Newton’s law of viscosity – correct functional relationship Newton’s law of viscosity expresses the relationship between which two quantities?

Introduction:
Newton’s law of viscosity defines the constitutive behaviour of Newtonian fluids and links internal friction to deformation rate. This law underpins laminar flow solutions and many turbulence models’ near-wall treatments.

Given Data / Assumptions:

• Newtonian fluid assumption.
• Simple shear with velocity gradient perpendicular to flow.
• Continuum hypothesis and steady conditions.

Concept / Approach:
The law states τ = μ * (du/dy), where τ is shear stress, μ is dynamic viscosity, and du/dy is the rate of shear strain. The proportionality μ characterizes resistance to layer-by-layer motion.

Step-by-Step Solution:
1) Consider two adjacent fluid layers moving with different velocities.2) Define the velocity gradient du/dy between them.3) The required tangential force per unit area to sustain this gradient is τ.4) For a Newtonian fluid, τ is directly proportional to du/dy with constant μ.

Verification / Alternative check:
Poiseuille flow in a circular pipe yields a linear τ–du/dy relation consistent with Newton’s law, producing a parabolic velocity profile.

Why Other Options Are Wrong:

• Pressure/temperature terms do not define the shear law.
• Bulk velocity alone does not determine τ without the gradient.

Common Pitfalls:
Confusing dynamic viscosity μ with kinematic viscosity ν = μ / ρ.

Final Answer:
shear stress and rate of shear strain