Prescribing Morse Scalar Curvatures: Blow-Up Analysis

We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais–Smale sequences, we determine precise blow-up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers, we aim to establish the sharpness of this result, proving a converse existence statement, together with a one-to-one correspondence of blowing-up subcritical solutions and critical points at infinity. This analysis will be then applied to deduce new existence results for the geometric problem.

A note on waveless subcritical flow past symmetric bottom topography

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.

Control of long-wavelength Marangoni–Bénard convection

A nonlinear feedback control strategy for delaying the onset and eliminating the subcritical nature of long-wavelength Marangoni–Bénard convection is investigated based on an evolution equation. A control temperature is applied to the lower wall in a gas–liquid layer otherwise heated uniformly from below. It is shown that, if the interface deflection is assumed to be known via sensing as a function of both horizontal coordinates and time, a control temperature with a cubic-order polynomial dependence on the deflection is capable of delaying the onset as well as eliminating the subcritical instability altogether, at least on the basis of a weakly nonlinear analysis. The analytical results are supported by direct numerical simulations. The control coefficients required for stabilization are O(1) for both delaying onset indefinitely and eliminating subcritical instability. In order to discuss the effects of control, a review is made of the dependence of the weakly nonlinear subcritical solutions without control upon the various governing parameters.

Withdrawal from a two-layer inviscid fluid in a duct

The steady simultaneous withdrawal of two inviscid fluids of different densities in a duct of finite height is considered. The flow is two-dimensional, and the fluids are removed by means of a line sink at some arbitrary position within the duct. It is assumed that the interface between the two fluids is drawn into the sink, and that the flow is uniform far upstream. A numerical method based on an integral equation formulation yields accurate solutions to the problem, and it is shown that under normal operating conditions, there is a solution for each value of the upstream interface height. Numerical solutions suggest that limiting configurations exist, in which the interface is drawn vertically into the sink. The appropriate hydraulic Froude number is derived for this situation, and it is shown that a continuum of solutions exists that are supercritical with respect to this Froude number. An isolated branch of subcritical solutions is also presented.

Weir flows and waterfalls

Two-dimensional free-surface flows, which are uniform far upstream in a channel of finite depth that ends suddenly, are computed numerically. The ending is in the form of a vertical wall, which may force the flow upward before it falls down forever as a jet under the effect of gravity. Both subcritical and supercritical solutions are presented. The subcritical solutions are a one-parameter family of solutions, the single parameter being the ratio between the height of the wall and the height of the uniform flow far upstream. On the other hand, the supercritical solutions are a two-parameter family of solutions, the second parameter being the Froude number. Moreover, for some combinations of the parameters, it is shown that the solution is not unique.