Linear Inviscid Analysis

Abstract

We begin this section by deriving the stability equations for infinitesimal disturbances when effects due to viscosity are negligible. Stability calculations of this sort were among the first in the field of hydrodynamic stability theory. We will assume parallel flow. Let U _i = U(y)δ_1i be the base flow, i.e., a flow in the x-direction that varies with y (see Figure 2.1). If this flow is substituted into the disturbance equations (1.6) and the nonlinear and viscous terms are omitted, the resulting equations can be written as

$$ \frac{{\partial u}} {{\partial t}} + U\frac{{\partial u}} {{\partial x}} + vU' = - \frac{{\partial p}} {{\partial x}} $$

(2.1)

$$ \frac{{\partial v}} {{\partial t}} + U\frac{{\partial v}} {{\partial x}} = - \frac{{\partial p}} {{\partial y}} $$

(2.2)

$$ \frac{{\partial w}} {{\partial t}} + U\frac{{\partial w}} {{\partial x}} = - \frac{{\partial p}} {{\partial z}} $$

(2.3)

and the continuity equation is

$$ \frac{{\partial u}} {{\partial x}} + \frac{{\partial v}} {{\partial y}} + \frac{{\partial w}} {{\partial z}} = 0. $$

(2.4)