Stability of the two dimensional jet

Abstract

The Orr-Sommerfeld equation can be written in the form

$$({{D}^{2}} - {{\alpha }^{2}})({{D}^{2}} - {{\omega }^{2}})\Phi = i\alpha R[(w{{D}^{2}} - {{\alpha }^{2}})\Phi - \omega ''\Phi ]$$

(1)

where ω ² = α²−i α R c, D=d/dy, ω = ω (y) = sech² y and otherwise the notation is standard (cf. Lin’s book). Vanishing velocity components at y = ± ∞ give for boundary conditions

$$\alpha \Phi (\pm \infty ) = \Phi '(\pm \infty ) = 0$$

(2)

and we are interested here in even eigenfunctions Ф (y). By regarding the right-hand side of (1) as an inhomogeneous term, one can solve the constant-coefficients differential equation that remains, using (2), to get a representation of Ф in terms of the right-hand side of (1). The term in Ф″ can be reduced by some integration by parts, again using (2), to one in Ф alone, and in this way one converts the original problem into the following integral equation (Re ω ≥ 0)

$$\begin{array}{*{20}{c}} {\Phi \left( y \right) = i\alpha R\int\limits_{{ - \infty }}^{\infty } {\left\{ { - \tfrac{1}{{2\omega }}{{e}^{{ - \omega |y - y'|w\left( {y'} \right)}}} + } \right.} } \\ {\left. { + \frac{1}{{{{\omega }^{2}} - {{\alpha }^{2}}}}\left( {{{e}^{{\alpha |y - y'|}}} - {{e}^{{ - \omega |y - y'|}}}} \right)\frac{{|y - y'|}}{{\left( {y - y'} \right)}}\omega '\left( {y'} \right)} \right\}\Phi \left( {y'} \right)dy'.} \\ \end{array}$$

(3)