Stability of the Laminar Parabolic Flow of a Viscous Fluid between Parallel Fixed Walls

Asymptotic expressions are derived for the characteristic values C of two-dimensional disturbances of the parabolic flow in the limit of small wave numbers α and large values of αR, where R denotes Reynolds number. For the first mode we get ${C}₁=1.617{e}^{{i{π}}{6}}{({α}R)}^{{-}1/3}+{{α}}²[.395{-}.883{e}^{{i{π}}{6}}{({α}R)}^{{-}1/3}+({i}{{α}R})]+0({{α}}⁴)$An excellent check on the constants in ${C}₁$ was obtained by a detailed numerical integration of the differential equation at ${α}R=5430$. The above result shows stability of the first mode at large ${α}R$ and small ${{α}}²$. The possible perturbations of the parabolic flow fall into two classes, one in which at ${{α}}²=0$, ${C}ᵣ{→}0$, as ${α}R{→}{∞}$, and another in which ${C}ᵣ{→}1$ as ${α}R{→}{∞}$. At large ${α}R$ the characteristic values of the modes in the latter class for finite ${{α}}²$ approach their values at ${{α}}²=0$, and are stable. All the modes of the first class are stable at large ${α}R$ as far as ${{α}}²$-terms in $C$ are concerned. In the even modes of this class the phase velocity becomes negative (propagation upstream) for ${{α}}⁷R>M$, where $M$ for the second mode has a value of 10700. The possible bearing of this result on stability experiments is discussed.