The Unsteady Navier-Stokes Problem

Part of the Springer Series in Computational Mathematics book series (SSCM, volume 23)


In this Chapter we turn our attention towards unsteady viscous flows, especially in the incompressible case. We therefore consider the time-dependent counterpart of the Navier-Stokes problem (10.1.1)-(10.1.3), that reads
$$ \left\{ {\begin{array}{*{20}c} {\frac{{\partial u}} {{\partial t}} - \nu \Delta u + \left( {u \cdot \nabla } \right)u + \nabla p = f in Q_T : = \left( {0,T} \right) \times \Omega } \\ \begin{gathered} div u = 0 in Q_T \hfill \\ u = 0 on \Sigma _{\rm T} : = \left( {0,T} \right) \times \partial \Omega \hfill \\ \end{gathered} \\ {u_{|t = 0} = u_0 on \Omega , } \\ \end{array} } \right. $$
where f=f(t, x) and u0=u0(x) are given data, ω is an open bounded domain of ℝ d , with d=2,3, and ηω is its boundary. One remarkable feature of (13.1) is the absence of an equation containing ηp/ηt. Indeed, in (13.1) the pressure p appears as a Lagrange multiplier associated to the divergence-free constraint div u=0.


Incompressible Flow Stokes Problem Inertial Manifold Trilinear Form Spectral Collocation Method 
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© Springer-Verlag Berlin Heidelberg 2008

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