Governing Equations

Newton's Second Law

The primary theoretical framework for the experiment is based on Newton's second law of motion, applied to the fluid inside the straw. The forces acting on the fluid are:

• Gravitational force acting downward.

• Hydrostatic pressure from the surrounding fluid pushing the water up into the straw.

• Damping force, which accounts for energy losses due to factors like fluid viscosity and surface friction.

Where p=m(dz/dt) is the momentum of the fluid, mmm is the mass of the fluid inside the straw, and (dz/dt) is the velocity of the fluid. The forces in the equation are:

• Fgravity=−mg, where g is the acceleration due to gravity and mmm is the mass of the fluid column.

• Fpressure=ρghA, where ρ is the density of the fluid, h is the depth of the straw submerged below the surface of the bath, and A=πr² is the cross-sectional area of the straw.

• Fdamping=−b0(dz/dt), where b0​ is a phenomenological damping coefficient accounting for energy dissipation due to viscosity and surface friction.

By applying these forces to Newton's second law, the equation of motion becomes:

Since the mass of the liquid in the straw changes as the fluid level rises or falls, the mass can be expressed as:

Substituting this into the equation of motion and simplifying, the governing second-order differential equation becomes:

Where b=b0/(ρA)​​ is the modified damping coefficient. This equation can be solved numerically to model the oscillatory motion of the fluid.

In the experiment, the initial conditions are:

• The fluid level starts near the bottom of the straw, z0=0.2 cm.

• The initial velocity of the fluid is zero (dz0/dt)=0 cm/s.

The solution of the equation predicts the fluid's oscillatory motion, where it rises above the bath level. Then it oscillates around the equilibrium position due to the balance between gravitational force and hydrostatic pressure.

The fluid oscillations are damped, meaning that over time, the oscillations decrease in amplitude due to energy losses caused by fluid friction and damping forces. The damping coefficient b plays a significant role in this, and the damping term b(dz/dt) in the equation accounts for the gradual reduction in the amplitude of the oscillations over time.

For small oscillations about the equilibrium point, the behavior of the system can be approximated using Hooke’s law. In this limit, where the displacement y=z−h is small, the equation of motion simplifies to:

The corresponding natural frequency f0 of oscillation is:

For the experimental parameters used in the study, with g=9.8 m/s² and h=10.2 cm, the calculated natural frequency is approximately f0=1.56 Hz.

The equation can be solved numerically using techniques such as the odeint function from Python's SciPy library, with the experimental parameters and initial conditions as input. The results are compared to the experimental data obtained by tracking the fluid level over time using video analysis software. The comparison shows excellent agreement between the theoretical model and experimental observations.

Damping and Fourier Transform Analysis

To quantify the damping behavior, a discrete Fourier transform (DFT) of the oscillation data is performed. The DFT provides the natural oscillation frequency and the full width at half maximum (FWHM) of the frequency peak, which can be used to calculate the decay time of the oscillations. The decay time Tdecay​ is given by:

Where Δf is the FWHM of the frequency peak. In this experiment, the damping is observed to have a decay time of approximately 3 seconds.

Temperature can significantly affect the oscillations by influencing the properties of the fluid, particularly its viscosity and density. As temperature increases, fluids typically become less viscous and less dense, which affects how they oscillate. Here's how temperature can influence the behavior:

1. Viscosity and Damping

• Viscosity decreases with increasing temperature, leading to less resistance to fluid motion. This would reduce the damping effect, allowing the fluid to oscillate more freely and for a longer period.

• Lower temperatures would increase viscosity, enhancing damping and causing the oscillations to settle more quickly.

2. Density and Frequency

• As temperature increases, the density of most fluids decreases. A lower density would mean less inertia in the fluid, which could slightly increase the frequency of oscillations.

• At lower temperatures, the fluid is denser, which increases inertia and may reduce the oscillation frequency.

3. Surface Tension

• Surface tension also decreases with temperature. A lower surface tension could reduce the resistance to the fluid entering the straw, possibly affecting the initial dynamics of the oscillation.

Temperature-dependent Parameters

If we want to simulate the impact of temperature, we can incorporate approximate empirical relations for how viscosity and density change with temperature. These relations are often specific to the type of fluid, but in general:

• Viscosity (μ\muμ) typically decreases exponentially with temperature:

Where μ0​ is the viscosity at a reference temperature, and k is a constant depending on the fluid.

 Density (ρ) decreases linearly with temperature (to a first approximation):

Where α is the thermal expansion coefficient of the fluid, and ρ0​ is the density at a reference temperature.