Critical Velocity Formula: v_c = Rₑη/(ρD), Reynolds Number

The critical velocity formula for flow in a pipe is $v_c = \frac{R_e\,\eta}{\rho D}$ : multiply the critical Reynolds number by the viscosity of the fluid, then divide by its density and the pipe diameter. Below this speed the fluid slides along in smooth layers (laminar or streamline flow). Above it, eddies form and the flow becomes turbulent.

For a straight round pipe the critical Reynolds number is usually taken as about 2 × 10³. Some Class 11 books instead write $v_c = \frac{K\eta}{\rho r}$ with r the radius and K a dimensionless constant; that is the same result with the factor of two folded into K. The idea comes from viscosity, so this page builds on the formula of viscosity and sits on the fluids formula sheet.

Quick facts

Formula
$v_c = \frac{R_e\,\eta}{\rho D}$
SI unit
metre per second (m/s)
Dimensional formula
[M⁰ L¹ T⁻¹]
Reynolds number
$R_e = \frac{\rho v D}{\eta}$ , no unit
Pipe limits
laminar below about 2 × 10³, turbulent above about 3 × 10³ to 4 × 10³
Trend
thicker fluid or narrower pipe gives a higher critical speed

Each symbol in the critical velocity formula

Quantities in v_c = Rₑη / (ρD)
Symbol	Meaning	SI unit
$v_c$	Critical velocity	m/s
$R_e$	Critical Reynolds number (pure number)	none
$\eta$	Viscosity (coefficient of viscosity)	Pa s
$\rho$	Density of the fluid	kg/m³
$D$	Inner diameter of the pipe	m

The formula is the Reynolds number rearranged. The Reynolds number $R_e = \frac{\rho v D}{\eta}$ compares the fluid's tendency to keep moving (its inertia, ρvD) with the viscous drag holding the layers together (η). When it passes the critical value, inertia wins and the flow breaks up. Setting $R_e$ to that value and solving for v gives $v_c$ .

Derivation from dimensions

Class 11 students are often asked to show the form by dimensional analysis. Assume $v_c$ depends on η, ρ and r: $v_c = K \eta^a \rho^b r^c$ . Writing dimensions,

$[\text{L T}^{-1}] = [\text{M L}^{-1} \text{T}^{-1}]^a [\text{M L}^{-3}]^b [\text{L}]^c$

Matching powers: M gives a + b = 0, T gives −a = −1, L gives −a − 3b + c = 1. So a = 1, b = −1, c = −1, and $v_c = \frac{K\eta}{\rho r}$ . Dimensional analysis cannot find K; experiment gives it. Using the diameter instead of the radius changes K into the Reynolds number form above.

Worked examples

Example 1: water in a garden pipe

Water (η = 1.0 × 10⁻³ Pa s, ρ = 1000 kg/m³) flows in a pipe of diameter 2.0 cm. Taking the critical Reynolds number as 2.0 × 10³, find the critical velocity.

Convert: D = 2.0 cm = 0.020 m.

$v_c = \frac{2.0\times10^3 \times 1.0\times10^{-3}}{1000 \times 0.020} = \frac{2.0}{20} = 0.10\ \text{m/s}$

Water in a household pipe usually moves faster than 10 cm/s, so its flow is often turbulent.

Example 2: a thick oil

Oil with η = 0.10 Pa s and ρ = 900 kg/m³ flows in a 5.0 cm pipe. Find the critical velocity, with Rₑ = 2.0 × 10³.

$v_c = \frac{2.0\times10^3 \times 0.10}{900 \times 0.050} = \frac{200}{45} = 4.4\ \text{m/s}$

A hundred times the viscosity of water lets the oil stay laminar at much higher speeds.

Example 3: is the flow laminar?

Water moves at 0.50 m/s through a tube of diameter 1.0 cm. Is the flow laminar or turbulent?

$R_e = \frac{1000 \times 0.50 \times 0.010}{1.0\times10^{-3}} = 5000$

This is well above the turbulent limit, so the flow is turbulent.

Use the same Reynolds number limits your book uses. Some quote laminar flow below 10³ and turbulent above 2 × 10³.

Other meanings of critical velocity

The phrase also appears in other chapters, so read the question carefully.

Satellites: some Indian books call the orbital speed $\sqrt{GM/r}$ the critical velocity of a satellite. See the orbital velocity formula.
Vertical circles: the least speed at the top of a vertical circle, $\sqrt{gr}$ , is sometimes called the critical speed.

In the fluids chapter it always means the laminar to turbulent limit described here. Turbulence matters because it sharply increases drag and energy loss, which is why it is linked to the terminal velocity formula for fast falling objects.

Common mistakes

Using radius where the formula needs diameter. Check which form your book uses.
Leaving viscosity in poise. Divide by 10 to get Pa s.
Thinking a wider pipe raises v_c. D is in the denominator, so a wider pipe turns turbulent at a lower speed.
Giving the Reynolds number a unit. It is a pure number.

Frequently asked questions

What is the formula of critical velocity?

In fluids, v_c = Rₑη / (ρD), where Rₑ is the critical Reynolds number (about 2 × 10³ for a pipe), η the viscosity, ρ the density and D the pipe diameter.

What happens above the critical velocity?

The flow stops being steady and layered. Eddies and swirls form, the flow becomes turbulent, and the resistance to flow rises sharply.

How does viscosity affect critical velocity?

v_c is proportional to η. A more viscous liquid, such as oil or honey, can flow faster before turning turbulent.

Which unit is critical velocity measured in?

Like any speed, in metres per second, with dimensional formula [M⁰ L¹ T⁻¹].