Fluid statics at a point: The normal stress is equal in all directions at a point in a fluid under which condition?

Introduction / Context:
Pascal’s law states that in a fluid at rest, the pressure at a point acts equally in all directions. This equality of normal stresses underpins hydrostatics and the definition of scalar pressure. In moving fluids, shear stresses can be present, complicating the stress state.

Given Data / Assumptions:

• We are considering a point in a fluid continuum.
• We distinguish between hydrostatic and general motion conditions.
• “No relative motion” implies no velocity gradients and thus no viscous shear.

Concept / Approach:
For a fluid at rest (or a fluid moving as a rigid body with no shear), the only stresses are normal stresses, and these are equal in all directions, defining pressure as a scalar. Viscosity, compressibility, or absolute pressure do not alone guarantee isotropy of normal stress if shear exists.

Step-by-Step Solution:
Identify condition for hydrostatic state: zero shear stress.Zero shear occurs when there is no relative motion between fluid layers.Therefore, normal stress (pressure) is equal in all directions under that condition.

Verification / Alternative check:
From the Cauchy stress tensor, in the absence of shear components, the tensor reduces to −p I (isotropic), confirming equal normal stresses.

Why Other Options Are Wrong:

• Non-viscous or incompressible alone: a fluid can be inviscid or incompressible yet still have shear if velocity gradients exist.
• Both together: still not sufficient without the “no shear” condition.
• High pressure: magnitude of pressure is irrelevant to isotropy.

Common Pitfalls:
Assuming “inviscid” implies no shear; shear stress can be zero in the inviscid model, but real fluids with motion have shear unless velocity is uniform.

Final Answer:
When there is no relative motion between adjacent fluid layers (no shear)