Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation

We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data

Investigation of Effects of Surface Temperature and Single Roughness Elements on Boundary-Layer Transition

The laminar boundaxy layer and the position of the transition point were investigated on a heated flat plate. It was found that the Reynolds number of transition decreases as the temperature of the plate is increased. It is shown from simple qualitative analytical considerations that the effect of variable viscosity in the boundary layer due to the temperature difference produces a velocity profile with an inflection point if the wall temperature is higher than the free-stream temperature. This profile is confirmed by measurements. Furthermore, it is confirmed that even with large deviation from the Blasius condition, the velocity and temperature profiles are very nearly identical, as predictable theoretically for a Prandtl number [sigma] of the order of 1.0 (for air, [sigma] = 0.76). The instability of inflection-point profiles is discussed. Studies of the flow in the wake of large, two-dimensional roughness elements are presented. It is shown that a boundary layer can separate and reattach itself to the wall without having transition take place

Investigations of Effects of Surface Temperature and Single Roughness Elements on Boundary-Layer Transition

The laminar boundary layer and the position of the transition point are investigated on a heated flat plate. It was found that the Reynolds number of transition decreases as the temperature of the plate is increased. It is shown from simple qualitative analytical considerations that the effect of variable viscosity in the boundary layer due to the temperature diference produces a velocity profile with an inflection point if the wall temperature is higher than the free-stream temperature. This profile is confirmed by measurements. Furthermore, it is confirmed that, even with large deviation from the Blasius condition, the velocity and temperature profiles are very nearly identical, as predictable theoretically for a Prandtl number [sigma] of the order of 1.0 (for air, [sigma]=0.76). The instability of injection-point profiles is discussed. Studies of the flow in the wake of large, two-dimensional roughness elements are presented. It is shown that a boundary laysr can separate and reattach itself to the wall without having transition take place

Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems