Bulk modulus (compressibility of a fluid):\nSelect the correct definition of bulk modulus K from the choices below.

Introduction / Context:
The bulk modulus K quantifies a fluid’s resistance to uniform compression. It is essential in acoustics (speed of sound), water hammer calculations, and compressible-flow corrections even for liquids that are nearly incompressible. Knowing its definition helps distinguish between different elastic and viscous properties in fluids and solids.

Given Data / Assumptions:

• Small, uniform changes in pressure and volume.
• Volumetric strain defined as ΔV/V (dimensionless).
• Isothermal or adiabatic context determines the numerical value but not the definition.

Concept / Approach:
By definition, bulk modulus K = Δp / (ΔV/V) = −V * dp/dV (the negative sign indicates that volume decreases as pressure increases). The larger the K, the less compressible the fluid. Water has a high K compared to gases, thereby supporting the approximation of incompressibility in many hydraulic problems.

Step-by-Step Solution:

Verification / Alternative check:
Speed of sound a in a fluid satisfies a = sqrt(K/ρ). Higher K yields higher a, as seen by comparing liquids (fast) to gases (slower), confirming consistency with physical intuition about compressibility.

Why Other Options Are Wrong:

• Shear stress to shear strain: That is shear modulus (solids), not bulk modulus.
• Increase in volume to viscosity: Mixes incompatible quantities.
• Critical velocity to viscosity: Unrelated ratio with mismatched dimensions.

Common Pitfalls:
Confusing bulk modulus with modulus of rigidity; forgetting that fluids cannot sustain shear in static conditions, so K concerns volumetric changes only.

Final Answer:
increase in pressure to the volumetric strain