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A modification of Homann’s axisymmetric outer potential stagnation-point flow of strain rate $a$ is obtained by adding periodic radial and azimuthal velocities of the form $b\hspace{0.167em} r\sin 2\theta $ and $b\hspace{0.167em} r\cos 2\theta $, respectively, where $b$ is a shear rate. This leads to the discovery of a new family of asymmetric viscous stagnation-point flows depending on the shear-to-strain-rate ratio $\gamma = b/ a$ that exist over the range $\ensuremath{-} \infty \lt \gamma \lt \infty $. Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large-$\gamma $ asymptotic behaviours. Sample similarity velocity profiles are also presented. [ABSTRACT FROM AUTHOR]

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results. [ABSTRACT FROM AUTHOR]