Problem 10 IYPT 2027: 10. Air vortex

Problem Description & Analysis

Place a small rod-shaped magnet on the bottom of a vessel filled with liquid and set it into rotation using a magnetic stirrer. Above a certain rotation speed, a thin air channel forms, extending from the liquid surface down to the rotating magnet itself. Investigate and explain this phenomenon.

Interactive Simulation (Python)

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فشار پرتاب P (Pa) 2000 Pa

شعاع حلقه R (m) 0.05 m

شعاع مقطع حلقه a (m) 0.01 m

ضریب پسا Cd 0.2

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Python Code Lab

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Python — Editable

# آزمایش گردابهٔ هوا — رابطهٔ برد با فشار پرتاب
import numpy as np
import matplotlib.pyplot as plt

rho = 1.2
R = 0.05; a = 0.01; Cd = 0.2
A = np.pi*R**2
m_ring = rho * (2*np.pi*R) * (np.pi*a**2) * 2

# --- بخش ۱: داده‌های شما ---
# فشار پرتاب اندازه‌گیری‌شده (Pa) و برد قابل مشاهدهٔ حلقه (m)
P_Pa   = np.array([500, 1000, 2000, 3500, 5000, 7500])
range_m= np.array([1.6, 2.4, 3.5, 4.7, 5.8, 7.4])

# --- بخش ۲: نظریه ---
def theoretical_range(P):
    v0 = np.sqrt(2*P/rho); dt = 0.001; v = v0; x = 0
    for i in range(6000):
        acc = -Cd*rho*A*v**2/(2*m_ring)
        v = max(v+acc*dt, 0); x += v*dt
        if v < 0.1*v0: return x
    return x

r_th = np.array([theoretical_range(P) for P in P_Pa])

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(1, 2, figsize=(11, 4.5))
Ps = np.linspace(200, 8000, 60)
rs = [theoretical_range(P) for P in Ps]
ax[0].plot(Ps, rs, 'b-', lw=2, label='theory')
ax[0].plot(P_Pa, range_m, 'ro', ms=9, label='measured')
ax[0].set_xlabel('P (Pa)'); ax[0].set_ylabel('range (m)')
ax[0].set_title('Vortex ring range vs launch pressure')
ax[0].legend(); ax[0].grid(True, alpha=0.3)

# fit: range ∝ sqrt(P)?
slope = np.polyfit(np.log(P_Pa), np.log(range_m), 1)[0]
ax[1].loglog(P_Pa, range_m, 'ro', ms=9)
ax[1].loglog(Ps, rs, 'b-')
ax[1].set_xlabel('P (Pa)'); ax[1].set_ylabel('range (m)')
ax[1].set_title(f'log-log slope = {slope:.2f}  (expected 0.5)')
ax[1].grid(True, which='both', alpha=0.3)

plt.tight_layout()

# --- بخش ۴: نتایج ---
print(" P(Pa)  r_th(m)  r_meas(m)")
for P, rt, rm in zip(P_Pa, r_th, range_m):
    print(f" {P:5.0f}  {rt:5.2f}    {rm:5.2f}")
print(f"nlog-log slope = {slope:.2f}  (ideal sqrt(P) => 0.5)")

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Problem Description & Analysis

Interactive Simulation (Python)

Python Code Lab

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