Stability and Transition in Shear Flows pp 15-53 | Cite as

# Linear Inviscid Analysis

Chapter

## 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
and the continuity equation is

*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)

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

(2.4)

## Keywords

Velocity Profile Couette Flow Boundary Layer Flow Modal Solution Rayleigh Equation
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## Copyright information

© Springer Science+Business Media New York 2001