Dimensionless Numbers

Dimensionless Numbers – a vital concept in fluid mechanics, heat transfer, and engineering modeling, especially important for competitive exams like SSC JE, RRB JE, GATE, IES, and various PSUs.

🔢 What Are Dimensionless Numbers?

Dimensionless numbers are ratios of quantities that have no physical units. They help compare different physical effects and are essential for modeling, analysis, and similarity in engineering problems.

They often arise from the non-dimensionalization of governing equations (like Navier-Stokes or energy equations) and reveal dominant physical forces in a system.

📦 Why Are They Important?

Help simplify complex problems
Used in model testing (e.g., wind tunnel, hydraulic models)
Allow scaling from model to prototype
Help determine regimes: laminar vs. turbulent, conduction vs. convection, etc.
Appear in correlation equations (e.g., heat transfer coefficients)

📘 Common Dimensionless Numbers (With Physical Meaning)

No.	Name	Symbol	Physical Meaning
1	Reynolds Number	Re	Ratio of inertia force to viscous force
2	Froude Number	Fr	Ratio of inertia force to gravity force
3	Weber Number	We	Ratio of inertia force to surface tension
4	Mach Number	Ma	Ratio of flow velocity to speed of sound
5	Prandtl Number	Pr	Ratio of momentum diffusivity to thermal diffusivity
6	Nusselt Number	Nu	Ratio of convective to conductive heat transfer
7	Grashof Number	Gr	Ratio of buoyancy to viscous force (free convection)
8	Stanton Number	St	Ratio of heat transferred to thermal capacity
9	Biot Number	Bi	Ratio of internal conduction to surface convection
10	Peclet Number	Pe	Product of Reynolds and Prandtl: flow effectiveness in heat transfer

🧮 Detailed Formulas and Significance

1️⃣ Reynolds Number (Re)

Re = \frac{\rho v L}{\mu} = \frac{v L}{\nu}

$\rho$ : density
$v$ : velocity
$L$ : characteristic length
$\mu$ : dynamic viscosity
$\nu = \mu/\rho$ : kinematic viscosity

Use: Determines flow regime

Re < 2000 → Laminar
2000 < Re < 4000 → Transitional
Re > 4000 → Turbulent

2️⃣ Froude Number (Fr)

Fr = \frac{v}{\sqrt{gL}}

$v$ : velocity
$g$ : gravity
$L$ : characteristic length

Use: Important in open channel and free surface flows.

3️⃣ Weber Number (We)

We = \frac{\rho v^2 L}{\sigma}

$\sigma$ : surface tension

Use: Dominates in droplets, bubbles, and jet breakup.

4️⃣ Mach Number (Ma)

Ma = \frac{v}{c}

$c$ : speed of sound in the medium

Use:

Ma < 1: Subsonic
Ma = 1: Sonic
Ma > 1: Supersonic
Ma >> 1: Hypersonic

5️⃣ Prandtl Number (Pr)

Pr = \frac{\mu C_p}{k} = \frac{\nu}{\alpha}

$\alpha$ : thermal diffusivity
$k$ : thermal conductivity
$C_p$ : specific heat

Use: Compares velocity and thermal boundary layers

6️⃣ Nusselt Number (Nu)

Nu = \frac{hL}{k}

$h$ : convective heat transfer coefficient

Use:

Nu = 1: Pure conduction
Nu > 1: Convection is present

7️⃣ Grashof Number (Gr)

Gr = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2}

$\beta$ : coefficient of thermal expansion
Related to free/natural convection

8️⃣ Biot Number (Bi)

Bi = \frac{hL}{k}

$L$ : characteristic length (volume/surface area)

Use:

Bi << 1 → Uniform internal temperature
Bi > 0.1 → Temperature gradients exist inside

9️⃣ Stanton Number (St)

St = \frac{h}{\rho C_p v}

Use: Used in convective heat transfer calculations.

🔟 Peclet Number (Pe)

Pe = Re \cdot Pr = \frac{vL}{\alpha}

Use: Compares advection to diffusion in heat transfer.

📌 Exam Tip

Remember Re, Fr, We, Ma: All are inertia force ratios.
Pr, Nu, Bi, Pe: Related to heat transfer
Common values and thresholds often appear in MCQs.

📚 Sample MCQ Questions

Q1. Reynolds number is the ratio of:
A. Inertia force to gravity force
B. Inertia force to viscous force
C. Viscous force to inertia force
D. Pressure force to viscous force
👉 Answer: B

Q2. Prandtl number for air is approximately:
A. 0.01
B. 0.7
C. 1.0
D. 7.0
👉 Answer: B

Q3. Biot number less than 0.1 indicates:
A. Convection is dominant
B. Uniform temperature inside body
C. High thermal conductivity
D. Large internal resistance
👉 Answer: B

Fluid mechanics

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