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Navier-Stokes and Euler equations in 2D

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.

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.

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

ρ(∂u∂t+u∂u∂x+v∂u∂y)=ρgx−∂p∂x+μ(∂2u∂x2+∂2u∂y2)ρ(∂v∂t+u∂v∂x+v∂v∂y)=ρgy−∂p∂y+μ(∂2v∂x2+∂2v∂y2)\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 μ=0\mu = 0 in the above:

∂u∂t+u∂u∂x+v∂u∂y=gx−1ρ∂p∂x∂v∂t+u∂v∂x+v∂v∂y=gy−1ρ∂p∂y\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 / termMeaning
u,vu, vVelocity components along x,yx, y
∂u/∂t\partial u/\partial tLocal acceleration (unsteadiness)
u ∂u/∂x+v ∂u/∂yu\,\partial u/\partial x + v\,\partial u/\partial yConvective acceleration
ρ\rhoFluid density
gx,gyg_x, g_yBody force (gravity) components per unit mass
∂p/∂x, ∂p/∂y\partial p/\partial x,\ \partial p/\partial yPressure gradient (pressure force)
μ(∂2u/∂x2+∂2u/∂y2)\mu(\partial^2u/\partial x^2 + \partial^2u/\partial y^2)Viscous force; μ\mu is dynamic viscosity

Together with the continuity equation ∂u/∂x+∂v/∂y=0\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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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