# Redefine parameters to simulate the surface liquid level

g = 9.8  # Acceleration due to gravity (m/s^2)

h = 0.102  # Submerged depth of straw (m)

b = 0.05  # Damping coefficient

y0 = 0.002  # Initial position relative to the bath surface (m)

y_dot0 = 0.0  # Initial velocity (m/s)

# Define the differential equation for surface liquid level position

def surface_model(y, t, g, h, b):

    y, y_dot = y  # Unpack surface position and its time derivative (velocity)

    dydt = y_dot  # First derivative (velocity)

    dydotdt = - (g / h) * y - b * y_dot  # Second derivative (acceleration)

    return [dydt, dydotdt]

# Time points where solution is computed

t = np.linspace(0, 10, 1000)  # Time range (0 to 10 seconds)

# Initial conditions for surface liquid level

y0 = [y0, y_dot0]

# Solve the ODE for surface liquid level

sol_surface = odeint(surface_model, y0, t, args=(g, h, b))

# Extract surface liquid level position y(t) from the solution

y = sol_surface[:, 0]

# Plot the results for surface liquid level

plt.figure(figsize=(10, 6))

plt.plot(t, y, label='Surface Liquid Level Position (m)')

plt.xlabel('Time (s)')

plt.ylabel('Surface Liquid Level Position (m)')

plt.title('Oscillations of Surface Liquid Level (y vs Time)')

plt.grid(True)

plt.legend()

plt.show()

# Increase the damping coefficient for more damping

b_heavy_damping = 0.5  # Increased damping coefficient

# Solve the ODE for surface liquid level with heavier damping

sol_surface_damped = odeint(surface_model, y0, t, args=(g, h, b_heavy_damping))

# Extract surface liquid level position y(t) from the solution

y_damped = sol_surface_damped[:, 0]

# Plot the results for surface liquid level with increased damping

plt.figure(figsize=(10, 6))

plt.plot(t, y_damped, label='Surface Liquid Level Position with Heavy Damping (m)', color='red')

plt.xlabel('Time (s)')

plt.ylabel('Surface Liquid Level Position (m)')

plt.title('Oscillations of Surface Liquid Level (Heavy Damping, y vs Time)')

plt.grid(True)

plt.legend()

plt.show()

   We can simulate the oscillations using different fluids by adjusting the fluid properties like density and viscosity, which impact the damping and oscillatory behavior. For this simulation, we will focus on how the fluid density affects the system, assuming the fluid viscosity influences the damping coefficient.

    Let me modify the simulation to show how different fluids, such as water, oil, and glycerin, would behave. I'll adjust the damping and fluid density for each fluid type.

Properties of Fluids:

1. Water: Density = 1000 kg/m³, Moderate damping

2. Oil: Density = 900 kg/m³, Higher damping

3. Glycerin: Density = 1260 kg/m³, Significantly higher damping

# Define different fluid densities (kg/m^3) and damping factors for each fluid

fluids = {

    'Water': {'density': 1000, 'damping': 0.05},     # Water: typical damping

    'Oil': {'density': 900, 'damping': 0.15},        # Oil: more viscous, higher damping

    'Glycerin': {'density': 1260, 'damping': 0.4}    # Glycerin: very viscous, much higher damping

}

# Prepare to plot all three fluids

plt.figure(figsize=(10, 6))

# Time points where solution is computed

t = np.linspace(0, 10, 1000)

# Simulate for each fluid

for fluid, properties in fluids.items():

    damping = properties['damping']

    # Solve the ODE for each fluid's surface liquid level

    sol_fluid = odeint(surface_model, y0, t, args=(g, h, damping))

    # Extract the surface liquid level for each fluid

    y_fluid = sol_fluid[:, 0]

    # Plot the results for each fluid

    plt.plot(t, y_fluid, label=f'{fluid} (Damping: {damping})')

# Customize the plot

plt.xlabel('Time (s)')

plt.ylabel('Surface Liquid Level Position (m)')

plt.title('Oscillations of Surface Liquid Level for Different Fluids')

plt.grid(True)

plt.legend()

plt.show()

Let's simulate the oscillations for water while adjusting its viscosity and density as a function of temperature. For this, we will consider the following:

 Viscosity as a Function of Temperature:

 For water, viscosity decreases approximately exponentially with temperature:

Where:

• μ0​ = 0.001 Pa·s (viscosity at 20°C)

• k ≈ 0.02 1/°C (empirical constant for water)

Density as a Function of Temperature:

 Density decreases linearly with temperature:

Where:

• ρ0​ = 1000 kg/m³ (density at 0°C)

• α = 0.000214 1/°C (thermal expansion coefficient for water)

We'll simulate the oscillations for a temperature range, say from 10°C to 50°C, and observe how the oscillations change with temperature. The damping coefficient will depend on the viscosity, and the oscillation frequency will be influenced by the density.

# Constants for water

mu_0 = 0.001  # Viscosity at 20°C (Pa·s)

k = 0.02  # Temperature dependence of viscosity (1/°C)

rho_0 = 1000  # Density at 0°C (kg/m³)

alpha = 0.000214  # Thermal expansion coefficient (1/°C)

# Define a temperature range

temperatures = np.linspace(10, 50, 5)  # Temperatures from 10°C to 50°C

# Time points for simulation

t = np.linspace(0, 10, 1000)  # Time range (0 to 10 seconds)

# Prepare to plot oscillations for different temperatures

plt.figure(figsize=(10, 6))

# Loop over each temperature to simulate

for T in temperatures:

    # Calculate the viscosity and density at this temperature

    mu_T = mu_0 * np.exp(-k * T)

    rho_T = rho_0 * (1 - alpha * T)

    # Adjust damping coefficient as a function of viscosity (simplified relation)

    b_T = 0.05 * (mu_T / mu_0)  # Adjust damping based on viscosity

    # Solve the ODE for surface liquid level at this temperature

    sol_temp = odeint(surface_model, y0, t, args=(g, h, b_T))

    # Extract surface liquid level position y(t)

    y_temp = sol_temp[:, 0]

    # Plot the results for this temperature

    plt.plot(t, y_temp, label=f'Temperature {T}°C')

# Customize the plot

plt.xlabel('Time (s)')

plt.ylabel('Surface Liquid Level Position (m)')

plt.title('Oscillations of Surface Liquid Level for Water at Different Temperatures')

plt.grid(True)

plt.legend()

plt.show()