Eigensolutions to the Viscous Problem

We will consider the governing equations for infinitesimal disturbances in parallel flows. Let U _i = U(y)δ_1i be the parallel base flow, i.e., a flow in the x-direction that only depends on the wall-normal direction y (see Figure 2.1 defining the coordinate system and base flow). If this mean velocity profile is introduced into the disturbance equations (1.6) and the nonlinear terms are neglected, the resulting equations can be written: 3.1$$ \frac{{\partial u}}{{\partial t}} + U\frac{{\partial u}}{{\partial x}} + vU' = - \frac{{\partial p}}{{\partial x}} + \frac{1}{{{\mathop{\rm Re}\nolimits} }}\nabla ^2 u $$3.2$$ \frac{{\partial v}}{{\partial t}} + U\frac{{\partial v}}{{\partial x}} = - \frac{{\partial p}}{{\partial y}} + \frac{1}{{{\mathop{\rm Re}\nolimits} }}\nabla ^2 v $$3.3$$ \frac{{\partial w}}{{\partial t}} + U\frac{{\partial w}}{{\partial x}} = - \frac{{\partial p}}{{\partial z}} + \frac{1}{{{\mathop{\rm Re}\nolimits} }}\nabla ^2 w. $$