MHT-CET Physics · Mechanical Properties of Fluids

Viscosity, Stokes' Law and Terminal Velocity

Viscosity is the internal friction between layers of a moving fluid, acting along the layers; a sphere falling through a fluid feels the drag 6πηrv of Stokes' law, and stops accelerating when drag plus upthrust equals its weight — the terminal velocity, which grows as the square of the radius.

All Mechanical Properties of Fluids notes MHT-CET Physics strategy

Why this matters

21 PYQs, 4 HARD. Seventeen are terminal velocity — drops of the same size merging into one, spheres of different radius or density in the same liquid, the viscous force at terminal speed, a ball rising at constant speed, and the height from which a ball must fall to enter water already at its terminal speed; four are about layers — the direction of the viscous force, flow speed rising with height in a river, Reynolds number and critical velocity. Two cards.

Concept 1 of 2: Viscous Force Between Layers, Reynolds Number and Critical Velocity

Adjacent layers of a flowing liquid slide over each other and drag on each other, so the viscous force is TANGENTIAL to the layers. In steady flow over a river bed the speed rises steadily with height, so v/h is constant. Whether flow stays smooth is set by the Reynolds number ρvd/η: small means streamline, large means turbulent. The critical velocity at which the change happens is N·η/(ρd), so a more viscous fluid can flow faster before turning turbulent.

Definition

Viscous force acts tangentially to the layers (F = ηA dv/dx).
Steady flow with speed proportional to height: $\dfrac{v_A}{h_A} = \dfrac{v_B}{h_B}$ (12 cm/s at 40 cm ⇒ 27 cm/s at 90 cm).
Reynolds number $R_e = \dfrac{\rho v d}{\eta}$ for a pipe of diameter d.
Critical velocity $v_c = \dfrac{R_e\,\eta}{\rho d}$ : directly proportional to η, inversely to ρ and d.

Reynolds number and critical velocity

R_e = \frac{\rho v d}{\eta}, \qquad v_c = \frac{R_e\,\eta}{\rho d}

Worked example

Water (ρ = 1000 kg/m³, η = 10⁻³ Pa s) flows at 0.1 m/s in a pipe 2 cm across. Reynolds number, and is the flow streamline if the change happens near 2000?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 19 April Shift II · Q26Moderate

Example 1 · Mechanical Properties of Fluids · Viscosity, Stokes' Law, Terminal Velocity, and Reynolds

Water is flowing steadily in a river. A and B are the two layers of water at heights 40 cm and 90 cm from the bottom. The velocity of the layer A is

12\,cm/s

. The velocity of the layer B is

Thinking a viscous liquid turns turbulent sooner

Viscosity damps disturbances. The critical velocity is proportional to η: honey stays streamline at speeds where water would be turbulent.

Drill 3 more on viscous force between layers, reynolds number and critical velocity

Concept 2 of 2: Terminal Velocity

At terminal velocity nothing accelerates: weight = upthrust + drag. With Stokes' drag that gives v = 2r²(ρ − σ)g/(9η) — so v goes as r² and as (ρ − σ). Merging drops keeps the volume: n drops make one of radius n^(1/3)r, whose terminal velocity is n^(2/3) times as large — two drops give 2^(2/3), 125 drops give 25. The viscous force at terminal speed is just weight minus upthrust, mg(1 − σ/ρ). A light ball rising at constant speed in a liquid three times denser feels a drag twice its weight.

Definition

$v_t = \dfrac{2r^2(\rho - \sigma)g}{9\eta}$ , so $v_t \propto r^2$ and $v_t \propto (\rho - \sigma)$ .
Merging n equal drops: $R = n^{1/3}r$ , $v = n^{2/3}v_0$ . Same material, mass × 8 ⇒ radius × 2 ⇒ $v \times 4$ . Terminal velocities 9 : 4 ⇒ radii 3 : 2 ⇒ volumes 27 : 8.
Viscous force at terminal speed $= mg\left(1 - \dfrac{\sigma}{\rho}\right)$ .
Rising ball in a liquid k times denser: drag $= (k - 1)$ × weight.
Equal speeds, different metals: $\dfrac{r_1^2}{r_2^2} = \dfrac{\rho_2 - \sigma}{\rho_1 - \sigma}$ .
Enter water already at terminal speed: $\sqrt{2gh} = v_t \Rightarrow h = \dfrac{v_t^2}{2g}$ .
If the drag is $Kv^2$ instead: $v_t = \sqrt{\dfrac{Vg(\rho_1 - \rho_2)}{K}}$ .

Terminal velocity (Stokes)

v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}

Worked example

Eight equal raindrops, each falling at 3 cm/s, merge into one. Its terminal velocity?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 9th May Shift 2 · Q49Moderate

Example 2 · Mechanical Properties of Fluids · Viscosity, Stokes' Law, Terminal Velocity, and Reynolds

Two identical drops of water are falling through air with steady velocity 'V'. If the two drops come together to form a single drop. The new velocity of the single drop is

Scaling terminal velocity with volume

v_t ∝ r², not r³. Two drops merging double the VOLUME, raise the radius by 2^(1/3), and raise the speed by 2^(2/3) ≈ 1.59, not 2.

Drill 16 more on terminal velocity

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Viscous Force Between Layers, Reynolds Number and Critical Velocity
Reynolds number and critical velocity
$R_e = \frac{\rho v d}{\eta}, \qquad v_c = \frac{R_e\,\eta}{\rho d}$
Terminal Velocity
Terminal velocity (Stokes)
$v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$

Watch out for (2)

Thinking a viscous liquid turns turbulent sooner
→ Viscous Force Between Layers, Reynolds Number and Critical Velocity
Scaling terminal velocity with volume
→ Terminal Velocity

Test yourself on Mechanical Properties of Fluids

20 past MHT-CET questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mechanical Properties of Fluids chapter test Drill all 21 questions

Related notes

Pressure and Buoyancy — the upthrust in the force balance →