Fluid Properties & Viscosity

Why does honey pour slowly while water gushes? Why does a pump need more power on a cold morning? The answer is viscosity — the property that measures a fluid's internal resistance to flow. Understanding viscosity and related properties is essential for everything from lubricating an engine to designing a pipeline.

In this lesson, we'll quantify viscosity, model how it changes with temperature, and explore surface tension, vapor pressure, and compressibility.

Newton's Law of Viscosity

Imagine fluid sheared between two plates: the bottom plate is fixed and the top moves at velocity $U$. Because of the no-slip condition, the fluid velocity varies linearly from $0$ at the bottom to $U$ at the top. The shear stress required to maintain this motion is:

$$\tau = \mu \frac{du}{dy}$$

Where:

• $\tau$ = shear stress (Pa)
• $\mu$ = dynamic (absolute) viscosity (Pa·s)
• $du/dy$ = velocity gradient or shear rate (1/s)

This linear relationship defines a Newtonian fluid.

Dynamic vs Kinematic Viscosity

$$\nu = \frac{\mu}{\rho}$$

It has units of m²/s and represents the ratio of momentum diffusion to inertia.

Newtonian vs Non-Newtonian Fluids

For Newtonian fluids, viscosity is constant — the stress vs shear-rate line is straight through the origin. Many real fluids deviate:

A common model is the power law: $\tau = K\dot\gamma^{\,n}$, where $n<1$ is shear-thinning, $n=1$ is Newtonian, and $n>1$ is shear-thickening. A Bingham plastic follows $\tau = \tau_0 + \mu_p\dot\gamma$ once $\tau > \tau_0$.

Plotting Couette Flow Velocity Profile and Shear Stress

import numpy as np
import matplotlib.pyplot as plt

# Couette flow: fluid between a fixed bottom plate and a top plate
# moving at velocity U. Gap height h.
mu = 1.0e-3       # water dynamic viscosity (Pa.s)
U = 2.0           # top plate velocity (m/s)
h = 0.005         # gap (m)

y = np.linspace(0, h, 100)
u = U * y / h                 # linear velocity profile
tau = mu * (U / h)            # constant shear stress

print(f"Velocity gradient du/dy = {U/h:.1f} 1/s")
print(f"Shear stress tau = {tau:.4f} Pa")

fig, ax = plt.subplots(1, 2, figsize=(11, 5))
ax[0].plot(u, y * 1000, 'b-', linewidth=2)
ax[0].set_xlabel('Velocity u (m/s)')
ax[0].set_ylabel('Height y (mm)')
ax[0].set_title('Couette Flow Velocity Profile')
ax[0].grid(True, alpha=0.3)

ax[1].axvline(tau, color='r', linewidth=2)
ax[1].set_xlabel('Shear stress tau (Pa)')
ax[1].set_ylabel('Height y (mm)')
ax[1].set_title('Shear Stress (constant across gap)')
ax[1].set_ylim(0, h * 1000)
ax[1].grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Velocity gradient du/dy = 400.0 1/s
Shear stress tau = 0.4000 Pa

Temperature Dependence of Viscosity

Viscosity changes strongly with temperature — and in opposite directions for liquids and gases:

• Liquids: viscosity decreases as temperature rises (molecules move past each other more easily).
• Gases: viscosity increases as temperature rises (more molecular momentum exchange).

For gases, Sutherland's law gives an accurate fit:

$$\mu = \mu_{ref}\left(\frac{T}{T_{ref}}\right)^{3/2}\frac{T_{ref}+S}{T+S}$$

For air, $\mu_{ref} = 1.716\times10^{-5}$ Pa·s at $T_{ref}=273.15$ K and $S=110.4$ K.

Viscosity vs Temperature Plot

import numpy as np
import matplotlib.pyplot as plt

# Sutherland's law for air
mu_ref = 1.716e-5   # Pa.s
T_ref = 273.15      # K
S = 110.4           # K

def sutherland(T):
    return mu_ref * (T / T_ref)**1.5 * (T_ref + S) / (T + S)

T = np.linspace(250, 1000, 200)
mu_air = sutherland(T)

# Sample values
for Tc in [0, 20, 100, 500]:
    Tk = Tc + 273.15
    print(f"Air at {Tc:3d} C: mu = {sutherland(Tk)*1e5:.3f} x10^-5 Pa.s")

plt.figure(figsize=(8, 5))
plt.plot(T - 273.15, mu_air * 1e5, 'r-', linewidth=2)
plt.xlabel('Temperature (C)')
plt.ylabel('Dynamic viscosity (x10^-5 Pa.s)')
plt.title("Air Viscosity vs Temperature (Sutherland's Law)")
plt.grid(True, alpha=0.3)
plt.show()

Air at   0 C: mu = 1.716 x10^-5 Pa.s
Air at  20 C: mu = 1.813 x10^-5 Pa.s
Air at 100 C: mu = 2.174 x10^-5 Pa.s
Air at 500 C: mu = 3.574 x10^-5 Pa.s

For accurate liquid and gas properties at any state, the CoolProp library is ideal: PropsSI('V', 'T', 293, 'P', 101325, 'Water') returns water's viscosity directly.

Surface Tension and Capillary Rise

The capillary rise in a circular tube of radius $r$ is:

$$h = \frac{2\sigma\cos\theta}{\rho g r}$$

Where $\theta$ is the contact angle ($\approx 0°$ for water on clean glass).

Capillary Rise Calculation

import numpy as np

sigma = 0.0728     # surface tension of water (N/m)
theta = 0.0        # contact angle (rad) for water-glass
rho = 998.0        # kg/m^3
g = 9.81           # m/s^2

for d_mm in [0.5, 1.0, 2.0, 5.0]:
    r = (d_mm / 1000) / 2          # tube radius (m)
    h = 2 * sigma * np.cos(theta) / (rho * g * r)
    print(f"Tube diameter {d_mm:.1f} mm -> capillary rise = {h*1000:6.2f} mm")

Tube diameter 0.5 mm -> capillary rise =  59.55 mm
Tube diameter 1.0 mm -> capillary rise =  29.77 mm
Tube diameter 2.0 mm -> capillary rise =  14.89 mm
Tube diameter 5.0 mm -> capillary rise =   5.95 mm

The rise is inversely proportional to tube diameter — that's why capillary effects dominate in tiny tubes but vanish in large pipes.

Vapor Pressure and Cavitation

# Check for cavitation risk at a pump inlet
p_local = 2.0e3    # local absolute pressure (Pa)
p_vapor = 2.34e3   # water vapor pressure at 20 C (Pa)

if p_local <= p_vapor:
    print(f"CAVITATION RISK: p_local ({p_local/1e3:.2f} kPa) "
          f"<= p_vapor ({p_vapor/1e3:.2f} kPa)")
else:
    print("No cavitation: local pressure above vapor pressure")

CAVITATION RISK: p_local (2.00 kPa) <= p_vapor (2.34 kPa)

Bulk Modulus and Compressibility

The bulk modulus $K$ measures resistance to compression:

$$K = -V\frac{dp}{dV} = \rho\frac{dp}{d\rho}$$

For water, $K \approx 2.2$ GPa — enormous, which is why liquids are usually treated as incompressible. The speed of sound follows $a = \sqrt{K/\rho}$.

import numpy as np

K = 2.2e9      # bulk modulus of water (Pa)
rho = 998.0    # kg/m^3

a = np.sqrt(K / rho)
print(f"Speed of sound in water: {a:.1f} m/s")

# Fractional volume change under 10 MPa pressure rise
dp = 10e6
dV_over_V = -dp / K
print(f"Volume change at 10 MPa: {dV_over_V*100:.3f} %")

Speed of sound in water: 1484.9 m/s
Volume change at 10 MPa: -0.455 %

Even a 10 MPa pressure rise compresses water by less than half a percent — confirming the incompressible assumption for most liquid flows.

Common Pitfalls

• Confusing $\mu$ and $\nu$. Dynamic viscosity is Pa·s; kinematic is m²/s. The Reynolds number uses $\nu$ (or $\mu/\rho$), not $\mu$ alone.
• Using liquid temperature trends for gases. Heating a liquid lowers its viscosity; heating a gas raises it.
• Ignoring units in old data. Viscosity is often tabulated in poise (1 P = 0.1 Pa·s) or centipoise (1 cP = $10^{-3}$ Pa·s).
• Forgetting cavitation uses absolute pressure. Compare local absolute pressure to vapor pressure, never gauge.

Key Takeaways

• Newton's law of viscosity: $\tau = \mu\,du/dy$ defines Newtonian fluids.
• Kinematic viscosity $\nu = \mu/\rho$ controls momentum diffusion and appears in the Reynolds number.
• Non-Newtonian fluids (shear-thinning, shear-thickening, Bingham) have variable or yield-dependent viscosity.
• Sutherland's law models gas viscosity vs temperature; liquids trend the opposite way.
• Surface tension, vapor pressure/cavitation, and bulk modulus complete the property picture for design.

In the next lesson, we'll apply these properties to fluids at rest — hydrostatics and buoyancy.