Newtonian fluids – what is shear stress directly proportional to? In a Newtonian fluid under simple shear, identify the quantity to which the shear stress is directly proportional (assume constant temperature and composition).

Introduction / Context:
In fluid mechanics, correctly relating shear stress to motion is essential for modeling laminar flow between plates, in pipes, and within lubrication films. For Newtonian fluids (water, air, many oils in moderate ranges), a simple linear law connects stress to the rate at which adjacent fluid layers slide past each other.

Given Data / Assumptions:

• Continuum, incompressible Newtonian fluid.
• Simple shear flow with velocity u varying in the transverse direction y.
• Temperature and composition are constant so viscosity is constant.

Concept / Approach:

The Newtonian constitutive relation states that shear stress tau is proportional to the rate of shear strain. In one-dimensional simple shear: tau = mu * (du/dy), where mu is dynamic viscosity and du/dy is the velocity gradient normal to the flow direction. This captures how faster relative sliding of adjacent layers increases internal frictional resistance linearly for Newtonian behavior.

Step-by-Step Solution:

Verification / Alternative check:

Between parallel plates separated by gap h, moving-plate speed U, the profile is linear, du/dy = U/h, giving tau = mu * U/h. Doubling U doubles tau; halving h doubles tau, as observed experimentally for Newtonian fluids.

Why Other Options Are Wrong:

(a) Velocity at a point is not the cause; spatial variation of velocity is. (b) Shear strain is a geometric angle; rate of change (per time) matters in fluids. (d) Viscosity sets the proportionality but motion is required; tau is zero if du/dy = 0. (e) Square dependence appears in some turbulent drag laws, not in Newtonian constitutive law.

Common Pitfalls:

Confusing laminar constitutive law with empirical turbulent wall-shear correlations; mixing up strain (angle) with strain rate (time rate of angle change).

Final Answer:

the rate of shear strain (velocity gradient du/dy)