Microfluidics: Modeling, Mechanics and Mathematics

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

$\frac{\partial g}{\partial t}\left(x,y,t\right)+\frac{\partial F_x}{\partial x}\left(x,y,t\right)+\frac{\partial F_y}{\partial y}\left(x,y,t\right)=0$

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 ...