The flow of water (mass density = 1000 kg/m3 and kinematic

The flow of water (mass density = 1000 kg/m³ and kinematic viscosity = 10-6 m² /s) in a commercial pipe, having equivalent roughness k_s as 0.12 mm, yields an average shear stress at the pipe boundary = 600 N/m². The value of k_s (being the thickness of laminar sub-layer) for this pipe is

Re =	VD
	v

τ = μ	dv
	dy

	du	=	u	(Linear ,commercial pipe)
	dy		y

∴ τ_o = μ	μ
	y

	τ_o	=	μ	×	u	= v	u
	e		ρ		y		y

Shear velocity = V₁

V₁² =	τ_o
	ρ

= v	u
	y

	u	=	V₁ × y	= Re
	V₁		v

	V¹y¹	= 11.6 at y = δ¹(experimental result)
	v

	V¹δ¹	= 11.6
	v

V¹ × δ¹ = 11.6 × 10^-6 m² / s

δ¹ =	11.6 × 10^-6	= 1.5 × 10^-5 m
	√0.6

= 1.5 × 10^-2 mm

	k	=	0.12	= 8
	δ¹		1.5 × 10^-2