Euler and Navier—Stokes Equations for Incompressible Fluids

Part of the Applied Mathematical Sciences book series (AMS, volume 117)


This chapter deals with equations describing motion of an incompressible fluid moving in a fixed compact space M, which it fills completely. We consider two types of fluid motion, with or without viscosity, and two types of compact space, a compact smooth Riemannian manifold with or without boundary. The two types of fluid motion are modeled by the Euler equation
$$\frac{\partial u} {\partial t} + {\nabla }_{u}u = -\text{ grad }p,\qquad \text{ div}u = 0,$$
for the velocity field u, in the absence of viscosity, and the Navier–Stokes equation
$$\frac{\partial u} {\partial t} + {\nabla }_{u}u = \nu \mathcal{L}u -\text{ grad }p,\qquad \text{ div }u = 0,$$
in the presence of viscosity. In (0.2), ν is a positive constant and \(\mathcal{L}\) is the second-order differential operator
$$\mathcal{L}u = \text{ div Def }u,$$
which on flat Euclidean space is equal to Δu, when div u = 0. If there is a boundary, the Euler equation has boundary condition nu = 0, that is, u is tangent to the boundary, while for the Navier–Stokes equation one poses the no-slip boundary condition u = 0 on ∂M.


Vector Field Stokes Equation Euler Equation Vortex Tube Compact Riemannian Manifold 
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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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