Viscosity and Turbulence

Up until now, fluid dynamics models assumed an ideal fluid with zero friction. Real fluids exhibit internal friction that resists flow, known as viscosity.

Viscosity Definition and Shear Force

Viscosity ( $\eta$ ) measures a fluid's resistance to flow.

Consider a fluid sandwiched between two flat plates of area $A$ separated by a distance $L$ . The bottom plate is fixed, and a horizontal force $F$ moves the top plate at constant speed $v$ :

F = \frac{\eta v A}{L} \implies \eta = \frac{F L}{v A}

(1)

The SI unit for viscosity is $\text{Pa} \cdot \text{s}$ or $\text{N} \cdot \text{s/m}^2$ .

Liquid viscosity decreases as temperature increases because molecules gain thermal energy to overcome cohesive forces. Gas viscosity increases with temperature because faster-moving gas molecules collide more frequently across fluid layers.

Poiseuille’s Law and Flow Resistance

In a viscous fluid flowing through a tube, speed is highest at the center and zero at the walls due to drag. The pressure difference $p_2 - p_1$ drives flow against resistance $R$ :

Q = \frac{p_2 - p_1}{R}

(2)

For laminar flow of an incompressible fluid through a horizontal tube of radius $r$ and length $l$ , the fluid resistance is:

R = \frac{8 \eta l}{\pi r^4}

(3)

Substituting $R$ yields Poiseuille’s Law:

Q = \frac{(p_2 - p_1) \pi r^4}{8 \eta l}

(4)

Flow rate depends on the fourth power of the radius ( $r^4$ ). Cutting the radius of a tube in half ( $r \to r/2$ ) increases resistance by a factor of $2^4 = 16$ , reducing flow rate to $6.25\%$ of its original value for a constant pressure difference.

Rearranging Equation 2 shows that flow and resistance cause pressure drops throughout a system:

p_2 - p_1 = R Q

(5)

Reynolds Number and Turbulence

Turbulence occurs when fluid layers mix via eddies and swirls, caused by high fluid speeds or sharp obstructions. Turbulence greatly increases flow resistance.

The Reynolds number ( $N_R$ ) is a dimensionless parameter used to predict whether flow in a tube of radius $r$ is laminar or turbulent:

N_R = \frac{2 \rho v r}{\eta}

(6)

where $\rho$ is fluid density, $v$ is average fluid speed, and $\eta$ is viscosity.

$N_R < 2000$ : Flow is laminar.
$2000 \le N_R \le 3000$ : Flow is unstable (chaotic) and oscillates between laminar and turbulent states.
$N_R > 3000$ : Flow is turbulent.

Example 1

An air conditioning duct has a diameter of $18.0 \text{ cm}$ ( $r = 0.090 \text{ m}$ ) and a length of $20.0 \text{ m}$ . It delivers air ( $\rho = 1.23 \text{ kg/m}^3$ , $\eta = 1.81 \cdot 10^{-5} \text{ Pa} \cdot \text{s}$ ) at a volume flow rate of $Q = 3.84 \cdot 10^{-3} \text{ m}^3/\text{s}$ . Determine if the air flow is laminar or turbulent.

Hint Click to unblur

First calculate fluid speed $v = \frac{Q}{\pi r^2}$ , then evaluate the Reynolds number using Equation 6.

Solution Click to unblur

Calculate the fluid velocity inside the duct:

v = \frac{Q}{\pi r^2} = \frac{3.84 \cdot 10^{-3} \text{ m}^3/\text{s}}{\pi (0.090 \text{ m})^2} \approx 0.151 \text{ m/s}

Calculate the Reynolds number:

N_R = \frac{2 \rho v r}{\eta} = \frac{2 (1.23 \text{ kg/m}^3)(0.151 \text{ m/s})(0.090 \text{ m})}{1.81 \cdot 10^{-5} \text{ Pa} \cdot \text{s}} \approx 1840

Since $N_R = 1840 < 2000$ , the flow is laminar.