Viscosity - Fluid Mechanics - Laboratory Report | EGR 365, Lab Reports of Fluid Mechanics

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Padnos School of Engineering

Grand Valley State University

EGR – 365

Fluid Mechanics

Dr. Blekhman

Laboratory Report

Viscosity

Matt Reimink

Grading Rubric

Theory

T r W r W

s s

s

3

side

h

μR ω2πLL

T 

b

4

bottom

2h

μR ωπL

T 

 

 



 t

2Ih h

μRπL4Lh Rh

b s

3

s s b sb

b s

3

1 e

μRπL4Lh Rh

2rWh h

ω t

2 πLπLL ω

r Wh

μ

3

s s



Present Major Results

Since the values of the torques are small to begin with, it was reasonable to neglect the

value of the bottom torque.

The distance needed to reach 90% of the terminal was determined to be 0.0195m or

19.5mm.

Mass

(kg)

Distance

(m)

Time

(s)

Velocity

(m/s)

Angular Velocity

(rad/s)

Viscosity (Ns/*

m

2

Average Viscosity 0.

Average Corrected

Viscosity 0.

Error Discussion

Measurement Value Tolerance

Weights, W 0.15 ± 0.0kg

Timing, t 14.68 ± 0.19s

Liquid Length, L 0.8863 ± 0.0005m

Thickness on Side hs 0.00437 ± 0.0005m

Spool Radius, rs 0.03 ± 0.0005m

Cylinder Radius, R 0.05771 ± 0.0005m

Falling Mass Length, d 0.9144 ± 0.02m

Mass of Cylinder 1.125 ± 0.00kg

Uncertainty

Viscosity 0.

Conclusions

1.) Glycerin is a newtonian fluid.

2.) The viscosity of glycerin is 0.8754 Ns/m

2

3.) The distance required before timing is 19.5mm.

4.) There was 0.2233% uncertainty in our viscosity value.

and the torque on the bottom is:

b

4

bottom

2h

μR ωπL

T 

Derivations of both equations can be seen in Appendix A. Dimensions for the test apparatus are

contained in Table 1.

Table 1 : Apparatus Dimensions

Lengths (m)

Liquid Height, L 0.

Large Cylinder ID 0.

Small Cylinder OD 0.

Small Cylinder Radius, R 0.

Side Fluid, hs 0.

Bottom Fluid, hb 0.

Spool Radius, rs 0.

Mass (kg)

Cylinder Assembly 1.

Theoretically, neglecting air resistance, the torques cancel each other out, and the following force

diagram can be setup.

Figure 2 : FBD of Cylinder

Figure 2 showed that there needs to be a bottom torque and a side torque acting against the

torque created by the falling weight. Laying out the force balance equations yields:

ΣAppliedApplied Torques rW T T 0

s side bottom

Inserting Equations (1) and (2) into Equation (3) allowed us to solve for viscosity. Equation (4)

is the formula for viscosity.

2 πLπLL ω

r Wh

μ

3

s s



If a ratio of the two torques is set up, it can be easily noticed that the bottom torque can be

neglected. The derivation of this assumption is shown in Appendix A. Setting up our definition

of applied torques yields:

dt

dω

ΣAppliedApplied Torques rW T T I

s side bottom

Inserting Equations (1) and (2) into Equation (5) yields the first order differential equation:

 

 



 t

2Ih h

μRπL4Lh Rh

b s

3

s s b sb

b s

3

1 e

μR πL4Lh Rh

2rWh h

ω t

Equation (6) yields the angular velocity of the cylinder in terms of time. However, we will be

measuring the velocity of the falling weight. Equation (7) links angular and linear velocity

together.

v  t r ω

The acceleration of the falling weight was not to be included; therefore, the weight had to fall far

enough to reach its terminal velocity. Theoretically, it need only reach roughly 90% of its

terminal velocity in order for acceleration not to be an issue. The time and length it takes to

reach the set amount are shown below:

3

s

terminal

2 LR

ln(0.1)h I

TIME 0.90ω

 

 

2

3

2

s

2

s

90%

2 πLμL R

ln(0.1)r Wh I

LENGTH L

Our assumption is to let the weight fall 20 inches in order to eliminate the acceleration of the

mass.

Discussion of Results

apparatus. With such a small weight, the data could easily have been skewered by the friction in

the pulley or the friction in the bearings of the cylinder or the estimating in our moment of inertia

equation. The rest of the data indicates that glycerin has a viscosity of roughly 0.8754 Ns/m

2

and

is a newtonian fluid. This was determined because the data in Table 2 shows that viscosity does

not vary with velocity, therefore making glycerin a newtonian fluid.

Looking up published values of the viscosity of glycerin, it appears that our value is relatively

close. The viscosity chart in the back of our Fluid Mechanics book lists glycerin at room

temperature at roughly 0.83 Ns/m

2

Error Discussion

In every engineering application, error persists and this laboratory exercise is no different. For

every measurement taken, a certain level of error was present and all of that error needs to be

accounted for. The error that will be present in our measurements of viscosity is error

propagation. The way to deal with error propagation is to determine the uncertainty in every

variable used when solving for another variable. If f was a function of x

1

and x

2

, f (x1, x2) the

uncertainty of f would be:

2

x

2

2

x

2

2

u

x

f

u

x

f

f

% uncertaintyin f

Using Equation (11) it will be possible to determine the propagated error in all our

measurements. Table 3 tabulates all of the measurements taken along with a reasonable

assessment of the corresponding error and then all of the propagated errors. Derivations for the

error equation can be seen in Appendix A.

Table 3 : Estimate Error Values

Measurement Value Tolerance

Weights, W 0.15 ± 0.0kg

Timing, t 14.68 ± 0.19s

Liquid Length, L 0.8863 ± 0.0005m

Thickness on Side hs 0.00437 ± 0.0005m

Spool Radius, rs 0.03 ± 0.0005m

Cylinder Radius, R 0.05771 ± 0.0005m

Falling Mass Length, d 0.9144 ± 0.02m

Uncertainty

Viscosity 0.2233%

There was a 0.2233% of uncertainty in our viscosity value. This is reasonable because our errors

for measured values were relatively small compared to the actual measurement, and our viscosity

is close to the publish value.

Conclusions

1.) Glycerin is a newtonian fluid.

2.) The viscosity of glycerin is 0.8754 Ns/m

2

3.) The distance required before timing is 19.5mm.

4.) There was 0.2233% uncertainty in our viscosity value.