Flow of Liquids

Unit: kg m⁻¹ s⁻¹ ; Dimensions: [M L⁻¹ T⁻¹]

From the formula η = F / [A × (dv/dz)]: Step 1: Dimensions of F = [M L T⁻²]. Step 2: Dimensions of A = [L²]. Step 3: Dimensions of dv/dz = [L T⁻¹ / L] = [T⁻¹]. Step 4: Therefore η = [M L T⁻²] / ([L²][T⁻¹]) = [M L T⁻²] / [L² T⁻¹] = [M L⁻¹ T⁻¹]. The SI unit is kg m⁻¹ s⁻¹, also called 1 decapoise = 10 poise. Option B is wrong (dimensions of momentum per unit area). Option C is wrong (these are dimensions of pressure). Option D has an extra negative power of L.

F = 6π η r v

Stokes' law states that the viscous (drag) force on a small sphere moving through a homogeneous medium is F = 6π η r v. Here η is the coefficient of viscosity of the medium, r is the radius of the sphere, and v is the velocity of the sphere. Option A is wrong (coefficient is 4π, not 6π). Option C is wrong (r should be to the power 1, not r²). Option D is wrong (v should be to the power 1, not v²). The number 6π comes from the mathematical derivation by Stokes for a sphere.

6 m/s

Principle of Continuity: A₁v₁ = A₂v₂. Step 1: Write given values — A₁ = 4 cm², v₁ = 3 m/s, A₂ = 2 cm². Step 2: Apply continuity equation — 4 × 3 = 2 × v₂. Step 3: Solve for v₂ — v₂ = 12/2 = 6 m/s. The velocity doubles when the area is halved. Option A (1.5 m/s) is wrong — this would be the answer if the student divided instead of multiplied (thinking velocity decreases when area decreases). Option B (3 m/s) is wrong — velocity is not constant throughout. Option D (12 m/s) is wrong — this comes from v₂ = A₁ × v₁ without dividing by A₂.

P = pressure energy, ½ρv² = kinetic energy, ρgh = potential energy

In Bernoulli's equation, each term represents energy per unit volume: (1) P is the pressure energy per unit volume (work done by pressure forces). (2) ½ρv² is the kinetic energy per unit volume (since KE per unit volume = ½(m/V)v² = ½ρv²). (3) ρgh is the potential energy per unit volume (since PE per unit volume = (m/V)gh = ρgh). Bernoulli's theorem is essentially the law of conservation of energy for a flowing ideal liquid. All other options incorrectly assign the terms.