Newtonian shear – dependence of shear stress on layer spacing The shear stress transmitted between two parallel layers of a Newtonian liquid moving with a velocity difference across a gap is __________ proportional to the distance (gap) between the layers, all else being equal.

Introduction / Context:
Newton's law of viscosity states that shear stress in a Newtonian fluid is proportional to the rate of shear strain (velocity gradient). Understanding how the gap between layers affects this gradient clarifies how shear stress scales in simple shear.

Given Data / Assumptions:

• Two large parallel layers (or plates) with a relative velocity difference Δu.
• Gap (distance) between layers is y.
• Fluid behaves as Newtonian with dynamic viscosity μ.

Concept / Approach:
The constitutive relation is τ = μ * (du/dy). For a linear velocity profile between plates, du/dy ≈ Δu / y. Therefore, for fixed Δu and μ, shear stress varies inversely with the separation y: increasing the gap reduces the gradient and thus the shear stress.

Step-by-Step Solution:
1) Write τ = μ * (du/dy).2) Replace du/dy by Δu / y for simple Couette flow.3) Obtain τ = μ * (Δu / y).4) Conclude τ ∝ 1 / y when Δu, μ are constants.

Verification / Alternative check:
Doubling the gap while holding plate speeds fixed halves du/dy and halves τ, which matches laboratory Couette apparatus observations.

Why Other Options Are Wrong:

• Directly: would require τ ∝ y for constant Δu, contradicting τ = μ * Δu / y.
• Exponential/logarithmic/unrelated: not supported by Newtonian shear theory in this configuration.

Common Pitfalls:
Forgetting that if the velocity difference is not held constant—e.g., fixed wall shear—the scaling changes; here Δu is taken as fixed to isolate the effect of gap size.

Final Answer:
indirectly (inversely)