09 / CHAPTER 10
Viscosity and Newtonian fluids
Introduce rate-of-deformation and the linear viscous law.
ANCHOR EQUATION
Newtonian viscosity
In symbols: τ = 2μD
Viscous stress is proportional to rate of deformation in a Newtonian fluid.
Newtonian viscosity links stress to deformation rate
A fluid at rest cannot sustain a static shear stress in the ideal model, but neighboring layers moving at different speeds exert tangential forces. Viscosity measures the stress required for a velocity gradient.
Derive shear from layer sliding
For parallel layers, shear strain rate is du/dy. Newtonian constitutive behavior is τ_xy = μ du/dy, where μ is dynamic viscosity in Pa·s. In steady planar flow, a constant pressure gradient balances the derivative of viscous stress, producing a velocity profile after integrating twice.
Water with μ = 1.0×10⁻³ Pa·s and velocity gradient du/dy = 500 s⁻¹ has shear stress τ = 0.50 Pa. Oil ten times more viscous under the same gradient needs 5 Pa.
Viscosity depends on temperature and material
Non-Newtonian fluids have μ that depends on shear rate or history. Turbulent momentum transport also creates an effective stress not captured by molecular μ alone.
Work it: A fluid layer 2 mm thick has velocity difference 0.6 m/s and μ = 0.05 Pa·s. Compute shear rate and wall stress. State the direction of the traction on the moving layer.