter along different directions and take number of oscillations.
the average value). Using these two mea- sure-
3 Observations
ments and Eq.(6) the moment of inertia I of
the pendulum can be calculated. The ring can Calculations are all done in python
now be removed and is not required in the rest Radius of the disk, r =6.005 cm
of the experiment. Thickness of the disk, d =1.03 cm
Outer radius of the ring, a =6.105 cm
2.2 Measuring λ0
Inner radius of the ring, b =5.01 cm
• To measure the logarithmic decrement, Mass of the ring, m = 955.45 g
the disk is again set into torsional oscillation. Temperature of water = 298K
When the amplitude has fallen to approxi- Time required for 10 oscillations in air=
mately the full scale reading, start the read- 96.5 s
ings by noting down the reading on the scale Time period in air, T =9.65 s
at one extreme position, B1 C1 . The very next Time required for 10 oscillations in air with
reading at the outer turning point B2 C2 is then ring = 180.75 s
recorded Time period in air with ring, T’ =18.075 s
• After 20 complete oscillations, again Viscosity of water ,
record the maximum amplitudes on both sides η = 8.531702980325829 ∗ 10− 4P as
B41 C41 and B42 C42 . The logarithmic decrement
in air can now be found by using these readings 3.1 Error Calculation
and Eq.(3) for 20 oscillations (i.e., for n=20) as,
Calc.V alue − T heoreticalV alue
Error% = ∗100
1 B1C 1 + B2C 2 T heoreticalV alue
λ0 = ln( )
2n B 2n+1 C 2n+1 + B 2n+2 C 2n+2 Theoretical Value of η at 298K =8.318*10-4 Pas
Error Percentage = 6.4334204132463695
2.3 Finding λ

A clean glass dish is now placed so as to

4 Acknowledgements
contain the disk, and water is poured into it
so as to cover the disk but not sub- merge the I am greatly thankful to IISER and
mirror (see Fig.1). The equilibrium position of Dr.Rajeev Kini for providing me an opportu-
the light spot is now adjusted (if necessary) so nity to perform this experiment.
Meyer’s Oscillating Disc Method 5

Table 1. Readings of maximum amplitude in air (λ0 )

No. of Oscillations Maximum Amplitude to the right Maximum Amplitude to the left
1 10 10
21 9 9
41 8.2 8.2
61 7.8 7.8
81 7.3 7.3
101 6.8 6.8

Table 2. Readings of maximum amplitudes in water (λ)

No. of Oscillations Maximum Amplitude to the right Maximum Amplitude to the left
1 19 19
3 15 15
5 12 12
7 8 8
9 6 6
11 4.5 4.5

References
[1] Viscosity , wikipedia.