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Microfluidics: Modeling, Mechanics and Mathematics by Bastian E. Rapp

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31.8 Two-Dimensional Problems of First Order in Time and Second-Order in Space

31.8.1 Introduction

The next case we want to study are differential equations, which are first order in time and second order in space. Obviously, the time-dependent Navier–Stokes equation for Poiseuille flow is a typical example of such an equation (see Eq. 18.1). It is given in analogy to Eq. 31.39 by

gtxyt+Fxxxyt+Fyyxyt=0

si146_e

where we restrict ourselves again to the homogeneous case. As stated, the inhomogeneous case only brings in additional terms on the right-hand side that must be rewritten and integrated as detailed in section 31.3.3. In this case, the vector ...

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