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Step 1: Calculate the Reynolds number for flow in a pipe. The formula for the Reynolds number ($Re$) for pipe flow is: $Re = (ρ \cdot v \cdot D) / μ$ where: $ρ$ = fluid density $v$ = average flow velocity $D$ = pipe diameter $μ$ = dynamic viscosity

Step 2: Substitute the given values. Given: $ρ = 1000$ kg/m³ $v = 2$ m/s $D = 0.05$ m $μ = 0.001$ Pa·s

$Re = (1000 kg/m³ \cdot 2 m/s \cdot 0.05 m) / 0.001 \text{ Pa·s}$

Step 3: Calculate the Reynolds number. $Re = (100) / 0.001$ $Re = 100000$

The Reynolds number is 100,000.

Step 4: Determine the flow regime. Since $Re = 100,000$, which is much greater than 4000, the flow is turbulent.

Step 5: Calculate the friction factor ($f$) using the Blasius correlation for turbulent flow in smooth pipes. The Blasius correlation is valid for $Re$ up to $100,000$: $f = 0.316 / Re^(1/4)$

Step 6: Substitute the Reynolds number into the Blasius correlation. $f = 0.316 / (100000)^(1/4)$

Step 7: Calculate the friction factor. $f = 0.316 / (10 \cdot 10000)^(1/4)$ $f = 0.316 / (10^5)^(1/4)$ $f = 0.316 / (10^(5/4))$ $f = 0.316 / (17.7828)$ $f ≈ 0.01777$

The friction factor is approximately 0.01777.

Step 8: Calculate the head loss ($h_f$) using the Darcy-Weisbach equation. The Darcy-Weisbach equation is: $h_f = f \cdot (L/D) \cdot (v²/2g)$ where: $L$ = pipe length $g$ = acceleration due to gravity (approximately 9.81 m/s²)

Step 9: Substitute the known values. Given: $f ≈ 0.01777$ $L = 100$ m $D = 0.05$ m $v = 2$ m/s $g = 9.81$ m/s²

$h_f = 0.01777 \cdot (100 m / 0.05 m) \cdot ((2 m/s)² / (2 \cdot 9.81 m/s²))$

Step 10: Calculate the head loss. $h_f = 0.01777 \cdot (2000) \cdot (4 m²/s² / 19.62 m/s²)$ $h_f = 35.54 \cdot (0.20387)$ $h_f ≈ 7.24$ m

The head loss due to friction is approximately 7.24 meters.