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Inviscid flow

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A flow in which viscous effects can be neglected is known as inviscid flow. At high Reynolds numbers, flow past slender bodies involve thin boundary layers. Viscous effects are important only inside the boundary layer and the flow outside it is nearly inviscid. If the boundary layer is not separated then the inviscid flow model can be used to predict the pressure distribution with reasonable accuracy. Although no practical flow is inviscid, the inviscid assumption is valid if the time scales for diffusion are much larger compared to the time scales for convection, which is measured by the Reynolds number.

Governing Equations

The governing equations for inviscid flow, also known as Euler equations, are obtained by discarding the viscous terms from the Navier-Stokes equations

  • Continuity equation

\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}(\rho u_j)= 0
  • Momentum equation

\frac{\partial}{\partial t}(\rho u_i) + \frac{\partial}{\partial x_j}(\rho u_i u_j + p \delta_{ij}) = 0
  • Energy equation

\frac{\partial}{\partial t}(\rho E) + \frac{\partial}{\partial x_j}[(\rho E + p)u_j] = 0


  • \rho is the density
  • u_i is the fluid velocity
  • p is the pressure
  • E is the total energy per unit mass of fluid

E = \frac{\gamma p}{\gamma - 1} + \frac{1}{2} \rho |u|^2

The above equations are closed by taking an equation of state, the simplest being the ideal gas

p = \rho R T


  • R is the gas constant
  • T is the absolute temperature

Euler equations as the limit of Navier-Stokes equations

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