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Hydrodynamic stability; shear flows; variable bottom; sea straits.
We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if $k>0$ is the wave number of a normal mode then $k>k_c$ (for some critical wave number $k_c$) implies the stability of the mode for a class of basic flows. Furthermore, if $K(z)=\frac{-({U''}_0-T_0{U'}_0)}{U_0-U_{0s}}$, where $U_0$ is the basic velocity, $T_0$ (a constant) the topography and prime denotes differentiation with respect to vertical coordinate 𝑧 then we prove that a sufficient condition for the stability of basic flow is $0 < K(z)≤\left(\frac{𝜋^2}{D^2}+\frac{T^2_0}{4}\right)$, where the flow domain is $0≤ z≤ D$.
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Volume 129 | Issue 2
April 2019
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