Rheology – shear stress versus shear strain rate for a Newtonian fluid For a Newtonian fluid, what is the shape of the graph when shear stress is plotted against shear strain rate (rate of shear deformation)?

Introduction / Context:
The constitutive relation for Newtonian fluids connects shear stress to shear strain rate. Recognizing this relationship helps identify flow behavior and select appropriate models in pipelines, boundary layers, and lubrication.

Given Data / Assumptions:

• Newtonian behavior: τ = μ (du/dy), where du/dy is the velocity gradient (shear strain rate).
• Constant dynamic viscosity μ under given temperature and pressure.

Concept / Approach:
Because τ is directly proportional to the strain rate with constant proportionality μ, a plot of τ versus du/dy is a straight line through the origin with slope μ.

Step-by-Step Reasoning:

Verification / Alternative check:
Non-Newtonian fluids deviate: shear-thinning (pseudoplastic) show concave-down curves; shear-thickening (dilatant) show concave-up; Bingham plastics have a yield intercept. None of these are straight through the origin with constant slope.

Why Other Options Are Wrong:

• Parabolic, hyperbolic, elliptical: do not represent τ ∝ du/dy with constant μ.

Common Pitfalls:
Confusing shear stress versus time or strain; the Newtonian linearity is with strain rate, not total strain.

Final Answer:
Straight line