---
title: "Navier-Stokes and Euler equations in 2D"
subject: "Fluid Mechanics"
chapter: 5
key: "semester_3/fluid_mechanics:navier-stokes-euler-2d-applications"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/navier-stokes-euler-2d-applications
site: BCT Engineering
---

# Navier-Stokes and Euler equations in 2D

Fluid Mechanics · Chapter 5: Hydrodynamics

- Asked in: 2075 Chaitra, 4 marks
- Video: [Description and Derivation of the Navier-Stokes Equations](https://www.youtube.com/watch?v=NjoMoH51UZc) — LearnMechE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/navier-stokes-euler-2d-applications
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-5

## Question

Write down the expression for Navier-Stokes equations and Euler equations of fluid motion in 2D with definition of each term. Also write their applications.

## Short answer

Navier-Stokes (2D) includes viscous terms μ∇²u and ∇²v; Euler omits viscosity. Applications: Navier-Stokes for laminar pipe/plate flows and boundary layers; Euler for inviscid potential flow, Bernoulli’s equation, and external aerodynamics.

## Answer

#### Navier-Stokes equations (2-D, incompressible, constant viscosity)

$$
\begin{aligned}
\rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right) &= \rho g_x - \frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2}\right)\\
\rho\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}\right) &= \rho g_y - \frac{\partial p}{\partial y} + \mu\left(\frac{\partial^2v}{\partial x^2} + \frac{\partial^2v}{\partial y^2}\right)
\end{aligned}
$$

#### Euler equations (2-D, inviscid)

Set $\mu = 0$ in the above:

$$
\begin{aligned}
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} &= g_x - \frac{1}{\rho}\frac{\partial p}{\partial x}\\
\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} &= g_y - \frac{1}{\rho}\frac{\partial p}{\partial y}
\end{aligned}
$$

#### Definition of terms

| Symbol / term | Meaning |
|---|---|
| $u, v$ | Velocity components along $x, y$ |
| $\partial u/\partial t$ | Local acceleration (unsteadiness) |
| $u\,\partial u/\partial x + v\,\partial u/\partial y$ | Convective acceleration |
| $\rho$ | Fluid density |
| $g_x, g_y$ | Body force (gravity) components per unit mass |
| $\partial p/\partial x,\ \partial p/\partial y$ | Pressure gradient (pressure force) |
| $\mu(\partial^2u/\partial x^2 + \partial^2u/\partial y^2)$ | Viscous force; $\mu$ is dynamic viscosity |

Together with the continuity equation $\partial u/\partial x + \partial v/\partial y = 0$ these form a closed system.

#### Applications

- **Navier-Stokes:** laminar flow in pipes and between plates (Hagen-Poiseuille and Couette flow), boundary layer analysis, flow around bodies, lubrication in bearings, weather and ocean modelling, CFD for turbulence (RANS) and aerodynamics.
- **Euler:** flow outside boundary layers where viscosity is negligible, derivation of Bernoulli's equation, potential flow around airfoils and bodies, flow through nozzles and open-channel waves, high-speed gas dynamics, and for a quick estimate of pressure distribution in non-viscous flow.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How are Euler equations obtained?
   A. Set density to 0
   B. Set pressure to 0
   C. Set viscosity mu = 0
   D. Remove gravity
   Answer: C. Euler equations are inviscid. (Key point: Euler from NS)

2. What does du/dt represent?
   A. Convective acceleration
   B. Viscous force
   C. Pressure force
   D. Local acceleration due to unsteadiness
   Answer: D. du/dt is the time rate at a fixed point. (Key point: Local acceleration)

3. Which problem is typical for Navier-Stokes?
   A. Laminar flow in pipes and between plates
   B. Ideal flow outside boundary layers
   C. Derivation of Bernoulli
   D. Potential flow
   Answer: A. Viscous problems need NS. (Key point: NS application)

4. Where is Euler used?
   A. Flow in lubrication films
   B. Flow outside boundary layers where viscosity is negligible
   C. Laminar pipe flow
   D. Boundary layer inside
   Answer: B. Euler suits inviscid regions. (Key point: Euler application)

## Related in Fluid Mechanics

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