An excursion into the physical applications of fundamental differential equations by Joshua Cuneo, Justin Melvin, Jennifer Lee, and Narendhra Seshadri.

An excursion into the physical applications of fundamental differential equations by Joshua Cuneo, Justin Melvin, Jennifer Lee, and Narendhra Seshadri.

OverviewLiquid placed into a cylindrical container and rotated about the container’s central axis will be subject to centripetal forces that push the liquid towards the center.

The result is that the liquid is molded into a rotationally symmetric shape dependent on the rotational velocity of the system, the density and volume of the liquid, gravitational forces, and shape of the container.

The objective of this experiment is to use these variables with differential equations and calculus of variations to derive a formula that models the shape of a particular liquid, in this case water.

u(x0)u(x0)

Experimental SetupBuilding the Apparatus

A secure and well-centered cylinder, to allow the cylinder and water to be spun at a fairly high rate of rotation without losing stability.

A wooden disk of radius 7 inches and approximately an inch thickness was affixed to a metal turntable using screws.

The cylinder that was used to hold the water—a plastic jar with radius 1.91 inches and 5.75 inches tall—was mounted to the center of the wooden disk.

The cylinder was filled with 3.375 inches of water, which was dyed blue with a few drops of food coloring to increase the contrast between the water and its surroundings, thereby easing observation.

A method of driving the rotation of the cylinder at a constant, high rate.

The cylinder was rotated using a Dremel tool and a large motor hooked up to a DC power supply in the ACE Lab.

To drive the rotation, the motor tip was placed flush against the surface of the wooden disk so that the rotation of the tip drove the rotation of the disk and cylinder.

A method of measuring the angular velocity of the cylinder’s rotation.

No attempt was made to control the speed of either the Dremel tool tip or the motor. Instead, a moderate power level was used, and the angular velocity was measured separately using the motion sensor provided with the edition of Science Workshop in the ACE Lab.

The rubber wheel on the sensor was held firmly against the surface of the wooden disk so that its rotation would be driven by the rotation of the disk.

A method of measuring the curve that the water surface forms upon rotation.

A ring support was clamped to two stands so that it was suspended 1.5 inches above the lip of the cylinder, and a block of foam was fitted firmly inside the ring.

11 pickup sticks were inserted into the foam at approximately equal distances apart so that they formed a line that spanned the diameter of the lip of the cylinder.

To determine the level of the water at the location of a stick, the stick was pushed down until it caused visible interference in the water. The stick was then pulled back up until the interference disappeared. This process was repeated for each stick

Running the Experiment

The apparatus was initialized before each trial by bringing all of the measuring sticks above the equilibrium level of the water.

One group member would begin driving the rotation of the cylinder. Because access to equipment was limited, both the motion sensor and the driving motor were handheld.

Trials were only considered valid if the measured angular velocity stayed relatively constant (plus or minus 50 degrees/sec) during the entire run.

When the water surface curve had stabilized, another group member would push each stick down until the point caused visible interference in the water. The stick was then pulled back up until the interference disappeared.

The ends of the 11 stick represented points on the water’s surface The foam supporting the sticks was removed so that the length of the sticks could be measured.

Calculations

Our objective is to derive a function u(x) that models the shape of the water by giving the height of the surface of the water above a given point on the x-axis

x

u(x 0)

u

x

u(x 0)

We can derive such a function by calculating the total energy from the system and minimizing it using the Euler-Lagrange equation.

x0

1. Calculate the total kinetic and potential energies of the system2. Convert each to cylindrical coordinates to turn it into a more solvable form3. Create the energy equation to be minimized by adding these energies together4. Factor in the constraint (the volume) using a Lagrange multiplier5. Substitute variables and plug into the Euler-Lagrange Equation6. Solve for the constraint7. Solve for u(x)

Overview:

Kinetic Energy Equation

•K is the rotational kinetic energy of the system

•I is the moment of inertial of the water

•ω is the rotational velocity the fluid (constant)

1. Calculate the total kinetic and potential energies of the system

I = ρx2

Potential Energy Equation

•U is the gravitational potential energy•ρ is the density of the water (constant)•g is the force of gravity on the water (constant)

•h is the height of each infinitesimally small cube of water dV above the bottom of the jar

2. Convert each to cylindrical coordinates to turn it into a more solvable form

3. Create the energy equation to be minimized by adding these energies together

4. Factor in the constraint (the volume) using a Lagrange multiplier

5. Substitute variables and plug into the Euler-Lagrange Equation

6. Solve for the constraint

πR2h0 =

7. Solve for u(x)

Second Question

Previously, we neglected to include surface tension into our calculations because we assumed that it was a negligible source of energy.

x

u(x 0)

u

x

u(x 0)

Now we will study the effect of surface tension on our calculations.

1. Add the equation for surface tension into the total energy equation2. Apply the revised equation to the Euler-Lagrange Equation3. Approximate u(x) with a power series and substitute it into the result from step 24. Solve for as many coefficients of the power series as possible5. Create a final approximation of u(x)

Overview:

1. Add the equation for surface tension into the total energy equation

Surface Tension Equation

2. Apply the revised equation to the Euler-Lagrange Equation

3. Approximate u(x) with a power series and substitute it into the result from step 2

4. Solve for as many coefficients of the power series as possible

Note: a1 = 0

x

u(x 0)

u

x

u’(0)

5. Create a final approximation of u(x)

Note: a0 depends on experimental values

x

u(x 0)

u

x

u(0)

Analysis & Conclusion

Initial observations would suggest a fair approximation of the surface of the water by both the set of calculations with and without surface tension.

However…

Observation 1:

Percent error of trial one

vs distance from axis

-20

-10

0

10

20

30

40

50

-3 -2 -1 0 1 2

Distance from axis of rotation

Per

cen

t er

ror

No surface tension

With surface tension

Percent error of trial eight vs. distance from axis

-80

-60

-40

-20

0

20

40

60

80

-3 -2 -1 0 1 2

Distance from axis

Per

cen

t er

ror

The percent error is less around the axis of rotation than the sides.

Observation 2:

Trial 6: w = 9.21 rad/s

0

1

2

3

4

5

6

7

8

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Calculated(no ST)

Actual

Calculated(ST)

Trial 7: w = 10.15 rad/s

0

1

2

3

4

5

6

7

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

There are obvious discrepancies between the theoretical values calculated using the derived formulae with and without surface tension.

In conclusion, although the calculations derived to model the shape of the spinning water provide a rough approximation of the actual shape, there exist both mathematical and experimental discrepancies that upset the precision of these calculations.

Finite Element Analysis

Fin