The pressure intensity is same in all directions at a point in a fluid :
Try it first: pick an option, then read the explanation.
Asked in SSC JE 2 MARCH 2017 Morning Shift
Detailed Explanation
Pascal's law requires the complete absence of shear stresses across all planes through a point. Shear stresses vanish either in an inviscid (frictionless) fluid or in any real fluid at rest (where no velocity gradients exist).
1. Stress tensor in a general fluid:
\[ \sigma_{ij} = -p \, \delta_{ij} + \tau_{ij} \]
Where \(p\) is the thermodynamic pressure, \(\delta_{ij}\) is the Kronecker delta, and \(\tau_{ij}\) is the viscous shear stress tensor.
2. For the normal stress (pressure) to be identical in all directions (\(\sigma_{xx} = \sigma_{yy} = \sigma_{zz} = -p\)):
\[ \tau_{ij} = 0 \quad (\text{zero shear stress on all planes}) \]
3. Viscous shear stress is given by Newton's law: \(\tau = \mu \frac{du}{dy}\). This becomes zero when:
- The fluid is inviscid/frictionless (\(\mu = 0\)), OR
- There is no relative motion between adjacent layers (\(\frac{du}{dy} = 0\), fluid at rest or moving as a rigid body).
Exam Note: The SSC JE book key designates option (b) ('Only when fluid is frictionless and is at rest') as the answer, though in standard continuum mechanics option (d) ('When there is no relative motion of one fluid layer relative to other') is the most comprehensive physical condition.
Why “Only when fluid is frictionless and is at rest” Is Correct
In the official exam key, option (b) is marked. A fluid at rest experiences zero velocity gradients (\(du/dy = 0\)) and therefore zero shear stress, satisfying Pascal's law.
Why the Other Options Are Wrong
- A. Only when fluid is frictionless and incompressible: Incompressibility is not a requirement for Pascal's law; compressible gases at rest also obey Pascal's law perfectly.
- C. Only when fluid is frictionless: A frictionless fluid in motion still experiences pressure gradients due to acceleration (Euler's equations).
- D. When there is no relative motion of one fluid layer relative to other: Option (d) is physically accurate in continuum mechanics, but option (b) is the specific answer designated in the official answer key.
Key Point
Pascal's law holds when shear stress is zero: fluids at rest or with no relative motion between layers.
Key Concepts for Students
- Static fluids cannot sustain shear stress; they deform until all shear stresses disappear.
- Gases (compressible) and liquids (incompressible) both obey Pascal's law when at rest.
- In rigid-body rotation or translation, fluid layers have no relative motion, so pressure is isotropic at any point.
Practice more: this is question 22 in Hydrostatic Pressure MCQs – Part 2. Try the whole set and check every answer instantly.