Problem 14 IYPT 2027: 14. Non-Newtonian worms

Problem Description & Analysis

If a dilatant non-Newtonian fluid (e.g. a suspension of water and cornflour) is vibrated vertically in a reservoir. When certain conditions are met, wormlike structures that rise from the surface may form. Explain and investigate this phenomenon.

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شاخص چگالی K (Pa·s^n) 5 Pa·s^n

توان n 0.5

شعاع لوله R (mm) 5 mm

طول لوله L (m) 0.5 m

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Python — Editable

# آزمایش «کرم‌های غیرنیوتنی» — منحنی جریان و فیت مدل قدرت
import numpy as np
import matplotlib.pyplot as plt

# --- بخش ۱: داده‌های شما ---
# نرخ برش اعمالی توسط رئومتر (1/s)
gdot = np.array([0.1, 0.3, 1.0, 3.0, 10.0, 30.0, 100.0, 300.0])
# تنش برشی اندازه‌گیری‌شده (Pa)
tau  = np.array([2.1, 3.7, 7.0, 12.3, 22.4, 39.2, 71.5, 128.7])

# --- بخش ۲: فیت مدل توانی τ = K γ̇^n روی داده‌ها ---
coef = np.polyfit(np.log(gdot), np.log(tau), 1)
n_fit = coef[0]
K_fit = np.exp(coef[1])
tau_fit = K_fit * gdot**n_fit

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(1, 2, figsize=(11, 4.5))
ax[0].plot(gdot, tau, 'ro', ms=9, label='measured')
gd = np.logspace(-1, 2.6, 100)
ax[0].plot(gd, K_fit*gd**n_fit, 'b-', lw=2,
           label=f'fit: K={K_fit:.2f}, n={n_fit:.2f}')
ax[0].set_xlabel('shear rate (1/s)'); ax[0].set_ylabel('stress (Pa)')
ax[0].set_title('Flow curve — power-law fit')
ax[0].legend(); ax[0].grid(True, alpha=0.3)

ax[1].loglog(gdot, tau, 'ro', ms=9)
ax[1].loglog(gd, K_fit*gd**n_fit, 'b-', lw=2)
ax[1].set_xlabel('shear rate (1/s)'); ax[1].set_ylabel('stress (Pa)')
ax[1].set_title(f'log-log; slope=n={n_fit:.3f}')
ax[1].grid(True, which='both', alpha=0.3)

plt.tight_layout()

# --- بخش ۴: نتایج ---
print(f"Fitted consistency K = {K_fit:.3f} Pa·s^n")
print(f"Fitted power index n = {n_fit:.3f}")
if n_fit < 0.95: print("  → shear-thinning (typical of worm fluids)")
elif n_fit > 1.05: print("  → shear-thickening (dilatant)")
else: print("  → nearly Newtonian")
# apparent viscosity vs shear rate
eta_app = K_fit * gdot**(n_fit-1)
print("nγ̇(1/s)   η_app(Pa·s)")
for g, e in zip(gdot, eta_app):
    print(f" {g:6.2f}   {e:8.3f}")

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