Newtonian Shear Stress

τ = μ \frac{d u}{d y}

For a Newtonian fluid, shear stress is proportional to the velocity gradient. μ is dynamic viscosity (Pa·s). Water and air are Newtonian. Between parallel plates a linear profile gives du/dy = V/h. Ideal inviscid flow has μ = 0.

$τ$ (Pa): Shear stress
$μ$ (Pa·s): Dynamic viscosity
$d u / d y$ (1/s): Velocity gradient (shear rate)

Step-by-step solved example

μ = 0.80 Pa·s and du/dy = 50 s⁻¹. Find τ.

τ = μ du/dy between two plates / in a pipe profile.

1. Newtonian law
τ = μ du/dy = 0.80 × 50 = 40 Pa.
$τ = 40 Pa$
2. Meaning
A linear constitutive law: double the shear rate, double τ.

Answer: 40 Pa

10 practice questions

0/10 correct

1.For a Newtonian fluid, τ is proportional to

2.Water at 20°C is

3.SI units of μ are

4.du/dy is

5.μ = 0.002 Pa·s, du/dy = 400 s⁻¹. τ =

6.The no-slip condition means

7.Couette flow between parallel plates: du/dy =

8.Ideal inviscid flow has

9.Gap 2.0 mm, top plate 0.80 m/s, μ = 0.10 Pa·s. τ =

10.Dynamic μ and kinematic ν are related by