Interdisciplinary Applied Mathematics

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In many microflow problems the nonlinear terms can be neglected, and in such cases the governing equations are the Stokes equations, which we can cast in the form



—v V2v + Vp/p = f in Q, Vv = 0 in Q,

along with appropriate boundary conditions for v.

2.1.2 Reduced Models

The mathematical nature of the Navier-Stokes equations varies depending on the flow that we model and the corresponding terms that dominate in the equations. For example, for an inviscid compressible flow, we obtain the Euler equations, which are ofhyperbolic nature, whereas the incompressible Euler equations are of hybrid type corresponding to both real and imaginary eigenvalues. The unsteady incompressible Navier-Stokes equations are of mixed parabolic/hyperbolic nature, but the steady incompressible Navier-Stokes equations are of elliptic/parabolic type. It is instructive, especially for a reader with not much experience in fluid mechanics, to follow a hierarchical approach in reducing the Navier-Stokes equations to simpler equations so that each introduces one new concept.

Taking as an example the incompressible Navier-Stokes equations (2.9), (2.10), a simpler model is the unsteady Stokes system. This retains all the complexity but not the nonlinear terms; that is,

—Ур/р + v y2v + f 0.

The Stokes system [equations (2.12a) and (2.12b)] is recovered by dropping the time derivative. Alternatively, we can drop the divergence-free constraint and study the purely parabolic scalar equation for a variable u, that is,

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